Bisection Method

Computational Math

Bracketing

No Derivatives

Guaranteed Convergence

Root Finding

Authors

Dhruv Azad

Apurva Nakade

Published

June 25, 2026

// VM is the shared utility library, loaded globally via _includes/head-scripts.html (js/**). This page uses VM.numerical.linearRegression, 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^3 - x - 2", placeholder: "Example: x^3 - x - 2, cos(x) - x"})
  fTextInput.classList.add("ojs-fill")
  return fTextInput
}

viewof a0 = {
  const params = new URLSearchParams(window.location.search)
  const a0Input = Inputs.text({label: "Left endpoint a", value: params.get("a") ?? "1", placeholder: "Example: 1, -pi/2"})
  return a0Input
}

viewof b0 = {
  const params = new URLSearchParams(window.location.search)
  const b0Input = Inputs.text({label: "Right endpoint b", value: params.get("b") ?? "2", placeholder: "Example: 2, pi"})
  return b0Input
}

// Plot commits the current function/endpoint 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/endpoint 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 a0, viewof b0]) {
    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/a0/b0 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 a0`/`viewof b0` (the stable DOM views) rather than
// `fText`/`a0`/`b0` (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 aText = (viewof a0).value
  const bText = (viewof b0).value
  const aVal = VM.expressions.makeNumber(math, aText)
  const bVal = VM.expressions.makeNumber(math, bText)

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

  return {fText: fVal, a0: aVal, b0: bVal}
}

// result: runs bisection with the current committed inputs.
//   rows — array of { n, a, b, m, fa, fb, fm, width }, one entry per iteration
//   f    — parsed function, reused for curve sampling in setup
result = {
  const math = window.math

  const f        = VM.expressions.makeFunction(math, committed.fText)
  const fallback = x => NaN

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

  let a = committed.a0
  let b = committed.b0

  if (!Number.isFinite(a) || !Number.isFinite(b) || a === b) return {rows: [], f}
  if (a > b) { const temp = a; a = b; b = temp }   // ensure a < b

  const fa = f(a)
  const fb = f(b)

  if (!Number.isFinite(fa) || !Number.isFinite(fb)) return {rows: [], f}

  // Bisection requires a sign change across [a, b].
  // If an endpoint is already a root, return it immediately.
  if (fa === 0) return {rows: [{n: 0, a, b, m: a, fa, fb, fm: fa, width: b - a}], f}
  if (fb === 0) return {rows: [{n: 0, a, b, m: b, fa, fb, fm: fb, width: b - a}], f}
  if (fa * fb > 0) return {rows: [], f}   // no sign change — can't bracket a root

  const rows = []
  let curA = a, curB = b, curFa = fa, curFb = fb
  let n = 0
  while (n < maxIterations) {
    const m  = 0.5 * (curA + curB)   // midpoint of the current bracket
    const fm = f(m)
  rows.push({n, a: curA, b: curB, m, fa: curFa, fb: curFb, fm, width: curB - curA})
  
  if (!Number.isFinite(fm)) return {rows, f}
  
  // If the midpoint is exactly a root, stop.
  // Otherwise the code would keep iterating even though the root was found.
  if (fm === 0) return {rows, f}
  
  // Keep the half that still contains a sign change.
  if (curFa * fm < 0) {
    curB = m
    curFb = fm
  } else {
    curA = m
    curFa = fm
  }
  n++
  }
  return {rows, f}
}

// 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)
  const currentRow  = visibleRows.length === 0 ? null : visibleRows[visibleRows.length - 1]

  // Axis ranges from ALL rows so the axes stay fixed while stepping.
  const xValues = [committed.a0, committed.b0]
  for (const row of result.rows) xValues.push(row.a, row.b, row.m)
  const xRange = VM.plotting.paddedRange(xValues, {emptyRange: [-5, 5], 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: [-10, 10], 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)
    }
  }

