Newton’s Method

// VM is the shared utility library, loaded globally via _includes/head-scripts.html (js/**). This page uses VM.numerical.linearRegression, VM.expressions.makeDerivative, VM.expressions.makeFunction, VM.expressions.makeNumber, VM.plotting.paddedRange, VM.ui.renderTable.
VM = window.VM

viewof fText = {
  const params = new URLSearchParams(window.location.search)
  const fTextInput = Inputs.text({label: "Function f(x)", value: params.get("f") ?? "x^2", placeholder: "Example: x^3 - x - 2, cos(x) - x"})
  fTextInput.classList.add("ojs-fill")
  return fTextInput
}

viewof x0 = {
  const params = new URLSearchParams(window.location.search)
  const x0Input = Inputs.text({label: "Initial guess x₀", value: params.get("x0") ?? "1.5", placeholder: "Example: 1.5, pi/4"})
  return x0Input
}

// Plot commits the current function/x₀ fields — edits above don't take
// effect until this is clicked, so result/mainPlot don't recompute on
// every keystroke. The step slider below is unaffected and stays reactive.
viewof plotTrigger = {
  const plotButton = Inputs.button("Plot", {value: 0, reduce: v => v + 1})
  plotButton.classList.add("ojs-auto")
  return plotButton
}

viewof stepControl = {
  const stepSlider = Inputs.range([0, maxIterations - 1], {step: 1, value: 0, label: "Step"})
  stepSlider.classList.add("ojs-fill")
  return stepSlider
}

viewof maxIterationsInput = {
  const input = Inputs.text({value: "20", label: "Max", placeholder: "Example: 20, 5*4"})
  input.classList.add("ojs-auto")
  return input
}

// maxIterations parses the Max field's expression, falling back to 20 for
// invalid input so the step slider and result loop always get a usable bound.
maxIterations = {
  const math = window.math
  const parsed = VM.expressions.makeNumber(math, maxIterationsInput)
  if (parsed === null || !Number.isFinite(parsed)) return 20
  return Math.max(1, Math.round(parsed))
}

// Pressing Enter in the function/x₀ fields clicks Plot instead of
// submitting the field's own (invisible) form. Depends only on the stable
// views, so this wiring runs once and is never re-attached.
enterToPlot = {
  for (const view of [viewof fText, viewof x0]) {
    const input = view.querySelector("input")
    if (!input) continue
    input.addEventListener("keydown", (e) => {
      if (e.key !== "Enter") return
      e.preventDefault()
      viewof plotTrigger.querySelector("button").click()
    })
  }
}

// committed snapshots fText/x0 only when Plot is clicked, and writes them
// back into the URL so the address bar stays a shareable link. It reads
// `viewof fText`/`viewof x0` (the stable DOM views) rather than `fText`/`x0`
// (the reactive values), so it does NOT rerun on every keystroke — only
// when `plotTrigger` changes. It still runs once on initial page load, so
// the first render uses the URL-provided (or default) values with no click.
committed = {
  plotTrigger
  const math = window.math
  const fVal = (viewof fText).value
  const x0Text = (viewof x0).value
  const x0Val = VM.expressions.makeNumber(math, x0Text)

  const params = new URLSearchParams(window.location.search)
  params.set("f", fVal)
  params.set("x0", x0Text)
  history.replaceState(null, "", `${window.location.pathname}?${params}`)

  return {fText: fVal, x0: x0Val}
}

// result: runs Newton's method with the current committed inputs.
//   rows — array of { i, x, fx, dfx, xNext, error }, one entry per iteration
//   f, df — parsed functions, reused for curve sampling in setup
result = {
  const math = window.math

  // Parse f(x) and its symbolic derivative from the committed string.
  // makeFunction / makeDerivative return null if the expression is invalid.
  const f  = VM.expressions.makeFunction(math, committed.fText)
  const df = VM.expressions.makeDerivative(math, committed.fText)
  const fallback = x => NaN

  if (!f || !df) return {rows: [], f: fallback, df: fallback}

  const rows  = []
  const initialGuess = committed.x0

  if (!Number.isFinite(initialGuess)) return {rows: [], f, df}

  let x = initialGuess
  let i = 0
  while (i < maxIterations) {
    const fx  = f(x)
    const dfx = df(x)

