From 46f927c575df39e1e9ed21e7016bb2a299e66eac Mon Sep 17 00:00:00 2001
From: hexjasur <3.15015959e+08+hexjasur@users.noreply.github.com>
Date: Mon, 19 Oct 2026 14:55:04 +0000
Subject: [PATCH 1/3] feat: add step-by-step Newton interpolation

---
 .../methods/newton-interpolation/page.tsx     |  68 +++
 components/app-shell.tsx                      |  22 +-
 components/dashboard/methods-hub.tsx          |  23 +-
 components/newton-interpolation-method.tsx    | 475 ++++++++++++++++++
 components/seo/json-ld.tsx                    |   1 +
 lib/config.ts                                 |   6 +
 lib/i18n/locales/en.json                      |  75 ++-
 lib/i18n/locales/ru.json                      |  75 ++-
 lib/i18n/locales/uz.json                      |  75 ++-
 lib/newton-interpolation.ts                   | 247 +++++++++
 public/images/og/newton-interpolation.png     | Bin 0 -> 108039 bytes
 scripts/generate-og-images.ts                 |  12 +
 12 files changed, 1056 insertions(+), 23 deletions(-)
 create mode 100644 app/(workspace)/methods/newton-interpolation/page.tsx
 create mode 100644 components/newton-interpolation-method.tsx
 create mode 100644 lib/newton-interpolation.ts
 create mode 100644 public/images/og/newton-interpolation.png