  // Cap row values at 1e8 so a near-singularity doesn't blow up the axis range.
  // Only samples within the initial view (xRange) count, so yRange — and
  // the initial view — stay unaffected by the wider sampling.
  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) {
    for (const v of [row.fa, row.fb, row.fm]) {
      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, currentRow, xRange, xBuffer, yRange, sampleXs, sampleYs}
}

// mainPlot draws f(x), the current bracket [a, b], and the midpoint.
//
// _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, currentRow, 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 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}}
    ]

    // Previous midpoints — all steps before the current one
    const prevRows = []
    for (const r of visibleRows.slice(0, -1)) {
      if (Number.isFinite(r.m) && Number.isFinite(r.fm)) prevRows.push(r)
    }
    if (prevRows.length > 0) {
      const xs = [], ys = [], labels = []
      for (const r of prevRows) {
        xs.push(r.m)
        ys.push(r.fm)
        labels.push(`m${r.n}`)
      }
      data.push({
        x: xs, y: ys,
        type: "scatter", mode: "markers+text", name: "Previous midpoints", showlegend: true,
        text: labels, textposition: "top center",
        marker: {color: "#f97316", size: 8, symbol: "circle", line: {color: "white", width: 1}}
      })
    }

    if (currentRow) {
      const row = currentRow
      data.push(
        // Endpoints a and b
        {x: [row.a, row.b], y: [row.fa, row.fb], type: "scatter", mode: "markers+text", name: "Endpoints",
          text: ["a", "b"], textposition: "top center", showlegend: true,
          marker: {color: "#2563eb", size: 11, symbol: "circle", line: {color: "white", width: 1}}},
        // Current midpoint m
        {x: [row.m], y: [row.fm], type: "scatter", mode: "markers+text", name: "Midpoint",
          text: [`m${row.n}`], textposition: "top center", showlegend: true,
          marker: {color: "#dc2626", size: 13, symbol: "circle", line: {color: "white", width: 1}}},
        // Dotted verticals from x-axis to each marker
        {x: [row.a, row.a], y: [0, row.fa], type: "scatter", mode: "lines", showlegend: false, hoverinfo: "skip", line: {color: "#94a3b8", width: 1.5, dash: "dot"}},
        {x: [row.b, row.b], y: [0, row.fb], type: "scatter", mode: "lines", showlegend: false, hoverinfo: "skip", line: {color: "#94a3b8", width: 1.5, dash: "dot"}},
        {x: [row.m, row.m], y: [0, row.fm], type: "scatter", mode: "lines", showlegend: false, hoverinfo: "skip", line: {color: "#dc2626", width: 1.5, dash: "dot"}}
      )
    }

    // Shaded rectangle showing the current bracket [a, b]
    let shapes = []
    if (currentRow) {
      shapes = [{
        type: "rect", xref: "x", yref: "paper",
        x0: currentRow.a, x1: currentRow.b, y0: 0, y1: 1,
        fillcolor: "rgba(37, 99, 235, 0.14)", line: {width: 0}
      }]
    }

    const layout = {
      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},
      shapes,
      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)
      new ResizeObserver(() => Plotly.Plots.resize(_plotDiv)).observe(_plotDiv)
    } else {
      Plotly.react(_plotDiv, data, layout, config)
    }
    return _plotDiv
  }
}

mainPlot(setup)