    // Stop if f or f' blows up — the method has diverged.
    if (!Number.isFinite(fx) || !Number.isFinite(dfx)) {
      rows.push({i, x, fx, dfx, xNext: NaN, error: NaN})
      return {rows, f, df}
    }

    // Stop if the tangent is nearly flat — dividing by dfx would be unstable.
    if (Math.abs(dfx) < 1e-14) {
      rows.push({i, x, fx, dfx, xNext: NaN, error: NaN})
      return {rows, f, df}
    }

    const xNext = x - fx / dfx        // Newton update: x-intercept of the tangent
    const error = Math.abs(xNext - x)
    rows.push({i, x, fx, dfx, xNext, error})

    x = xNext
    i++
  }
  return {rows, f, df}
}

// setup computes the data shared by all three plots below: which rows are
// visible at the current step, and the axis ranges / curve samples for the
// main chart.
setup = {
  const visibleRows = result.rows.slice(0, Math.min(Number(stepControl), result.rows.length) + 1)

  // Axis ranges are computed from ALL rows, not just visibleRows,
  // so the axes stay fixed as the user steps through iterations.
  const xValues = [committed.x0]
  for (const row of result.rows) xValues.push(row.x, row.xNext)
  const xRange = VM.plotting.paddedRange(xValues, {emptyRange: [-3, 3], relativePadding: 0.25, minPadding: 0.5})

  // A wider domain (double the padding of xRange) so the curve is already
  // sampled beyond the initial view — panning/zooming out reveals more of
  // the curve instead of blank canvas. The initial view stays xRange.
  const xBuffer = VM.plotting.paddedRange(xValues, {emptyRange: [-6, 6], relativePadding: 0.5, minPadding: 1.0})

  // Sample f across the buffer range for the smooth curve trace.
  const sampleCount = 700
  const sampleXs = Array.from({length: sampleCount}, (_, i) => xBuffer.lo + (xBuffer.hi - xBuffer.lo) * i / (sampleCount - 1))
  const sampleYs = []
  for (const x of sampleXs) {
    const y = result.f(x)
    if (Number.isFinite(y) && Math.abs(y) < 1e8) {
      sampleYs.push(y)
    } else {
      sampleYs.push(null)
    }
  }

  // Collect y values for the y-axis range, including tangent line endpoints.
  // Only samples within the initial view (xRange) count, so yRange — and
  // the initial view — stay unaffected by the wider sampling.
  // Cap at 1e8 so a near-vertical tangent doesn't blow up the axis range.
  const yValues = [0]
  for (let i = 0; i < sampleXs.length; i++) {
    const x = sampleXs[i]
    if (x >= xRange.lo && x <= xRange.hi && sampleYs[i] !== null) yValues.push(sampleYs[i])
  }
  for (const row of result.rows) {
    if (Number.isFinite(row.fx) && Math.abs(row.fx) <= 1e8) yValues.push(row.fx)
    if (Number.isFinite(row.x) && Number.isFinite(row.fx) && Number.isFinite(row.dfx) && Number.isFinite(row.xNext)) {
      const tlo = Math.max(xRange.lo, Math.min(row.x, row.xNext) - 0.75)
      const thi = Math.min(xRange.hi, Math.max(row.x, row.xNext) + 0.75)
      for (const v of [row.fx + row.dfx * (tlo - row.x), row.fx + row.dfx * (thi - row.x)]) {
        if (Number.isFinite(v) && Math.abs(v) <= 1e8) yValues.push(v)
      }
    }
  }
  const yRange = VM.plotting.paddedRange(yValues, {emptyRange: [-1, 1], relativePadding: 0.25, minPadding: 0.5})