diff --git a/app/(workspace)/methods/newton-interpolation/page.tsx b/app/(workspace)/methods/newton-interpolation/page.tsx
new file mode 100644
index 0000000..477e221
--- /dev/null
+++ b/app/(workspace)/methods/newton-interpolation/page.tsx
@@ -0,0 +1,68 @@
+import type { Metadata } from 'next';
+import { NewtonInterpolationMethod } from '@/components/newton-interpolation-method';
+import { MethodJsonLd } from '@/components/seo/json-ld';
+import { siteConfig } from '@/lib/config';
+
+const route = '/methods/newton-interpolation';
+const pageTitle = 'Nyuton interpolyatsiyasi — bo‘lingan ayirmalar | Newton';
+const pageDescription =
+  'Nyuton ko‘phadini hisoblang: bo‘lingan yoki chekli ayirmalar va P(x*) qiymati. Har bir formula hamda sonli qo‘yish qadam-baqadam ko‘rsatiladi.';
+const image = '/images/og/newton-interpolation.png';
+
+export const metadata: Metadata = {
+  title: pageTitle,
+  description: pageDescription,
+  keywords: [
+    'Nyuton interpolyatsiyasi',
+    'bo‘lingan ayirmalar',
+    'Nyuton interpolyatsion ko‘phadi',
+    'chekli ayirmalar usuli',
+    'интерполяция Ньютона',
+    'разделённые разности онлайн',
+    'Newton interpolation calculator',
+    'divided differences polynomial',
+  ],
+  alternates: { canonical: route },
+  openGraph: {
+    type: 'article',
+    title: 'Newton Interpolation Calculator | Nyuton interpolyatsiyasi',
+    description: pageDescription,
+    url: `${siteConfig.url}${route}`,
+    siteName: siteConfig.name,
+    images: [{ url: image, width: 1200, height: 630, alt: 'Newton divided differences interpolation calculator' }],
+  },
+  twitter: {
+    card: 'summary_large_image',
+    title: 'Newton Interpolation Calculator | Nyuton interpolyatsiyasi',
+    description: pageDescription,
+    images: [image],
+  },
+};
+
+const breadcrumbSchema = {
+  '@context': 'https://schema.org',
+  '@type': 'BreadcrumbList',
+  itemListElement: [
+    { '@type': 'ListItem', position: 1, name: siteConfig.name, item: siteConfig.url },
+    { '@type': 'ListItem', position: 2, name: 'Nyuton interpolyatsiyasi', item: `${siteConfig.url}${route}` },
+  ],
+};
+
+export default function NewtonInterpolationPage() {
+  return (
+    <>
+      <MethodJsonLd
+        name="Nyuton interpolyatsiyasi"
+        alternateName="Newton interpolation / Интерполяция Ньютона"
+        description={pageDescription}
+        path={route}
+        formula="P_{n-1}(x) = a_0 + a_1(x-x_0) + ... + a_{n-1}(x-x_0)...(x-x_{n-2})"
+      />
+      <script
+        type="application/ld+json"
+        dangerouslySetInnerHTML={{ __html: JSON.stringify(breadcrumbSchema).replace(/</g, '\\u003c') }}
+      />
+      <NewtonInterpolationMethod />
+    </>
+  );
+}
diff --git a/components/app-shell.tsx b/components/app-shell.tsx
index 16e8eae..4535b27 100644
--- a/components/app-shell.tsx
+++ b/components/app-shell.tsx
@@ -86,34 +86,40 @@ export function AppShell({
         code: '08',
         icon: Spline,
       },
+      {
+        href: '/methods/newton-interpolation',
+        label: t('nav.newtonInterpolation'),
+        code: '09',
+        icon: Spline,
+      },
       {
         href: '/methods/rectangles',
         label: t('nav.rectangles'),
-        code: '09',
+        code: '10',
         icon: Sigma,
       },
       {
         href: '/methods/trapezoid',
         label: t('nav.trapezoid'),
-        code: '10',
+        code: '11',
         icon: Sigma,
       },
       {
         href: '/methods/simpson',
         label: t('nav.simpson'),
-        code: '11',
+        code: '12',
         icon: Sigma,
       },
       {
         href: '/methods/euler',
         label: t('nav.euler'),
-        code: '12',
+        code: '13',
         icon: Activity,
       },
       {
         href: '/methods/runge-kutta',
         label: t('nav.rungeKutta'),
-        code: '13',
+        code: '14',
         icon: Activity,
       },
     ],
@@ -145,6 +151,8 @@ export function AppShell({
             ? t('nav.learn')
           : pathname.startsWith('/methods/bisection')
             ? t('nav.bisection')
+          : pathname.startsWith('/methods/newton-interpolation')
+            ? t('nav.newtonInterpolation')
             : pathname.startsWith('/methods/newton')
               ? t('nav.newton')
               : pathname.startsWith('/methods/sweep')
@@ -215,7 +223,9 @@ export function AppShell({
               const active =
                 item.href === '/'
                   ? pathname === '/'
-                  : pathname.startsWith(item.href);
+                  : item.href === '/methods/newton'
+                    ? pathname === item.href
+                    : pathname === item.href || pathname.startsWith(`${item.href}/`);
               return (
                 <Link
                   key={item.href}
diff --git a/components/dashboard/methods-hub.tsx b/components/dashboard/methods-hub.tsx
index 6d6ea88..4abb3c9 100644
--- a/components/dashboard/methods-hub.tsx
+++ b/components/dashboard/methods-hub.tsx
@@ -85,9 +85,22 @@ const METHODS: MethodDef[] = [
     icon: Spline,
     accentClass: 'text-indigo-600 dark:text-indigo-400 bg-indigo-500/10 border-indigo-500/25',
   },
+  {
+    id: 'newton-interpolation',
+    code: '05 / INT',
+    category: 'interpolation',
+    categoryKey: 'dashboard.cat_interpolation',
+    titleKey: 'nav.newtonInterpolation',
+    descKey: 'dashboard.desc_newtonInterpolation',
+    href: '/methods/newton-interpolation',
+    formulaTex: 'f[x_0,\\ldots,x_n] = \\frac{f[x_1,\\ldots,x_n]-f[x_0,\\ldots,x_{n-1}]}{x_n-x_0}',
+    orderTex: 'O(n^2)',
+    icon: Spline,
+    accentClass: 'text-indigo-600 dark:text-indigo-400 bg-indigo-500/10 border-indigo-500/25',
+  },
   {
     id: 'rectangles',
-    code: '05 / QUA',
+    code: '06 / QUA',
     category: 'quadrature',
     categoryKey: 'dashboard.cat_quadrature',
     titleKey: 'nav.rectangles',
@@ -100,7 +113,7 @@ const METHODS: MethodDef[] = [
   },
   {
     id: 'trapezoid',
-    code: '06 / QUA',
+    code: '07 / QUA',
     category: 'quadrature',
     categoryKey: 'dashboard.cat_quadrature',
     titleKey: 'nav.trapezoid',
@@ -113,7 +126,7 @@ const METHODS: MethodDef[] = [
   },
   {
     id: 'simpson',
-    code: '07 / QUA',
+    code: '08 / QUA',
     category: 'quadrature',
     categoryKey: 'dashboard.cat_quadrature',
     titleKey: 'nav.simpson',
@@ -126,7 +139,7 @@ const METHODS: MethodDef[] = [
   },
   {
     id: 'euler',
-    code: '08 / ODE',
+    code: '09 / ODE',
     category: 'odes',
     categoryKey: 'dashboard.cat_odes',
     titleKey: 'nav.euler',
@@ -139,7 +152,7 @@ const METHODS: MethodDef[] = [
   },
   {
     id: 'runge-kutta',
-    code: '09 / ODE',
+    code: '10 / ODE',
     category: 'odes',
     categoryKey: 'dashboard.cat_odes',
     titleKey: 'nav.rungeKutta',
diff --git a/components/newton-interpolation-method.tsx b/components/newton-interpolation-method.tsx
new file mode 100644
index 0000000..8954027
--- /dev/null
+++ b/components/newton-interpolation-method.tsx
@@ -0,0 +1,475 @@
+'use client';
+
+import { useState, type ReactNode } from 'react';
+import katex from 'katex';
+import 'katex/dist/katex.min.css';
+import { ArrowUpRight, Info, Minus, Plus } from 'lucide-react';
+import { useI18n } from '@/components/i18n-provider';
+import {
+  PsHero,
+  PsMap,
+  PsNote,
+  PsPanel,
+  PsPanelHead,
+  PsStrip,
+} from '@/components/ui/print-stream';
+import { ResultActions } from '@/components/ui/result-actions';
+import {
+  calculateNewtonInterpolation,
+  formatNewtonValue,
+  type NewtonInputPoint,
+  type NewtonInterpolationMode,
+  type NewtonInterpolationResult,
+} from '@/lib/newton-interpolation';
+
+const DEFAULT_POINTS: NewtonInputPoint[] = [
+  { x: '1', y: '1' },
+  { x: '2', y: '4' },
+  { x: '3', y: '9' },
+];
+const DEFAULT_TARGET = '1.5';
+const MAX_POINTS = 12;
+
+const EXAMPLES: {
+  labelKey: string;
+  points: NewtonInputPoint[];
+  targetX: string;
+  mode: NewtonInterpolationMode;
+}[] = [
+  {
+    labelKey: 'newtonInterpolation.example_1',
+    points: DEFAULT_POINTS,
+    targetX: DEFAULT_TARGET,
+    mode: 'divided',
+  },
+  {
+    labelKey: 'newtonInterpolation.example_2',
+    points: [
+      { x: '0', y: '0' },
+      { x: '1', y: '1' },
+      { x: '3', y: '10' },
+    ],
+    targetX: '2',
+    mode: 'divided',
+  },
+  {
+    labelKey: 'newtonInterpolation.example_3',
+    points: [
+      { x: '-1', y: '0' },
+      { x: '0', y: '2' },
+      { x: '1/2', y: '1.125' },
+      { x: '1', y: '0' },
+    ],
+    targetX: '1/4',
+    mode: 'divided',
+  },
+];
+
+function MathOutput({ tex, display = false }: { tex: string; display?: boolean }) {
+  let html: string;
+  try {
+    html = katex.renderToString(tex, { throwOnError: false, displayMode: display });
+  } catch {
+    html = tex;
+  }
+  return <span dangerouslySetInnerHTML={{ __html: html }} />;
+}
+
+function texNumber(value: number): string {
+  const text = formatNewtonValue(value, 8);
+  const match = /^(-?\d+(?:\.\d+)?)e([+-]?\d+)$/i.exec(text);
+  if (match) return `${match[1]}\\times 10^{${Number(match[2])}}`;
+  return text;
+}
+
+function dividedDifferenceLabel(start: number, end: number): string {
+  return start === end
+    ? `f[x_{${start}}]`
+    : `f[x_{${start}},\\ldots,x_{${end}}]`;
+}
+
+function newtonBasisPolynomialTex(
+  result: NewtonInterpolationResult,
+  termCount: number,
+): string {
+  const terms: string[] = [];
+  for (let order = 0; order < termCount; order += 1) {
+    const coefficient = result.coefficients[order];
+    const magnitude = texNumber(Math.abs(coefficient));
+    const factors = result.basisNodes[order]
+      .map((node) => `(x - (${texNumber(node)}))`)
+      .join('\\,');
+    const term = order === 0
+      ? magnitude
+      : `${magnitude}${factors ? `\\,${factors}` : ''}`;
+    if (terms.length === 0) {
+      terms.push(coefficient < 0 ? `-${term}` : term);
+    } else {
+      terms.push(`${coefficient < 0 ? '-' : '+'} ${term}`);
+    }
+  }
+  return terms.join(' ') || '0';
+}
+
+function expandedPolynomialTex(coefficients: number[]): string {
+  const terms: string[] = [];
+  for (let power = coefficients.length - 1; power >= 0; power -= 1) {
+    const coefficient = coefficients[power];
+    if (coefficient === 0) continue;
+    const absolute = Math.abs(coefficient);
+    const coefficientText = absolute === 1 && power > 0 ? '' : texNumber(absolute);
+    const variable = power === 0
+      ? ''
+      : power === 1
+        ? 'x'
+        : `x^{${power}}`;
+    const term = `${coefficientText}${variable}`;
+    if (terms.length === 0) {
+      terms.push(coefficient < 0 ? `-${term}` : term);
+    } else {
+      terms.push(`${coefficient < 0 ? '-' : '+'} ${term}`);
+    }
+  }
+  return terms.join(' ') || '0';
+}
+
+function PointInputTable({
+  points,
+  onChange,
+}: {
+  points: NewtonInputPoint[];
+  onChange: (index: number, axis: 'x' | 'y', value: string) => void;
+}) {
+  const { t } = useI18n();
+  return (
+    <div className="overflow-hidden rounded-lg border border-line">
+      <table className="w-full table-fixed border-collapse font-mono text-sm">
+        <thead className="bg-bg-2 text-muted">
+          <tr>
+            <th className="w-12 border-b border-line px-2 py-2 text-center text-xs font-semibold">i</th>
+            <th className="border-b border-line px-2 py-2 text-center font-semibold">xᵢ</th>
+            <th className="border-b border-line px-2 py-2 text-center font-semibold">yᵢ</th>
+          </tr>
+        </thead>
+        <tbody>
+          {points.map((point, index) => (
+            <tr key={index} className="border-b border-line last:border-b-0 hover:bg-bg-2/30">
+              <td className="px-2 py-2 text-center text-xs font-bold text-brand">{index}</td>
+              <td className="p-1">
+                <input
+                  aria-label={t('newtonInterpolation.point_x', { index: index + 1 })}
+                  value={point.x}
+                  onChange={(event) => onChange(index, 'x', event.target.value)}
+                  className="w-full min-w-0 rounded border border-line-2 bg-transparent px-2 py-2 text-center font-mono text-sm text-ink outline-none focus:border-brand focus:bg-bg"
+                  inputMode="decimal"
+                />
+              </td>
+              <td className="p-1">
+                <input
+                  aria-label={t('newtonInterpolation.point_y', { index: index + 1 })}
+                  value={point.y}
+                  onChange={(event) => onChange(index, 'y', event.target.value)}
+                  className="w-full min-w-0 rounded border border-line-2 bg-transparent px-2 py-2 text-center font-mono text-sm text-ink outline-none focus:border-brand focus:bg-bg"
+                  inputMode="decimal"
+                />
+              </td>
+            </tr>
+          ))}
+        </tbody>
+      </table>
+    </div>
+  );
+}
+
+function StepBlock({
+  number,
+  title,
+  children,
+}: {
+  number: string;
+  title: string;
+  children: ReactNode;
+}) {
+  return (
+    <section className="rounded-lg border border-line bg-bg-2 p-3 sm:p-4">
+      <h3 className="mb-3 flex items-start gap-2 font-mono text-xs font-bold uppercase tracking-wider text-brand">
+        <span className="shrink-0">{number} ·</span>
+        <span>{title}</span>
+      </h3>
+      {children}
+    </section>
+  );
+}
+
+export function NewtonInterpolationMethod() {
+  const { t } = useI18n();
+  const [points, setPoints] = useState<NewtonInputPoint[]>(DEFAULT_POINTS.map((point) => ({ ...point })));
+  const [targetX, setTargetX] = useState(DEFAULT_TARGET);
+  const [mode, setMode] = useState<NewtonInterpolationMode>('divided');
+  const [result, setResult] = useState<NewtonInterpolationResult | null>(() =>
+    calculateNewtonInterpolation(DEFAULT_POINTS, DEFAULT_TARGET, 'divided'),
+  );
+  const [error, setError] = useState('');
+
+  function calculate(inputPoints = points, inputTarget = targetX, inputMode = mode) {
+    setError('');
+    try {
+      setResult(calculateNewtonInterpolation(inputPoints, inputTarget, inputMode));
+    } catch (cause) {
+      setResult(null);
+      setError(
+        cause instanceof Error
+          ? t(cause.message)
+          : t('newtonInterpolation.err_general'),
+      );
+    }
+  }
+
+  function updatePoint(index: number, axis: 'x' | 'y', value: string) {
+    setPoints((current) =>
+      current.map((point, pointIndex) =>
+        pointIndex === index ? { ...point, [axis]: value } : point,
+      ),
+    );
+    setResult(null);
+    setError('');
+  }
+
+  function applyExample(index: number) {
+    const example = EXAMPLES[index];
+    const nextPoints = example.points.map((point) => ({ ...point }));
+    setPoints(nextPoints);
+    setTargetX(example.targetX);
+    setMode(example.mode);
+    calculate(nextPoints, example.targetX, example.mode);
+  }
+
+  function addPoint() {
+    if (points.length >= MAX_POINTS) return;
+    setPoints((current) => [...current, { x: '', y: '' }]);
+    setResult(null);
+    setError('');
+  }
+
+  function removePoint() {
+    if (points.length <= 2) return;
+    setPoints((current) => current.slice(0, -1));
+    setResult(null);
+    setError('');
+  }
+
+  const anchorIndex = result?.mode === 'finite' && result.direction === 'backward'
+    ? (result.points.length - 1)
+    : 0;
+
+  return (
+    <div className="relative mx-auto max-w-[1160px] px-4 pb-12">
+      <PsStrip eyebrow="INTERPOLATION / 010" id="NEWTON" name={t('newtonInterpolation.strip')} />
+      <PsHero
+        eyebrow={t('newtonInterpolation.hero_eyebrow')}
+        title={t('newtonInterpolation.hero_title')}
+        titleHighlight={t('newtonInterpolation.hero_highlight')}
+        lead={t('newtonInterpolation.hero_lead')}
+        conditionMain={<MathOutput tex={String.raw`P_{n-1}(x)=a_0+a_1(x-x_0)+\cdots+a_{n-1}\prod_{j=0}^{n-2}(x-x_j)`} />}
+        conditionSub={t('newtonInterpolation.hero_condition')}
+      />
+      <PsMap
+        eyebrow={t('newtonInterpolation.map_eyebrow')}
+        title={t('newtonInterpolation.map_title')}
+        steps={[
+          { n: '01', t: t('newtonInterpolation.map_1_title'), s: t('newtonInterpolation.map_1_desc') },
+          { n: '02', t: t('newtonInterpolation.map_2_title'), s: <MathOutput tex={String.raw`f[x_i,\ldots,x_{i+j}]=\frac{f[x_{i+1},\ldots,x_{i+j}]-f[x_i,\ldots,x_{i+j-1}]}{x_{i+j}-x_i}`} /> },
+          { n: '03', t: t('newtonInterpolation.map_3_title'), s: <MathOutput tex={String.raw`P_n(x)=a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots`} /> },
+          { n: '04', t: t('newtonInterpolation.map_4_title'), s: t('newtonInterpolation.map_4_desc') },
+        ]}
+      />
+
+      <div className="grid grid-cols-1 items-start gap-[18px] lg:grid-cols-[minmax(300px,.78fr)_minmax(0,1.4fr)]">
+        <PsPanel>
+          <PsPanelHead eyebrow={t('newtonInterpolation.input_eyebrow')} title={t('newtonInterpolation.input_title')} idx="01" />
+          <PsNote icon={<Info size={18} />}>
+            <p>{t('newtonInterpolation.note_points')} <code className="font-mono text-ink">1/2</code>, <code className="font-mono text-ink">-1</code>. {t('newtonInterpolation.note_unique')}</p>
+          </PsNote>
+          <form
+            onSubmit={(event) => {
+              event.preventDefault();
+              calculate();
+            }}
+            className="mt-4"
+          >
+            <PointInputTable points={points} onChange={updatePoint} />
+            <div className="my-3 flex items-center gap-2">
+              <span className="flex-1 text-xs text-muted">{t('newtonInterpolation.point_controls')}</span>
+              <button type="button" disabled={points.length >= MAX_POINTS} onClick={addPoint} aria-label={t('newtonInterpolation.add_point')} className="grid h-9 w-9 place-items-center rounded-md border border-line-2 bg-bg-2 text-ink hover:border-brand hover:text-brand disabled:opacity-40"><Plus size={17} /></button>
+              <button type="button" disabled={points.length <= 2} onClick={removePoint} aria-label={t('newtonInterpolation.remove_point')} className="grid h-9 w-9 place-items-center rounded-md border border-line-2 bg-bg-2 text-ink hover:border-brand hover:text-brand disabled:opacity-40"><Minus size={17} /></button>
+            </div>
+            <div className="grid gap-3 sm:grid-cols-2">
+              <label className="block text-[13px] font-semibold text-ink">
+                {t('newtonInterpolation.mode')}
+                <select value={mode} onChange={(event) => { setMode(event.target.value as NewtonInterpolationMode); setResult(null); setError(''); }} className="mt-2 w-full rounded-lg border border-line-2 bg-bg-2 p-3 font-mono text-sm text-ink outline-none focus:border-brand">
+                  <option value="divided">{t('newtonInterpolation.mode_divided')}</option>
+                  <option value="finite">{t('newtonInterpolation.mode_finite')}</option>
+                </select>
+              </label>
+              <label className="block text-[13px] font-semibold text-ink">
+                {t('newtonInterpolation.target_x')}
+                <input value={targetX} onChange={(event) => { setTargetX(event.target.value); setResult(null); setError(''); }} className="mt-2 w-full rounded-lg border border-line-2 bg-bg-2 p-3 font-mono text-sm text-ink outline-none focus:border-brand" inputMode="decimal" />
+              </label>
+            </div>
+            <div className="my-4 grid gap-2 sm:grid-cols-1">
+              {EXAMPLES.map((example, index) => (
+                <button key={example.labelKey} type="button" onClick={() => applyExample(index)} className="min-h-10 rounded-md border border-line bg-bg-2 px-3 py-2 text-left text-xs font-semibold text-muted transition-colors hover:border-brand hover:text-brand">
+                  {t(example.labelKey)}
+                </button>
+              ))}
+            </div>
+            <button type="submit" className="flex min-h-[50px] w-full items-center justify-center gap-2 rounded-lg border border-ink bg-ink px-3 py-3 text-sm font-bold text-panel transition hover:-translate-y-0.5 hover:border-brand hover:bg-brand">
+              {t('newtonInterpolation.calculate')} <ArrowUpRight size={17} />
+            </button>
+            {error && <div className="mt-4 rounded-lg border border-neg/30 bg-neg/10 p-3.5 font-mono text-xs font-semibold text-neg" role="alert">{error}</div>}
+          </form>
+        </PsPanel>
+
+        <div className="grid min-w-0 gap-[18px]">
+          {result ? (
+            <>
+              <ResultActions
+                summaryText={`P(${formatNewtonValue(result.targetX, 8)}) = ${formatNewtonValue(result.interpolatedValue, 10)}`}
+                csvData={result.differenceSteps.map((step) => ({
+                  order: step.order,
+                  i: step.index,
+                  left_value: formatNewtonValue(step.leftValue),
+                  right_value: formatNewtonValue(step.rightValue),
+                  denominator: step.denominator === null ? '' : formatNewtonValue(step.denominator),
+                  result: formatNewtonValue(step.result),
+                }))}
+                csvFileName="newton-interpolation-steps.csv"
+                imageFileName="newton-interpolation-steps.png"
+              />
+              <PsPanel>
+                <PsPanelHead eyebrow={t('newtonInterpolation.steps_eyebrow')} title={t('newtonInterpolation.steps_title')} idx="02" />
+                <div className="space-y-4">
+                  <StepBlock number="01" title={t('newtonInterpolation.step_points')}>
+                    <p className="mb-3 text-sm leading-relaxed text-muted">{t('newtonInterpolation.step_points_desc', { count: result.points.length })}</p>
+                    <div className="overflow-x-auto rounded-lg border border-line bg-panel">
+                      <table className="w-full min-w-[360px] border-collapse font-mono text-sm">
+                        <thead className="bg-bg-2 text-muted"><tr><th className="px-3 py-2 text-left">i</th><th className="px-3 py-2 text-right">xᵢ</th><th className="px-3 py-2 text-right">yᵢ</th></tr></thead>
+                        <tbody>{result.points.map((point) => <tr key={point.index} className="border-t border-line"><td className="px-3 py-2 text-brand">{point.index}</td><td className="px-3 py-2 text-right">{formatNewtonValue(point.x)}</td><td className="px-3 py-2 text-right">{formatNewtonValue(point.y)}</td></tr>)}</tbody>
+                      </table>
+                    </div>
+                    <p className="mt-3 rounded-md border border-cyan/20 bg-cyan-soft px-3 py-2 text-xs leading-relaxed text-ink-2">{t('newtonInterpolation.step_points_check')}</p>
+                  </StepBlock>
+
+                  <StepBlock number="02" title={t('newtonInterpolation.step_differences')}>
+                    <p className="mb-3 text-sm leading-relaxed text-muted">{result.mode === 'divided' ? t('newtonInterpolation.divided_intro') : t('newtonInterpolation.finite_intro', { direction: t(`newtonInterpolation.${result.direction}`), h: formatNewtonValue(result.spacing ?? 0) })}</p>
+                    <div className="mb-4 overflow-x-auto rounded-lg border border-dashed border-line-2 bg-panel p-3 text-center text-brand">
+                      {result.mode === 'divided'
+                        ? <MathOutput display tex={String.raw`f[x_i,\ldots,x_{i+j}]=\frac{f[x_{i+1},\ldots,x_{i+j}]-f[x_i,\ldots,x_{i+j-1}]}{x_{i+j}-x_i}`} />
+                        : result.direction === 'backward'
+                          ? <MathOutput display tex={String.raw`\nabla^j y_i=\nabla^{j-1}y_i-\nabla^{j-1}y_{i-1}`} />
+                          : <MathOutput display tex={String.raw`\Delta^j y_i=\Delta^{j-1}y_{i+1}-\Delta^{j-1}y_i`} />}
+                    </div>
+                    <div className="space-y-2.5">
+                      {result.differenceSteps.map((step) => {
+                        const isBackward = result.mode === 'finite' && result.direction === 'backward';
+                        const formulaTex = result.mode === 'divided'
+                          ? `${dividedDifferenceLabel(step.index, step.index + step.order)}=\\frac{${dividedDifferenceLabel(step.index + 1, step.index + step.order)}-${dividedDifferenceLabel(step.index, step.index + step.order - 1)}}{x_{${step.index + step.order}}-x_{${step.index}}}=\\frac{${texNumber(step.leftValue)}-(${texNumber(step.rightValue)})}{${texNumber(step.endX)}-(${texNumber(step.startX)})}=${texNumber(step.result)}`
+                          : isBackward
+                            ? `\\nabla^{${step.order}}y_{${step.index}}=${step.order === 1 ? `y_{${step.index}}-y_{${step.index - 1}}` : `\\nabla^{${step.order - 1}}y_{${step.index}}-\\nabla^{${step.order - 1}}y_{${step.index - 1}}`}=${texNumber(step.leftValue)}-(${texNumber(step.rightValue)})=${texNumber(step.result)}`
+                            : `${step.order === 1 ? '\\Delta y' : `\\Delta^{${step.order}}y`}_{${step.index}}=${step.order === 1 ? `y_{${step.index + 1}}-y_{${step.index}}` : `\\Delta^{${step.order - 1}}y_{${step.index + 1}}-\\Delta^{${step.order - 1}}y_{${step.index}}`}=${texNumber(step.leftValue)}-(${texNumber(step.rightValue)})=${texNumber(step.result)}`;
+                        return (
+                          <div key={`${step.order}-${step.index}`} className="rounded-lg border border-line bg-bg-2 p-3">
+                            <p className="mb-1 text-[11px] font-bold uppercase tracking-wide text-muted">{t('newtonInterpolation.difference_cell', { order: step.order, index: step.index })}</p>
+                            <div className="overflow-x-auto text-center text-ink"><MathOutput display tex={formulaTex} /></div>
+                          </div>
+                        );
+                      })}
+                    </div>
+                    <details className="mt-4 rounded-lg border border-line bg-panel">
+                      <summary className="cursor-pointer px-3 py-2.5 text-sm font-semibold text-brand">{t('newtonInterpolation.table_summary')}</summary>
+                      <div className="overflow-x-auto p-3 pt-1">
+                        <table className="w-full min-w-[440px] border-collapse font-mono text-xs">
+                          <thead><tr className="border-b border-line text-muted"><th className="px-2 py-2 text-left">i</th><th className="px-2 py-2 text-right">xᵢ</th><th className="px-2 py-2 text-right">yᵢ</th>{result.points.slice(1).map((_, index) => <th key={index} className="px-2 py-2 text-right">{result.mode === 'divided' ? `f[·] ${index + 1}` : `${result.direction === 'backward' ? '∇' : 'Δ'}${index + 1}y`}</th>)}</tr></thead>
+                          <tbody>{result.points.map((point, row) => <tr key={point.index} className="border-t border-line"><td className="px-2 py-2 text-brand">{point.index}</td><td className="px-2 py-2 text-right">{formatNewtonValue(point.x)}</td><td className="px-2 py-2 text-right">{formatNewtonValue(point.y)}</td>{result.points.slice(1).map((_, orderIndex) => { const order = orderIndex + 1; const validCell = result.mode === 'finite' && result.direction === 'backward' ? row >= order : row < result.points.length - order; return <td key={order} className="px-2 py-2 text-right">{validCell ? formatNewtonValue(result.differenceTable[row][order]) : '—'}</td>; })}</tr>)}</tbody>
+                        </table>
+                      </div>
+                    </details>
+                  </StepBlock>
+
+                  <StepBlock number="03" title={t('newtonInterpolation.step_polynomial')}>
+                    <p className="mb-3 text-sm leading-relaxed text-muted">{t('newtonInterpolation.polynomial_intro')}</p>
+                    <div className="mb-3 overflow-x-auto rounded-lg border border-dashed border-line-2 bg-panel p-3 text-center text-brand"><MathOutput display tex={String.raw`P_{n-1}(x)=a_0+a_1(x-x_{m_0})+a_2(x-x_{m_0})(x-x_{m_1})+\cdots`} /></div>
+                    <div className="space-y-3">
+                      {result.coefficients.map((coefficient, order) => {
+                        let coefficientTex: string;
+                        if (order === 0) {
+                          coefficientTex = `a_0=y_{${anchorIndex}}=${texNumber(coefficient)}`;
+                        } else if (result.mode === 'divided') {
+                          coefficientTex = `a_${order}=${dividedDifferenceLabel(0, order)}=${texNumber(coefficient)}`;
+                        } else {
+                          const symbol = result.direction === 'forward' ? '\\Delta' : '\\nabla';
+                          const nodeIndex = result.direction === 'forward' ? 0 : result.points.length - 1;
+                          const differenceRow = result.direction === 'forward' ? 0 : result.points.length - 1;
+                          coefficientTex = `a_${order}=\\frac{${symbol}^{${order}}y_{${nodeIndex}}}{${order}!\\,h^{${order}}}=\\frac{${texNumber(result.differenceTable[differenceRow][order])}}{${order}!\\,(${texNumber(result.spacing ?? 0)})^{${order}}}=${texNumber(coefficient)}`;
+                        }
+                        return (
+                          <div key={order} className="rounded-lg border border-line bg-bg-2 p-3">
+                            <div className="overflow-x-auto text-center text-ink"><MathOutput display tex={coefficientTex} /></div>
+                            <div className="mt-2 overflow-x-auto border-t border-dashed border-line pt-2 text-center text-brand"><MathOutput display tex={`P_${order}(x)=${newtonBasisPolynomialTex(result, order + 1)}`} /></div>
+                          </div>
+                        );
+                      })}
+                    </div>
+                    <div className="mt-4 rounded-lg border border-brand/30 bg-brand-soft p-3">
+                      <p className="mb-1 text-xs font-bold uppercase tracking-wider text-muted">{t('newtonInterpolation.expanded_polynomial')}</p>
+                      <div className="overflow-x-auto text-center text-lg font-bold text-brand"><MathOutput display tex={`P_{${result.points.length - 1}}(x)=${expandedPolynomialTex(result.expandedCoefficients)}`} /></div>
+                    </div>
+                  </StepBlock>
+
+                  <StepBlock number="04" title={t('newtonInterpolation.step_evaluate')}>
+                    <p className="mb-3 text-sm leading-relaxed text-muted">{t('newtonInterpolation.evaluate_intro', { x: formatNewtonValue(result.targetX) })}</p>
+                    <div className="space-y-2.5">
+                      {result.evaluationTerms.map((term) => {
+                        const factors = term.nodes.map((node) => `(${texNumber(result.targetX)}-(${texNumber(node)}))`).join('\\cdot ') || '1';
+                        const termFormula = `T_${term.order}(${texNumber(result.targetX)})=${texNumber(term.coefficient)}\\cdot ${factors}=${texNumber(term.value)}`;
+                        const previous = term.order === 0 ? '0' : texNumber(result.evaluationTerms[term.order - 1].cumulative);
+                        return (
+                          <div key={term.order} className="rounded-lg border border-line bg-bg-2 p-3">
+                            <div className="overflow-x-auto text-center text-ink"><MathOutput display tex={termFormula} /></div>
+                            <div className="mt-2 overflow-x-auto border-t border-dashed border-line pt-2 text-center text-brand"><MathOutput display tex={`P_${term.order}(${texNumber(result.targetX)})=${previous}+(${texNumber(term.value)})=${texNumber(term.cumulative)}`} /></div>
+                          </div>
+                        );
+                      })}
+                    </div>
+                    <div className="mt-4 rounded-lg border border-cyan/30 bg-cyan-soft p-4 text-center">
+                      <p className="mb-1 text-xs font-bold uppercase tracking-wider text-muted">{t('newtonInterpolation.final_value')}</p>
+                      <MathOutput display tex={`P_{${result.points.length - 1}}(${texNumber(result.targetX)})=${texNumber(result.interpolatedValue)}`} />
+                    </div>
+                  </StepBlock>
+                </div>
+              </PsPanel>
+            </>
+          ) : (
+            <PsPanel>
+              <PsPanelHead eyebrow={t('newtonInterpolation.steps_eyebrow')} title={t('newtonInterpolation.steps_title')} idx="02" />
+              <p className="rounded-lg border border-line bg-bg-2 p-4 text-sm leading-relaxed text-muted">{error || t('newtonInterpolation.waiting')}</p>
+            </PsPanel>
+          )}
+        </div>
+      </div>
+      <PsPanel className="mt-[18px]">
+        <PsPanelHead eyebrow={t('newtonInterpolation.theory_eyebrow')} title={t('newtonInterpolation.theory_title')} idx="03" />
+        <div className="grid gap-4 md:grid-cols-2">
+          <div className="rounded-lg border border-line bg-bg-2 p-4">
+            <h2 className="mb-2 text-sm font-bold text-ink">{t('newtonInterpolation.theory_divided_title')}</h2>
+            <p className="text-sm leading-relaxed text-muted">{t('newtonInterpolation.theory_divided_desc')}</p>
+          </div>
+          <div className="rounded-lg border border-line bg-bg-2 p-4">
+            <h2 className="mb-2 text-sm font-bold text-ink">{t('newtonInterpolation.theory_finite_title')}</h2>
+            <p className="text-sm leading-relaxed text-muted">{t('newtonInterpolation.theory_finite_desc')}</p>
+          </div>
+        </div>
+      </PsPanel>
+    </div>
+  );
+}
diff --git a/components/seo/json-ld.tsx b/components/seo/json-ld.tsx
index ed4f9b1..e9bfc86 100644
--- a/components/seo/json-ld.tsx
+++ b/components/seo/json-ld.tsx
@@ -45,6 +45,7 @@ export function RootJsonLd() {
       'Eyler usuli (Oddiy differensial tenglamalarni yechish)',
       'Runge-Kutta 4-tartibli usuli (Yuqori aniqlikdagi ODU yechish)',
       'Lagranj interpolyatsiya koʻphadi',
+      'Nyuton interpolyatsiya ko‘phadi (bo‘lingan ayirmalar)',
       'Simpson parabolalar formulasi bilan integrallash',
       'Trapetsiyalar usuli bilan integrallash',
       'Toʻgʻri toʻrtburchaklar usuli bilan integrallash',
diff --git a/lib/config.ts b/lib/config.ts
index 74b2528..446fe5b 100644
--- a/lib/config.ts
+++ b/lib/config.ts
@@ -34,6 +34,8 @@ export const siteConfig = {
     'trapetsiya usuli',
     'togri tortburchaklar usuli',
     'lagranj interpolyatsiyasi',
+    'nyuton interpolyatsiyasi',
+    'nyutonning bo‘lingan ayirmalari',
     'progonka usuli',
     'quvib otish usuli',
     'tenglama ildizini topish',
@@ -55,6 +57,8 @@ export const siteConfig = {
     'метод трапеций',
     'метод прямоугольников',
     'интерполяция лагранжа',
+    'интерполяция ньютона',
+    'разделённые разности ньютона',
     'метод прогонки',
     'решение дифференциальных уравнений онлайн',
     'численное интегрирование',
@@ -71,6 +75,8 @@ export const siteConfig = {
     'trapezoidal rule',
     'rectangles rule',
     'lagrange interpolation',
+    'newton interpolation',
+    'newton divided differences',
     'thomas algorithm',
     'tridiagonal matrix algorithm',
     'math solver online',
diff --git a/lib/i18n/locales/en.json b/lib/i18n/locales/en.json
index b7fbba9..b8b89a5 100644
--- a/lib/i18n/locales/en.json
+++ b/lib/i18n/locales/en.json
@@ -16,7 +16,8 @@
     "v2_tooltip": "Open legacy v2.sonli-usullar.uz version",
     "sweep": "Thomas Method",
     "rungeKutta": "Runge–Kutta 4th Order",
-    "learn": "Uzbek learning guides"
+    "learn": "Uzbek learning guides",
+    "newtonInterpolation": "Newton Interpolation"
   },
   "about": {
     "strip": "DOC-002 / ABOUT",
@@ -77,7 +78,7 @@
     "hero_title_2": ", step by step.",
     "hero_lead": "Learn numerical methods through real interactive computation, not just theory.",
     "stat1_label": "Active methods",
-    "stat1_val": "09 Methods",
+    "stat1_val": "10 Methods",
     "stat2_label": "Domains",
     "stat2_val": "5 Categories",
     "stat3_label": "Compute environment",
@@ -85,7 +86,7 @@
     "overview": "Workspace status",
     "catalog_eyebrow": "02 / METHODS HUB",
     "catalog_title": "Interactive Numerical Methods Catalog",
-    "catalog_desc": "Explore all 9 core numerical computation algorithms, their mathematical formulas, and live interactive labs.",
+    "catalog_desc": "Explore all 10 core numerical computation algorithms, their mathematical formulas, and live interactive labs.",
     "cat_all": "All",
     "cat_equations": "Equations",
     "cat_interpolation": "Interpolation",
@@ -109,7 +110,8 @@
     "learn_cta": "Read the guides",
     "desc_sweep": "Solve tridiagonal linear systems with forward and backward sweeps.",
     "cat_linear": "Linear systems",
-    "desc_rungeKutta": "Solve a first-order initial value problem with the classical fourth-order Runge–Kutta method."
+    "desc_rungeKutta": "Solve a first-order initial value problem with the classical fourth-order Runge–Kutta method.",
+    "desc_newtonInterpolation": "Build the Newton polynomial from divided differences, compare finite differences, and evaluate P(x*) step by step."
   },
   "actions": {
     "tools_badge": "EXPORT / TOOLS",
@@ -724,5 +726,70 @@
   },
   "a11y": {
     "backToTop": "Back to top"
+  },
+  "newtonInterpolation": {
+    "strip": "Newton Interpolation",
+    "hero_eyebrow": "INTERPOLATION / NEWTON",
+    "hero_title": "Newton interpolation",
+    "hero_highlight": "polynomial",
+    "hero_lead": "Enter data nodes. Construct the polynomial with divided differences or equally spaced finite differences, then see exactly how the value at any x is obtained.",
+    "hero_condition": "Divided differences accept any distinct x nodes. Finite differences require equally spaced x nodes.",
+    "map_eyebrow": "CALCULATION PATH",
+    "map_title": "How is the answer built?",
+    "map_1_title": "Validate the nodes",
+    "map_1_desc": "Read each (xᵢ, yᵢ) point and make sure no x value is repeated.",
+    "map_2_title": "Calculate differences step by step",
+    "map_3_title": "Build the polynomial",
+    "map_4_title": "Evaluate at x*",
+    "map_4_desc": "Evaluate every factor and term separately, then add it to the running sum.",
+    "input_eyebrow": "INPUT / NODES",
+    "input_title": "Enter interpolation points",
+    "note_points": "Enter xᵢ and yᵢ as real numbers or fractions such as 1/2.",
+    "note_unique": "Every xᵢ value must be distinct.",
+    "point_x": "x value for point {index}",
+    "point_y": "y value for point {index}",
+    "point_controls": "Add a node or remove the last one",
+    "add_point": "Add point",
+    "remove_point": "Remove last point",
+    "mode": "Difference method",
+    "mode_divided": "Divided differences",
+    "mode_finite": "Finite differences",
+    "target_x": "Evaluate at x*",
+    "calculate": "Calculate step by step",
+    "example_1": "Example 1 · y=x², equal spacing (1, 2, 3), x*=1.5",
+    "example_2": "Example 2 · nonuniform nodes (divided), x*=2",
+    "example_3": "Example 3 · classic v2 nodes, x*=1/4",
+    "steps_eyebrow": "PRINT STREAM / CALCULATION",
+    "steps_title": "Step-by-step solution",
+    "step_points": "Input nodes",
+    "step_points_desc": "The calculation uses {count} nodes. Their x coordinates must be distinct.",
+    "step_points_check": "Why check this? Divided differences divide by xⱼ−xᵢ; repeated x values make the denominator zero.",
+    "step_differences": "Fill the difference table by hand",
+    "divided_intro": "Each higher-order difference uses two neighboring differences from the previous order. Divide their difference by the span between the corresponding x nodes.",
+    "finite_intro": "The nodes have equal spacing h={h}. Each new {direction} finite difference is the difference of two neighboring entries from the previous order.",
+    "forward": "forward",
+    "backward": "backward",
+    "difference_cell": "Order {order} · i={index}",
+    "table_summary": "View the complete difference table",
+    "step_polynomial": "Construct the Newton polynomial",
+    "polynomial_intro": "Each coefficient aᵢ comes from the difference table. Add one coefficient and its new product factor at each stage to build the polynomial.",
+    "expanded_polynomial": "Expanded polynomial",
+    "step_evaluate": "Evaluate at x*",
+    "evaluate_intro": "Substitute x*={x} into the polynomial basis and calculate each term in sequence.",
+    "final_value": "Final interpolated value",
+    "theory_eyebrow": "THEORY / GUIDE",
+    "theory_title": "Which difference method should I use?",
+    "theory_divided_title": "Divided differences",
+    "theory_divided_desc": "Nodes do not need equal spacing. Newton divided differences work for any distinct x values and make it straightforward to add another node.",
+    "theory_finite_title": "Finite differences",
+    "theory_finite_desc": "Use only with equally spaced nodes. The table gives finite differences; the forward or backward formula is selected based on whether x* is closer to the start or end.",
+    "waiting": "The initial example is shown. After changing nodes or the method, calculate again to get an updated solution.",
+    "err_points_min": "Enter at least 2 points.",
+    "err_points_max": "You can enter at most 12 points.",
+    "err_real": "x, y, and x* values must be finite real numbers or fractions such as 1/2.",
+    "err_duplicate": "The x coordinates must be distinct.",
+    "err_spacing": "Finite differences require equal spacing between all x nodes. Choose divided differences instead.",
+    "err_overflow": "The calculation exceeded the supported numeric range. Try smaller values or fewer nodes.",
+    "err_general": "Could not calculate. Check the entered points."
   }
 }
diff --git a/lib/i18n/locales/ru.json b/lib/i18n/locales/ru.json
index 084aeb5..8e3b8b3 100644
--- a/lib/i18n/locales/ru.json
+++ b/lib/i18n/locales/ru.json
@@ -16,7 +16,8 @@
     "v2_tooltip": "Открыть старую версию v2.sonli-usullar.uz",
     "sweep": "Метод прогонки",
     "rungeKutta": "Метод Рунге — Кутты 4-го порядка",
-    "learn": "Учебник на узбекском"
+    "learn": "Учебник на узбекском",
+    "newtonInterpolation": "Интерполяция Ньютона"
   },
   "about": {
     "strip": "DOC-002 / О ПРОЕКТЕ",
@@ -77,7 +78,7 @@
     "hero_title_2": ", понимая каждый шаг.",
     "hero_lead": "Изучайте численные методы не только в теории, но и с помощью интерактивных вычислений.",
     "stat1_label": "Активные методы",
-    "stat1_val": "09 методов",
+    "stat1_val": "10 методов",
     "stat2_label": "Области",
     "stat2_val": "5 категорий",
     "stat3_label": "Среда вычислений",
@@ -85,7 +86,7 @@
     "overview": "Состояние среды",
     "catalog_eyebrow": "02 / КАТАЛОГ МЕТОДОВ",
     "catalog_title": "Каталог интерактивных численных методов",
-    "catalog_desc": "Все 9 ключевых алгоритмов вычислений, их математические формулы и интерактивные лаборатории.",
+    "catalog_desc": "Все 10 ключевых алгоритмов вычислений, их математические формулы и интерактивные лаборатории.",
     "cat_all": "Все",
     "cat_equations": "Уравнения",
     "cat_interpolation": "Интерполяция",
@@ -109,7 +110,8 @@
     "learn_cta": "Читать учебник",
     "desc_sweep": "Решение трехдиагональных систем прямым и обратным ходом.",
     "cat_linear": "Линейные системы",
-    "desc_rungeKutta": "Решение задачи Коши для ОДУ классическим методом Рунге — Кутты четвертого порядка."
+    "desc_rungeKutta": "Решение задачи Коши для ОДУ классическим методом Рунге — Кутты четвертого порядка.",
+    "desc_newtonInterpolation": "Постройте многочлен Ньютона по разделённым разностям, сравните конечные разности и пошагово вычислите P(x*)."
   },
   "actions": {
     "tools_badge": "ЭКСПОРТ / ИНСТРУМЕНТЫ",
@@ -724,5 +726,70 @@
   },
   "a11y": {
     "backToTop": "Вернуться в начало страницы"
+  },
+  "newtonInterpolation": {
+    "strip": "Интерполяция Ньютона",
+    "hero_eyebrow": "ИНТЕРПОЛЯЦИЯ / НЬЮТОН",
+    "hero_title": "Интерполяционный",
+    "hero_highlight": "многочлен Ньютона",
+    "hero_lead": "Введите узловые точки. Постройте многочлен по разделённым разностям или конечным разностям при равноотстоящих узлах и проследите, откуда берётся значение в любой точке x.",
+    "hero_condition": "Разделённые разности допускают любые различные узлы x. Для конечных разностей шаг между узлами должен быть одинаковым.",
+    "map_eyebrow": "ХОД РАСЧЁТА",
+    "map_title": "Как получается ответ?",
+    "map_1_title": "Проверить узлы",
+    "map_1_desc": "Считайте каждую точку (xᵢ, yᵢ) и убедитесь, что значения x не повторяются.",
+    "map_2_title": "Пошагово вычислить разности",
+    "map_3_title": "Построить многочлен",
+    "map_4_title": "Вычислить при x*",
+    "map_4_desc": "Отдельно вычислите каждый множитель и член, затем прибавьте его к текущей сумме.",
+    "input_eyebrow": "ВВОД / УЗЛЫ",
+    "input_title": "Введите точки интерполяции",
+    "note_points": "Введите xᵢ и yᵢ как действительные числа или дроби, например 1/2.",
+    "note_unique": "Все значения xᵢ должны различаться.",
+    "point_x": "Значение x точки {index}",
+    "point_y": "Значение y точки {index}",
+    "point_controls": "Добавить узел или удалить последний",
+    "add_point": "Добавить точку",
+    "remove_point": "Удалить последнюю точку",
+    "mode": "Метод разностей",
+    "mode_divided": "Разделённые разности",
+    "mode_finite": "Конечные разности",
+    "target_x": "Вычислить при x*",
+    "calculate": "Вычислить пошагово",
+    "example_1": "Пример 1 · y=x², равный шаг (1, 2, 3), x*=1.5",
+    "example_2": "Пример 2 · неравномерные узлы (разделённые), x*=2",
+    "example_3": "Пример 3 · классические узлы v2, x*=1/4",
+    "steps_eyebrow": "PRINT STREAM / РАСЧЁТ",
+    "steps_title": "Пошаговое решение",
+    "step_points": "Введённые узлы",
+    "step_points_desc": "Используются {count} узла. Их координаты x должны различаться.",
+    "step_points_check": "Зачем это нужно? В разделённых разностях делим на xⱼ−xᵢ; при совпадающих x знаменатель равен нулю.",
+    "step_differences": "Последовательно заполнить таблицу разностей",
+    "divided_intro": "Каждая разность следующего порядка строится по двум соседним разностям предыдущего порядка. Их разность делится на расстояние между соответствующими узлами x.",
+    "finite_intro": "Узлы равноотстоящие с шагом h={h}. Каждая новая {direction} конечная разность равна разности двух соседних элементов предыдущего порядка.",
+    "forward": "прямая",
+    "backward": "обратная",
+    "difference_cell": "Порядок {order} · i={index}",
+    "table_summary": "Показать полную таблицу разностей",
+    "step_polynomial": "Построить многочлен Ньютона",
+    "polynomial_intro": "Каждый коэффициент aᵢ берётся из таблицы разностей. На каждом этапе к предыдущему многочлену добавляются коэффициент и новый множитель.",
+    "expanded_polynomial": "Развёрнутый многочлен",
+    "step_evaluate": "Вычислить при x*",
+    "evaluate_intro": "Подставьте x*={x} в базис многочлена и вычислите каждый член по порядку.",
+    "final_value": "Итоговое значение интерполяции",
+    "theory_eyebrow": "ТЕОРИЯ / СПРАВКА",
+    "theory_title": "Какой метод разностей выбрать?",
+    "theory_divided_title": "Разделённые разности",
+    "theory_divided_desc": "Расстояния между узлами могут быть разными. Разделённые разности Ньютона подходят для любых различных x и позволяют удобно добавлять новые узлы.",
+    "theory_finite_title": "Конечные разности",
+    "theory_finite_desc": "Используйте только для равноотстоящих узлов. Таблица даёт прямые или обратные разности; формула выбирается автоматически по близости x* к началу или концу таблицы.",
+    "waiting": "Показан начальный пример. После изменения узлов или метода выполните расчёт повторно.",
+    "err_points_min": "Введите не менее 2 точек.",
+    "err_points_max": "Можно ввести не более 12 точек.",
+    "err_real": "x, y и x* должны быть конечными действительными числами или дробями, например 1/2.",
+    "err_duplicate": "Координаты x должны различаться.",
+    "err_spacing": "Для конечных разностей шаг между всеми узлами x должен быть одинаковым. Вместо этого выберите разделённые разности.",
+    "err_overflow": "Расчёт вышел за поддерживаемый числовой диапазон. Попробуйте меньшие значения или меньшее число узлов.",
+    "err_general": "Не удалось выполнить расчёт. Проверьте введённые точки."
   }
 }
diff --git a/lib/i18n/locales/uz.json b/lib/i18n/locales/uz.json
index 76c0a4a..4f77981 100644
--- a/lib/i18n/locales/uz.json
+++ b/lib/i18n/locales/uz.json
@@ -16,7 +16,8 @@
     "v2_tooltip": "Eski v2.sonli-usullar.uz talqiniga o'tish",
     "sweep": "Haydash usuli",
     "rungeKutta": "Runge–Kutta 4-tartibli usuli",
-    "learn": "O‘zbekcha qo‘llanma"
+    "learn": "O‘zbekcha qo‘llanma",
+    "newtonInterpolation": "Nyuton interpolyatsiyasi"
   },
   "about": {
     "strip": "DOC-002 / HAQIDA",
@@ -77,7 +78,7 @@
     "hero_title_2": ", har qadamni tushuning.",
     "hero_lead": "Sonli usullarni nazariya bilan emas, real interaktiv hisoblash orqali o'rganing.",
     "stat1_label": "Faol usullar",
-    "stat1_val": "09 ta metod",
+    "stat1_val": "10 ta metod",
     "stat2_label": "Hisoblash sohalari",
     "stat2_val": "5 ta toifa",
     "stat3_label": "Hisoblash muhiti",
@@ -85,7 +86,7 @@
     "overview": "Workspace holati",
     "catalog_eyebrow": "02 / METODLAR KATALOGI",
     "catalog_title": "Interaktiv sonli usullar laboratoriyasi",
-    "catalog_desc": "Barcha 9 ta asosiy hisoblash algoritmlari, ularning matematik formulalari va interaktiv parametrlari.",
+    "catalog_desc": "Barcha 10 ta asosiy hisoblash algoritmlari, ularning matematik formulalari va interaktiv parametrlari.",
     "cat_all": "Barchasi",
     "cat_equations": "Tenglamalar",
     "cat_interpolation": "Interpolyatsiya",
@@ -109,7 +110,8 @@
     "learn_cta": "Qo‘llanmani o‘qish",
     "desc_sweep": "Uch diagonalli chiziqli tenglamalar sistemasini to‘g‘ri va teskari yurish bilan yechish.",
     "cat_linear": "Chiziqli sistemalar",
-    "desc_rungeKutta": "Birinchi tartibli Koshi masalasini klassik to‘rtinchi tartibli Runge–Kutta usuli bilan yeching."
+    "desc_rungeKutta": "Birinchi tartibli Koshi masalasini klassik to‘rtinchi tartibli Runge–Kutta usuli bilan yeching.",
+    "desc_newtonInterpolation": "Bo‘lingan ayirmalar orqali Nyuton ko‘phadini tuzing, chekli ayirmalarni taqqoslang va P(x*) qiymatini qadam-baqadam hisoblang."
   },
   "actions": {
     "tools_badge": "EXPORT / TOOLS",
@@ -724,5 +726,70 @@
   },
   "a11y": {
     "backToTop": "Sahifa boshiga qaytish"
+  },
+  "newtonInterpolation": {
+    "strip": "Nyuton interpolyatsiyasi",
+    "hero_eyebrow": "INTERPOLYATSIYA / NYUTON",
+    "hero_title": "Nyuton interpolyatsion",
+    "hero_highlight": "ko‘phadi",
+    "hero_lead": "Tugun nuqtalarni kiriting. Bo‘lingan ayirmalar yoki teng oraliqli chekli ayirmalar orqali ko‘phadni quring va istalgan x dagi qiymat qayerdan kelganini ko‘ring.",
+    "hero_condition": "TBA usuli takrorlanmaydigan ixtiyoriy x tugunlarida ishlaydi. TCHA usuli uchun x tugunlar orasidagi masofa teng bo‘lishi kerak.",
+    "map_eyebrow": "HISOBLASH YO‘LI",
+    "map_title": "Javob qanday hosil bo‘ladi?",
+    "map_1_title": "Tugunlarni tekshirish",
+    "map_1_desc": "Har bir (xᵢ, yᵢ) nuqtani o‘qing va takroriy x qiymat yo‘qligini tekshiring.",
+    "map_2_title": "Ayirmalarni bosqichma-bosqich topish",
+    "map_3_title": "Ko‘phadni qurish",
+    "map_4_title": "x* da hisoblash",
+    "map_4_desc": "Har bir ko‘paytuvchi va hadni alohida hisoblab, qisman yig‘indiga qo‘shing.",
+    "input_eyebrow": "KIRITISH / TUGUNLAR",
+    "input_title": "Nuqtalarni kiriting",
+    "note_points": "xᵢ va yᵢ qiymatiga haqiqiy son yoki 1/2 kabi kasr kiriting.",
+    "note_unique": "xᵢ qiymatlar takrorlanmasligi kerak.",
+    "point_x": "{index}-nuqtaning x qiymati",
+    "point_y": "{index}-nuqtaning y qiymati",
+    "point_controls": "Tugun qo‘shish yoki oxirgisini o‘chirish",
+    "add_point": "Nuqta qo‘shish",
+    "remove_point": "Oxirgi nuqtani o‘chirish",
+    "mode": "Ayirma usuli",
+    "mode_divided": "Bo‘lingan ayirmalar (TBA)",
+    "mode_finite": "Chekli ayirmalar (TCHA)",
+    "target_x": "Hisoblanadigan x*",
+    "calculate": "Qadam-baqadam hisoblash",
+    "example_1": "Misol 1 · y=x², teng qadamlar (1, 2, 3), x*=1.5",
+    "example_2": "Misol 2 · teng bo‘lmagan tugunlar (TBA), x*=2",
+    "example_3": "Misol 3 · v2 dagi klassik nuqtalar, x*=1/4",
+    "steps_eyebrow": "PRINT STREAM / HISOBLASH",
+    "steps_title": "Qadam-baqadam yechim",
+    "step_points": "Kiritilgan tugunlar",
+    "step_points_desc": "Avval {count} ta tugun olindi. Ularning x koordinatalari turlicha bo‘lishi shart.",
+    "step_points_check": "Nega tekshiramiz? Bo‘lingan ayirmada xⱼ−xᵢ maxrajga kiradi; x lar teng bo‘lsa, maxraj nol bo‘ladi.",
+    "step_differences": "Ayirmalar jadvalini qo‘lda to‘ldirish",
+    "divided_intro": "Har bir yangi daraja oldingi darajadagi ikkita qo‘shni ayirma bilan topiladi. Ularning farqini x koordinatalari oralig‘iga bo‘lamiz.",
+    "finite_intro": "Tugunlar teng h={h} masofada. {direction} chekli ayirmalar jadvalining har bir yangi katagi yuqoridagi ikkita katak farqidan topiladi.",
+    "forward": "oldinga",
+    "backward": "orqaga",
+    "difference_cell": "{order}-tartib · i={index}",
+    "table_summary": "Barcha hisoblangan ayirmalarni jadvalda ko‘rish",
+    "step_polynomial": "Nyuton ko‘phadini qurish",
+    "polynomial_intro": "aᵢ koeffitsiyenti ayirmalar jadvalidan olinadi. Keyingi had har safar oldingi ko‘phadga bitta koeffitsiyent va yangi ko‘paytuvchini qo‘shib hosil qilinadi.",
+    "expanded_polynomial": "Ko‘phadning yoyilgan ko‘rinishi",
+    "step_evaluate": "x* da qiymatni hisoblash",
+    "evaluate_intro": "Endi x*={x} qiymatini ko‘phad asoslariga qo‘yib, har bir hadni ketma-ket hisoblaymiz.",
+    "final_value": "Yakuniy interpolatsion qiymat",
+    "theory_eyebrow": "NAZARIYA / QO‘LLANMA",
+    "theory_title": "Qaysi ayirma usulini tanlayman?",
+    "theory_divided_title": "Bo‘lingan ayirmalar (TBA)",
+    "theory_divided_desc": "Tugunlar oralig‘i teng bo‘lishi shart emas. Nyutonning bo‘lingan ayirmalari turlicha x qiymatlarida ishlaydi; yangi tugun qo‘shish ham qulay.",
+    "theory_finite_title": "Chekli ayirmalar (TCHA)",
+    "theory_finite_desc": "Faqat teng qadamli tugunlarda ishlaydi. Jadval chekli ayirmalarni beradi; x* jadval boshiga yaqin bo‘lsa oldinga, oxiriga yaqin bo‘lsa orqaga formula tanlanadi.",
+    "waiting": "Dastlabki misol ko‘rsatilgan. Tugun yoki usulni o‘zgartirsangiz, yangi javobni qayta hisoblang.",
+    "err_points_min": "Kamida 2 ta tugun kerak.",
+    "err_points_max": "Ko‘pi bilan 12 ta tugun kiritish mumkin.",
+    "err_real": "x, y va x* qiymatlari chekli haqiqiy son yoki 1/2 kabi kasr bo‘lishi kerak.",
+    "err_duplicate": "x koordinatalari takrorlanmasligi kerak.",
+    "err_spacing": "Chekli ayirmalar usuli uchun barcha x tugunlar orasidagi qadam teng bo‘lishi shart. Bo‘lingan ayirmalarni tanlang.",
+    "err_overflow": "Hisoblashda sonlar diapazoni oshib ketdi. Kichikroq sonlar yoki kamroq tugun kiriting.",
+    "err_general": "Hisoblab bo‘lmadi. Kiritilgan nuqtalarni tekshiring."
   }
 }
diff --git a/lib/newton-interpolation.ts b/lib/newton-interpolation.ts
new file mode 100644
index 0000000..5ef4679
--- /dev/null
+++ b/lib/newton-interpolation.ts
@@ -0,0 +1,247 @@
+import { math } from '@/lib/math';
+
+export type NewtonInterpolationMode = 'divided' | 'finite';
+export type NewtonDirection = 'forward' | 'backward';
+export type NewtonInputPoint = { x: string; y: string };
+export type NewtonPoint = { index: number; x: number; y: number };
+
+export interface NewtonDifferenceStep {
+  index: number;
+  order: number;
+  leftValue: number;
+  rightValue: number;
+  startX: number;
+  endX: number;
+  denominator: number | null;
+  result: number;
+}
+
+export interface NewtonEvaluationTerm {
+  order: number;
+  coefficient: number;
+  nodes: number[];
+  productAtTarget: number;
+  value: number;
+  cumulative: number;
+}
+
+export interface NewtonInterpolationResult {
+  mode: NewtonInterpolationMode;
+  direction: NewtonDirection | null;
+  spacing: number | null;
+  points: NewtonPoint[];
+  targetX: number;
+  differenceTable: number[][];
+  differenceSteps: NewtonDifferenceStep[];
+  coefficients: number[];
+  basisNodes: number[][];
+  evaluationTerms: NewtonEvaluationTerm[];
+  expandedCoefficients: number[];
+  interpolatedValue: number;
+}
+
+const MIN_POINTS = 2;
+const MAX_POINTS = 12;
+const MAX_INPUT_MAGNITUDE = 1e6;
+
+function parseReal(input: string): number {
+  try {
+    const value = math.evaluate(input);
+    if (
+      typeof value !== 'number' ||
+      !Number.isFinite(value) ||
+      Math.abs(value) > MAX_INPUT_MAGNITUDE
+    ) {
+      throw new Error('invalid');
+    }
+    return value;
+  } catch {
+    throw new Error('newtonInterpolation.err_real');
+  }
+}
+
+function factorial(value: number): number {
+  let result = 1;
+  for (let i = 2; i <= value; i += 1) result *= i;
+  return result;
+}
+
+function multiplyByLinearFactor(coefficients: number[], node: number): number[] {
+  const result = Array.from({ length: coefficients.length + 1 }, () => 0);
+  coefficients.forEach((coefficient, power) => {
+    result[power] -= coefficient * node;
+    result[power + 1] += coefficient;
+  });
+  return result;
+}
+
+function buildExpandedCoefficients(
+  coefficients: number[],
+  basisNodes: number[][],
+): number[] {
+  const expanded = Array.from({ length: coefficients.length }, () => 0);
+  coefficients.forEach((coefficient, index) => {
+    let basis = [1];
+    basisNodes[index].forEach((node) => {
+      basis = multiplyByLinearFactor(basis, node);
+    });
+    basis.forEach((value, power) => {
+      expanded[power] += coefficient * value;
+    });
+  });
+  const scale = Math.max(...expanded.map((value) => Math.abs(value)));
+  const tolerance = Number.EPSILON * scale * 16;
+  return expanded.map((value) => (Math.abs(value) <= tolerance ? 0 : value));
+}
+
+export function calculateNewtonInterpolation(
+  inputPoints: NewtonInputPoint[],
+  targetXInput: string,
+  mode: NewtonInterpolationMode,
+): NewtonInterpolationResult {
+  if (inputPoints.length < MIN_POINTS) {
+    throw new Error('newtonInterpolation.err_points_min');
+  }
+  if (inputPoints.length > MAX_POINTS) {
+    throw new Error('newtonInterpolation.err_points_max');
+  }
+
+  const points: NewtonPoint[] = inputPoints.map((point, index) => ({
+    index,
+    x: parseReal(point.x),
+    y: parseReal(point.y),
+  }));
+  if (new Set(points.map((point) => point.x)).size !== points.length) {
+    throw new Error('newtonInterpolation.err_duplicate');
+  }
+  const targetX = parseReal(targetXInput);
+  const count = points.length;
+  let spacing: number | null = null;
+  let direction: NewtonDirection | null = null;
+
+  if (mode === 'finite') {
+    spacing = points[1].x - points[0].x;
+    const equallySpaced = points.slice(1).every((point, index) => {
+      const step = point.x - points[index].x;
+      return Math.abs(step - spacing!) <= 1e-9 * Math.max(1, Math.abs(spacing!));
+    });
+    if (!equallySpaced) {
+      throw new Error('newtonInterpolation.err_spacing');
+    }
+    direction = Math.abs(targetX - points[0].x) <= Math.abs(targetX - points[count - 1].x)
+      ? 'forward'
+      : 'backward';
+  }
+
+  const differenceTable = Array.from({ length: count }, () =>
+    Array.from({ length: count }, () => 0),
+  );
+  points.forEach((point, index) => {
+    differenceTable[index][0] = point.y;
+  });
+
+  const differenceSteps: NewtonDifferenceStep[] = [];
+  const useBackwardDifferences = mode === 'finite' && direction === 'backward';
+  for (let order = 1; order < count; order += 1) {
+    const firstIndex = useBackwardDifferences ? order : 0;
+    const lastIndex = useBackwardDifferences ? count : count - order;
+    for (let index = firstIndex; index < lastIndex; index += 1) {
+      const leftValue = useBackwardDifferences
+        ? differenceTable[index][order - 1]
+        : differenceTable[index + 1][order - 1];
+      const rightValue = useBackwardDifferences
+        ? differenceTable[index - 1][order - 1]
+        : differenceTable[index][order - 1];
+      const numerator = leftValue - rightValue;
+      const denominator = mode === 'divided'
+        ? points[index + order].x - points[index].x
+        : null;
+      const value = denominator === null
+        ? numerator
+        : numerator / denominator;
+      if (!Number.isFinite(value)) {
+        throw new Error('newtonInterpolation.err_overflow');
+      }
+      differenceTable[index][order] = value;
+      differenceSteps.push({
+        index,
+        order,
+        leftValue,
+        rightValue,
+        startX: points[useBackwardDifferences ? index - order : index].x,
+        endX: points[useBackwardDifferences ? index : index + order].x,
+        denominator,
+        result: value,
+      });
+    }
+  }
+
+  const anchorIndex = mode === 'finite' && direction === 'backward'
+    ? count - 1
+    : 0;
+  const coefficients = [points[anchorIndex].y];
+  const basisNodes: number[][] = [[]];
+
+  for (let order = 1; order < count; order += 1) {
+    if (mode === 'divided') {
+      coefficients.push(differenceTable[0][order]);
+      basisNodes.push(points.slice(0, order).map((point) => point.x));
+    } else {
+      const rowIndex = direction === 'forward' ? 0 : count - 1;
+      const denominator = factorial(order) * Math.pow(spacing!, order);
+      coefficients.push(differenceTable[rowIndex][order] / denominator);
+      const nodes = direction === 'forward'
+        ? points.slice(0, order).map((point) => point.x)
+        : points.slice(count - order).map((point) => point.x).reverse();
+      basisNodes.push(nodes);
+    }
+  }
+
+  const evaluationTerms: NewtonEvaluationTerm[] = [];
+  let interpolatedValue = 0;
+  coefficients.forEach((coefficient, order) => {
+    const nodes = basisNodes[order];
+    const productAtTarget = nodes.reduce(
+      (product, node) => product * (targetX - node),
+      1,
+    );
+    const value = coefficient * productAtTarget;
+    interpolatedValue += value;
+    if (![productAtTarget, value, interpolatedValue].every(Number.isFinite)) {
+      throw new Error('newtonInterpolation.err_overflow');
+    }
+    evaluationTerms.push({
+      order,
+      coefficient,
+      nodes,
+      productAtTarget,
+      value,
+      cumulative: interpolatedValue,
+    });
+  });
+
+  const expandedCoefficients = buildExpandedCoefficients(coefficients, basisNodes);
+  if (!expandedCoefficients.every(Number.isFinite)) {
+    throw new Error('newtonInterpolation.err_overflow');
+  }
+
+  return {
+    mode,
+    direction,
+    spacing,
+    points,
+    targetX,
+    differenceTable,
+    differenceSteps,
+    coefficients,
+    basisNodes,
+    evaluationTerms,
+    expandedCoefficients,
+    interpolatedValue,
+  };
+}
+
+export function formatNewtonValue(value: number, digits = 8): string {
+  if (!Number.isFinite(value)) return '—';
+  return Number(value.toPrecision(digits)).toString();
+}
diff --git a/public/images/og/newton-interpolation.png b/public/images/og/newton-interpolation.png
new file mode 100644
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literal 0
HcmV?d00001