Convergence plots

// iteratesPlot draws the midpoint mₙ vs n for the entire run — it doesn't
// depend on the current step, so it's just plotted once per run.
iteratesPlot = {
  const midpointAccessor = row => {
    if (Number.isFinite(row.m)) return row.m
    return null
  }
  return Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "n"},
    y: {label: "mₙ"},
    marks: [
      Plot.lineY(result.rows, {x: "n", y: midpointAccessor, stroke: "#2563eb"}),
      Plot.dot(result.rows, {x: "n", y: midpointAccessor, fill: "#2563eb", r: 4})
    ]
  })
}
// convergencePlot draws log₁₀|f(mₙ)| vs n (semi-log) with a linear
// regression line, for the entire run — it doesn't depend on the current
// step, so it's just plotted once per run.
convergencePlot = {
  const logRows = []
  for (const r of result.rows) {
    if (Number.isFinite(r.fm) && r.fm !== 0) {
      logRows.push({n: r.n, y: Math.log10(Math.abs(r.fm))})
    }
  }

  const pointsName = "log₁₀|f(mₙ)|"
  const marks = [Plot.dot(logRows, {x: "n", 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 slope, not the raw
  // point series, to keep it uncluttered.
  const legendDomain = [], legendRange = []

  const regressionPoints = []
  for (const row of logRows) regressionPoints.push({x: row.n, y: row.y})
  const fit = VM.numerical.linearRegression(regressionPoints)
  if (fit) {
    const lineName = `slope ≈ ${fit.slope.toFixed(2)}`
    marks.push(Plot.line(
      [{n: fit.xlo, y: fit.slope * fit.xlo + fit.intercept}, {n: fit.xhi, y: fit.slope * fit.xhi + fit.intercept}],
      {x: "n", 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: "n"},
    y: {label: "log₁₀|f(mₙ)|"},
    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
}

Iteration 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 aCell = "NaN"
    if (Number.isFinite(row.a)) aCell = row.a.toPrecision(10)

    let bCell = "NaN"
    if (Number.isFinite(row.b)) bCell = row.b.toPrecision(10)

    let mCell = "NaN"
    if (Number.isFinite(row.m)) mCell = row.m.toPrecision(10)

    let faCell = "NaN"
    if (Number.isFinite(row.fa)) faCell = row.fa.toExponential(5)

    let fbCell = "NaN"
    if (Number.isFinite(row.fb)) fbCell = row.fb.toExponential(5)

    let fmCell = "NaN"
    if (Number.isFinite(row.fm)) fmCell = row.fm.toExponential(5)

    let widthCell = "NaN"
    if (Number.isFinite(row.width)) widthCell = row.width.toExponential(4)

    formattedRows.push([row.n, aCell, bCell, mCell, faCell, fbCell, fmCell, widthCell])
  }

  return VM.ui.renderTable({
    html,
    headers: ["n", tex`a_n`, tex`b_n`, tex`m_n`, tex`f(a_n)`, tex`f(b_n)`, tex`f(m_n)`, "width"],
    csvHeaders: ["n", "a_n", "b_n", "m_n", "f(a_n)", "f(b_n)", "f(m_n)", "width"],
    filename: "bisection-method-iterations.csv",
    rows: formattedRows
  })
}

The bisection method finds a root of \(f(x)=0\) by repeatedly halving an interval \([a,b]\) where \(f\) changes sign, keeping whichever half still brackets the root.

Examples

The bisection method is slower than Newton’s method or the secant method, but it has one major advantage: if \(f\) is continuous and changes sign on the interval \([a,b]\), then a root is guaranteed to remain inside the bracket.

The examples below illustrate ordinary convergence, multiple possible roots, endpoint roots, invalid brackets, and roots near singular-looking behavior.

Click any example to load it into the app.

Standard guaranteed convergence

f(x) = x³ − x − 2, [a,b] = [1,2]

A classic example with a sign change across the interval. Bisection steadily shrinks the bracket around the real root near 1.521.

Square root of 2

f(x) = x² − 2, [a,b] = [1,2]

The method converges to √2. This is a simple example where the interval width halves at every step.

Transcendental equation

f(x) = cos(x) − x, [a,b] = [0,1]

Finds the root of cos(x) = x. This is the same solution that appears in fixed point iteration, but here the bracket gives a guarantee.

Bracket around a negative root

f(x) = x³ − 2x + 2, [a,b] = [−2,−1]