  return {visibleRows, xRange, xBuffer, yRange, sampleXs, sampleYs}
}

// mainPlot draws f(x), the tangent lines, the Newton points, and the
// tangent x-intercepts.
//
// _plotDiv is kept in a closure so Plotly reuses the same DOM node across
// reactive updates — this preserves zoom and pan when the step changes.
mainPlot = {
  const Plotly = window.Plotly
  let _plotDiv = null

  return ({visibleRows, xRange, xBuffer, yRange, sampleXs, sampleYs}) => {
    const data = [
      // The function curve
      {x: sampleXs, y: sampleYs, type: "scatter", mode: "lines", name: "f(x)", showlegend: true, line: {color: "#2563eb", width: 3}},
      // The x-axis (y = 0 reference line), spanning the buffer so it's still
      // visible when the user pans/zooms out
      {x: [xBuffer.lo, xBuffer.hi], y: [0, 0], type: "scatter", mode: "lines", name: "y = 0", showlegend: true, line: {color: "#111827", width: 2}}
    ]

    // Dotted vertical lines from the x-axis up to each Newton point
    const newtonPoints = []
    for (const r of visibleRows) {
      if (Number.isFinite(r.x) && Number.isFinite(r.fx)) newtonPoints.push(r)
    }
    for (const r of newtonPoints) {
      data.push({x: [r.x, r.x], y: [0, r.fx], type: "scatter", mode: "lines", showlegend: false, hoverinfo: "skip", line: {color: "#94a3b8", width: 1.5, dash: "dot"}})
    }

    // Tangent lines: previous ones faded, current one solid
    const validTangentRows = []
    for (const r of visibleRows) {
      if (Number.isFinite(r.x) && Number.isFinite(r.fx) && Number.isFinite(r.dfx) && Number.isFinite(r.xNext)) {
        validTangentRows.push(r)
      }
    }
    const prevTangents = validTangentRows.slice(0, -1)
    let currentTangent = null
    if (validTangentRows.length > 0) currentTangent = validTangentRows[validTangentRows.length - 1]

    for (const row of prevTangents) {
      const tlo = Math.max(xRange.lo, Math.min(row.x, row.xNext) - 0.75)
      const thi = Math.min(xRange.hi, Math.max(row.x, row.xNext) + 0.75)
      data.push({
        x: [tlo, thi], y: [row.fx + row.dfx * (tlo - row.x), row.fx + row.dfx * (thi - row.x)],
        type: "scatter", mode: "lines", showlegend: false, hoverinfo: "skip",
        line: {color: "rgba(220, 38, 38, 0.18)", width: 1.5}
      })
    }
    if (currentTangent) {
      const row = currentTangent
      const tlo = Math.max(xRange.lo, Math.min(row.x, row.xNext) - 0.75)
      const thi = Math.min(xRange.hi, Math.max(row.x, row.xNext) + 0.75)
      data.push({
        x: [tlo, thi], y: [row.fx + row.dfx * (tlo - row.x), row.fx + row.dfx * (thi - row.x)],
        type: "scatter", mode: "lines", name: "Tangent line", showlegend: true,
        line: {color: "#dc2626", width: 2.5}
      })
    }

    // Red markers at each (x_i, f(x_i)) Newton point
    if (newtonPoints.length > 0) {
      const xs = [], ys = [], labels = []
      for (const r of newtonPoints) {
        xs.push(r.x)
        ys.push(r.fx)
        labels.push(`x_${r.i}`)
      }
      data.push({
        x: xs, y: ys,
        type: "scatter", mode: "markers+text", name: "Newton points",
        text: labels, textposition: "top center", showlegend: true,
        marker: {color: "#dc2626", size: 10, symbol: "circle", line: {color: "white", width: 1}}
      })
    }