diff --git a/scripts/generate-og-images.ts b/scripts/generate-og-images.ts
index 6c3cbc4..3664898 100644
--- a/scripts/generate-og-images.ts
+++ b/scripts/generate-og-images.ts
@@ -314,6 +314,18 @@ export const ALL_OG_CONFIGS: MethodOgConfig[] = [
     badge: 'POLINOMIAL FIT',
     accentColor: '#ff6a2b',
   },
+  {
+    slug: 'newton-interpolation',
+    methodId: 'METHOD // 010',
+    category: 'APPROKSIMATSIYA // INTERPOLATION',
+    titleUz: 'Nyuton interpolyatsiyasi',
+    titleAlt: 'Интерполяция Ньютона • Newton Interpolation',
+    descriptionLine1: 'Bo‘lingan ayirmalar jadvalidan Newton ko‘phadini',
+    descriptionLine2: 'tuzing va P(x*) qiymatini to‘liq qadamlar bilan toping.',
+    formulaLatex: 'f[x_0,\\ldots,x_n] = \\frac{f[x_1,\\ldots,x_n] - f[x_0,\\ldots,x_{n-1}]}{x_n - x_0}',
+    badge: 'DIVIDED DIFFERENCES',
+    accentColor: '#ff6a2b',
+  },
   {
     slug: 'rectangles',
     methodId: 'METHOD // 006',

From 230da06c33759a73801e912d7439e0dce45ade02 Mon Sep 17 00:00:00 2001
From: hexjasur <hexjasur@gmail.com>
Date: Mon, 19 Oct 2026 20:28:34 +0500
Subject: [PATCH 2/3] =?UTF-8?q?feat(learn):=20add=20sourced=20academic=20n?=
 =?UTF-8?q?umerical-method=20guides=20=E2=9C=A8?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
---
 app/(workspace)/learn/[slug]/page.tsx |  415 +++++++--
 app/(workspace)/learn/page.tsx        |  420 ++++++++--
 app/layout.tsx                        |   12 +-
 lib/learning-guides.ts                | 1114 +++++++++++++++++++++----
 4 files changed, 1624 insertions(+), 337 deletions(-)

diff --git a/app/(workspace)/learn/[slug]/page.tsx b/app/(workspace)/learn/[slug]/page.tsx
index 5a172e4..c946e0b 100644
--- a/app/(workspace)/learn/[slug]/page.tsx
+++ b/app/(workspace)/learn/[slug]/page.tsx
@@ -1,7 +1,21 @@
 import type { Metadata } from 'next';
 import Link from 'next/link';
 import { notFound } from 'next/navigation';
-import { ArrowLeft, ArrowUpRight } from 'lucide-react';
+import {
+  ArrowLeft,
+  ArrowUpRight,
+  BookOpen,
+  BookOpenText,
+  Calculator,
+  CheckCircle2,
+  HelpCircle,
+  Library,
+  Lightbulb,
+  ShieldAlert,
+  Sparkles,
+} from 'lucide-react';
+import katex from 'katex';
+import 'katex/dist/katex.min.css';
 import { LearningGuideJsonLd } from '@/components/seo/json-ld';
 import { getLearningGuide, learningGuides } from '@/lib/learning-guides';
 import { siteConfig } from '@/lib/config';
@@ -49,6 +63,17 @@ export async function generateMetadata({
   };
 }
 
+function renderKatex(tex: string, displayMode = false): string {
+  try {
+    return katex.renderToString(tex, {
+      throwOnError: false,
+      displayMode,
+    });
+  } catch {
+    return tex;
+  }
+}
+
 export default async function LearningGuidePage({ params }: LearningGuidePageProps) {
   const { slug } = await params;
   const guide = getLearningGuide(slug);
@@ -59,111 +84,341 @@ export default async function LearningGuidePage({ params }: LearningGuidePagePro
     .slice(0, 3);
 
   return (
-    <article className="mx-auto max-w-4xl">
+    <article className="mx-auto max-w-4xl space-y-8 pb-12">
       <LearningGuideJsonLd guide={guide} />
-      <nav aria-label="Sahifa yo‘li" className="mb-6">
+
+      <nav aria-label="Sahifa yo‘li">
         <Link
           href="/learn"
           className="inline-flex min-h-10 items-center gap-2 text-sm font-semibold text-brand hover:underline"
         >
           <ArrowLeft aria-hidden="true" size={16} />
-          Barcha qo‘llanmalar
+          Barcha akademik qo‘llanmalar
         </Link>
       </nav>
 
-      <header className="rounded-xl border border-line bg-panel p-7 shadow-[var(--shadow)] sm:p-10">
-        <p className="font-mono text-xs font-bold uppercase tracking-[.14em] text-brand">
-          {guide.area} / O‘zbekcha qo‘llanma
-        </p>
+      {/* Hero Header */}
+      <header className="rounded-2xl border border-line bg-panel p-6 shadow-[var(--shadow)] sm:p-10">
+        <div className="flex flex-wrap items-center gap-2.5">
+          <span className="inline-flex items-center gap-1.5 rounded-md bg-brand-soft px-3 py-1 font-mono text-xs font-bold uppercase tracking-wider text-brand">
+            <BookOpenText aria-hidden="true" size={14} />
+            {guide.area}
+          </span>
+          <span className="inline-flex items-center gap-1 rounded-md border border-line bg-bg-2 px-2.5 py-1 text-xs font-medium text-muted">
+            <Library aria-hidden="true" size={13} />
+            Akademik darsliklar asosida
+          </span>
+        </div>
+
         <h1 className="mt-4 text-3xl font-black tracking-tight text-ink sm:text-5xl">
           {guide.title}
         </h1>
-        <p className="mt-5 text-base leading-7 text-muted sm:text-lg">{guide.summary}</p>
-        <Link
-          href={guide.calculatorHref}
-          className="mt-7 inline-flex min-h-11 items-center gap-2 rounded-lg bg-ink px-5 py-3 text-sm font-bold text-panel transition-colors hover:bg-brand hover:text-white"
-        >
-          Interaktiv hisoblagichni ochish
-          <ArrowUpRight aria-hidden="true" size={17} />
-        </Link>
+
+        <p className="mt-5 text-base leading-relaxed text-muted sm:text-lg">
+          {guide.summary}
+        </p>
+
+        <div className="mt-7 flex flex-wrap items-center gap-4">
+          <Link
+            href={guide.calculatorHref}
+            className="inline-flex min-h-11 items-center gap-2 rounded-xl bg-ink px-5 py-3 text-sm font-bold text-panel transition-all hover:bg-brand hover:text-white"
+          >
+            <Calculator aria-hidden="true" size={17} />
+            Interaktiv hisoblagichni ochish
+            <ArrowUpRight aria-hidden="true" size={17} />
+          </Link>
+          <a
+            href="#manbalar"
+            className="inline-flex min-h-11 items-center gap-2 rounded-xl border border-line bg-bg-2 px-4 py-3 text-sm font-semibold text-ink transition-colors hover:border-brand/40"
+          >
+            <Library aria-hidden="true" size={16} className="text-brand" />
+            Adabiyotlar ({guide.sources.length})
+          </a>
+        </div>
       </header>
 
-      <div className="mt-8 grid gap-6">
-        <section aria-labelledby="guide-formula" className="rounded-xl border border-line bg-bg-2 p-6 sm:p-8">
-          <h2 id="guide-formula" className="text-xl font-bold text-ink">Asosiy formula</h2>
-          <p className="mt-3 overflow-x-auto rounded-lg border border-line bg-panel p-4 font-mono text-sm leading-7 text-cyan">
-            {guide.formula}
-          </p>
-        </section>
+      {/* Asosiy formula */}
+      <section aria-labelledby="guide-formula" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <div className="flex items-center gap-2">
+          <Sparkles aria-hidden="true" className="text-cyan" size={20} />
+          <h2 id="guide-formula" className="text-xl font-bold text-ink">
+            Asosiy formula
+          </h2>
+        </div>
+        <div
+          className="mt-4 overflow-x-auto rounded-xl border border-dashed border-line-2 bg-bg-2 p-5 text-center text-cyan [&_.katex-display]:my-0 [&_.katex]:text-lg sm:[&_.katex]:text-xl"
+          dangerouslySetInnerHTML={{ __html: renderKatex(guide.formulaTex, true) }}
+        />
+        <p className="mt-3 font-mono text-xs text-muted">
+          Matnli ifoda: {guide.formula}
+        </p>
+      </section>
 
-        <section aria-labelledby="guide-steps" className="rounded-xl border border-line bg-panel p-6 sm:p-8">
-          <h2 id="guide-steps" className="text-xl font-bold text-ink">Qadam-baqadam algoritm</h2>
-          <ol className="mt-4 grid gap-3">
-            {guide.steps.map((step, index) => (
-              <li key={step} className="flex gap-3 text-sm leading-6 text-muted">
-                <span className="grid size-7 shrink-0 place-items-center rounded-full bg-brand-soft font-mono text-xs font-bold text-brand">
-                  {index + 1}
-                </span>
-                <span>{step}</span>
-              </li>
-            ))}
-          </ol>
+      {/* Nazariy asoslar va teoremalar */}
+      <section aria-labelledby="guide-theory" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <div className="flex items-center gap-2">
+          <BookOpen aria-hidden="true" className="text-brand" size={20} />
+          <h2 id="guide-theory" className="text-xl font-bold text-ink">
+            Nazariy asoslar va teoremalar
+          </h2>
+        </div>
+        <div className="mt-4 space-y-3">
+          {guide.theory.map((item, idx) => (
+            <div
+              key={idx}
+              className="flex gap-3.5 rounded-xl border border-line bg-bg-2 p-4 text-sm leading-relaxed text-muted"
+            >
+              <span className="grid size-6 shrink-0 place-items-center rounded-full bg-brand-soft font-mono text-xs font-bold text-brand">
+                {idx + 1}
+              </span>
+              <p className="text-ink-2">{item}</p>
+            </div>
+          ))}
+        </div>
+      </section>
+
+      {/* Yaqinlashish sharti va xatolik bahosi */}
+      {(guide.convergenceTex || guide.errorEstimateTex) && (
+        <section aria-labelledby="guide-convergence" className="grid gap-6 md:grid-cols-2">
+          {guide.convergenceTex && (
+            <div className="rounded-2xl border border-line bg-panel p-6">
+              <h3 id="guide-convergence" className="text-base font-bold text-ink">
+                Yaqinlashish sharti (Teorema)
+              </h3>
+              <p className="mt-2 text-xs text-muted leading-relaxed">
+                {guide.convergenceCondition}
+              </p>
+              <div
+                className="mt-3 overflow-x-auto rounded-lg border border-dashed border-line bg-bg-2 p-3 text-center text-sm text-cyan [&_.katex]:text-[1.05em]"
+                dangerouslySetInnerHTML={{ __html: renderKatex(guide.convergenceTex, true) }}
+              />
+            </div>
+          )}
+
+          {guide.errorEstimateTex && (
+            <div className="rounded-2xl border border-line bg-panel p-6">
+              <h3 className="text-base font-bold text-ink">
+                Xatolikni baholash formulasi
+              </h3>
+              <p className="mt-2 text-xs text-muted leading-relaxed">
+                Nazariy aprior va aposterior baholar:
+              </p>
+              <div
+                className="mt-3 overflow-x-auto rounded-lg border border-dashed border-line bg-bg-2 p-3 text-center text-sm text-cyan [&_.katex]:text-[1.05em]"
+                dangerouslySetInnerHTML={{ __html: renderKatex(guide.errorEstimateTex, true) }}
+              />
+            </div>
+          )}
         </section>
+      )}
+
+      {/* Platformadagi qabul qilingan standart va konvensiyalar (Ziddiyatlar yechimi) */}
+      <section
+        aria-labelledby="guide-conventions"
+        className="rounded-2xl border-2 border-brand/30 bg-brand-soft/20 p-6 sm:p-8"
+      >
+        <div className="flex items-center gap-2.5 text-brand">
+          <Lightbulb aria-hidden="true" size={22} />
+          <h2 id="guide-conventions" className="text-xl font-bold text-ink">
+            💡 Platformadagi qabul qilingan standart va kiritish qoidalari
+          </h2>
+        </div>
+        <p className="mt-2 text-sm leading-relaxed text-muted">
+          Klassik universitet darsliklarida turli mualliflar indekslar, formulalar va to‘xtash shartlarini
+          turlicha belgilashgan bo‘lishi mumkin. Quyida ushbu tafovutlar{' '}
+          <strong className="text-ink">Sonli Usullar interaktiv platformasida</strong> qanday ko‘rinishda kiritilishi
+          va hisoblanishi aniq tushuntirilgan:
+        </p>
+
+        <div className="mt-5 space-y-4">
+          {guide.platformConventions.map((conv, idx) => (
+            <div
+              key={idx}
+              className="rounded-xl border border-line bg-panel p-5 shadow-xs"
+            >
+              <h3 className="text-sm font-bold uppercase tracking-wider text-brand">
+                {conv.topic}
+              </h3>
+              <div className="mt-3 grid gap-3 text-sm sm:grid-cols-2">
+                <div className="rounded-lg border border-line bg-bg-2 p-3">
+                  <span className="block text-xs font-bold text-muted">Adabiyotlardagi an’anaviy holat:</span>
+                  <p className="mt-1 text-xs leading-relaxed text-ink-2">
+                    {conv.textbookStandard}
+                  </p>
+                </div>
+                <div className="rounded-lg border border-brand/30 bg-brand-soft/30 p-3">
+                  <span className="block text-xs font-bold text-brand">Platformadagi yechim va kiritish:</span>
+                  <p className="mt-1 text-xs leading-relaxed text-ink font-medium">
+                    {conv.platformImplementation}
+                  </p>
+                </div>
+              </div>
+              <p className="mt-3 text-xs leading-relaxed text-muted">
+                <strong className="text-ink">Nega shunday?</strong> {conv.whyChosen}
+              </p>
+            </div>
+          ))}
+        </div>
+      </section>
 
-        <section aria-labelledby="guide-example" className="rounded-xl border border-line bg-panel p-6 sm:p-8">
-          <h2 id="guide-example" className="text-xl font-bold text-ink">Ishlangan misol</h2>
-          <p className="mt-4 text-sm leading-6 text-muted">{guide.example.problem}</p>
-          <p className="mt-3 overflow-x-auto rounded-lg bg-bg-2 p-4 font-mono text-sm leading-6 text-ink">
+      {/* Qadam-baqadam algoritm */}
+      <section aria-labelledby="guide-steps" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <h2 id="guide-steps" className="text-xl font-bold text-ink">
+          Qadam-baqadam hisoblash algoritmi
+        </h2>
+        <ol className="mt-4 grid gap-3">
+          {guide.steps.map((step, index) => (
+            <li key={index} className="flex gap-3 text-sm leading-6 text-muted">
+              <span className="grid size-7 shrink-0 place-items-center rounded-full bg-brand-soft font-mono text-xs font-bold text-brand">
+                {index + 1}
+              </span>
+              <span className="text-ink-2">{step}</span>
+            </li>
+          ))}
+        </ol>
+      </section>
+
+      {/* Ishlangan namunaviy misol */}
+      <section aria-labelledby="guide-example" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <h2 id="guide-example" className="text-xl font-bold text-ink">
+          Ishlangan namunaviy misol (Darslikdan)
+        </h2>
+        <div className="mt-4 rounded-xl border border-line bg-bg-2 p-4">
+          <p className="text-xs font-mono font-bold uppercase tracking-wider text-brand">Masalaning qo‘yilishi</p>
+          <p className="mt-1 text-sm font-semibold text-ink">{guide.example.problem}</p>
+        </div>
+        <div className="mt-4">
+          <p className="text-xs font-mono font-bold uppercase tracking-wider text-muted">Hisob-kitob jarayoni</p>
+          <p className="mt-1 overflow-x-auto rounded-lg bg-bg-2 p-4 font-mono text-sm leading-relaxed text-ink">
             {guide.example.calculation}
           </p>
-          <p className="mt-3 text-sm font-semibold leading-6 text-ink">{guide.example.result}</p>
-        </section>
+        </div>
+        <div className="mt-4 rounded-xl border border-ok/30 bg-ok/5 p-4">
+          <p className="text-xs font-mono font-bold uppercase tracking-wider text-ok">Natija va tahlil</p>
+          <p className="mt-1 text-sm font-semibold text-ink">{guide.example.result}</p>
+        </div>
+      </section>
 
-        <section className="grid gap-6 md:grid-cols-2">
-          <div className="rounded-xl border border-line bg-panel p-6">
-            <h2 className="text-lg font-bold text-ink">Qachon ishlatiladi?</h2>
-            <ul className="mt-3 list-disc space-y-2 pl-5 text-sm leading-6 text-muted">
-              {guide.whenToUse.map((item) => <li key={item}>{item}</li>)}
-            </ul>
+      {/* Qo‘llanish va cheklovlar */}
+      <section className="grid gap-6 md:grid-cols-2">
+        <div className="rounded-2xl border border-line bg-panel p-6">
+          <div className="flex items-center gap-2 text-ok">
+            <CheckCircle2 aria-hidden="true" size={18} />
+            <h2 className="text-base font-bold text-ink">Qachon ishlatiladi?</h2>
           </div>
-          <div className="rounded-xl border border-line bg-panel p-6">
-            <h2 className="text-lg font-bold text-ink">Cheklovlari</h2>
-            <ul className="mt-3 list-disc space-y-2 pl-5 text-sm leading-6 text-muted">
-              {guide.limitations.map((item) => <li key={item}>{item}</li>)}
-            </ul>
+          <ul className="mt-3 list-disc space-y-2 pl-5 text-sm leading-relaxed text-muted">
+            {guide.whenToUse.map((item, idx) => (
+              <li key={idx}>{item}</li>
+            ))}
+          </ul>
+        </div>
+        <div className="rounded-2xl border border-line bg-panel p-6">
+          <div className="flex items-center gap-2 text-neg">
+            <ShieldAlert aria-hidden="true" size={18} />
+            <h2 className="text-base font-bold text-ink">Kamchiliklari va cheklovlari</h2>
           </div>
-        </section>
+          <ul className="mt-3 list-disc space-y-2 pl-5 text-sm leading-relaxed text-muted">
+            {guide.limitations.map((item, idx) => (
+              <li key={idx}>{item}</li>
+            ))}
+          </ul>
+        </div>
+      </section>
 
-        <section aria-labelledby="guide-questions" className="rounded-xl border border-line bg-panel p-6 sm:p-8">
-          <h2 id="guide-questions" className="text-xl font-bold text-ink">Ko‘p so‘raladigan savollar</h2>
-          <div className="mt-4 divide-y divide-line">
-            {guide.questions.map((item) => (
-              <div key={item.question} className="py-4 first:pt-0 last:pb-0">
-                <h3 className="font-semibold text-ink">{item.question}</h3>
-                <p className="mt-2 text-sm leading-6 text-muted">{item.answer}</p>
+      {/* Foydalanilgan adabiyotlar va manbalar */}
+      <section id="manbalar" aria-labelledby="guide-sources" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <div className="flex items-center gap-2.5">
+          <Library aria-hidden="true" className="text-brand" size={20} />
+          <h2 id="guide-sources" className="text-xl font-bold text-ink">
+            Foydalanilgan akademik adabiyotlar va manbalar
+          </h2>
+        </div>
+        <p className="mt-2 text-sm text-muted">
+          Ushbu qo‘llanmadagi nazariya, teoremalar va misollar loyihadagi quyidagi tasdiqlangan darsliklardan olindi:
+        </p>
+        <div className="mt-4 divide-y divide-line">
+          {guide.sources.map((src, idx) => (
+            <div key={idx} className="py-4 first:pt-2 last:pb-0">
+              <div className="flex flex-wrap items-baseline justify-between gap-2">
+                <h3 className="font-bold text-ink text-sm sm:text-base">
+                  {idx + 1}. {src.authors} ({src.year})
+                </h3>
+                <span className="font-mono text-xs text-brand font-semibold">
+                  {src.pages}
+                </span>
               </div>
-            ))}
-          </div>
-        </section>
+              <p className="mt-1 text-sm font-medium italic text-ink-2">
+                «{src.title}». {src.publisher}
+              </p>
+              <p className="mt-1 text-xs text-muted">
+                <strong>Bo‘lim:</strong> {src.chapterSection}
+              </p>
+              {src.note && (
+                <p className="mt-1 text-xs text-muted">
+                  <strong>Izoh:</strong> {src.note}
+                </p>
+              )}
+            </div>
+          ))}
+        </div>
+      </section>
+
+      {/* Ko‘p so‘raladigan savollar */}
+      <section aria-labelledby="guide-questions" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+        <div className="flex items-center gap-2.5">
+          <HelpCircle aria-hidden="true" className="text-cyan" size={20} />
+          <h2 id="guide-questions" className="text-xl font-bold text-ink">
+            Ko‘p so‘raladigan savollar va muammoli vaziyatlar
+          </h2>
+        </div>
+        <div className="mt-4 divide-y divide-line">
+          {guide.questions.map((item, idx) => (
+            <div key={idx} className="py-4 first:pt-2 last:pb-0">
+              <h3 className="font-semibold text-ink text-sm sm:text-base">
+                {item.question}
+              </h3>
+              <p className="mt-2 text-sm leading-relaxed text-muted">
+                {item.answer}
+              </p>
+            </div>
+          ))}
+        </div>
+      </section>
 
-        {relatedGuides.length > 0 && (
-          <nav aria-label="Tegishli qo‘llanmalar" className="rounded-xl border border-line bg-bg-2 p-6 sm:p-8">
-            <h2 className="text-xl font-bold text-ink">Tegishli mavzular</h2>
-            <ul className="mt-4 flex flex-wrap gap-3">
-              {relatedGuides.map((related) => (
-                <li key={related.slug}>
-                  <Link
-                    href={`/learn/${related.slug}`}
-                    className="inline-flex min-h-10 items-center rounded-lg border border-line bg-panel px-4 text-sm font-semibold text-brand hover:border-brand/50"
-                  >
-                    {related.title}
-                  </Link>
-                </li>
-              ))}
-            </ul>
-          </nav>
-        )}
+      {/* Bottom CTA to calculator */}
+      <div className="rounded-2xl border border-line bg-bg-2 p-6 sm:p-8 text-center">
+        <h3 className="text-xl font-bold text-ink">Nazariyani amalda sinab ko‘ring</h3>
+        <p className="mt-2 text-sm text-muted max-w-xl mx-auto">
+          Ushbu mavzu bo‘yicha istalgan parametrlaringizni interaktiv kalkulyatorga kiriting va qadam-baqadam natijalarni oling.
+        </p>
+        <Link
+          href={guide.calculatorHref}
+          className="mt-5 inline-flex min-h-11 items-center gap-2 rounded-xl bg-brand px-6 py-3 text-sm font-bold text-white transition-opacity hover:opacity-90"
+        >
+          <Calculator aria-hidden="true" size={17} />
+          {guide.title} kalkulyatorini ochish
+          <ArrowUpRight aria-hidden="true" size={17} />
+        </Link>
       </div>
+
+      {/* Related Guides */}
+      {relatedGuides.length > 0 && (
+        <nav aria-label="Tegishli qo‘llanmalar" className="rounded-2xl border border-line bg-panel p-6 sm:p-8">
+          <h2 className="text-lg font-bold text-ink">Mavzuga doir boshqa qo‘llanmalar</h2>
+          <ul className="mt-4 flex flex-wrap gap-3">
+            {relatedGuides.map((related) => (
+              <li key={related.slug}>
+                <Link
+                  href={`/learn/${related.slug}`}
+                  className="inline-flex min-h-10 items-center rounded-xl border border-line bg-bg-2 px-4 text-sm font-semibold text-brand transition-colors hover:border-brand/50 hover:bg-panel"
+                >
+                  {related.title}
+                </Link>
+              </li>
+            ))}
+          </ul>
+        </nav>
+      )}
     </article>
   );
 }
diff --git a/app/(workspace)/learn/page.tsx b/app/(workspace)/learn/page.tsx
index e42f5ef..af6f128 100644
--- a/app/(workspace)/learn/page.tsx
+++ b/app/(workspace)/learn/page.tsx
@@ -1,117 +1,381 @@
 import type { Metadata } from 'next';
 import Link from 'next/link';
-import { ArrowRight, BookOpenText } from 'lucide-react';
-import { learningGuides } from '@/lib/learning-guides';
+import {
+  ArrowRight,
+  ArrowUpRight,
+  BookOpen,
+  BookOpenText,
+  Calculator,
+  CheckCircle2,
+  GraduationCap,
+  Library,
+  Lightbulb,
+  Sparkles,
+} from 'lucide-react';
+import katex from 'katex';
+import 'katex/dist/katex.min.css';
+import { academicBibliography, learningGuides } from '@/lib/learning-guides';
 
 export const metadata: Metadata = {
-  title: 'Sonli usullar qo‘llanmasi: formulalar va misollar',
+  title: 'Sonli usullar akademik qo‘llanmasi: formulalar, manbalar va misollar',
   description:
-    'Sonli usullarni o‘zbek tilida o‘rganing: biseksiya, Nyuton, integrallash, interpolyatsiya va differensial tenglamalar bo‘yicha formulalar, ishlangan misollar va kalkulyatorlar.',
+    'O‘zbekiston darsliklari va ilmiy adabiyotlar asosida sonli usullarni o‘rganing: biseksiya, Nyuton, Simpson, trapetsiya, Lagranj, Eyler, RK4 va haydash usuli bo‘yicha nazariya, formulalar va interaktiv kalkulyatorlar.',
   keywords: [
     'sonli usullar qo‘llanmasi',
-    'sonli usullar misollar bilan',
+    'hisoblash usullari kitoblar',
+    'sonli usullar darsliklari',
     'hisoblash matematikasi',
-    'sonli usullar darslari',
+    'sonli usullar formulalari',
+    'sonli usullar adabiyotlar',
   ],
   alternates: {
     canonical: '/learn',
   },
   openGraph: {
     type: 'website',
-    title: 'Sonli usullar qo‘llanmasi | Sonli Usullar',
+    title: 'Sonli usullar akademik qo‘llanmasi | Sonli Usullar',
     description:
-      'O‘zbek tilidagi sonli usullar qo‘llanmasi: formulalar, ishlangan misollar va interaktiv hisoblash vositalari.',
+      'O‘zbekiston oliy ta’lim darsliklari asosidagi sonli usullar nazariyasi, formulalari, ishlangan misollari va interaktiv hisoblash vositalari.',
     url: '/learn',
     siteName: 'Sonli Usullar',
   },
 };
 
+function renderKatex(tex: string, displayMode = false): string {
+  try {
+    return katex.renderToString(tex, {
+      throwOnError: false,
+      displayMode,
+    });
+  } catch {
+    return tex;
+  }
+}
+
 export default function LearningGuideIndexPage() {
   const guideGroups = [...new Set(learningGuides.map((guide) => guide.area))];
 
   return (
-    <div className="mx-auto max-w-5xl space-y-10">
-      <header className="rounded-xl border border-line bg-panel p-7 shadow-[var(--shadow)] sm:p-10">
-        <p className="font-mono text-xs font-bold uppercase tracking-[.14em] text-brand">
-          O‘zbekcha nazariya / formulalar / misollar
-        </p>
+    <div className="mx-auto max-w-5xl space-y-12 pb-16">
+      {/* Hero Header */}
+      <header className="rounded-2xl border border-line bg-panel p-7 shadow-[var(--shadow)] sm:p-12">
+        <div className="flex flex-wrap items-center gap-2">
+          <span className="inline-flex items-center gap-1.5 rounded-md bg-brand-soft px-3 py-1 font-mono text-xs font-bold uppercase tracking-wider text-brand">
+            <GraduationCap aria-hidden="true" size={14} />
+            Akademik nazariya &amp; Amaliyot
+          </span>
+          <span className="inline-flex items-center gap-1 rounded-md border border-line bg-bg-2 px-2.5 py-1 text-xs font-medium text-muted">
+            <Library aria-hidden="true" size={13} />
+            O‘zbekiston darsliklari bazasi
+          </span>
+        </div>
+
         <h1 className="mt-4 text-3xl font-black tracking-tight text-ink sm:text-5xl">
           Sonli usullar qo‘llanmasi
         </h1>
-        <p className="mt-5 max-w-3xl text-base leading-7 text-muted sm:text-lg">
-          Sonli usullar matematik masalalarni kompyuterda taqribiy yechishga
-          yordam beradi. Quyidagi qo‘llanmalarda algoritmning g‘oyasi, formulasi,
-          qadam-baqadam misoli va qachon ishlatish mumkinligi sodda o‘zbek tilida
-          tushuntiriladi.
-        </p>
-        <p className="mt-3 max-w-3xl text-sm leading-6 text-muted">
-          Mavzuni o‘qib bo‘lgach, shu sahifadagi interaktiv kalkulyatorda
-          parametrlarni o‘zgartirib natijaga ta’sirini tekshirishingiz mumkin.
+
+        <p className="mt-5 max-w-3xl text-base leading-relaxed text-muted sm:text-lg">
+          Ushbu bo‘limda hisoblash matematikasining asosiy usullari O‘zbekiston oliy o‘quv yurtlari
+          darsliklari va ilmiy o‘quv qo‘llanmalari (Ismoilov, Irisqulov, Ismatullayev, Isroilov)
+          asosida tizimlashtirilgan. Har bir usul uchun nazariy teoremalar, formulalar, qadam-baqadam
+          algoritmlar va platformamizdagi hisoblash standartlari keltirilgan.
         </p>
+
+        <div className="mt-8 flex flex-wrap gap-4">
+          <a
+            href="#usullar-katalogi"
+            className="inline-flex min-h-11 items-center gap-2 rounded-xl bg-ink px-5 py-3 text-sm font-bold text-panel transition-all hover:bg-brand hover:text-white"
+          >
+            <BookOpen aria-hidden="true" size={17} />
+            Usullar katalogiga o‘tish
+          </a>
+          <a
+            href="#adabiyotlar-kutubxonasi"
+            className="inline-flex min-h-11 items-center gap-2 rounded-xl border border-line bg-bg-2 px-5 py-3 text-sm font-semibold text-ink transition-colors hover:border-brand/40"
+          >
+            <Library aria-hidden="true" size={17} className="text-brand" />
+            Adabiyotlar kutubxonasi ({academicBibliography.length})
+          </a>
+          <a
+            href="#standartlar-va-konvensiyalar"
+            className="inline-flex min-h-11 items-center gap-2 rounded-xl border border-line bg-bg-2 px-5 py-3 text-sm font-semibold text-ink transition-colors hover:border-brand/40"
+          >
+            <Lightbulb aria-hidden="true" size={17} className="text-cyan" />
+            Platforma standartlari
+          </a>
+        </div>
       </header>
 
-      <section aria-labelledby="choose-method" className="rounded-xl border border-line bg-bg-2 p-6 sm:p-8">
-        <h2 id="choose-method" className="text-xl font-bold text-ink sm:text-2xl">
-          Masalangizga mos usulni tanlang
-        </h2>
-        <p className="mt-2 max-w-3xl leading-7 text-muted">
-          Avval masala turini aniqlang: tenglama ildizini qidiryapsizmi,
-          integralni hisoblaysizmi yoki differensial tenglama yechyapsizmi?
+      {/* Masalangizga mos usulni tanlang */}
+      <section aria-labelledby="choose-method" className="rounded-2xl border border-line bg-bg-2 p-6 sm:p-8">
+        <div className="flex items-center gap-2.5">
+          <Sparkles aria-hidden="true" className="text-brand" size={20} />
+          <h2 id="choose-method" className="text-xl font-bold text-ink sm:text-2xl">
+            Masalangizga mos usulni tanlang
+          </h2>
+        </div>
+        <p className="mt-2 max-w-3xl text-sm leading-relaxed text-muted sm:text-base">
+          Avval masala turini aniqlang: algebraik tenglama ildizini qidiryapsizmi, aniq integralni
+          hisoblaysizmi, matritsali sistemani yechasizmi yoki Koshi masalasini diskretlashtirasizmi?
         </p>
-        <ul className="mt-5 grid gap-3 sm:grid-cols-2">
-          <li className="rounded-lg border border-line bg-panel p-4 text-sm leading-6 text-muted">
-            <strong className="text-ink">Tenglama ildizi:</strong> ishonchli
-            oraliq uchun <Link className="text-brand underline" href="/learn/biseksiya-usuli">biseksiya</Link>,
-            hosila va yaxshi boshlang‘ich taxmin bo‘lsa <Link className="text-brand underline" href="/learn/nyuton-usuli">Nyuton usuli</Link>.
-          </li>
-          <li className="rounded-lg border border-line bg-panel p-4 text-sm leading-6 text-muted">
-            <strong className="text-ink">Aniq integral:</strong> sodda baho
-            uchun <Link className="text-brand underline" href="/learn/trapetsiyalar-usuli">trapetsiyalar</Link>,
-            silliq funksiya va juft bo‘linish uchun <Link className="text-brand underline" href="/learn/simpson-usuli">Simpson usuli</Link>.
-          </li>
-          <li className="rounded-lg border border-line bg-panel p-4 text-sm leading-6 text-muted">
-            <strong className="text-ink">Differensial tenglama:</strong>{' '}
-            boshlang‘ich va sodda qadamlar uchun <Link className="text-brand underline" href="/learn/eyler-usuli">Eyler</Link>,
-            yuqoriroq tartibli baho uchun <Link className="text-brand underline" href="/learn/runge-kutta-4-usuli">RK4</Link>.
-          </li>
-          <li className="rounded-lg border border-line bg-panel p-4 text-sm leading-6 text-muted">
-            <strong className="text-ink">Jadval nuqtalari yoki sistema:</strong>{' '}
-            qiymatlar orasini to‘ldirish uchun <Link className="text-brand underline" href="/learn/lagranj-interpolyatsiyasi">Lagranj</Link>,
-            uch diagonalli sistemalar uchun <Link className="text-brand underline" href="/learn/haydash-usuli">haydash usuli</Link>.
-          </li>
-        </ul>
+
+        <div className="mt-6 grid gap-4 sm:grid-cols-2">
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <span className="font-mono text-xs font-bold uppercase tracking-wider text-brand">1. Tenglama ildizlari</span>
+            <h3 className="mt-1 text-base font-bold text-ink">Biseksiya va Nyuton usullari</h3>
+            <p className="mt-2 text-sm leading-relaxed text-muted">
+              Ishonchli va kafolatlangan oraliq uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/biseksiya-usuli">biseksiya</Link>,
+              hosila mavjud bo‘lib tezkor kvadratik yaqinlashish kerak bo‘lsa{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/nyuton-usuli">Nyuton (urinmalar)</Link> usuli.
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <span className="font-mono text-xs font-bold uppercase tracking-wider text-cyan">2. Aniq integrallar</span>
+            <h3 className="mt-1 text-base font-bold text-ink">Kvadraturalar (Trapetsiya va Simpson)</h3>
+            <p className="mt-2 text-sm leading-relaxed text-muted">
+              Tezkor va barqaror baho uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/trapetsiyalar-usuli">trapetsiyalar</Link> yoki{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/togri-tortburchaklar-usuli">to‘g‘ri to‘rtburchaklar</Link>,
+              silliq funksiyalar va 4-tartibli yuqori aniqlik uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/simpson-usuli">Simpson usuli</Link>.
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <span className="font-mono text-xs font-bold uppercase tracking-wider text-ok">3. Differensial tenglamalar</span>
+            <h3 className="mt-1 text-base font-bold text-ink">Koshi masalasi (Eyler va RK4)</h3>
+            <p className="mt-2 text-sm leading-relaxed text-muted">
+              Birinchi tartibli qadamlar uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/eyler-usuli">Eyler usuli</Link>,
+              muhandislik va ilmiy hisoblashlardagi standart yuqori aniqlik uchun esa{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/runge-kutta-4-usuli">Runge-Kutta 4 (RK4)</Link>.
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <span className="font-mono text-xs font-bold uppercase tracking-wider text-purple-600">4. Interpolyatsiya &amp; Matritsalar</span>
+            <h3 className="mt-1 text-base font-bold text-ink">Lagranj va Progonka usullari</h3>
+            <p className="mt-2 text-sm leading-relaxed text-muted">
+              Jadval nuqtalari orasini silliq to‘ldirish uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/lagranj-interpolyatsiyasi">Lagranj ko‘phadi</Link>,
+              uch diagonalli katta sistemalarni O(n) da yechish uchun{' '}
+              <Link className="text-brand font-semibold hover:underline" href="/learn/haydash-usuli">haydash (progonka)</Link> usuli.
+            </p>
+          </div>
+        </div>
       </section>
 
-      {guideGroups.map((area, groupIndex) => (
-        <section key={area} aria-labelledby={`guide-group-${groupIndex}`}>
-          <div className="mb-4 flex items-center gap-3">
-            <BookOpenText aria-hidden="true" className="text-brand" size={20} />
-            <h2 id={`guide-group-${groupIndex}`} className="text-xl font-bold text-ink">
-              {area}
+      {/* Usullar katalogi (Grouped by area) */}
+      <div id="usullar-katalogi" className="space-y-10">
+        <div className="border-b border-line pb-4">
+          <p className="font-mono text-xs font-bold uppercase tracking-wider text-brand">Qo‘llanmalar to‘plami</p>
+          <h2 className="text-2xl font-black text-ink sm:text-3xl">Usullar bo‘yicha to‘liq darslar</h2>
+        </div>
+
+        {guideGroups.map((area, groupIndex) => (
+          <section key={area} aria-labelledby={`guide-group-${groupIndex}`} className="space-y-4">
+            <div className="flex items-center gap-3">
+              <BookOpenText aria-hidden="true" className="text-brand" size={22} />
+              <h3 id={`guide-group-${groupIndex}`} className="text-xl font-bold text-ink">
+                {area}
+              </h3>
+            </div>
+
+            <div className="grid gap-5 md:grid-cols-2">
+              {learningGuides
+                .filter((guide) => guide.area === area)
+                .map((guide) => (
+                  <article
+                    key={guide.slug}
+                    className="flex flex-col justify-between rounded-2xl border border-line bg-panel p-6 transition-all hover:border-brand/50 hover:shadow-md"
+                  >
+                    <div>
+                      <div className="flex items-start justify-between gap-3">
+                        <h4 className="text-lg font-bold text-ink">
+                          <Link
+                            className="hover:text-brand focus-visible:text-brand"
+                            href={`/learn/${guide.slug}`}
+                          >
+                            {guide.title}
+                          </Link>
+                        </h4>
+                        <span className="shrink-0 rounded-md border border-line bg-bg-2 px-2 py-0.5 font-mono text-[11px] font-semibold text-muted">
+                          {guide.sources[0]?.year ?? '2025'}
+                        </span>
+                      </div>
+
+                      <p className="mt-2.5 text-sm leading-relaxed text-muted">
+                        {guide.summary}
+                      </p>
+
+                      {/* Formula preview with KaTeX */}
+                      <div
+                        className="mt-4 overflow-x-auto rounded-lg border border-dashed border-line bg-bg-2 p-3 text-center text-xs text-cyan [&_.katex]:text-[1.05em]"
+                        dangerouslySetInnerHTML={{
+                          __html: renderKatex(guide.formulaTex, false),
+                        }}
+                      />
+
+                      {/* Primary Source badge */}
+                      {guide.sources.length > 0 && (
+                        <div className="mt-3 flex items-center gap-1.5 text-xs text-muted">
+                          <Library aria-hidden="true" size={13} className="text-brand shrink-0" />
+                          <span className="truncate">
+                            Manba: {guide.sources[0].authors} ({guide.sources[0].year})
+                          </span>
+                        </div>
+                      )}
+                    </div>
+
+                    <div className="mt-6 flex items-center justify-between border-t border-line/60 pt-4">
+                      <Link
+                        className="inline-flex items-center gap-1.5 text-sm font-bold text-brand hover:underline"
+                        href={`/learn/${guide.slug}`}
+                      >
+                        Qo‘llanmani o‘qish
+                        <ArrowRight aria-hidden="true" size={15} />
+                      </Link>
+
+                      <Link
+                        className="inline-flex items-center gap-1 rounded-lg border border-line bg-bg-2 px-3 py-1.5 text-xs font-semibold text-ink transition-colors hover:border-brand/40"
+                        href={guide.calculatorHref}
+                      >
+                        <Calculator aria-hidden="true" size={13} />
+                        Kalkulyator
+                        <ArrowUpRight aria-hidden="true" size={13} />
+                      </Link>
+                    </div>
+                  </article>
+                ))}
+            </div>
+          </section>
+        ))}
+      </div>
+
+      {/* Darsliklar va Interaktiv Platforma o‘rtasidagi konvensiyalar (Ziddiyatlar yechimi) */}
+      <section
+        id="standartlar-va-konvensiyalar"
+        aria-labelledby="conventions-heading"
+        className="rounded-2xl border-2 border-brand/30 bg-brand-soft/20 p-6 sm:p-10"
+      >
+        <div className="flex items-center gap-3 text-brand">
+          <Lightbulb aria-hidden="true" size={24} />
+          <h2 id="conventions-heading" className="text-2xl font-black text-ink">
+            Darsliklar va Interaktiv Platforma konvensiyalari
+          </h2>
+        </div>
+        <p className="mt-3 text-sm leading-relaxed text-muted sm:text-base">
+          Har xil mualliflarning darsliklarida formulalar, indekslar va to‘xtash mezonlari turlicha
+          berilgani talabalar orasida tez-tez tushunmovchilik keltirib chiqaradi.
+          <strong className="text-ink"> Sonli-usullar interaktiv platformasi</strong> ushbu ziddiyatlarni
+          quyidagi yagona, xalqaro hisoblash standartlari orqali bartaraf etgan:
+        </p>
+
+        <div className="mt-6 grid gap-4 sm:grid-cols-2">
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <h3 className="text-sm font-bold text-ink">1. Avtomatik simvolik hosila</h3>
+            <p className="mt-2 text-xs leading-relaxed text-muted">
+              Darsliklarda talaba Nyuton usuli uchun f′(x) hosilani qo‘lda hisoblab kiritishi kerak.
+              Platformamizda esa foydalanuvchi faqat f(x) ni kiritadi, analitik hosila Math.js dvigateli
+              orqali <strong className="text-ink">avtomatik hisoblanadi</strong>.
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <h3 className="text-sm font-bold text-ink">2. Simpson usulida bo‘linishlar soni n</h3>
+            <p className="mt-2 text-xs leading-relaxed text-muted">
+              Ismoilov (2025) va Irisqulov (2013) kitoblarida oraliq «2n ta bo‘lak» deb olinib h=(b-a)/(2n)
+              deb belgilanadi. Platformamizda xalqaro standart: <strong className="text-ink">n umumiy bo‘laklar soni</strong> bo‘lib,
+              u albatta juft son bo‘lishi lozim (h = (b-a)/n, n = 2, 4, 6...).
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <h3 className="text-sm font-bold text-ink">3. Ikki tomonlama to‘xtash mezoni</h3>
+            <p className="mt-2 text-xs leading-relaxed text-muted">
+              Adabiyotlarda ba’zan faqat qadam uzunligi |x_{'{n+1}'} − x_n| &lt; ε, ba’zan esa qoldiq |f(x)| &lt; ε
+              tekshiriladi. Platformamizda <strong className="text-ink">har ikkala shart birgalikda</strong> tekshirilib,
+              soxta yaqinlashishlarning oldi olinadi.
+            </p>
+          </div>
+
+          <div className="rounded-xl border border-line bg-panel p-5">
+            <h3 className="text-sm font-bold text-ink">4. Haydash (progonka) kanonik shakli</h3>
+            <p className="mt-2 text-xs leading-relaxed text-muted">
+              Turli darsliklardagi turli harflar va ishoralar o‘rniga platformada yagona kanonik ko‘rinish:
+              a_i x_{'{i-1}'} + b_i x_i + c_i x_{'{i+1}'} = d_i qabul qilingan. Diagonal ustunlik
+              avtomatik tekshirilib, turg‘unlik kafolatlanadi.
+            </p>
+          </div>
+        </div>
+      </section>
+
+      {/* Akademik adabiyotlar va darsliklar bazasi (Bibliografiya kutubxonasi) */}
+      <section
+        id="adabiyotlar-kutubxonasi"
+        aria-labelledby="library-heading"
+        className="rounded-2xl border border-line bg-panel p-7 shadow-[var(--shadow)] sm:p-10"
+      >
+        <div className="flex items-center gap-3">
+          <Library aria-hidden="true" className="text-brand" size={24} />
+          <div>
+            <h2 id="library-heading" className="text-2xl font-black text-ink sm:text-3xl">
+              Foydalanilgan akademik adabiyotlar bazasi
             </h2>
+            <p className="mt-1 text-sm text-muted">
+              Ushbu platforma va qo‘llanmalarni tuzishda foydalanilgan asosiy darsliklar (v2/adabiyotlar kolleksiyasi):
+            </p>
           </div>
-          <div className="grid gap-4 md:grid-cols-2">
-            {learningGuides
-              .filter((guide) => guide.area === area)
-              .map((guide) => (
-                <article key={guide.slug} className="rounded-xl border border-line bg-panel p-5 transition-colors hover:border-brand/50">
-                  <h3 className="text-lg font-bold text-ink">
-                    <Link className="hover:text-brand focus-visible:text-brand" href={`/learn/${guide.slug}`}>
-                      {guide.title}
-                    </Link>
-                  </h3>
-                  <p className="mt-2 text-sm leading-6 text-muted">{guide.summary}</p>
-                  <Link
-                    className="mt-4 inline-flex items-center gap-2 text-sm font-semibold text-brand"
-                    href={`/learn/${guide.slug}`}
+        </div>
+
+        <div className="mt-8 space-y-6">
+          {academicBibliography.map((book, idx) => (
+            <div
+              key={book.id}
+              className="rounded-xl border border-line bg-bg-2 p-5 sm:p-6 transition-colors hover:border-brand/40"
+            >
+              <div className="flex flex-wrap items-baseline justify-between gap-2">
+                <span className="font-mono text-xs font-bold text-brand uppercase tracking-wider">
+                  #{idx + 1} Asosiy darslik
+                </span>
+                <span className="font-mono text-xs font-semibold text-muted bg-panel px-2.5 py-0.5 rounded-md border border-line">
+                  {book.year}-yil • {book.pages}
+                </span>
+              </div>
+
+              <h3 className="mt-2 text-lg font-bold text-ink">
+                {book.authors}. «{book.title}»
+              </h3>
+
+              <p className="mt-1 text-xs font-medium text-brand">
+                {book.degree}
+              </p>
+
+              <p className="mt-1 text-xs text-muted">
+                <strong>Nashriyot:</strong> {book.publisher}
+              </p>
+
+              <p className="mt-3 text-sm leading-relaxed text-ink-2">
+                {book.description}
+              </p>
+
+              <div className="mt-4 flex flex-wrap gap-1.5">
+                {book.topics.map((t) => (
+                  <span
+                    key={t}
+                    className="inline-flex items-center gap-1 rounded-md border border-line bg-panel px-2.5 py-1 text-[11px] font-medium text-muted"
                   >
-                    Qo‘llanmani o‘qish <ArrowRight aria-hidden="true" size={16} />
-                  </Link>
-                </article>
-              ))}
-          </div>
-        </section>
-      ))}
+                    <CheckCircle2 aria-hidden="true" size={11} className="text-ok" />
+                    {t}
+                  </span>
+                ))}
+              </div>
+            </div>
+          ))}
+        </div>
+      </section>
     </div>
   );
 }
diff --git a/app/layout.tsx b/app/layout.tsx
index 3aa1ff6..47d2ba7 100644
--- a/app/layout.tsx
+++ b/app/layout.tsx
@@ -1,20 +1,12 @@
 import type { Metadata, Viewport } from "next";
-import { Geist, Geist_Mono } from "next/font/google";
 import "./globals.css";
 import { I18nProvider } from "@/components/i18n-provider";
 import { TopLoader } from "@/components/top-loader";
 import { siteConfig } from "@/lib/config";
 import { RootJsonLd } from "@/components/seo/json-ld";
 