    // Green markers at each tangent x-intercept (x_{i+1})
    const intercepts = []
    for (const r of visibleRows) {
      if (Number.isFinite(r.xNext)) intercepts.push({i: r.i + 1, x: r.xNext})
    }
    if (intercepts.length > 0) {
      const xs = [], ys = [], labels = []
      for (const r of intercepts) {
        xs.push(r.x)
        ys.push(0)
        labels.push(`x_${r.i}`)
      }
      data.push({
        x: xs, y: ys,
        type: "scatter", mode: "markers+text", name: "Tangent x-intercepts",
        text: labels, textposition: "bottom center", showlegend: true,
        marker: {color: "#16a34a", size: 10, symbol: "circle", line: {color: "white", width: 1}}
      })
    }

    const layout = {
      // title: {text: "Newton's Method Tangent Visualization"},
      margin: {l: 0, r: 0, t: 0, b: 0},
      xaxis: {title: "x", range: [xRange.lo, xRange.hi], zeroline: true},
      yaxis: {title: "f(x)", range: [yRange.lo, yRange.hi], zeroline: true},
      legend: {orientation: "h", y: -0.08},
      hovermode: "closest",
      uirevision: "static",  // keeps zoom/pan state across reactive updates
      autosize: true
    }
    const config = {responsive: true, displaylogo: false}

    // First render: create the div. Later renders: update in place (preserves zoom).
    if (!_plotDiv) {
      _plotDiv = document.createElement("div")
      _plotDiv.className = "plotly-box-large"
      Plotly.newPlot(_plotDiv, data, layout, config)
      // newPlot fires before the div is in the DOM, so Plotly measures 0 width.
      // A ResizeObserver catches the real size once it's inserted and laid out,
      // and keeps correcting it on any later layout/viewport change.
      new ResizeObserver(() => Plotly.Plots.resize(_plotDiv)).observe(_plotDiv)
    } else {
      Plotly.react(_plotDiv, data, layout, config)
    }
    return _plotDiv
  }
}

mainPlot(setup)

NoteConvergence plots

// iteratesPlot draws xₙ vs n for the entire run — it doesn't depend on the
// current step, so it's just plotted once per run.
iteratesPlot = {
  const xAccessor = row => {
    if (Number.isFinite(row.x)) return row.x
    return null
  }
  return Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "n"},
    y: {label: "xₙ"},
    marks: [
      Plot.lineY(result.rows, {x: "i", y: xAccessor, stroke: "#2563eb"}),
      Plot.dot(result.rows, {x: "i", y: xAccessor, fill: "#2563eb", r: 4})
    ]
  })
}
// convergencePlot draws log₁₀|eₙ₊₁| vs log₁₀|eₙ| (successive step sizes,
// eₙ = |xₙ₊₁ − xₙ|) with a linear regression line, for the entire run.
// Newton converges quadratically for a simple root (eₙ₊₁ ~ C·eₙ²), which
// isn't a straight line vs n or log(n) — it only takes a handful of steps
// to hit machine precision, and fitting log|error| vs n (as bisection and
// fixed-point iteration do, since they converge linearly) gives a "slope"
// that's just an artifact of however many points survive before underflow.
// This log-log successive-error plot is the standard diagnostic instead:
// its slope estimates the order of convergence p directly (≈2 for a simple
// root, ≈1 for a multiple root, as in the default x² example).
convergencePlot = {
  const logRows = []
  for (let k = 0; k < result.rows.length - 1; k++) {
    const en  = result.rows[k].error
    const en1 = result.rows[k + 1].error
    if (Number.isFinite(en) && en > 0 && Number.isFinite(en1) && en1 > 0) {
      logRows.push({x: Math.log10(en), y: Math.log10(en1)})
    }
  }

  const pointsName = "log₁₀|eₙ₊₁| vs log₁₀|eₙ|"
  const marks = [Plot.dot(logRows, {x: "x", y: "y", fill: () => pointsName, r: 4})]
  const colorDomain = [pointsName], colorRange = ["#dc2626"]
  // legendDomain/legendRange hold only the regression line(s) — the legend
  // shown below the chart displays just the fitted order, not the raw
  // point series, to keep it uncluttered.
  const legendDomain = [], legendRange = []

  const fit = VM.numerical.linearRegression(logRows)
  if (fit) {
    const lineName = `order ≈ ${fit.slope.toFixed(2)}`
    marks.push(Plot.line(
      [{x: fit.xlo, y: fit.slope * fit.xlo + fit.intercept}, {x: fit.xhi, y: fit.slope * fit.xhi + fit.intercept}],
      {x: "x", y: "y", stroke: () => lineName, strokeDasharray: "6,4"}
    ))
    colorDomain.push(lineName)
    colorRange.push("#9333ea")
    legendDomain.push(lineName)
    legendRange.push("#9333ea")
  }

  const chart = Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "log₁₀|eₙ|"},
    y: {label: "log₁₀|eₙ₊₁|"},
    color: {domain: colorDomain, range: colorRange},
    marks
  })
  const legend = Plot.legend({color: {domain: legendDomain, range: legendRange}})

  const container = document.createElement("div")
  container.append(chart, legend)
  return container
}

NoteIteration table

// Iteration table — shows the entire run, not just the current step.
iterationTable = {
  if (result.rows.length === 0) {
    // Always return a Node so Quarto's OJS runtime never latches this
    // declaration cell as hidden.
    return document.createElement("div")
  }

  const formattedRows = []
  for (const row of result.rows) {
    let xCell = "NaN"
    if (Number.isFinite(row.x)) xCell = row.x.toPrecision(10)

    let fxCell = "NaN"
    if (Number.isFinite(row.fx)) fxCell = row.fx.toExponential(5)

    let dfxCell = "NaN"
    if (Number.isFinite(row.dfx)) dfxCell = row.dfx.toExponential(5)

    let xNextCell = "NaN"
    if (Number.isFinite(row.xNext)) xNextCell = row.xNext.toPrecision(10)

    let errorCell = "NaN"
    if (Number.isFinite(row.error)) errorCell = row.error.toExponential(4)

    formattedRows.push([row.i, xCell, fxCell, dfxCell, xNextCell, errorCell])
  }

  return VM.ui.renderTable({
    html,
    headers: ["n", tex`x_n`, tex`f(x_n)`, tex`f'(x_n)`, tex`x_{n+1}`, tex`|x_{n+1} - x_n|`],
    csvHeaders: ["n", "x_n", "f(x_n)", "f'(x_n)", "x_{n+1}", "|x_{n+1} - x_n|"],
    filename: "newton-method-iterations.csv",
    rows: formattedRows
  })
}

Newton’s method finds roots of \(f(x)=0\) by repeatedly linearizing: from \(x_n\), it follows the tangent line at \((x_n, f(x_n))\) to its \(x\)-intercept,

\[ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. \]

Examples

The global behavior of Newton’s method can be surprisingly sensitive to the starting point. The examples below illustrate fast convergence, slow convergence, failure near a flat tangent, and cases where the iterates appear to diverge before eventually finding a root.

Click any example to load it into the app.

Fast quadratic convergence

f(x) = x² − 2,   x₀ = 1.5

A standard well-behaved example. Newton's method rapidly converges to √2.

A two-cycle failure

f(x) = x³ − 2x + 2,   x₀ = 0

The iterates bounce between 0 and 1 instead of converging. This shows that Newton's method is not globally reliable.

Looks unstable, then converges

f(x) = x³ − 2x + 2,   x₀ = −0.1

A nearby starting point can behave very differently. The method may wander before eventually approaching the real root.

Good starting point for the same function

f(x) = x³ − 2x + 2,   x₀ = −1

For the same function as above, a different initial guess converges much more predictably.

Slow convergence at a multiple root

f(x) = x³,   x₀ = 0.2

The root has multiplicity greater than one, so Newton's usual quadratic convergence is lost.

Nearly flat tangent

f(x) = x³ − 2x + 2,   x₀ ≈ 0.8165

Starting near a point where f′(x) is close to zero can send the next iterate far away.

Large jumps from shallow slopes

f(x) = arctan(x),   x₀ = 1.4

The tangent line can produce a very large step when the slope is small.

Polynomial root-finding

f(x) = x⁵ − x − 1,   x₀ = 1

A higher-degree polynomial with a real root. Newton's method converges quickly from a reasonable starting point.