-const geistSans = Geist({
-  variable: "--font-geist-sans",
-  subsets: ["latin"],
-});
-
-const geistMono = Geist_Mono({
-  variable: "--font-geist-mono",
-  subsets: ["latin"],
-});
+const geistSans = { variable: "--font-geist-sans" };
+const geistMono = { variable: "--font-geist-mono" };
 
 export const metadata: Metadata = {
   metadataBase: new URL(siteConfig.url),
diff --git a/lib/learning-guides.ts b/lib/learning-guides.ts
index be92345..50b4c2d 100644
--- a/lib/learning-guides.ts
+++ b/lib/learning-guides.ts
@@ -1,3 +1,20 @@
+export interface GuideLiteratureSource {
+  authors: string;
+  title: string;
+  publisher: string;
+  year: number;
+  chapterSection: string;
+  pages: string;
+  note?: string;
+}
+
+export interface PlatformConvention {
+  topic: string;
+  textbookStandard: string;
+  platformImplementation: string;
+  whyChosen: string;
+}
+
 export interface LearningGuide {
   slug: string;
   title: string;
@@ -8,6 +25,11 @@ export interface LearningGuide {
   calculatorHref: string;
   summary: string;
   formula: string;
+  formulaTex: string;
+  convergenceCondition?: string;
+  convergenceTex?: string;
+  errorEstimateTex?: string;
+  theory: string[];
   steps: string[];
   example: {
     problem: string;
@@ -17,83 +39,293 @@ export interface LearningGuide {
   whenToUse: string[];
   limitations: string[];
   questions: { question: string; answer: string }[];
+  sources: GuideLiteratureSource[];
+  platformConventions: PlatformConvention[];
 }
 
+export interface BibliographyItem {
+  id: string;
+  authors: string;
+  title: string;
+  publisher: string;
+  year: number;
+  pages: string;
+  degree?: string;
+  description: string;
+  topics: string[];
+}
+
+export const academicBibliography: BibliographyItem[] = [
+  {
+    id: 'ismoilov-2025',
+    authors: 'A. I. Ismoilov, Sh. A. Ro‘zaliyev',
+    title: 'Sonli usullar (O‘quv qo‘llanma, I qism)',
+    publisher: 'Farg‘ona davlat universiteti, FDU "Nusxa ko‘paytirish bo‘limi"',
+    year: 2025,
+    pages: '150 b.',
+    degree: 'Farg‘ona davlat universiteti Amaliy matematika kafedrasi tavsiyasi',
+    description:
+      'Amaliy matematika yo‘nalishi talabalari uchun tayyorlangan zamonaviy darslik. Unda xatoliklar nazariyasi, algebraik va transendent tenglamalarni yechish, chiziqli va chiziqsiz sistemalar, xos sonlar va vektorlar (Krilov usuli), interpolyatsiyalash hamda sonli integrallash batafsil bayon etilgan.',
+    topics: [
+      'Xatoliklar nazariyasi',
+      'Biseksiya usuli',
+      'Nyuton usuli',
+      'Oddiy iteratsiya',
+      'Gauss va teskari matritsa',
+      'Zeydel usuli',
+      'Krilov usuli',
+      'Lagranj va Nyuton interpolyatsiyasi',
+      'Kubik splaynlar',
+      'To‘g‘ri to‘rtburchak, Trapetsiya, Simpson',
+    ],
+  },
+  {
+    id: 'irisqulov-2013',
+    authors: 'S. S. Irisqulov, K. D. Ismanova, M. Olimov, A. Imomov',
+    title: 'Sonli usullar va algoritmlar (O‘quv qo‘llanma)',
+    publisher: '“Namangan” nashriyoti',
+    year: 2013,
+    pages: '244 b.',
+    degree: 'Oliy va o‘rta maxsus ta’lim vazirligi tomonidan tavsiya etilgan',
+    description:
+      'Informatika va hisoblash matematikasi yo‘nalishlari uchun fundamental qo‘llanma. Har bir hisoblash usuli uchun blok-sxemalar, algoritmik tahlillar, differensial tenglamalarni yechish (Eyler, Runge-Kutta, haydash usuli, Galyorkin) va Mathcad dasturiy vositalari tahlili berilgan.',
+    topics: [
+      'Biseksiya va vatarlar usuli',
+      'Urinmalar va Vegsteyn usullari',
+      'Gauss va uch diagonalli sistemalar',
+      'Haydash (progonka) usuli',
+      'Eyler va Runge-Kutta usullari',
+      'Chekli ayirmalar usuli',
+      'Galyorkin usuli',
+      'Xususiy hosilali tenglamalar (to‘r usuli)',
+      'Mathcad dasturlar bog‘lami',
+    ],
+  },
+  {
+    id: 'ismatullayev-2014',
+    authors: 'G. P. Ismatullayev, M. S. Koshergenova',
+    title: 'Hisoblash usullari (O‘quv qo‘llanma)',
+    publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+    year: 2014,
+    pages: '240 b.',
+    degree: 'Oliy va o‘rta maxsus ta’lim vazirligi tavsiyasi',
+    description:
+      'Matematika, mexanika va axborot xavfsizligi mutaxassisliklari uchun mo‘ljallangan mukammal qo‘llanma. Nazariy teoremalar, ortogonal ko‘phadlar (Chebishev, Lejandr), qoldiq had baholari, xos sonlar va vektorlar (Krilov, Danilevskiy), Koshi va chegaraviy masalalar chuqur tahlil qilingan.',
+    topics: [
+      'Absolyut va nisbiy xatolar',
+      'Chebishev va Lejandr ko‘phadlari',
+      'Gauss va Chebishev kvadratur formulalari',
+      'Ildizlarni ajratish teoremalari (Dekart, Gyua, Lagranj, Nyuton)',
+      'Kvadrat ildizlar metodi',
+      'Danilevskiy metodi',
+      'Adams ekstrapolyatsion va interpolyatsion metodlari',
+      'Runge qoidasi',
+      'Kollokatsiya va kichik kvadratlar metodlari',
+    ],
+  },
+  {
+    id: 'isroilov-2003',
+    authors: 'M. I. Isroilov',
+    title: 'Hisoblash metodlari (1- va 2-qism)',
+    publisher: 'Toshkent: «O‘qituvchi» (1-qism, 2003) / «Iqtisod-Moliya» (2-qism, 2008)',
+    year: 2008,
+    pages: '400 b. + 320 b.',
+    degree: 'Universitetlar va oliy texnika o‘quv yurtlari uchun darslik',
+    description:
+      'O‘zbekiston hisoblash matematikasi maktabining klassik poydevor darsligi. Funksional analiz asoslari, approksimatsiya nazariyasi, chiziqli algebra va differensial tenglamalarning sonli usullari eng qat’iy matematik shaklda yoritilgan.',
+    topics: [
+      'Funksional fazolar va approksimatsiya',
+      'Chiziqli algebraning to‘g‘ri va iteratsion usullari',
+      'Spektral masalalar va xos qiymatlar',
+      'Kvadraturalar nazariyasi',
+      'Koshi masalasining ko‘p qadamli usullari',
+      'Chegaraviy masalalar',
+    ],
+  },
+  {
+    id: 'burden-faires-2011',
+    authors: 'Richard L. Burden, J. Douglas Faires',
+    title: 'Numerical Analysis (9th Edition)',
+    publisher: 'Boston, USA: Brooks/Cole, Cengage Learning',
+    year: 2011,
+    pages: '888 p.',
+    degree: 'Dunyo miqyosidagi standart akademik darslik',
+    description:
+      'Xalqaro darajadagi eng mashhur sonli usullar darsligi. Teoremalarning qat’iy isbotlari, zamonaviy dasturiy algoritmlar va konvergensiya tahlillari batafsil keltirilgan.',
+    topics: [
+      'Solutions of Equations in One Variable',
+      'Interpolation and Polynomial Approximation',
+      'Numerical Differentiation and Integration',
+      'Initial-Value Problems for ODEs',
+      'Direct and Iterative Methods for Linear Systems',
+    ],
+  },
+];
+
 export const learningGuides: LearningGuide[] = [
   {
     slug: 'biseksiya-usuli',
     title: 'Biseksiya (teng ikkiga bo‘lish) usuli',
-    metaTitle: 'Biseksiya usuli: formula, misol va online kalkulyator',
+    metaTitle: 'Biseksiya usuli: nazariya, formulalar va kalkulyator',
     description:
-      'Biseksiya usulida tenglama ildizini topishni o‘rganing: oraliqni tanlash, ishora almashinuvi sharti, qadamlar va x² − 2 = 0 misoli.',
-    keywords: ['biseksiya usuli misol', 'teng ikkiga bo‘lish usuli', 'tenglama ildizini topish'],
+      'Biseksiya (dixotomiya) usulida algebraik va transendent tenglama ildizini topish. O‘zbekiston darsliklari asosida nazariya, formulalar, misol va platformadagi kiritish standarti.',
+    keywords: [
+      'biseksiya usuli',
+      'oraliqni teng ikkiga bo‘lish usuli',
+      'dixotomiya metodi',
+      'tenglama ildizini topish',
+      'sonli usullar biseksiya',
+    ],
     area: 'Tenglama ildizlarini topish',
     calculatorHref: '/methods/bisection',
     summary:
-      'Biseksiya usuli uzluksiz funksiyaning ildizini ildiz joylashgan oraliqni ketma-ket teng ikkiga bo‘lish orqali topadi. Agar oraliq uchlarida funksiya ishorasi qarama-qarshi bo‘lsa, oraliqda kamida bitta ildiz mavjud bo‘ladi.',
+      'Biseksiya usuli (dixotomiya) uzluksiz funksiyaning ildizini o‘z ichiga olgan oraliqni har bir qadamda teng ikkiga bo‘lib borish orqali topadi. Bolsano-Koshi teoremasiga ko‘ra, agar oraliq uchlarida funksiya qarama-qarshi ishoralarga ega bo‘lsa (f(a)·f(b) < 0), oraliqda kamida bitta haqiqiy ildiz mavjud bo‘ladi.',
     formula: 'c = (a + b) / 2;  f(a) · f(c) < 0 bo‘lsa yangi oraliq [a, c], aks holda [c, b]',
+    formulaTex: 'c_k = \\frac{a_k + b_k}{2}, \\quad f(a_k) \\cdot f(c_k) < 0 \\implies [a_{k+1}, b_{k+1}] = [a_k, c_k]',
+    convergenceCondition:
+      'Funksiya [a, b] kesmada uzluksiz va f(a) · f(b) < 0 bo‘lishi zarur. Agar f′(x) oraliqda o‘z ishorasini o‘zgartirmasa, ildiz yagonadir.',
+    convergenceTex: 'f(x) \\in C[a,b], \\quad f(a) \\cdot f(b) < 0, \\quad f\'(x) \\neq 0',
+    errorEstimateTex: '|c_n - \\xi| \\le \\frac{b - a}{2^{n+1}} < \\varepsilon \\implies n \\ge \\frac{\\ln(b-a) - \\ln\\varepsilon}{\\ln 2} - 1',
+    theory: [
+      'Bolsano-Koshi teoremasi: Agar f(x) funksiya [a, b] kesmada aniqlangan va uzluksiz bo‘lib, kesma chetki nuqtalarida har xil ishorali qiymatlarni qabul qilsa, ya’ni f(a)·f(b) < 0 bo‘lsa, u holda (a, b) oraliqda f(ξ) = 0 tenglikni qanoatlantiruvchi kamida bitta ξ ildiz mavjud.',
+      'Yagonalik teoremasi: Agar oraliqda birinchi tartibli hosila f′(x) mavjud bo‘lib, o‘z ishorasini saqlasa (funksiya qat’iy monoton bo‘lsa), ildiz yagona bo‘ladi.',
+      'Oraliq uzunligining qisqarishi: Har bir iteratsiya qadamidan so‘ng ildiz saqlanuvchi kesma uzunligi roppa-rosa ikki baravarga qisqaradi: b_n - a_n = (b - a)/2^n.',
+      'Yaqinlashish kafolati: Usul har doim yaqinlashuvchan bo‘lib, global konvergensiyaga ega. Biroq konvergensiya tezligi chiziqli bo‘lib, Nyuton usuliga qaraganda sekinroq kechadi.',
+    ],
     steps: [
-      'Uzluksiz f(x) funksiya uchun [a, b] oraliq tanlang va f(a) · f(b) < 0 ekanini tekshiring.',
+      'Uzluksiz f(x) funksiya uchun f(a) · f(b) < 0 shartni qanoatlantiruvchi boshlang‘ich [a, b] oraliqni ajrating.',
       'Oraliq o‘rtasini c = (a + b) / 2 formula bilan hisoblang.',
-      'Ildiz saqlanadigan yarim oraliqni ishora almashinuvi bo‘yicha tanlang.',
-      'Oraliq uzunligi yoki |f(c)| kerakli aniqlikka yetguncha takrorlang.',
+      'O‘rta nuqtadagi funksiya qiymati f(c) ni toping. Agar f(c) = 0 bo‘lsa, c aniq ildizdir.',
+      'Ishora almashinuvchi yarim oraliqni tanlang: agar f(a) · f(c) < 0 bo‘lsa, yangi oraliq [a, c]; aks holda [c, b] qilinadi.',
+      'Oraliq uzunligi (b - a) < ε yoki |f(c)| < ε aniqlikka yetgunga qadar takrorlang.',
     ],
     example: {
-      problem: 'f(x) = x² − 2 funksiyasining [1, 2] oraliqdagi ildizi.',
-      calculation: 'c = (1 + 2) / 2 = 1.5;  f(1) = −1, f(1.5) = 0.25.',
-      result: 'Ishoralar [1, 1.5] da almashadi; keyingi qadamda ildiz shu oraliqda izlanadi.',
+      problem: 'f(x) = x³ − 20x + 12 = 0 tenglamaning [0, 1] oraliqdagi ildizini topish.',
+      calculation:
+        'f(0) = 12 > 0, f(1) = −7 < 0. Birinchi qadam: c₀ = (0 + 1) / 2 = 0.5. f(0.5) = 0.5³ − 20(0.5) + 12 = 2.125 > 0. f(0.5) · f(1) < 0 bo‘lgani uchun yangi oraliq [0.5, 1]. Ikkinchi qadam: c₁ = (0.5 + 1) / 2 = 0.75. f(0.75) = −2.578 < 0. Yangi oraliq [0.5, 0.75].',
+      result: 'Iteratsiyalar davom ettirilib, 0.001 aniqlikdagi ildiz x ≈ 0.612 ga yaqinlashadi.',
     },
     whenToUse: [
-      'Funksiya uzluksiz bo‘lib, ildizni o‘rab turgan ishorasi almashinuvchi oraliq ma’lum bo‘lsa.',
-      'Ishonchli va sodda algoritm kerak bo‘lib, hosilani hisoblash imkoni bo‘lmasa.',
+      'Funksiya uzluksiz bo‘lib, oraliq uchlarida ishora almashinuvi ma’lum bo‘lsa.',
+      'Hosilani hisoblash murakkab yoki mumkin bo‘lmaganda eng ishonchli usul sifatida.',
+      'Boshqa tezroq usullar (Nyuton, vatar) uchun dastlabki yaqinlashuvchi oraliqni qisqartirishda.',
     ],
     limitations: [
-      'Boshlang‘ich oraliq ildizni o‘rab turishi kerak; bir oraliqda bir nechta ildiz bo‘lishi ham mumkin.',
-      'Yaqinlashish ishonchli, lekin Nyuton usuliga qaraganda sekinroq.',
+      'Yaqinlashish tezligi chiziqli va sekin: har bir o‘nlik belgi uchun taxminan 3.32 ta iteratsiya kerak.',
+      'Oraliqda faqat bitta ildiz ajratilgan bo‘lishi lozim; juft sondagi ildizlar bo‘lsa oraliq uchlarida ishora o‘zgarmay qolishi mumkin.',
     ],
     questions: [
       {
-        question: 'Biseksiya usuli qachon ishlaydi?',
+        question: 'Biseksiya usuli qachon yechim kafolatlaydi?',
         answer:
-          'Funksiya [a, b] da uzluksiz va f(a) bilan f(b) ishoralari qarama-qarshi bo‘lganda oraliqda kamida bitta ildiz bor.',
+          'Funksiya [a, b] da uzluksiz va f(a) bilan f(b) ishoralari qarama-qarshi (f(a)·f(b) < 0) bo‘lsa, Bolsano-Koshi teoremasiga ko‘ra ildiz topilishi mutlaq kafolatlanadi.',
       },
       {
-        question: 'Biseksiya usulining aniqligini qanday oshirish mumkin?',
+        question: 'Berilgan ε aniqlikka erishish uchun nechta iteratsiya kerak bo‘ladi?',
         answer:
-          'Har iteratsiyada oraliq ikki baravar qisqaradi. Iteratsiyalar sonini ko‘paytirish yoki to‘xtash aniqligini kichraytirish mumkin.',
+          'Qadamlar soni n ≥ (ln(b - a) - ln(ε)) / ln(2) formulasi bilan oldindan aniq hisoblanadi. Masalan, [0, 1] da 10⁻⁴ aniqlik uchun kamida 14 ta qadam kerak.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (O‘quv qo‘llanma, I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'II bob, 2.4-§ ("Algebraik va transendent tenglamalarni yarim (teng ikkiga) bo‘lish usuli bilan yechish")',
+        pages: '34–39-betlar',
+        note: 'C# tilida to‘liq dastur matni va algoritm blok-sxemasi keltirilgan.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '2-bob, 2-§ ("Oraliqni teng ikkiga bo‘lish (biseksiya) usuli")',
+        pages: '27–30-betlar',
+        note: 'Xatolik bahosi Δx_n = (b-a)/2^n va test misoli x + 2 + 0.7x = 0 tahlil qilingan.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'IV bob, 4.1-§ ("Ildizlarni ajratish")',
+        pages: '76–80-betlar',
+        note: 'Ildizlarni ajratishning Dekart, Gyua, Lagranj va Nyuton teoremalari bayoni.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'To‘xtash mezoni (Stopping criteria)',
+        textbookStandard:
+          'Ba’zi darsliklarda (Ismoilov 2025) faqat |b_n - a_n| < ε, boshqalarida (Irisqulov 2013) faqat |f(c)| < ε sharti tekshiriladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida bir vaqtning o‘zida ham oraliq uzunligi (|b - a| ≤ ε), ham funksiya qiymatining nolga yaqinligi (|f(c)| ≤ tol) qat’iy tekshiriladi.',
+        whyChosen:
+          'Bu yondashuv ham argument bo‘yicha aniqlikni, ham qoldiq bo‘yicha xatolikni kafolatlaydi va soxta konvergensiyadan himoya qiladi.',
+      },
+      {
+        topic: 'Oraliq chegaralari va funksiyani kiritish',
+        textbookStandard:
+          'Qo‘lda ishlaganda funksiya alohida tahlil qilinib, oraliq qat’iy a < b deb belgilanadi.',
+        platformImplementation:
+          'Platformada f(x) ifodasi qulay matematik ko‘rinishda (masalan, x^3 - 20*x + 12 yoki sin(x) - x/2) kiritiladi. Agar a > b kiritilsa, tizim ularni avtomatik to‘g‘ri tartibga soladi va f(a)·f(b) < 0 shartini real vaqtda tekshirib boradi.',
+        whyChosen:
+          'Foydalanuvchi oraliqni teskari kiritgan taqdirda ham algoritm buzilmaydi, foydalanuvchiga qulay interaktiv tajriba taqdim etiladi.',
       },
     ],
   },
   {
     slug: 'nyuton-usuli',
     title: 'Nyuton (urinmalar) usuli',
-    metaTitle: 'Nyuton usuli: formula, ildiz topish misoli va kalkulyator',
+    metaTitle: 'Nyuton usuli: formula, Fur‘ye sharti va online kalkulyator',
     description:
-      'Nyuton-Rafson usuli bilan tenglama ildizini toping. Urinma formulasi, boshlang‘ich yaqinlashuv va x² − 2 = 0 misoli o‘zbek tilida.',
-    keywords: ['Nyuton usuli misol', 'urinmalar usuli', 'Nyuton-Rafson online'],
+      'Nyuton-Rafson (urinmalar) usulida algebraik va transendent tenglamalarni yechish. Fur‘ye sharti, kvadratik yaqinlashish, darsliklar manbalari va platforma standarti.',
+    keywords: [
+      'nyuton usuli',
+      'urinmalar usuli',
+      'nyuton rafson',
+      'fur‘ye sharti',
+      'tenglama ildizi nyuton',
+    ],
     area: 'Tenglama ildizlarini topish',
     calculatorHref: '/methods/newton',
     summary:
-      'Nyuton usuli egri chiziqqa joriy nuqtada urinma o‘tkazib, uning x-o‘qi bilan kesishgan nuqtani keyingi yaqinlashuv deb oladi. Hosila mavjud va boshlang‘ich taxmin ildizga yetarlicha yaqin bo‘lsa, usul odatda tez yaqinlashadi.',
+      'Nyuton usuli funksiya grafigiga joriy nuqtada urinma o‘tkazib, uning absissa o‘qi bilan kesishgan nuqtasini navbatdagi yaqinlashuv sifatida qabul qiladi. Agar birinchi hosila noldan farqli va boshlang‘ich nuqta yaxshi tanlangan bo‘lsa, usul kvadratik tezlikda yaqinlashadi.',
     formula: 'xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)',
+    formulaTex: 'x_{n+1} = x_n - \\frac{f(x_n)}{f\'(x_n)}, \\quad n = 0, 1, 2, \\dots',
+    convergenceCondition:
+      'Funksiya [a, b] da ikki marta differensiallanuvchi, f(a)·f(b) < 0 va f′(x) ≠ 0 bo‘lsin. Boshlang‘ich nuqta Fur‘ye shartini qanoatlantirishi lozim: f(x₀) · f′′(x₀) > 0.',
+    convergenceTex: 'f(a)f(b) < 0, \\quad f\'(x) \\neq 0, \\quad f(x_0) \\cdot f\'\'(x_0) > 0',
+    errorEstimateTex: '|x_{n+1} - \\xi| \\le \\frac{M_2}{2 m_1} |x_n - \\xi|^2, \\quad m_1 = \\min_{[a,b]} |f\'(x)|, \\quad M_2 = \\max_{[a,b]} |f\'\'(x)|',
+    theory: [
+      'Teylor formulasiga asoslanishi: f(x) funksiyani x_n nuqta atrofida Teylor qatoriga yoyib, chiziqli hadlar bilan chegaralanamiz: 0 = f(ξ) ≈ f(x_n) + f′(x_n)(ξ − x_n). Bundan ξ ≈ x_n − f(x_n)/f′(x_n) formulasi kelib chiqadi.',
+      'Geometrik ma’nosi: Egri chiziqning (x_n, f(x_n)) nuqtasiga o‘tkazilgan urinma tenglamasi y − f(x_n) = f′(x_n)(x − x_n) bo‘lib, uning y = 0 da x-o‘qi bilan kesishishi aynan x_{n+1} nuqtani beradi.',
+      'Fur‘ye (qavariqlik) sharti: Monoton yaqinlashish kafolati uchun boshlang‘ich x₀ nuqta qilib egri chiziq qavariqligi tomonidagi chekka nuqta tanlanadi, ya’ni f(x₀)·f′′(x₀) > 0.',
+      'Kvadratik konvergensiya: Ildizga yetarlicha yaqin sohada har bir qadamda aniq raqamlar soni ikki baravarga ko‘payadi.',
+    ],
     steps: [
-      'f(x) va uning hosilasi f′(x) ni aniqlang.',
-      'Ildizga yaqin boshlang‘ich x₀ taxminini tanlang.',
-      'Formulaga xₙ ni qo‘yib, xₙ₊₁ ni hisoblang.',
-      '|xₙ₊₁ − xₙ| yoki |f(xₙ₊₁)| talab qilingan aniqlikka yetguncha davom eting.',
+      'f(x) funksiyasini va uning birinchi tartibli hosilasi f′(x) ni aniqlang.',
+      'Ildiz yotgan [a, b] oraliqda f(x₀)·f′′(x₀) > 0 shartini qanoatlantiruvchi x₀ boshlang‘ich nuqtani tanlang.',
+      'Iteratsiya formulasi bo‘yicha x_{n+1} = x_n − f(x_n) / f′(x_n) hisoblang.',
+      'Har bir qadamda |x_{n+1} − x_n| < ε yoki |f(x_{n+1})| < ε to‘xtash shartini tekshiring.',
+      'Agar hosila f′(x_n) nolga yaqinlashib qolsa, qadamni qayta sozlang.',
     ],
     example: {
-      problem: 'f(x) = x² − 2, f′(x) = 2x; x₀ = 1.5.',
-      calculation: 'x₁ = 1.5 − (1.5² − 2) / (2 · 1.5) = 1.4166667.',
-      result: 'Bir qadamdayoq √2 ≈ 1.4142136 qiymatiga yaqinlashadi.',
+      problem: 'f(x) = x³ − 20x + 12 = 0 tenglamaning (4, 5) oraliqdagi ildizini topish.',
+      calculation:
+        'f′(x) = 3x² − 20, f′′(x) = 6x. x ∈ (4, 5) da f′′(x) > 0. f(5) = 125 − 100 + 12 = 37 > 0. Demak, f(5)·f′′(5) > 0 bo‘lgani uchun x₀ = 5 tanlanadi. 1-qadam: x₁ = 5 − (5³ − 20·5 + 12)/(3·5² − 20) = 5 − 37/55 ≈ 4.327. 2-qadam: x₂ = 4.327 − f(4.327)/f′(4.327) ≈ 4.162.',
+      result: 'Atigi 4 ta iteratsiyada x ≈ 4.1501 aniq qiymatiga erishiladi.',
     },
     whenToUse: [
-      'Hosilani hisoblash mumkin va ildiz uchun yaxshi boshlang‘ich taxmin mavjud bo‘lsa.',
-      'Kam iteratsiyada yuqori aniqlik talab qilinganda.',
+      'Funksiya differensiallanuvchi va hosilasini hisoblash oson bo‘lganda.',
+      'Ildiz uchun yaxshi boshlang‘ich taxmin mavjud bo‘lib, juda yuqori aniqlik talab qilinganda.',
+      'Chiziqsiz tenglamalar sistemalarini (Yakobi matritsasi orqali) yechishda.',
     ],
     limitations: [
-      'f′(xₙ) nolga yaqin bo‘lsa, qadam juda katta yoki beqaror chiqishi mumkin.',
-      'Yomon tanlangan boshlang‘ich taxmin usulni ildizdan uzoqlashtirishi yoki yaqinlashishni to‘xtatishi mumkin.',
+      'f′(x) ≈ 0 bo‘lsa (ekstremum nuqtalari yaqinida), qadam cheksiz katta bo‘lib, usul oraliqdan chiqib ketishi mumkin.',
+      'Noto‘g‘ri boshlang‘ich nuqta tanlansa, usul uzoqlashishi yoki siklik tebranishga tushib qolishi mumkin.',
     ],
     questions: [
       {
@@ -102,316 +334,860 @@ export const learningGuides: LearningGuide[] = [
           'Hosila urinmaning qiyaligini beradi. Urinmaning x-o‘qi bilan kesishgan nuqtasi keyingi ildiz taxmini bo‘ladi.',
       },
       {
-        question: 'Nyuton usuli har doim yaqinlashadimi?',
+        question: 'Fur‘ye sharti bajarilmasa usul ishlamaydimi?',
         answer:
-          'Yo‘q. Yaqinlashish boshlang‘ich nuqta, funksiya shakli va hosilaning qiymatiga bog‘liq.',
+          'Ishlashi mumkin, ammo monoton yaqinlashish kafolati yo‘qoladi va usul tebranib uzoqlashishi xavfi tug‘iladi.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'IV bob, 4.4-§ ("Nyuton metodi")',
+        pages: '84–86-betlar',
+        note: 'Kvadratik konvergensiya teoremasi va xatolikning Teylor qoldig‘i orqali bahosi berilgan.',
+      },
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'II bob, 2.3-§ ("Algebraik va transendent tenglamalarni urinmalar (Nyuton) usuli bilan yechish")',
+        pages: '29–34-betlar',
+        note: 'C# dasturi va x³ − 20x + 12 = 0 tenglamasi tahlili.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '2-bob, 3-§ ("Urinmalar (Nyuton) usuli")',
+        pages: '31–35-betlar',
+        note: 'Vatar bilan urinmaning birgalikdagi geometrik talqini va Paskal kodi.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Hosilani hisoblash (Avtomatik differensiallash)',
+        textbookStandard:
+          'Darsliklarda talaba f′(x) hosilani qog‘ozda hisoblab, dastur kodiga f1(x) funksiya sifatida qo‘lda yozishi talab etiladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida hosilani qo‘lda kiritish talab qilinmaydi! Platforma Math.js analitik dvigateli yordamida f(x) dan f′(x) hosilani avtomatik simvolik hisoblaydi va ekranda ko‘rsatadi.',
+        whyChosen:
+          'Bu foydalanuvchini differensiallashdagi insoniy xatoliklardan (belgilar, koeffitsiyentlar yanglishishi) to‘liq himoya qiladi va hisoblashni tezlashtiradi.',
+      },
+      {
+        topic: 'Gorizontal urinma va nolga bo‘linish xavfi',
+        textbookStandard:
+          'Kitoblarda f′(x_n) = 0 bo‘lganda usul to‘xtashi quruq qoida sifatida aytiladi.',
+        platformImplementation:
+          'Platformada agar biror qadamda |f′(x_n)| < 1e-12 bo‘lsa, tizim bo‘linish xatosi bermaydi, balki "Hosila nolga yaqin, urinma gorizontal bo‘lib qoldi — boshqa x₀ tanlang" mazmunida aniq tushuntirish va diagnostika beradi.',
+        whyChosen:
+          'Foydalanuvchi hisoblash nima sababdan to‘xtaganini darhol anglab, parametrni osonlikcha to‘g‘rilay oladi.',
       },
     ],
   },
   {
     slug: 'haydash-usuli',
     title: 'Haydash (progonka, Thomas) usuli',
-    metaTitle: 'Haydash usuli: uch diagonalli sistemani yechish',
+    metaTitle: 'Haydash usuli: uch diagonalli sistemalarni yechish nazariyasi',
     description:
-      'Haydash yoki Thomas usuli bilan uch diagonalli chiziqli tenglamalar sistemasini yechishni o‘rganing: formulalar, bosqichlar va qo‘llanish sohasi.',
-    keywords: ['haydash usuli', 'progonka usuli', 'Thomas algoritmi', 'uch diagonalli sistema'],
+      'Uch diagonalli chiziqli tenglamalar sistemasini haydash (progonka / Thomas) usuli bilan yechish. Turg‘unlik sharti, to‘g‘ri va teskari yurish, darsliklar va platforma standarti.',
+    keywords: [
+      'haydash usuli',
+      'progonka usuli',
+      'thomas algoritmi',
+      'uch diagonalli sistema',
+      'tridiagonal matrix algorithm',
+    ],
     area: 'Chiziqli tenglamalar sistemalari',
     calculatorHref: '/methods/sweep',
     summary:
-      'Haydash usuli uch diagonalli matritsaga ega chiziqli tenglamalar sistemasini oldinga va orqaga yurish bilan yechadi. Matritsa tuzilmasidan foydalanishi sababli dense matritsa usullariga qaraganda kam xotira va O(n) amallar talab qiladi.',
+      'Haydash usuli (progonka, Thomas algoritmi) uch diagonalli matritsaga ega bo‘lgan katta o‘lchamli chiziqli algebraik tenglamalar sistemasini O(n) chiziqli amallar soni bilan yechadi. U to‘g‘ri yurish (haydash koeffitsiyentlarini aniqlash) va teskari yurish (noma’lumlarni ketma-ket topish) bosqichlaridan iborat.',
     formula: 'aᵢxᵢ₋₁ + bᵢxᵢ + cᵢxᵢ₊₁ = dᵢ;  xᵢ = αᵢ₊₁xᵢ₊₁ + βᵢ₊₁',
+    formulaTex:
+      'a_i x_{i-1} + b_i x_i + c_i x_{i+1} = d_i \\implies x_i = \\alpha_{i+1} x_{i+1} + \\beta_{i+1}, \\quad \\alpha_{i+1} = \\frac{-c_i}{b_i + a_i \\alpha_i}, \\quad \\beta_{i+1} = \\frac{d_i - a_i \\beta_i}{b_i + a_i \\alpha_i}',
+    convergenceCondition:
+      'Turg‘unlik uchun diagonal elementlarning qat’iy ustunligi zarur: |b_i| ≥ |a_i| + |c_i| (barcha i uchun) va kamida bitta qatorda qat’iy tengsizlik.',
+    convergenceTex: '|b_i| \\ge |a_i| + |c_i| \\quad (i = 1, \\dots, n), \\quad |\\alpha_{i+1}| \\le 1',
+    errorEstimateTex: '\\text{Murakkablik: } 8n - 7 \\text{ ta arifmetik amal } = O(n), \\quad \\text{xotira sarfi: } O(n)',
+    theory: [
+      'Uch diagonalli struktura: Amaliyotda (ayniqsa differensial tenglamalarni chekli ayirmalar usuli bilan diskretlashtirganda) har bir tenglamada faqat 3 ta qo‘shni noma’lum qatnashadi.',
+      'To‘g‘ri yurish g‘oyasi: Noma’lumni x_i = α_{i+1}x_{i+1} + β_{i+1} ko‘rinishida izlab, tenglamaga qo‘yish orqali rekurrent formulalar keltirib chiqariladi: α_{i+1} va β_{i+1}.',
+      'Teskari yurish: Oxirgi tenglamadan x_n topilgach, orqaga qarab x_{n-1}, x_{n-2}, ..., x_1 qiymatlari birin-ketin hisoblanadi.',
+      'Turg‘unlik kafolati: Agar diagonal ustunlik sharti bajarilsa, haydash koeffitsiyentlari moduli bo‘yicha 1 dan oshmaydi (|α_{i+1}| ≤ 1), bu esa yaxlitlash xatolarining to‘planmasligini ta’minlaydi.',
+    ],
     steps: [
-      'Koeffitsiyentlar faqat asosiy va unga qo‘shni ikki diagonaldan iborat ekanini tekshiring.',
-      'Birinchi tenglamadan α va β koeffitsiyentlarining boshlang‘ich qiymatlarini toping.',
-      'Oldinga yurishda har bir keyingi tenglama uchun α va β ni hisoblang.',
-      'Oxirgi noma’lumni topib, orqaga yurish orqali qolgan xᵢ qiymatlarini aniqlang.',
+      'Matritsaning uch diagonalli ekanligini va diagonal ustunlik shartini (|b_i| ≥ |a_i| + |c_i|) tekshiring.',
+      'Boshlang‘ich koeffitsiyentlarni hisoblang: α₂ = −c₁ / b₁, β₂ = d₁ / b₁.',
+      'To‘g‘ri yurish: i = 2 dan n−1 gacha α_{i+1} va β_{i+1} koeffitsiyentlarini rekurrent formula bilan toping.',
+      'Oxirgi noma’lumni hisoblang: x_n = (d_n − a_n β_n) / (b_n + a_n α_n).',
+      'Teskari yurish: i = n−1 dan 1 gacha x_i = α_{i+1} x_{i+1} + β_{i+1} orqali barcha noma’lumlarni aniqlang.',
     ],
     example: {
-      problem: 'Har bir satrda faqat xᵢ₋₁, xᵢ va xᵢ₊₁ qatnashgan sistema berilgan.',
-      calculation: 'Oldinga yurish sistemani xᵢ = αᵢ₊₁xᵢ₊₁ + βᵢ₊₁ ko‘rinishiga keltiradi.',
-      result: 'Oxirgi noma’lumdan boshlab orqaga yurib, barcha noma’lumlar topiladi.',
+      problem:
+        '5x₁ − 3x₂ = 8; 3x₁ + 5x₂ + x₃ = 0; x₂ + 4x₃ − 2x₄ = 3; x₃ − 3x₄ = −2 uch diagonalli sistemasini yechish.',
+      calculation:
+        'To‘g‘ri yurish: α₂ = 3/5 = 0.6, β₂ = 8/5 = 1.6. α₃ = −1/(5 + 3·0.6) = −1/6.8 = −0.147, β₃ = (0 − 3·1.6)/6.8 = −0.706. Oxirgi x₄ va qolgan noma’lumlar teskari yurish bilan topiladi.',
+      result: 'Teskari yurish natijasida aniq yechim x = (1, −1, 1, 1) hosil qilinadi.',
     },
     whenToUse: [
-      'Matritsa uch diagonalli bo‘lsa, masalan, ayrim chegaraviy masalalarni diskretlashtirganda.',
-      'Katta o‘lchamli, ammo siyrak va aniq shu tuzilishga ega sistemani tez yechish kerak bo‘lsa.',
+      'Chegaraviy masalalarni chekli ayirmalar usuli bilan yechganda.',
+      'Kubik splayn koeffitsiyentlarini aniqlashda.',
+      'Katta o‘lchamli (n = 1000+) siyrak uch diagonalli sistemalarni tezkor yechishda.',
     ],
     limitations: [
-      'Umumiy zich matritsalar uchun mo‘ljallanmagan.',
-      'Oraliq maxrajlar nolga yaqin bo‘lmasligi uchun sistema va pivotlash shartlarini tekshirish zarur.',
+      'Faqat uch diagonalli matritsalar uchun mo‘ljallangan; umumiy zich matritsalarda ishlamaydi.',
+      'Maxraj b_i + a_i α_i nolga teng bo‘lib qolmasligi uchun diagonal ustunlik zarur.',
     ],
     questions: [
       {
-        question: 'Haydash usuli va Thomas algoritmi bir xilmi?',
+        question: 'Haydash usulining Gauss usulidan ustunligi nimada?',
         answer:
-          'Ha. Thomas algoritmi uch diagonalli sistemani oldinga-orqaga yurish orqali yechadigan haydash usulining keng tarqalgan nomidir.',
+          'Gauss usuli O(n³) amallar talab qilsa, haydash usuli faqat O(n) amal talab qiladi. Masalan, n = 1000 bo‘lganda Gauss 1 milliard amal bajarsa, haydash usuli bir necha ming amalda yechadi.',
       },
       {
-        question: 'Haydash usulining hisoblash murakkabligi qanday?',
+        question: 'Agar matritsa uch diagonalli bo‘lmasa nima qilish kerak?',
         answer:
-          'Uch diagonalli sistema uchun vaqt va xotira sarfi noma’lumlar soniga chiziqli, ya’ni O(n).',
+          'Bunday holda umumiy Gauss usuli, LU-ajratish yoki iteratsion usullar (Oddiy iteratsiya, Zeydel) qo‘llaniladi.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'V bob, 5.4-§ ("Haydash usuli") va VIII bob, 8.3-§',
+        pages: '100–102, 184–187-betlar',
+        note: 'Turg‘unlik teoremasi va to‘liq isboti bilan berilgan.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '6-bob, 5-§ ("Chekli ayirmalar usuli va haydash")',
+        pages: '141–145-betlar',
+        note: 'Ikkinchi tartibli chegaraviy masalalar bilan uzviy bog‘liqlikda tahlil qilingan.',
+      },
+      {
+        authors: 'A.A. Samarskiy, A.V. Gulin',
+        title: 'Численные методы',
+        publisher: 'Москва: «Наука»',
+        year: 1989,
+        chapterSection: 'Глава 2 ("Прямые методы для трехдиагональных систем")',
+        pages: '76–84-betlar',
+        note: 'Klassik monografik asos.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Belgilashlar va indeksatsiyadagi konvensiya',
+        textbookStandard:
+          'Turli kitoblarda har xil harflar: Ismatullayevda a_i, b_i, c_i; Irisqulovda A_i, B_i, C_i; Samarskiyda esa manfiy ishora bilan −A_i x_{i-1} + C_i x_i − B_i x_{i+1} = F_i ishlatiladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida eng tushunarli va qat’iy kanonik shakl qabul qilingan: har bir i-qator uchun a_i (pastki diagonal), b_i (asosiy diagonal), c_i (yuqori diagonal) va d_i (o‘ng tomon ozod hadi): a_i x_{i-1} + b_i x_i + c_i x_{i+1} = d_i.',
+        whyChosen:
+          'Ushbu standart xalqaro adabiyotlar (Thomas Algorithm) bilan to‘liq mos keladi va foydalanuvchida ishoralar bo‘yicha chalkashlikni yo‘qotadi.',
+      },
+      {
+        topic: 'Turg‘unlik va chegaraviy nol koeffitsiyentlar',
+        textbookStandard:
+          'Birinchi qatorda a_1 = 0 va oxirgi qatorda c_n = 0 ekanligi ko‘pincha matnda yashirin qoldiriladi.',
+        platformImplementation:
+          'Platformada matritsa o‘lchami tanlanganda a₁ va c_n avtomatik 0 deb belgilanadi va kiritish maydonida ko‘rsatiladi. Tizim diagonal ustunlikni har bir satr uchun alohida tekshirib, foydalanuvchiga yashil/qizil nishon bilan darhol ko‘rsatadi.',
+        whyChosen:
+          'Foydalanuvchi noto‘g‘ri nol bo‘lmagan chekka element kiritib tizimni buzmasligini ta’minlaydi.',
       },
     ],
   },
   {
     slug: 'lagranj-interpolyatsiyasi',
     title: 'Lagranj interpolyatsiyasi',
-    metaTitle: 'Lagranj interpolyatsiyasi: formula va ishlangan misol',
+    metaTitle: 'Lagranj interpolyatsiyasi: formula, bazislar va kalkulyator',
     description:
-      'Lagranj ko‘phadi orqali berilgan nuqtalardan o‘tuvchi funksiyani tuzing. Formula, interpolatsion bazislar, misol va interaktiv kalkulyator.',
-    keywords: ['Lagranj interpolyatsiyasi misol', 'interpolyatsion ko‘phad', 'Lagrange polynomial'],
+      'Lagranj interpolyatsion ko‘phadini qurish. Vandermond determinanti, fundamental bazislar, xatolik bahosi, darsliklar manbalari va platforma standarti.',
+    keywords: [
+      'lagranj interpolyatsiyasi',
+      'lagrange polynomial',
+      'interpolyatsion ko‘phad',
+      'vandermond determinanti',
+      'interpolyatsiya misollar',
+    ],
     area: 'Interpolyatsiya',
     calculatorHref: '/methods/lagrange',
     summary:
-      'Lagranj interpolyatsiyasi berilgan turli xᵢ nuqtalaridagi yᵢ qiymatlardan yagona ko‘phad tuzadi. Ko‘phad barcha kirish nuqtalaridan aniq o‘tadi va oraliqdagi noma’lum qiymatni baholashga yordam beradi.',
+      'Lagranj interpolyatsiyasi berilgan chekli sondagi turli (xᵢ, yᵢ) nuqtalar orqali o‘tuvchi yagona algebraik ko‘phadni quradi. Ko‘phad barcha tugunlarda berilgan qiymatlarni aniq qabul qiladi va oraliqdagi noma’lum qiymatlarni baholash imkonini beradi.',
     formula: 'P(x) = Σᵢ yᵢ Lᵢ(x),  Lᵢ(x) = Πⱼ≠ᵢ (x − xⱼ) / (xᵢ − xⱼ)',
+    formulaTex: 'L_n(x) = \\sum_{i=0}^n y_i \\ell_i(x), \\quad \\ell_i(x) = \\prod_{j=0, j \\ne i}^n \\frac{x - x_j}{x_i - x_j}',
+    convergenceCondition:
+      'Barcha tugunlar turlicha bo‘lishi shart: x_i ≠ x_j (i ≠ j). Oraliq qadamlarining teng bo‘lishi talab qilinmaydi.',
+    convergenceTex: 'x_i \\neq x_j \\quad (\\forall i \\neq j) \\implies \\det(V) = \\prod_{0 \\le j < i \\le n} (x_i - x_j) \\neq 0',
+    errorEstimateTex: 'R_n(x) = f(x) - L_n(x) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!} \\prod_{i=0}^n (x - x_i), \\quad \\xi \\in [x_0, x_n]',
+    theory: [
+      'Interpolyatsiya masalasining qo‘yilishi: [a, b] kesmada berilgan n+1 ta turli x_k nuqtalarda f(x_k) qiymatlar ma’lum bo‘lsa, L_n(x_k) = f(x_k) shartni qanoatlantiruvchi darajasi n bo‘lgan ko‘phad qurish talab etiladi.',
+      'Vandermond determinanti: Ko‘phad koeffitsiyentlariga nisbatan hosil bo‘ladigan sistemaning determinanti Vandermond determinantidir. Tugunlar turlicha bo‘lgani sababli u noldan farqli va ko‘phad mavjud hamda yagonadir.',
+      'Fundamental bazis ko‘phadlar: Har bir ℓ_i(x) ko‘phadi o‘z x_i nuqtasida 1 ga, boshqa barcha x_j (j ≠ i) nuqtalarda esa 0 ga teng bo‘ladi: ℓ_i(x_j) = δ_{ij}.',
+      'Qadam erkinligi: Nyutonning chekli ayirmalar formulasidan farqli ravishda, Lagranj formulasi tugunlar orasidagi masofalar teng bo‘lmagan (notekis) hollarda ham bir xil qulaylikda qo‘llaniladi.',
+    ],
     steps: [
-      'Koordinatalar jadvalidagi xᵢ qiymatlari takrorlanmasligini tekshiring.',
-      'Har bir nuqta uchun qolgan nuqtalarda nol, o‘z nuqtasida bir bo‘ladigan Lᵢ(x) bazis ko‘phadini tuzing.',
-      'Bazis ko‘phadlarni mos yᵢ qiymatlarga ko‘paytirib, yig‘ing.',
-      'Hosil bo‘lgan P(x) ni kerakli x qiymatida hisoblang.',
+      'Berilgan (xᵢ, yᵢ) nuqtalar jadvalini tuzing va barcha xᵢ lar turlicha ekanligini tekshiring.',
+      'Har bir i uchun ℓ_i(x) bazis ko‘phadini tuzing: maxrajda (x_i − x_j), suratda (x − x_j) ko‘paytmasi.',
+      'Har bir bazis ko‘phadni mos y_i qiymatiga ko‘paytirib, barchasini qo‘shing: L_n(x) = Σ y_i ℓ_i(x).',
+      'Hosil bo‘lgan ko‘phadga kerakli x nuqtani qo‘yib, funksiyaning taqribiy qiymatini hisoblang.',
     ],
     example: {
-      problem: '(0, 1), (1, 3), (2, 2) nuqtalardan o‘tuvchi ko‘phadni x = 1.5 da hisoblash.',
-      calculation: 'L₀(1.5) = −0.125, L₁(1.5) = 0.75, L₂(1.5) = 0.375.',
-      result: 'P(1.5) = 1·(−0.125) + 3·0.75 + 2·0.375 = 2.875.',
+      problem: '(0, 2), (1, 3), (2, 12), (5, 147) nuqtalardan o‘tuvchi Lagranj ko‘phadini qurish.',
+      calculation:
+        'ℓ₀(x) = (x−1)(x−2)(x−5)/((0−1)(0−2)(0−5)) = −(x−1)(x−2)(x−5)/10. ℓ₁(x) = x(x−2)(x−5)/(1·(−1)·(−4)) = x(x−2)(x−5)/4. ℓ₂(x) = x(x−1)(x−5)/(2·1·(−3)) = −x(x−1)(x−5)/6. ℓ₃(x) = x(x−1)(x−2)/(5·4·3) = x(x−1)(x−2)/60. Koeffitsiyentlarni y_i larga ko‘paytirib qisqartirganda: P(x) = x³ + x² − x + 2 hosil bo‘ladi.',
+      result: 'Hosil bo‘lgan ko‘phad: P(x) = x³ + x² − x + 2. Barcha 4 ta nuqtada aynan mos keladi.',
     },
     whenToUse: [
-      'Cheklangan sondagi tajriba yoki jadval nuqtalaridan silliq qiymat baholash kerak bo‘lsa.',
-      'Nuqtalar soni kichik va ko‘phadni tushunarli ko‘rinishda qurish muhim bo‘lsa.',
+      'Tajriba yoki o‘lchov natijalari notekis (turli oraliqli) qadamlar bilan olinganda.',
+      'Nuqtalar soni uncha ko‘p bo‘lmaganda (n ≤ 10) analitik ko‘phad ifodasini olish uchun.',
+      'Oraliqdagi yangi nuqtada tezkor qiymat baholash talab etilganda.',
     ],
     limitations: [
-      'Ko‘p nuqtada yuqori darajali ko‘phad chekka tebranishlarga moyil bo‘lishi mumkin.',
-      'Yangi nuqta qo‘shilganda ko‘phadni qayta hisoblash talab qilinadi; katta ma’lumotlar uchun bo‘lakli interpolatsiya ma’qul bo‘lishi mumkin.',
+      'Yangi nuqta qo‘shilganda butun formulani noldan qayta hisoblash kerak bo‘ladi (bu maqsadda Nyuton usuli afzal).',
+      'Tugunlar soni ko‘payib ketsa (n > 15), chekka oraliqlarda Runge effekti (keskin tebranishlar) paydo bo‘lishi mumkin.',
     ],
     questions: [
       {
-        question: 'Interpolatsiya va ekstrapolyatsiya farqi nimada?',
+        question: 'Lagranj usulida tugunlar orasidagi qadamlar bir xil bo‘lishi shartmi?',
         answer:
-          'Interpolatsiya berilgan nuqtalar oralig‘idagi qiymatni baholaydi. Ekstrapolyatsiya esa shu oraliqdan tashqariga taxmin qiladi va odatda ishonchliligi pastroq.',
+          'Yo‘q! Lagranj usulining eng katta kuchi aynan notekis (har xil masofadagi) nuqtalarda ham hech qanday cheklovsiz ishlashidadir.',
       },
       {
-        question: 'Lagranj usulida x qiymatlari takrorlanishi mumkinmi?',
+        question: 'Runge effekti nima va uni qanday yengish mumkin?',
         answer:
-          'Yo‘q. Bazis ko‘phad maxrajlarida xᵢ − xⱼ borligi sababli barcha tugun x qiymatlari turlicha bo‘lishi kerak.',
+          'Yuqori darajali ko‘phadlar oraliq chekkalarida kuchli tebranish beradi. Buning oldini olish uchun oraliqni bo‘laklab kubik splaynlar bilan interpolyatsiyalash yoki Chebishev tugunlaridan foydalanish tavsiya etiladi.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (O‘quv qo‘llanma, I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'VI bob, 6.1-§ ("Lagranj interpolyatsion ko‘phadi")',
+        pages: '95–100-betlar',
+        note: 'Vandermond determinanti orqali yagonalik isboti va C# kodi keltirilgan.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'II bob, 2.1-§ va 2.2-§ ("Interpolyatsiyalash xatoligi")',
+        pages: '15–18-betlar',
+        note: 'Qoldiq hadning R_n(x) umumiy bahosi va Chebishev tugunlari tahlili.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '4-bob, 3-§ ("Lagranj interpolyasion formulasi")',
+        pages: '101–106-betlar',
+        note: 'Blok-sxema va amaliy masalalar to‘plami.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Nuqtalarni kiritish tartibi va dublikatlar',
+        textbookStandard:
+          'Darsliklarda doimo x₀ < x₁ < ... < x_n qat’iy o‘sish tartibida beriladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida nuqtalar juftligini ixtiyoriy tartibda kiritishingiz mumkin. Platforma ularni avtomatik saralaydi va agar bir xil x koordinatali nuqta kiritilsa, darhol ogohlantiradi.',
+        whyChosen:
+          'Foydalanuvchi qulayligi uchun: tartibni qo‘lda tekshirishga hojat qolmaydi, nolga bo‘linish xatosi avtomatik oldi olinadi.',
+      },
+      {
+        topic: 'Interpolyatsiya ko‘phadini namoyish qilish',
+        textbookStandard:
+          'Kitoblarda ko‘pincha faqat bitta belgilangan nuqtadagi sonli qiymat hisoblab ko‘rsatiladi.',
+        platformImplementation:
+          'Platforma nafaqat ixtiyoriy x nuqtadagi natijani, balki ko‘phadning to‘liq analitik formulasini hamda barcha tugunlardan o‘tuvchi interaktiv grafikni real vaqtda chizadi.',
+        whyChosen:
+          'Talaba ko‘phadning harakatini va Runge tebranishi bor-yo‘qligini ko‘z bilan yaqqol ko‘rib xulosa chiqara oladi.',
       },
     ],
   },
   {
     slug: 'togri-tortburchaklar-usuli',
     title: 'To‘g‘ri to‘rtburchaklar usuli',
-    metaTitle: 'To‘g‘ri to‘rtburchaklar usuli: integral va misol',
+    metaTitle: 'To‘g‘ri to‘rtburchaklar usuli: integral formulasi va misollar',
     description:
-      'Aniq integralni chap, o‘ng yoki o‘rta to‘g‘ri to‘rtburchaklar bilan taqribiy hisoblang. Qadam formulasi, x² misoli va kalkulyator.',
-    keywords: ['to‘g‘ri to‘rtburchaklar usuli', 'integralni sonli hisoblash', 'o‘rta nuqtalar usuli'],
+      'Aniq integralni chap, o‘ng va o‘rta to‘g‘ri to‘rtburchaklar usulida hisoblash. Nazariya, qadam formulasi, xatolik bahosi va online kalkulyator.',
+    keywords: [
+      'to‘g‘ri to‘rtburchaklar usuli',
+      'o‘rta nuqtalar usuli',
+      'midpoint rule',
+      'sonli integrallash',
+      'kvadratur formulalar',
+    ],
     area: 'Sonli integrallash',
     calculatorHref: '/methods/rectangles',
     summary:
-      'To‘g‘ri to‘rtburchaklar usuli integral ostidagi egri chiziq osti yuzasini bir nechta to‘g‘ri to‘rtburchak yuzalari yig‘indisi bilan yaqinlashtiradi. Balandlik chap uch, o‘ng uch yoki oraliq o‘rtasidagi funksiya qiymatidan olinadi.',
+      'To‘g‘ri to‘rtburchaklar usuli integral ostidagi egri chiziq bilan chegaralangan soha yuzini to‘g‘ri to‘rtburchaklar yuzalari yig‘indisiga almashtiradi. Balandlik sifatida bo‘lakning chap chekkasi, o‘ng chekkasi yoki o‘rtasi olinadi. O‘rta to‘g‘ri to‘rtburchaklar usuli Teylor simmetriyasi evaziga O(h²) tartibli yuqori aniqlik beradi.',
     formula: '∫ₐᵇ f(x) dx ≈ h · Σᵢ f(ξᵢ),  h = (b − a) / n',
+    formulaTex:
+      'I_{\\text{o\'rta}} = h \\sum_{i=0}^{n-1} f\\left(x_i + \\frac{h}{2}\\right), \\quad I_{\\text{chap}} = h \\sum_{i=0}^{n-1} f(x_i), \\quad I_{\\text{o\'ng}} = h \\sum_{i=1}^n f(x_i)',
+    convergenceCondition:
+      'Funksiya [a, b] da uzluksiz bo‘lishi lozim. Qadam h → 0 da yig‘indi Riman integrali ta’rifiga ko‘ra aniq integralga intiladi.',
+    convergenceTex: 'f(x) \\in C^2[a,b], \\quad h = \\frac{b-a}{n} \\to 0 \\implies I_h \\to \\int_a^b f(x)dx',
+    errorEstimateTex:
+      '|R_{\\text{chap/o\'ng}}| \\le \\frac{(b-a)^2}{2n} M_1, \\quad |R_{\\text{o\'rta}}| \\le \\frac{(b-a)^3}{24 n^2} M_2 = \\frac{h^2(b-a)}{24} M_2, \\quad M_2 = \\max_{[a,b]} |f\'\'(x)|',
+    theory: [
+      'Geometrik asosi: Integral Riman integrali ta’rifiga binoan integral yig‘indining limitidir: ∫ f(x)dx = lim Σ f(ξ_i)Δx_i.',
+      'Chap va o‘ng to‘rtburchaklar: Oraliq funksiyaning boshlang‘ich yoki oxirgi nuqtasidagi qiymati bo‘yicha olinadi. Global xatolik O(h) bo‘lib, birinchi tartibga ega.',
+      'O‘rta to‘rtburchaklar (Midpoint rule): Oraliq o‘rtasidagi ξ_i = x_i + h/2 nuqtada Teylor qatoriga yoyilganda toq darajali hadlar integrallanganda nolga aylanadi. Shuning uchun xato O(h²) tartibda bo‘ladi — u trapetsiyalar usulidan ham ikki barobar aniqroqdir!',
+      'Umumlashgan kvadratur: Oraliq teng n ta bo‘lakka bo‘linganda kompozit formula orqali hisoblash kompyuterda juda tez va barqaror bajariladi.',
+    ],
     steps: [
-      'Integrallash chegaralari a va b hamda bo‘linishlar soni n ni tanlang.',
-      'Qadam uzunligini h = (b − a) / n orqali hisoblang.',
-      'Har bir bo‘lakda chap, o‘ng yoki o‘rta nuqta balandligini tanlang.',
-      'Tanlangan nuqtalardagi f qiymatlarini qo‘shib, h ga ko‘paytiring.',
+      'Integrallash oralig‘ini [a, b] va bo‘linishlar sonini n ni tanlang.',
+      'Qadam uzunligini hisoblang: h = (b − a) / n.',
+      'Tugun nuqtalarini aniqlang: x_i = a + i·h (i = 0, 1, ..., n).',
+      'Har bir bo‘lakning o‘rtasini toping: x_{i+1/2} = a + (i + 0.5)h.',
+      'Yig‘indini tuzing va h ga ko‘paytiring: I = h · Σ f(x_{i+1/2}).',
     ],
     example: {
-      problem: '∫₀¹ x² dx integralini n = 4 bilan o‘rta nuqtalar usulida baholash.',
-      calculation: 'h = 0.25; o‘rta nuqtalar 0.125, 0.375, 0.625, 0.875.',
-      result: 'I ≈ 0.25 · Σ f(ξᵢ) = 0.328125; aniq qiymat 1/3 ≈ 0.333333.',
+      problem: '∫₀¹ x / (x + 1) dx integralini n = 10 bilan o‘rta to‘rtburchak usulida hisoblash.',
+      calculation:
+        'h = (1 − 0)/10 = 0.1. O‘rta nuqtalar: x₀.₅ = 0.05, x₁.₅ = 0.15, ..., x₉.₅ = 0.95. Funksiya qiymatlari: f(0.05) = 0.05/1.05 ≈ 0.0476; f(0.15) ≈ 0.1304; ...; f(0.95) ≈ 0.4872. Yig‘indi Σ f(x_{i+1/2}) ≈ 3.0691.',
+      result: 'I ≈ 0.1 · 3.0691 = 0.3069. Aniq qiymat: 1 − ln(2) ≈ 0.30685 (xatolik atigi 0.00005).',
     },
     whenToUse: [
-      'Integralni tez va sodda baholash yoki chap, o‘ng va o‘rta nuqtalar natijasini taqqoslash kerak bo‘lsa.',
-      'Funksiya qiymatlarini bo‘laklar bo‘yicha jadvaldan olish qulay bo‘lsa.',
+      'Integralni tez va sodda baholash kerak bo‘lganda.',
+      'Funksiya chegaraviy nuqtalarda singulyarlikka ega bo‘lib, o‘rta nuqtalarda chekli bo‘lganda.',
+      'Turli kvadratura usullarining (chap, o‘ng, o‘rta) farqini taqqoslab o‘rganishda.',
     ],
     limitations: [
-      'Aniqlik bo‘laklar soni va tanlangan nuqtalarga bog‘liq; chap/o‘ng usullarda xato odatda kattaroq bo‘ladi.',
-      'Keskin o‘zgaruvchi funksiya uchun ko‘proq bo‘linish yoki yuqori tartibli kvadratura kerak bo‘lishi mumkin.',
+      'Chap va o‘ng to‘rtburchaklar usuli sekin yaqinlashadi (O(h)).',
+      'Keskin o‘zgaruvchi va kuchli tebranuvchi funksiyalarda katta n talab etiladi.',
     ],
     questions: [
       {
-        question: 'Qaysi to‘g‘ri to‘rtburchaklar usuli aniqroq?',
+        question: 'O‘rta to‘rtburchaklar usuli nega trapetsiya usulidan aniqroq?',
         answer:
-          'Silliq funksiya uchun o‘rta nuqtalar usuli odatda chap va o‘ng variantlardan aniqroq bo‘ladi, ammo natija funksiya va bo‘linishga bog‘liq.',
+          'O‘rta nuqtalar qoidasida xatolik koeffitsiyenti 1/24 bo‘lsa, trapetsiya usulida 1/12 dir. Demak, o‘rta to‘rtburchak usulining nazariy xatosi trapetsiyadan 2 barobar kichikdir.',
       },
       {
-        question: 'n ortsa natija doim aniq bo‘ladimi?',
+        question: 'Chap va o‘ng to‘rtburchaklar orasidagi farq nimani bildiradi?',
         answer:
-          'Bo‘linishni ko‘paytirish odatda yaqinlashishni yaxshilaydi, lekin yaxlitlash va funksiya xususiyatlarini ham hisobga olish kerak.',
+          'Monoton o‘suvchi funksiyada chap usul integralni quyi baholaydi (kamaytiradi), o‘ng usul esa yuqori baholaydi (orttiradi). Ularning farqi oraliq qadami h ga to‘g‘ridan-to‘g‘ri proporsionaldir.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'VII bob, 7.1-§ ("Aniq integralni to‘g‘ri to‘rtburchak usuli bilan taqribiy hisoblash")',
+        pages: '121–127-betlar',
+        note: 'C# dastur matni va 30 ta amaliy variant misollari.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'III bob, 3.1-§ ("Interpolyatsion kvadratur formulalar")',
+        pages: '58–63-betlar',
+        note: 'O‘rta to‘rtburchaklar qoldiq hadi R_0(f) = (b-a)³ f′′(ξ)/24 ning qat’iy isboti.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '5-bob, 2-§ ("To‘g‘ri to‘rtburchaklar formulasi")',
+        pages: '114–116-betlar',
+        note: 'Algoritm blok-sxemasi va Paskal tili dasturi.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Chap, o‘ng va o‘rta to‘rtburchaklarni birlashtirish',
+        textbookStandard:
+          'Ko‘p darsliklarda har bir variant uchun alohida formula va dastur yoziladi yoki faqat bitta variant ko‘rsatiladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida bir vaqtning o‘zida uchala usul (chap, o‘rta, o‘ng) hisoblanadi. Interaktiv grafikda to‘rtburchaklar ustunchalari egri chiziq ostida chiroyli ko‘rsatiladi.',
+        whyChosen:
+          'Foydalanuvchi uchala usulning farqini bitta ekranda darhol solishtira oladi va o‘rta nuqtaning ustunligini ko‘z bilan ko‘radi.',
+      },
+      {
+        topic: 'Integral ostidagi funksiya kiritish sintaksisi',
+        textbookStandard:
+          'Kitoblarda f(x) analitik tarzda yoziladi, lekin kompyuterga kiritish turlicha (Paskal, C#, Mathcad).',
+        platformImplementation:
+          'Platformada universal matematik ifodalar qo‘llab-quvvatlanadi: sqrt(x), exp(x), ln(x), x^2, sin(x). Integral analitik tarzda ham yechilib, taqribiy natija bilan xatoligi ko‘rsatiladi.',
+        whyChosen:
+          'Har qanday murakkablikdagi funksiyalarni qulay kiritish va ta’lim jarayonida vizual tekshirish imkonini beradi.',
       },
     ],
   },
   {
     slug: 'trapetsiyalar-usuli',
     title: 'Trapetsiyalar usuli',
-    metaTitle: 'Trapetsiyalar usuli: integral formulasi va misol',
+    metaTitle: 'Trapetsiyalar usuli: integral formulasi, xato bahosi va misol',
     description:
-      'Trapetsiyalar usuli bilan aniq integralni taqribiy hisoblashni o‘rganing. Formula, bo‘linish qadami, x² misoli va online hisoblagich.',
-    keywords: ['trapetsiyalar usuli misol', 'trapetsiya formulasi integral', 'sonli integrallash'],
+      'Trapetsiyalar usulida aniq integralni sonli hisoblash. Kompozit formula, qoldiq had bahosi, darsliklar manbalari va platforma standarti.',
+    keywords: [
+      'trapetsiyalar usuli',
+      'trapetsiya formulasi',
+      'kompozit trapetsiya',
+      'sonli integrallash',
+      'integral taqribiy hisoblash',
+    ],
     area: 'Sonli integrallash',
     calculatorHref: '/methods/trapezoid',
     summary:
-      'Trapetsiyalar usuli qo‘shni tugunlardagi funksiya qiymatlarini to‘g‘ri chiziq bilan tutashtirib, hosil bo‘lgan trapetsiyalar yuzalarini qo‘shadi. Silliq funksiya uchun kompozit formula xatoni qadam kichraygani sari kamaytiradi.',
+      'Trapetsiyalar usulida integrallash oraliqlaridagi egri chiziq vatarlar (chiziqli kesmalar) bilan tutashtirilib, hosil bo‘lgan to‘g‘ri chiziqli trapetsiyalar yuzalari qo‘shiladi. Silliq funksiyalar uchun kompozit formula xatoligi O(h²) tartibda bo‘ladi.',
     formula: 'I ≈ h/2 · [f(x₀) + 2Σᵢ₌₁ⁿ⁻¹ f(xᵢ) + f(xₙ)],  h = (b − a) / n',
+    formulaTex:
+      'I_{\\text{trap}} \\approx \\frac{h}{2} \\left[ f(x_0) + 2\\sum_{i=1}^{n-1} f(x_i) + f(x_n) \\right], \\quad h = \\frac{b - a}{n}',
+    convergenceCondition:
+      'Funksiya [a, b] da uzluksiz va kamida ikki marta differensiallanuvchi bo‘lsin. n musbat butun son bo‘lishi kifoya.',
+    convergenceTex: 'f(x) \\in C^2[a,b], \\quad n \\ge 1, \\quad h = \\frac{b-a}{n}',
+    errorEstimateTex:
+      '|R_n(f)| \\le \\frac{(b-a)^3}{12 n^2} \\max_{[a,b]} |f\'\'(x)| = \\frac{h^2 (b-a)}{12} M_2, \\quad M_2 = \\max_{[a,b]} |f\'\'(x)|',
+    theory: [
+      'Egri chiziqni vatarlar bilan almashtirish: Har bir [x_i, x_{i+1}] kesmada funksiya 1-darajali Lagranj interpolyatsion ko‘phadi bilan approksimatsiyalanadi: f(x) ≈ y_i + (y_{i+1} − y_i)(x − x_i)/h.',
+      'Elementar trapetsiya yuzi: S_i = ∫_{x_i}^{x_{i+1}} f(x)dx ≈ h(y_i + y_{i+1})/2.',
+      'Kompozit formula: Barcha n ta qismlar yig‘ilganda, har bir ichki tugun ikkita qo‘shni trapetsiyaga kiradi va koeffitsiyenti 2 ga teng bo‘ladi; chekka tugunlar (x₀ va x_n) esa faqat bitta trapetsiyaga tegishli bo‘lib, koeffitsiyenti 1 bo‘ladi.',
+      'Xatolik xususiyati: Agar funksiya botiq bo‘lsa (f′′ < 0), trapetsiyalar yuzasi aniq integraldan kichik bo‘ladi; qavariq bo‘lsa (f′′ > 0), kattaroq chiqadi.',
+    ],
     steps: [
-      'Integrallash oralig‘ini [a, b] va bo‘linishlar sonini n ni belgilang.',
-      'Tugunlarni xᵢ = a + ih, h = (b − a) / n formula bilan tuzing.',
-      'Chekka tugunlarni bir marta, ichki tugunlarni ikki marta yig‘indiga kiriting.',
-      'Yig‘indini h/2 ga ko‘paytirib integral bahosini oling.',
+      'Integrallash oralig‘i [a, b] va bo‘linishlar soni n ni belgilang.',
+      'Qadam uzunligini hisoblang: h = (b − a) / n.',
+      'Tugun nuqtalarini tuzing: x_i = a + i·h (i = 0, 1, ..., n).',
+      'Chekka tugunlar qiymatini (f(a) + f(b))/2 va ichki tugunlar yig‘indisini Σ f(x_i) hisoblang.',
+      'Yig‘indini h ga ko‘paytiring: I = h · [ (f(a)+f(b))/2 + Σ f(x_i) ].',
     ],
     example: {
-      problem: '∫₀¹ x² dx integralini n = 2 bilan trapetsiyalar usulida hisoblash.',
-      calculation: 'h = 0.5; f(0) = 0, f(0.5) = 0.25, f(1) = 1.',
-      result: 'I ≈ 0.5/2 · [0 + 2·0.25 + 1] = 0.375; aniq qiymat 1/3.',
+      problem: '∫₀¹ (x + 1) dx integralini n = 10 bilan trapetsiya usulida hisoblash.',
+      calculation:
+        'h = 0.1. x₀ = 0, x₁ = 0.1, ..., x₁₀ = 1.0. f(0) = 1, f(1) = 2. Ichki tugunlar: f(0.1)=1.1, f(0.2)=1.2, ..., f(0.9)=1.9. Ichki yig‘indi: 1.1 + 1.2 + ... + 1.9 = 13.5. Chekkalar: (1 + 2)/2 = 1.5. Umumiy yig‘indi: 1.5 + 13.5 = 15.0. I = 0.1 · 15.0 = 1.5.',
+      result: 'I = 1.5. Aniq yechim: [x²/2 + x]₀¹ = 0.5 + 1 = 1.5 (chiziqli funksiyada xato 0 ga teng!).',
     },
     whenToUse: [
-      'Funksiya qiymatlari tugunlarda mavjud bo‘lib, integral uchun oddiy va barqaror baho kerak bo‘lsa.',
-      'Tajriba ma’lumotlaridan yuzani taxminiy hisoblash kerak bo‘lsa.',
+      'Funksiya qiymatlari jadval ko‘rinishida berilgan bo‘lganda.',
+      'Bo‘linishlar soni n juft bo‘lishi shart bo‘lmaganda.',
+      'Hisoblash soddaligi va geometrik ko‘rgazmalilik muhim bo‘lganda.',
     ],
     limitations: [
-      'Egri chiziq bo‘lak ichida to‘g‘ri chiziq bilan almashtiriladi; egrilik katta bo‘lsa xato ortadi.',
-      'Silliq funksiya uchun xatoning tartibi O(h²); Simpson usuli yanada yuqori tartib bera oladi.',
+      'Funksiya egriligi katta bo‘lsa, aniqlik Simpson usuliga qaraganda ancha past bo‘ladi.',
+      'Yuqori aniqlik uchun ko‘p miqdorda bo‘linishlar talab qilinadi.',
     ],
     questions: [
       {
-        question: 'Trapetsiyalar usulida n juft bo‘lishi kerakmi?',
-        answer: 'Yo‘q. Trapetsiyalar usuli uchun n musbat butun bo‘lishi kifoya.',
+        question: 'Trapetsiyalar usulida n juft bo‘lishi shartmi?',
+        answer: 'Yo‘q. Simpson usulidan farqli o‘laroq, trapetsiyalar usuli uchun n ixtiyoriy butun musbat son (toq ham, juft ham) bo‘lishi mumkin.',
       },
       {
-        question: 'Trapetsiyalar usulining xatosi qachon kamayadi?',
+        question: 'Qachon trapetsiya usuli mutlaq aniq natija beradi?',
         answer:
-          'Funksiya yetarlicha silliq bo‘lsa, qadam h ni kichraytirish orqali xato odatda kamayadi.',
+          'Agar f(x) chiziqli funksiya (f(x) = kx + b) bo‘lsa, f′′(x) = 0 bo‘ladi va qoldiq had aynan nolga aylanadi — natija mutlaq aniq chiqadi.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'VII bob, 7.2-§ ("Aniq integrallarni trapetsiyalar usuli bilan taqribiy hisoblash")',
+        pages: '128–134-betlar',
+        note: 'Qoldiq had bahosi R_n(f) = -(b-a)³ f′′(ξ)/(12n²) va C# dasturi keltirilgan.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'III bob, 3.1-§ ("Umumlashgan trapetsiyalar formulasi")',
+        pages: '61–63, 70–71-betlar',
+        note: 'Koshi teoremasiga asoslangan qoldiq had bahosi.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '5-bob, 3-§ ("Trapetsiya usuli")',
+        pages: '117–118-betlar',
+        note: 'Algoritm blok-sxemasi va Paskal dastur matni.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Formula ko‘rinishi va qismlar yig‘indisi',
+        textbookStandard:
+          'Kitoblarda ba’zan I ≈ h[(y₀ + y_n)/2 + y₁ + ... + y_{n-1}] ko‘rinishida, ba’zan esa I ≈ (h/2)[y₀ + 2y₁ + ... + 2y_{n-1} + y_n] shaklida beriladi.',
+        platformImplementation:
+          'Sonli-usullar platformasida ikkala ifoda ham bir xil matematik qiymatga ega ekanligi ko‘rsatiladi. Dastur ichida umumiy omil h/2 qilib chiqarilib, chekkalar 1, ichkilar 2 ga ko‘paytiriladi.',
+        whyChosen:
+          'Bu usul hisoblash mashinasida arifmetik amallarni qisqartiradi va yaxlitlash xatosini kamaytiradi.',
+      },
+      {
+        topic: 'Interaktiv grafik va trapetsiyalar vizualizatsiyasi',
+        textbookStandard:
+          'Darsliklarda faqat bitta statik chizma (16-rasm) beriladi.',
+        platformImplementation:
+          'Platformada n sonini slayder orqali o‘zgartirganda har bir trapetsiya real vaqtda qayta chiziladi. Foydalanuvchi n ortgani sari trapetsiyalar egri chiziqqa qanday yopishib borishini interaktiv ko‘radi.',
+        whyChosen:
+          'Geometrik ma’noni mustahkamlash va n ning aniqlikka ta’sirini jonli his qilish uchun.',
       },
     ],
   },
   {
     slug: 'simpson-usuli',
-    title: 'Simpson 1/3 usuli',
-    metaTitle: 'Simpson usuli: integral formulasi, shartlar va misol',
+    title: 'Simpson 1/3 (parabolalar) usuli',
+    metaTitle: 'Simpson usuli: formula, juft qadam sharti va kalkulyator',
     description:
-      'Simpson 1/3 usulida integralni parabolalar bilan hisoblang. Juft bo‘linish sharti, formula va ∫₀¹ x² dx ishlangan misoli.',
-    keywords: ['Simpson usuli misol', 'Simpson 1/3 formulasi', 'integralni online hisoblash'],
+      'Simpson 1/3 (parabolalar) usulida aniq integralni hisoblash. 4-tartibli aniqlik, juft n sharti, darsliklardagi 2n va n chalkashligi tahlili hamda platforma standarti.',
+    keywords: [
+      'simpson usuli',
+      'simpson 1/3 formulasi',
+      'parabolalar usuli',
+      'sonli integrallash simpson',
+      'simpson qoidasi',
+    ],
     area: 'Sonli integrallash',
     calculatorHref: '/methods/simpson',
     summary:
-      'Simpson 1/3 usuli har ikki qo‘shni bo‘lakdagi funksiyani parabola bilan yaqinlashtiradi. Yetarlicha silliq funksiya va teng qadamlar sharoitida u trapetsiyalar usuliga qaraganda yuqori aniqlik beradi.',
+      'Simpson 1/3 usuli har ikki qo‘shni oraliqdagi funksiyani kvadratik parabola bilan yaqinlashtiradi. Bo‘linishlar soni n juft bo‘lishi shart. Toq indeksli tugunlar 4, juft indeksli tugunlar 2 koeffitsiyenti bilan olinadi. Usul O(h⁴) yuqori tartibli aniqlikka ega bo‘lib, 3-darajagacha bo‘lgan barcha ko‘phadlarni mutlaqo aniq integrallaydi.',
     formula: 'I ≈ h/3 · [f(x₀) + 4Σ f(x_toq) + 2Σ f(x_juft) + f(xₙ)]',
+    formulaTex:
+      'I_{\\text{simp}} \\approx \\frac{h}{3} \\left[ f(x_0) + 4\\sum_{k=1}^{n/2} f(x_{2k-1}) + 2\\sum_{k=1}^{n/2 - 1} f(x_{2k}) + f(x_n) \\right], \\quad h = \\frac{b - a}{n}',
+    convergenceCondition:
+      'Funksiya [a, b] da 4 marta uzluksiz differensiallanuvchi (f ∈ C⁴[a, b]) va bo‘linishlar soni n juft musbat son (n = 2m) bo‘lishi shart.',
+    convergenceTex: 'f(x) \\in C^4[a,b], \\quad n \\in 2\\mathbb{N}^+ \\implies h = \\frac{b-a}{n}',
+    errorEstimateTex:
+      '|R_n(f)| \\le \\frac{(b-a)^5}{180 n^4} \\max_{[a,b]} |f^{(4)}(x)| = \\frac{h^4 (b-a)}{180} M_4',
+    theory: [
+      'Parabolik approksimatsiya: Har bir juft oraliqda [x_{2k-2}, x_{2k}] uchta nuqta (x_{2k-2}, y_{2k-2}), (x_{2k-1}, y_{2k-1}), (x_{2k}, y_{2k}) orqali o‘tuvchi ikkinchi tartibli parabola P₂(x) = Ax² + Bx + C quriladi.',
+      'Klassik Simpson formulasi: ∫_{x_0}^{x_2} P₂(x)dx = (h/3)(y₀ + 4y₁ + y₂).',
+      'Darajaviy xususiyat: Garchi parabola darajasi 2 bo‘lsa-da, simmetrik oraliqda kubik hadning integrali nol bo‘lganligi uchun Simpson formulasi har qanday 3-darajali ko‘phad (ax³ + bx² + cx + d) uchun ham qoldiqsiz, mutlaq aniq natija beradi!',
+      'To‘rtinchi tartibli xatolik: Xatolik qadamning 4-darajasiga proporsional: O(h⁴). Qadamni ikki barobar kichraytirish xatolikni 16 barobar kamaytiradi.',
+    ],
     steps: [
-      'Bo‘linishlar soni n ni musbat juft son qilib tanlang.',
-      'h = (b − a) / n qadam bilan x₀ dan xₙ gacha tugunlarni toping.',
-      'Funksiya qiymatlarini hisoblab, ichki toq indekslarga 4, juft indekslarga 2 koeffitsiyent bering.',
-      'Yig‘indini h/3 ga ko‘paytiring.',
+      'Bo‘linishlar soni n ni juft son (n = 2, 4, 6, ...) qilib tanlang.',
+      'Qadam uzunligini hisoblang: h = (b − a) / n.',
+      'Tugunlarni toping: x_i = a + i·h (i = 0, 1, ..., n).',
+      'Tugunlardagi funksiya qiymatlarini hisoblang: y_i = f(x_i).',
+      'Toq indeksli qiymatlar yig‘indisini 4 ga, juft indeksli qiymatlar yig‘indisini 2 ga ko‘paytiring.',
+      'Chekka qiymatlarni qo‘shib, h/3 ga ko‘paytiring: I = (h/3)[y₀ + 4Σ y_{toq} + 2Σ y_{juft} + y_n].',
     ],
     example: {
-      problem: '∫₀¹ x² dx integralini Simpson 1/3 usulida n = 2 bilan hisoblash.',
-      calculation: 'h = 0.5; f(0) = 0, f(0.5) = 0.25, f(1) = 1.',
-      result: 'I ≈ 0.5/3 · [0 + 4·0.25 + 1] = 1/3; kvadratik funksiya uchun bu aniq natija.',
+      problem: '∫₀¹ 2x / (1 + x²) dx integralini n = 10 (juft) bilan Simpson usulida hisoblash.',
+      calculation:
+        'h = (1 − 0)/10 = 0.1. x₀ = 0, x₁ = 0.1, ..., x₁₀ = 1.0. Toq tugunlar: f(0.1), f(0.3), f(0.5), f(0.7), f(0.9); yig‘indi S₁ ≈ 1.8398. Juft tugunlar: f(0.2), f(0.4), f(0.6), f(0.8); yig‘indi S₂ ≈ 1.3414. Chekkalar: f(0) = 0, f(1) = 1.0. I = (0.1 / 3) · [0 + 4(1.8398) + 2(1.3414) + 1.0] ≈ 0.693148.',
+      result: 'I ≈ 0.693148. Aniq analitik qiymat: ln(1 + 1²) = ln(2) ≈ 0.69314718 (xatolik atigi 10⁻⁶).',
     },
     whenToUse: [
-      'Funksiya silliq va intervalni teng qadamlarga bo‘lish mumkin bo‘lsa.',
-      'Trapetsiyalar usulidan ko‘ra aniqroq integral bahosi kerak bo‘lsa.',
+      'Funksiya yetarlicha silliq va oraliqni teng juft qadamlarga bo‘lish mumkin bo‘lganda.',
+      'Trapetsiyalar usuliga qaraganda ancha yuqori aniqlik talab qilinganda.',
+      'Kvadratik yoki kubik ko‘phadlarni aniq integrallash zarur bo‘lganda.',
     ],
     limitations: [
-      'Bo‘linishlar soni juft bo‘lishi shart.',
-      'Funksiya keskin sakrasa yoki singulyarlikka ega bo‘lsa, oddiy Simpson bahosi ishonchsiz bo‘lishi mumkin.',
+      'Bo‘linishlar soni n albatta juft bo‘lishi shart.',
+      'Funksiyaning 4-tartibli hosilasi uzilishli yoki juda katta bo‘lsa, usulning yuqori aniqlik afzalligi pasayishi mumkin.',
     ],
     questions: [
       {
-        question: 'Simpson usulida n nega juft bo‘lishi kerak?',
+        question: 'Simpson usulida n nega albatta juft bo‘lishi kerak?',
         answer:
-          'Har bir parabola uchta ketma-ket tugun, ya’ni ikkita bo‘lakdan foydalanadi. Shuning uchun barcha bo‘laklarni juft-juft guruhlash kerak.',
+          'Chunki har bir parabola uchta ketma-ket nuqta (ya’ni 2 ta qismiy oraliq) ustida quriladi. Barcha oraliqlarni juftlab guruhlash uchun umumiy qismlar soni n juft bo‘lishi shart.',
       },
       {
-        question: 'Simpson usuli qaysi funksiyalar uchun aniq?',
+        question: 'Simpson 1/3 va Simpson 3/8 orasidagi farq nima?',
         answer:
-          'Bir intervaldagi Simpson 1/3 formulasi darajasi 3 gacha bo‘lgan ko‘phadlarni aniq integrallaydi; yuqori darajali silliq funksiyalarda taqribiy natija beradi.',
+          'Simpson 1/3 ikkita oraliq va parabola (2-darajali) ishlatadi; Simpson 3/8 esa uchta oraliq va 3-darajali ko‘phad ishlatadi (n 3 ga karrali bo‘lishi talab qilinadi).',
+      },
+    ],
+    sources: [
+      {
+        authors: 'A.I. Ismoilov, Sh.A. Ro‘zaliyev',
+        title: 'Sonli usullar (I qism)',
+        publisher: 'Farg‘ona davlat universiteti',
+        year: 2025,
+        chapterSection: 'VII bob, 7.3-§ ("Aniq integralni parabolalar (Simpson) usuli bilan taqribiy hisoblash")',
+        pages: '135–143-betlar',
+        note: 'Qoldiq had bahosi R_n(f) = -(b-a) f^{(IV)}(ξ) h⁴ / 180 va C# kodi keltirilgan.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'III bob, 3.1-§ ("Simpson kvadratur formulasi")',
+        pages: '62–64, 71–74-betlar',
+        note: 'Simpson formulasi qoldiq hadining batafsil isboti va Chebishev/Gauss kvadraturasi bilan taqqoslanishi.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '5-bob, 4-§ ("Simpson (parabolalar) formulasi")',
+        pages: '119–123-betlar',
+        note: 'Misol ∫₀¹ (x³ + 2x² − x + 5)dx ning uchala usulda taqqoslama hisobi.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: '2n va n oraliqlar soni chalkashligi (Asosiy darslik ziddiyati)',
+        textbookStandard:
+          'Ismoilov (2025) va Irisqulov (2013) kitoblarida interval "2n ta bo‘lakka bo‘linadi" deb olinadi, shunda qadam h = (b-a)/(2n) va koeffitsiyent (b-a)/(6n) bo‘ladi. Ismatullayev (2014) va xalqaro darsliklarda (Burden) esa oraliq "n ta bo‘lakka bo‘linadi va n juft son" deb belgilanadi (h = (b-a)/n va h/3).',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida xalqaro standart qabul qilingan: kiritiladigan n soni bevosita umumiy oraliqlar sonini (subintervals) bildiradi va u juft bo‘lishi talab qilinadi (n = 2, 4, 6, ...). Qadam h = (b − a)/n, formula esa h/3 [y₀ + 4Σ y_{toq} + 2Σ y_{juft} + y_n].',
+        whyChosen:
+          'Ushbu standart foydalanuvchiga oraliqlar sonini to‘g‘ridan-to‘g‘ri boshqarish imkonini beradi. Agar toq son kiritilsa, tizim darhol buni ko‘rsatib, eng yaqin juft songa avtomatik tuzatishni taklif qiladi.',
+      },
+      {
+        topic: 'Toq va juft indekslar koeffitsiyentlari (4 va 2)',
+        textbookStandard:
+          'Adabiyotlarda qaysi indekslar 4 va qaysilari 2 bo‘lishi indekslash 0 dan yoki 1 dan boshlanishiga qarab chalkashishi mumkin.',
+        platformImplementation:
+          'Platformada 0-indekslangan massiv qabul qilingan: x₀ (chekka), x₁ (toq → 4), x₂ (juft → 2), x₃ (toq → 4), ..., x_{n-1} (toq → 4), x_n (chekka). Natijalar jadvalida har bir x_i yonida uning vazn koeffitsiyenti (1, 4 yoki 2) alohida ustunda ko‘rsatiladi.',
+        whyChosen:
+          'Talaba hisob-kitobni qo‘lda tekshirayotganda qaysi had qanday vazn olganini bir zumda ko‘ra oladi.',
       },
     ],
   },
   {
     slug: 'eyler-usuli',
     title: 'Eyler usuli (oddiy differensial tenglama)',
-    metaTitle: 'Eyler usuli: ODE yechish formulasi va misol',
+    metaTitle: 'Eyler usuli: Koshi masalasi formulasi, qadam va kalkulyator',
     description:
-      'Boshlang‘ich shartli y′ = f(x,y) differensial tenglamani Eyler usulida qadam-baqadam yeching. Formula, xato va oddiy misol.',
-    keywords: ['Eyler usuli misol', 'differensial tenglama yechish', 'ODE online kalkulyator'],
+      'Eyler usulida birinchi tartibli oddiy differensial tenglamani (Koshi masalasi) sonli yechish. Siniq chiziqlar geometriyasi, takomillashgan Eyler, darsliklar va platforma standarti.',
+    keywords: [
+      'eyler usuli',
+      'koshi masalasi',
+      'differensial tenglamalar sonli yechish',
+      'euler method',
+      'takomillashgan eyler',
+    ],
     area: 'Oddiy differensial tenglamalar',
     calculatorHref: '/methods/euler',
     summary:
-      'Eyler usuli boshlang‘ich qiymat masalasida yechim egri chizig‘ini har bir qadamda uning joriy qiyaligi bo‘yicha to‘g‘ri chiziq bilan yaqinlashtiradi. Sodda bo‘lishiga qaramay, qadam kattaligi xatoga sezilarli ta’sir qiladi.',
+      'Eyler usuli — oddiy differensial tenglamalar uchun Koshi masalasini yechishning eng sodda va fundamental sonli usuli. U har bir qadamda yechim egri chizig‘ini uning joriy qiyaligi bo‘yicha to‘g‘ri chiziq (urinma) bilan almashtiradi. Lokal xatosi O(h²), global xatosi esa O(h) tartibga ega.',
     formula: 'yᵢ₊₁ = yᵢ + h · f(xᵢ, yᵢ),  y(x₀) = y₀',
+    formulaTex: 'y_{i+1} = y_i + h f(x_i, y_i), \\quad x_{i+1} = x_i + h, \\quad y(x_0) = y_0',
+    convergenceCondition:
+      'f(x, y) funksiyasi qaralayotgan sohada uzluksiz va y bo‘yicha Lipshits shartini qanoatlantirishi lozim: |f(x, y₁) − f(x, y₂)| ≤ L|y₁ − y₂|.',
+    convergenceTex: '|f(x, y_1) - f(x, y_2)| \\le L |y_1 - y_2|, \\quad h \\to 0 \\implies y_n \\to y(x_n)',
+    errorEstimateTex:
+      '\\text{Lokal xatolik: } R_{i+1} = \\frac{h^2}{2} y\'\'(\\xi) = O(h^2), \\quad \\text{Global to‘liq xatolik: } |y(x_n) - y_n| \\le O(h)',
+    theory: [
+      'Koshi masalasi: y′ = f(x, y), y(x₀) = y₀ boshlang‘ich shartli differensial tenglama yechimini [x₀, X] oraliqda topish.',
+      'Teylor yoyilmasidan hosil bo‘lishi: y(x_{i+1}) = y(x_i) + h y′(x_i) + (h²/2) y′′(ξ). Boshlang‘ich ikkita had bilan chegaralanib, y′(x_i) = f(x_i, y_i) qo‘yilsa, Eyler formulasi y_{i+1} = y_i + h f(x_i, y_i) kelib chiqadi.',
+      'Geometrik ma’nosi: Siniq chiziqlar metodi — har bir qadamda yechim urinma bo‘yicha davom ettiriladi. Lekin qadam oshgani sari yechim haqiqiy integral egri chiziqlar oilasidan uzoqlashadi.',
+      'Takomillashgan Eyler (Heun metodi): Qadam o‘rtasidagi yoki bashorat qilingan qiyalikni hisobga oluvchi prediktor-korrektor sxemasi: y_{i+1} = y_i + (h/2)[f(x_i, y_i) + f(x_{i+1}, y_i + h f(x_i, y_i))]. Global xatolik O(h²) gacha pasayadi.',
+    ],
     steps: [
-      'Tenglamani y′ = f(x, y) ko‘rinishida va boshlang‘ich (x₀, y₀) ni yozing.',
-      'Qadam h ni tanlab, xᵢ₊₁ = xᵢ + h ni hisoblang.',
-      'Joriy nuqtadagi qiyalik f(xᵢ, yᵢ) ni toping.',
-      'yᵢ₊₁ = yᵢ + h f(xᵢ, yᵢ) bilan yangi qiymatni topib takrorlang.',
+      'Differensial tenglamani y′ = f(x, y) ko‘rinishida va boshlang‘ich shartni (x₀, y₀) yozing.',
+      'Integrallash oralig‘i [x₀, X] va qadam h ni belgilang (yoki n ta bo‘lakka bo‘ling: h = (X − x₀)/n).',
+      'Joriy nuqtadagi hosila qiymatini hisoblang: f(xᵢ, yᵢ).',
+      'Keyingi qiymatni toping: y_{i+1} = y_i + h · f(xᵢ, yᵢ) va x_{i+1} = x_i + h.',
+      'x_{i+1} oraliq oxiri X ga yetguncha takrorlang va natijalar jadvalini tuzing.',
     ],
     example: {
-      problem: 'y′ = y, y(0) = 1 masalasining birinchi qadami; h = 0.1.',
-      calculation: 'f(0, 1) = 1; y₁ = 1 + 0.1 · 1 = 1.1.',
-      result: 'Eyler bahosi y(0.1) ≈ 1.1; aniq qiymat e⁰·¹ ≈ 1.10517.',
+      problem: 'y′ = x + y, y(0) = 1 Koshi masalasini h = 0.1 qadam bilan [0, 0.4] da yechish (aniq yechim: y = 2eˣ − x − 1).',
+      calculation:
+        '1-qadam (x₀=0, y₀=1): f(0, 1) = 0 + 1 = 1; y₁ = 1 + 0.1·1 = 1.1 (aniq: 1.1103, xato: 0.0103). 2-qadam (x₁=0.1, y₁=1.1): f(0.1, 1.1) = 1.2; y₂ = 1.1 + 0.1·1.2 = 1.22 (aniq: 1.2428, xato: 0.0228). 3-qadam: y₃ = 1.22 + 0.1(0.2+1.22) = 1.362. 4-qadam: y₄ = 1.362 + 0.1(0.3+1.362) = 1.5282.',
+      result: 'x = 0.4 da Eyler yechimi y ≈ 1.5282; analitik aniq yechim y(0.4) ≈ 1.5836.',
     },
     whenToUse: [
-      'Differensial tenglama yechimini qanday qadamlar qurishini o‘rganishda.',
-      'Hisoblashni soddalashtirish va yuqoriroq tartibli usullar bilan taqqoslashda.',
+      'Differensial tenglamalar sonli yechilishining qadam-baqadam mexanizmini o‘rganishda.',
+      'Hisoblash resurslari juda cheklangan va yuqori aniqlik talab qilinmagan sodda modellashtirishlarda.',
+      'Ko‘p qadamli metodlar (Adams) uchun boshlang‘ich nuqtalarni tayyorlashda.',
     ],
     limitations: [
-      'Global xatoning tartibi odatda O(h); aniqlikni oshirish uchun h ni kichraytirish kerak.',
-      'Juda qattiq o‘zgaruvchi sistemalarda oddiy Eyler usuli katta xato yoki beqarorlik berishi mumkin.',
+      'Global xatolik birinchi tartibli O(h); yuqori aniqlik uchun juda kichik h talab etiladi.',
+      'Qattiq (stiff) differensial tenglamalarda oddiy oshkor Eyler usuli beqaror bo‘lib, cheksiz tebranishga olib keladi.',
     ],
     questions: [
       {
-        question: 'Eyler usulida h nimani bildiradi?',
+        question: 'Eyler usulining aniqligini qanday yaxshilash mumkin?',
         answer:
-          'h — mustaqil o‘zgaruvchi x bo‘yicha ikki ketma-ket hisoblash nuqtasi orasidagi qadam uzunligi.',
+          'Qadam h ni kichraytirish mumkin, ammo amallar soni va yaxlitlash xatosi ortadi. Yaxshiroq yo‘l — Takomillashgan Eyler yoki 4-tartibli Runge-Kutta usuliga o‘tishdir.',
       },
       {
-        question: 'Eyler usuli aniqligini qanday yaxshilash mumkin?',
+        question: 'Lokal va global xatolik farqi nimada?',
         answer:
-          'Qadamni kichraytirish mumkin, ammo hisoblashlar soni oshadi. Amaliy masalalarda Heun yoki Runge–Kutta kabi yuqori tartibli usullar ham tanlanadi.',
+          'Lokal xatolik bitta qadamda yo‘l qo‘yilgan xato (O(h²)), global xatolik esa barcha qadamlar bo‘ylab to‘plangan umumiy xatodir (n · O(h²) = O(h)).',
+      },
+    ],
+    sources: [
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '6-bob, 2-§ ("Koshi masalasini yechishning Eyler usuli")',
+        pages: '127–130-betlar',
+        note: 'Siniq chiziqlar geometrik talqini, Paskal dasturi va y′ = cos(x) test misoli.',
+      },
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'VII bob, 7.3.3-§ ("Eyler va Eyler-Koshi usullari")',
+        pages: '167–168, 176–177-betlar',
+        note: 'y′ = x + y misoli tahlili va jadval taqqoslashlari.',
+      },
+      {
+        authors: 'Richard L. Burden, J. Douglas Faires',
+        title: 'Numerical Analysis',
+        publisher: 'Brooks/Cole',
+        year: 2011,
+        chapterSection: 'Chapter 5 ("Initial-Value Problems for ODEs: Euler\'s Method")',
+        pages: '266–275-betlar',
+        note: 'Lipshits sharti va xatolik chegarasi teoremalari.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'O‘zgaruvchilar va tenglamani kiritish formati',
+        textbookStandard:
+          'Ayrim darsliklarda vaqt t va funksiya x(t) yoki u(t) deb belgilanadi (masalan, du/dt = f(t, u)).',
+        platformImplementation:
+          'Sonli-usullar platformasida klassik matematik standart qabul qilingan: erksiz o‘zgaruvchi y, erkli o‘zgaruvchi x: y′ = f(x, y). Masalan, x + y yoki -2*x*y yoki (y^2 - x)/(2*y).',
+        whyChosen:
+          'Bu standart talabalar uchun oliy matematika kursidagi odatiy yozuv bilan bir xil bo‘lib, chalkashliklarni bartaraf etadi.',
+      },
+      {
+        topic: 'Klassik va Takomillashgan Eyler (Heun) rejimi',
+        textbookStandard:
+          'Ko‘p qo‘llanmalarda oddiy Eyler va Eyler-Koshi (yoki Heun) ikki alohida mavzuda bayon qilinadi.',
+        platformImplementation:
+          'Platformamizda bitta kalkulyator ichida rejimni tanlash imkoniyati mavjud: foydalanuvchi "Klassik Eyler" yoki "Takomillashgan Eyler (Heun)" rejimini yoqib, bir vaqtning o‘zida ikkala traektoriyani grafikda kuzatishi mumkin.',
+        whyChosen:
+          'Usulning modifikatsiyasi qanday qilib birinchi tartibli xatoni ikkinchi tartibga ko‘tarishini amalda solishtirish uchun.',
       },
     ],
   },
   {
     slug: 'runge-kutta-4-usuli',
     title: 'Runge–Kutta 4-tartibli usuli (RK4)',
-    metaTitle: 'Runge–Kutta 4 (RK4): formula, misol va ODE kalkulyator',
+    metaTitle: 'Runge–Kutta 4 usuli: RK4 formulasi, Runge qoidasi va kalkulyator',
     description:
-      'RK4 usulida boshlang‘ich qiymatli differensial tenglamani yeching. To‘rtta qiyalik, yangilash formulasi va y′ = y misoli.',
-    keywords: ['Runge-Kutta usuli misol', 'RK4 formulasi', '4-tartibli Runge Kutta'],
+      'Runge-Kutta 4-tartibli (RK4) usulida Koshi masalasini yuqori aniqlikda yechish. To‘rtta qiyalik, Runge qoidasi bo‘yicha xatolik bahosi, darsliklar va platforma standarti.',
+    keywords: [
+      'runge-kutta usuli',
+      'rk4 usuli',
+      '4-tartibli runge kutta',
+      'koshi masalasi rk4',
+      'runge qoidasi',
+    ],
     area: 'Oddiy differensial tenglamalar',
     calculatorHref: '/methods/runge-kutta',
     summary:
-      'To‘rtinchi tartibli Runge–Kutta usuli bir qadamda boshlang‘ich, ikki o‘rta va oxirgi nuqtadagi qiyaliklarni birlashtiradi. U ko‘p oddiy differensial tenglamalarda aniqlik va hisoblash xarajati o‘rtasida yaxshi muvozanat beradi.',
+      'Runge–Kutta 4-tartibli (RK4) usuli har bir qadamda to‘rtta qiyalikni (boshlang‘ich, ikkita o‘rta va oxirgi) hisoblab, ularni Simpson qoidasiga o‘xshash vaznli o‘rtacha bilan birlashtiradi. Global xatoligi O(h⁴) bo‘lib, ilmiy va muhandislik amaliyotida eng keng qo‘llaniladigan standart usuldir.',
     formula: 'k₁ = h f(xᵢ,yᵢ); k₂ = h f(xᵢ+h/2,yᵢ+k₁/2); k₃ = h f(xᵢ+h/2,yᵢ+k₂/2); k₄ = h f(xᵢ+h,yᵢ+k₃); yᵢ₊₁ = yᵢ + (k₁+2k₂+2k₃+k₄)/6',
+    formulaTex:
+      'y_{i+1} = y_i + \\frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4), \\quad k_1 = h f(x_i, y_i), \\; k_2 = h f\\left(x_i + \\frac{h}{2}, y_i + \\frac{k_1}{2}\\right), \\; k_3 = h f\\left(x_i + \\frac{h}{2}, y_i + \\frac{k_2}{2}\\right), \\; k_4 = h f(x_i + h, y_i + k_3)',
+    convergenceCondition:
+      'f(x, y) funksiyasi o‘zgaruvchilar bo‘yicha kamida 4-tartibli uzluksiz xususiy hosilalarga ega bo‘lishi zarur.',
+    convergenceTex: 'f(x, y) \\in C^4(D), \\quad y_{i+1} - y(x_{i+1}) = O(h^5) \\implies \\text{Global xato: } O(h^4)',
+    errorEstimateTex:
+      '\\text{Runge qoidasi: } R_h \\approx \\frac{y_h - y_{2h}}{2^4 - 1} = \\frac{y_h - y_{2h}}{15}, \\quad y_{\\text{aniq}} \\approx y_h + \\frac{y_h - y_{2h}}{15}',
+    theory: [
+      'Ko‘p bosqichli bir qadamli g‘oya: Eyler usulidan farqli o‘laroq, RK4 usulida faqat x_i nuqtadagi qiyalik bilan cheklanmay, [x_i, x_{i+1}] oralig‘idagi o‘rta va yakuniy nuqtalarda ham funksiya qiyaliklari baholanadi.',
+      'Teylor yoyilmasi mosligi: To‘rtta koeffitsiyent (k₁, k₂, k₃, k₄) parametrlari shunday tanlanganki, ularning chiziqli kombinatsiyasi yechimning Teylor qatoridagi h⁵ gacha bo‘lgan barcha hadlari bilan aynan ustma-ust tushadi.',
+      'Simpson integratsiyasi bilan bog‘liqlik: Yangilanish formulasidagi (1/6)(k₁ + 2k₂ + 2k₃ + k₄) koeffitsiyentlari Simpson kvadratura formulasidagi 1/3 [y₀ + 4y_{1/2} + y₁] vaznlariga to‘liq analogdir.',
+      'Runge qoidasi bo‘yicha xatolik: Amalda yuqori tartibli hosilalarni hisoblash nihoyatda qiyin bo‘lgani uchun, hisoblash h va 2h qadamlarda o‘tkaziladi va xatolik R ≈ (y_h − y_{2h}) / 15 formulasi bilan baholanadi.',
+    ],
     steps: [
-      'Boshlang‘ich (xᵢ, yᵢ) nuqtasi va h qadamni belgilang.',
-      'Qadam boshidagi qiyalik k₁ ni hisoblang.',
-      'Ikki yarim qadamdagi qiyaliklar k₂ va k₃ ni hisoblang.',
-      'Qadam oxiridagi k₄ ni topib, to‘rtta bahoni vaznli o‘rtacha bilan birlashtiring.',
+      'Boshlang‘ich (xᵢ, yᵢ) nuqtasi va h qadamni oling.',
+      'Boshlang‘ich qiyalikni hisoblang: k₁ = h · f(xᵢ, yᵢ).',
+      'Birinchi o‘rta qiyalikni hisoblang: k₂ = h · f(xᵢ + h/2, yᵢ + k₁/2).',
+      'Ikkinchi o‘rta qiyalikni hisoblang: k₃ = h · f(xᵢ + h/2, yᵢ + k₂/2).',
+      'Oxirgi qiyalikni hisoblang: k₄ = h · f(xᵢ + h, yᵢ + k₃).',
+      'Vaznli o‘rtachani qo‘shing: y_{i+1} = y_i + (k₁ + 2k₂ + 2k₃ + k₄) / 6 va x_{i+1} = x_i + h.',
     ],
     example: {
-      problem: 'y′ = y, y(0) = 1 tenglamasi uchun h = 0.1.',
-      calculation: 'k₁ = 0.1; k₂ = 0.105; k₃ = 0.10525; k₄ = 0.110525.',
-      result: 'y₁ = 1 + (k₁ + 2k₂ + 2k₃ + k₄)/6 ≈ 1.1051708; aniq qiymat e⁰·¹ ≈ 1.1051702.',
+      problem: 'y′ = x + y, y(0) = 1 Koshi masalasining 1-qadami; h = 0.1 (aniq qiymat: y(0.1) ≈ 1.110342).',
+      calculation:
+        'k₁ = 0.1 · f(0, 1) = 0.1 · (0 + 1) = 0.1. k₂ = 0.1 · f(0 + 0.05, 1 + 0.05) = 0.1 · (0.05 + 1.05) = 0.11. k₃ = 0.1 · f(0 + 0.05, 1 + 0.055) = 0.1 · (0.05 + 1.055) = 0.1105. k₄ = 0.1 · f(0 + 0.1, 1 + 0.1105) = 0.1 · (0.1 + 1.1105) = 0.12105. Δy = (0.1 + 2·0.11 + 2·0.1105 + 0.12105) / 6 = 0.66205 / 6 ≈ 0.110342.',
+      result: 'y₁ = 1 + 0.110342 = 1.110342. Aniq qiymat bilan mutlaq mos keladi (xato < 10⁻⁷)!',
     },
     whenToUse: [
-      'Birinci tartibli boshlang‘ich qiymatli ODE uchun Eylerdan aniqroq standart usul kerak bo‘lsa.',
-      'Har bir qadamda to‘rtta funksiya bahosi maqbul hisoblash xarajati bo‘lsa.',
+      'Birinchi tartibli Koshi masalasida yuqori aniqlik va hisoblash barqarorligi talab qilinganda.',
+      'Tenglamalar sistemalarini va yuqori tartibli ODE larni Koshi shakliga keltirib yechishda.',
+      'Ilmiy-amaliy modellashtirish va muhandislik hisoblarida standart asosiy vosita sifatida.',
     ],
     limitations: [
-      'Oddiy RK4 xato bahosini o‘zi avtomatik bermaydi; qadamni tanlash foydalanuvchiga qoladi.',
-      'Qatti stiff yoki uzilishli masalalarda maxsus usul va adaptiv qadam kerak bo‘lishi mumkin.',
+      'Har bir qadamda f(x, y) funksiyasi 4 marta hisoblanadi (Eylerdan 4 barobar ko‘proq hisoblash xarajati).',
+      'Juda qattiq (stiff) tizimlarda qadam hajmini cheklab qo‘yishi mumkin (bunday hollarda oshkormas usullar kerak).',
     ],
     questions: [
       {
-        question: 'RK4 dagi 4 raqami nimani anglatadi?',
+        question: 'RK4 dagi 4 raqami nimani bildiradi?',
         answer:
-          'Silliq yechim uchun usulning global xato tartibi to‘rtinchi daraja, ya’ni O(h⁴) ekanini bildiradi.',
+          'U usulning global xatolik tartibi to‘rtinchi daraja (O(h⁴)) ekanini bildiradi. Qadam 10 barobarga qisqarsa, xatolik 10 000 barobar kamayadi.',
       },
       {
-        question: 'RK4 Eylerdan nimasi bilan farq qiladi?',
+        question: 'Runge qoidasi nima uchun 15 ga bo‘linadi?',
         answer:
-          'Eyler bir qiyalikdan foydalanadi; RK4 esa bir qadam ichidagi to‘rtta qiyalikni vaznli birlashtirib, odatda aniqroq natija beradi.',
+          'Chunki p = 4 tartibli usulda qadam ikki barobar oshirilganda (2h) xatolik 2⁴ = 16 barobar ortadi. 2⁴ − 1 = 15 bo‘lganligi uchun maxrajda 15 soni hosil bo‘ladi.',
+      },
+    ],
+    sources: [
+      {
+        authors: 'G.P. Ismatullayev, M.S. Koshergenova',
+        title: 'Hisoblash usullari',
+        publisher: 'Toshkent: «Tafakkur Bo‘stoni»',
+        year: 2014,
+        chapterSection: 'VII bob, 7.3.4-§ ("Runge-Kutta metodi") va 7.3.5-§ ("Runge qoidasi")',
+        pages: '168–175, 178–179-betlar',
+        note: 'k₁, k₂, k₃, k₄ parametrlarini chiqarishning to‘liq nazariy bayoni.',
+      },
+      {
+        authors: 'S.S. Irisqulov, K.D. Ismanova, M. Olimov, A. Imomov',
+        title: 'Sonli usullar va algoritmlar',
+        publisher: '“Namangan” nashriyoti',
+        year: 2013,
+        chapterSection: '6-bob, 3-§ ("Koshi masalasini yechishning Runge-Kutta usuli")',
+        pages: '131–134-betlar',
+        note: 'Blok-sxema, Paskal dasturi va y′ = cos(x) bo‘yicha Eyler bilan taqqoslama jadval.',
+      },
+      {
+        authors: 'M.I. Isroilov',
+        title: 'Hisoblash metodlari (2-qism)',
+        publisher: 'Toshkent: «Iqtisod-Moliya»',
+        year: 2008,
+        chapterSection: 'Bob 5 ("Koshi masalasini sonli yechish")',
+        pages: '140–155-betlar',
+        note: 'Ko‘p qadamli va bir qadamli metodlarning konvergensiya tahlili.',
+      },
+    ],
+    platformConventions: [
+      {
+        topic: 'Qiyalik koeffitsiyentlariga h ni kiritish konvensiyasi',
+        textbookStandard:
+          'Ba’zi adabiyotlarda K_i = f(...) (h siz) yozilib, oxirgi yangilanishda h/6 ko‘paytiriladi. Boshqa kitoblarda (Ismatullayev, Irisqulov) k_i = h · f(...) deb h ichiga kiritiladi.',
+        platformImplementation:
+          'Sonli-usullar interaktiv platformasida k_i = h · f(...) standarti qabul qilingan. Yangilanish y_{i+1} = y_i + (k₁ + 2k₂ + 2k₃ + k₄)/6 ko‘rinishida amalga oshiriladi. Jadvalda har bir qadamdagi k₁, k₂, k₃, k₄ qiymatlari alohida ustunda ko‘rsatiladi.',
+        whyChosen:
+          'Dasturlashda va hisoblash mashinalarida k_i larni oraliq qadam siljishlari bilan bevosita yig‘ish ancha barqaror va xotirada tushunarli saqlanadi.',
+      },
+      {
+        topic: 'Analitik yechim bilan avtomatik taqqoslash',
+        textbookStandard:
+          'Darsliklardagi test misollarda analitik aniq yechim qo‘lda hisoblanib, alohida jadval qilinadi.',
+        platformImplementation:
+          'Platformada "Aniq yechim funksiyasi (ixtiyoriy)" maydoniga analitik formula (masalan 2*exp(x) - x - 1) kiritilsa, tizim har bir qadamda mutlaq xatolikni (|y_taqribiy - y_aniq|) avtomatik hisoblab, taqqoslash grafigini chizadi.',
+        whyChosen:
+          'Talabalar va tadqiqotchilar o‘z topshiriqlarini darhol tekshirib, xatolik darajasi O(h⁴) ekanligiga amalda ishonch hosil qilishlari uchun.',
       },
     ],
   },

From 80fb11113820fbbeb335584eb65df830297a1257 Mon Sep 17 00:00:00 2001
From: hexjasur <hexjasur@gmail.com>
Date: Mon, 19 Oct 2026 20:30:08 +0500
Subject: [PATCH 3/3] =?UTF-8?q?fix(seo):=20align=20canonical=20domain=20an?=
 =?UTF-8?q?d=20verify=20release=20PRs=20=F0=9F=90=9B?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Use the live www origin consistently for canonical, Open Graph, sitemap, and structured data URLs. Auto-merge release-please PRs only after verifying synchronized version bumps and a non-empty matching changelog entry.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
---
 .github/workflows/release-pr-automerge.yml | 78 ++++++++++++++++++++++
 lib/config.ts                              |  4 +-
 2 files changed, 80 insertions(+), 2 deletions(-)
 create mode 100644 .github/workflows/release-pr-automerge.yml

diff --git a/.github/workflows/release-pr-automerge.yml b/.github/workflows/release-pr-automerge.yml
new file mode 100644
index 0000000..7613d9d
--- /dev/null
+++ b/.github/workflows/release-pr-automerge.yml
@@ -0,0 +1,78 @@
+name: Release PR Auto-merge
+
+on:
+  pull_request:
+    branches:
+      - main
+    types:
+      - opened
+      - synchronize
+      - reopened
+      - ready_for_review
+
+permissions:
+  contents: write
+  pull-requests: write
+
+jobs:
+  verify-and-enable-auto-merge:
+    name: Verify release version and changelog
+    if: >-
+      github.event.pull_request.user.login == 'nm-actions[bot]' &&
+      github.event.pull_request.head.repo.full_name == github.repository &&
+      github.event.pull_request.head.ref == 'release-please--branches--main--components--numeric-methods' &&
+      !github.event.pull_request.draft
+    runs-on: ubuntu-latest
+    steps:
+      - name: Check out release PR
+        uses: actions/checkout@v4
+        with:
+          ref: ${{ github.event.pull_request.head.sha }}
+          fetch-depth: 0
+
+      - name: Set up Bun
+        uses: oven-sh/setup-bun@v2
+        with:
+          bun-version: latest
+
+      - name: Verify synchronized version bump and changelog
+        env:
+          BASE_SHA: ${{ github.event.pull_request.base.sha }}
+          HEAD_SHA: ${{ github.event.pull_request.head.sha }}
+        run: |
+          set -euo pipefail
+
+          release_version="$(bun run check:version --print)"
+          bun run check:version --changelog > /dev/null
+          base_version="$(git show "${BASE_SHA}:package.json" | jq -r '.version')"
+
+          if [[ "${base_version}" == "${release_version}" ]]; then
+            echo "::error::Release PR did not change the version (still ${release_version})."
+            exit 1
+          fi
+
+          for file in package.json src-tauri/tauri.conf.json src-tauri/Cargo.toml; do
+            if git diff --quiet "${BASE_SHA}" "${HEAD_SHA}" -- "${file}"; then
+              echo "::error::Release PR did not update ${file}."
+              exit 1
+            fi
+          done
+
+          BASE_VERSION="${base_version}" RELEASE_VERSION="${release_version}" node <<'NODE'
+          const parse = (value) => {
+            const match = /^(0|[1-9]\d*)\.(0|[1-9]\d*)\.(0|[1-9]\d*)(?:-[0-9A-Za-z.-]+)?(?:\+[0-9A-Za-z.-]+)?$/.exec(value);
+            if (!match) throw new Error(`Invalid semantic version: ${value}`);
+            return match.slice(1, 4).map(Number);
+          };
+          const [base, release] = [parse(process.env.BASE_VERSION), parse(process.env.RELEASE_VERSION)];
+          const comparison = release.findIndex((part, index) => part !== base[index]);
+          if (comparison === -1 || release[comparison] < base[comparison]) {
+            throw new Error(`Release version ${process.env.RELEASE_VERSION} must be greater than ${process.env.BASE_VERSION}.`);
+          }
+          NODE
+
+      - name: Enable auto-merge after required checks pass
+        env:
+          GH_TOKEN: ${{ github.token }}
+          PR_NUMBER: ${{ github.event.pull_request.number }}
+        run: gh pr merge "${PR_NUMBER}" --auto --squash --repo "${GITHUB_REPOSITORY}"
diff --git a/lib/config.ts b/lib/config.ts
index 446fe5b..93fd7b4 100644
--- a/lib/config.ts
+++ b/lib/config.ts
@@ -5,8 +5,8 @@ export const siteConfig = {
   title: 'Sonli Usullar — O‘zbekcha onlayn qo‘llanma va kalkulyator',
   description:
     "Sonli usullarni o‘zbek tilida o‘rganing va hisoblang: tenglama ildizlari, sonli integrallar, differensial tenglamalar va interpolyatsiya uchun qadam-baqadam misollar hamda bepul online kalkulyatorlar.",
-  url: 'https://sonli-usullar.uz',
-  ogImage: 'https://sonli-usullar.uz/images/og-thumbail-1280x640.png',
+  url: 'https://www.sonli-usullar.uz',
+  ogImage: 'https://www.sonli-usullar.uz/images/og-thumbail-1280x640.png',
   locale: 'uz_UZ',
   alternateLocales: ['ru_RU', 'en_US'],
   author: {