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Numerical Methods · Module 3 of 10

Roots of Equations: Open Methods

Open methods drop the safety of a bracket for speed. Newton-Raphson follows the tangent to the root and doubles its correct digits each step, when it converges, which is the catch worth understanding.

01

Readiness check

This module finds roots without a bracket. Tick only what you can do closed-notes.

  • Differentiate a simple function.
  • Recall the equation of a tangent line.
  • Evaluate the approximate relative error.
  • Recall how bracketing guarantees convergence.
  • Iterate a formula from a starting guess.
0 or 1 weak itemsContinue with this module.
2 weak itemsRevisit bracketing and stopping criteria in Module 2.
3 or more weak itemsReview derivatives in Mathematics, Module 4.
02

The core idea

Open methods iterate from a guess without keeping a bracket. Newton-Raphson follows the tangent to its axis crossing; near a root it converges quadratically, doubling the correct digits each step, but a poor start can diverge.

xi+1 = xi − f(xi)/f′(xi)secant: f′ ≈ (fi − fi−1)/(xi − xi−1)quadratic convergence near the root

Open methods use one or two starting values and a formula that may move outside any bracket. The Newton-Raphson method is the workhorse: from a guess xi it follows the tangent line to where it crosses the axis, giving xi+1 = xi − f(xi)/f′(xi). Close to a simple root it converges quadratically, roughly doubling the number of correct digits each iteration, far faster than bisection. The secant method replaces the derivative with a finite-difference slope from the two latest points, useful when f′ is awkward to compute, at a slightly slower rate. The price for this speed is reliability: a bad starting guess, a near-zero derivative, or an inflection point can send the iteration away from the root or oscillate. Open methods are fast but not fail-safe, so they are often started from a bracket.

The skill works when: you start near the root and watch the approximate error fall quadratically.
The skill breaks down when: the derivative is near zero, or the guess is far from the root and the iteration diverges.
The concept. Newton-Raphson draws the tangent at the current guess and takes its axis crossing as the next estimate. Near a root the corrections shrink quadratically toward it.
03

The skills, taught in order

Five skills build from simple iteration to Newton-Raphson, the secant method, and their failure modes.

3.1 Open iteration

An open method starts from a guess and applies a formula repeatedly, with no bracket to hold the root. Fixed-point iteration rewrites f(x) = 0 as x = g(x) and iterates xi+1 = g(xi); it converges only if g is sufficiently flat near the root, the first hint that openness costs reliability.

3.2 The Newton-Raphson method

Newton-Raphson uses the tangent: xi+1 = xi − f(xi)/f′(xi). Geometrically it slides down the slope to the axis. It needs the derivative but rewards you with the fastest convergence of the common methods.

MethodNeedsConvergenceRisk
Newton-Raphsonf and f′quadraticdiverges if f′ ≈ 0
Secantf only, two pointssuperlinear (≈1.6)diverges, slower than Newton
Bracketingsign changelinearalways converges

Open methods buy speed with the risk of divergence; bracketing trades speed for certainty.

3.3 Quadratic convergence

Near a simple root, Newton-Raphson's error roughly squares each step: the number of correct digits doubles. This is why a method that takes ten bisections can take three or four Newton steps once it is close.

3.4 The secant method

When the derivative is hard to find, the secant method approximates it from the two latest points: f′ ≈ (fi − fi−1)/(xi − xi−1). It needs two starting values but no analytic derivative, converging only a little slower than Newton.

3.5 Divergence and safeguards

Open methods can fail: a near-zero derivative throws the estimate far away, an inflection point causes oscillation, and a distant guess may converge to the wrong root. Practical solvers safeguard them, falling back to bisection whenever a step leaves a known bracket.

Engineering connection: Newton-Raphson is the engine inside most equation solvers and is the local step in many optimisation and FEM routines.

04

Worked example 1: Newton-Raphson and quadratic convergence

Find the real root of f(x) = x3 − x − 1 (near 1.3247) by Newton-Raphson from x0 = 1.5. Carry out three iterations and watch the error fall.

Figure 1. The approximate error collapses from 11 percent to 0.04 percent in three steps, the signature of quadratic convergence: each step roughly squares the previous error.
  1. ProblemFind the root of f(x) = x3 − x − 1 by Newton from x0 = 1.5, as in Figure 1.
  2. Given / findf(x) = x3 − x − 1, f′(x) = 3x2 − 1, x0 = 1.5. Find x1, x2, x3 and εa.
  3. AssumptionsThe starting guess is near the simple real root; the derivative stays nonzero.
  4. Modelxi+1 = xi − f(xi)/f′(xi); εa = |(xi+1 − xi)/xi+1| × 100%.
  5. Equationsxi+1 = xi − (xi3 − xi − 1)/(3xi2 − 1)
  6. Solvex1 = 1.5 − 0.875/5.75 = 1.347826 (εa = 11.3%). x2 = 1.325200 (εa = 1.71%). x3 = 1.324718 (εa = 0.036%).
  7. Checkf(1.324718) ≈ 0.0000, and the error roughly squares each step (11% → 1.7% → 0.04%), confirming quadratic convergence to the root 1.324718.
  8. ConclusionThree Newton steps reach five-figure accuracy from a rough guess. Quadratic convergence is the payoff for using the derivative.
Result. Root 1.324718 after three iterations, εa down to 0.036%.
05

Worked example 2: the secant method

Find √2, the root of f(x) = x2 − 2, by the secant method from the two guesses x0 = 1 and x1 = 2, with no derivative. Carry out three steps.

Figure 2. The secant method joins the two latest points with a line and takes its axis crossing, approximating the tangent without a derivative. It closes on √2 in a few steps.
  1. ProblemFind √2 by the secant method from x0 = 1, x1 = 2, as in Figure 2.
  2. Given / findf(x) = x2 − 2, x0 = 1, x1 = 2. Find x2, x3, x4.
  3. AssumptionsThe two guesses are near the root; the secant slope approximates the derivative.
  4. Modelxi+1 = xi − f(xi)(xi − xi−1)/(f(xi) − f(xi−1)).
  5. Equationsxi+1 = xi − fi(xi − xi−1)/(fi − fi−1)
  6. Solvex2 = 2 − 2(2 − 1)/(2 − (−1)) = 2 − 2/3 = 1.3333. x3 = 1.4000. x4 = 1.41463, approaching √2 = 1.41421.
  7. CheckThe estimates close on 1.41421 without any derivative, a little slower than Newton would but using only function values, which is the secant method's advantage.
  8. ConclusionThe secant method reaches √2 in a few steps using function values alone, trading a small amount of speed for not needing f′.
Result. x4 = 1.41463, converging to √2 = 1.41421.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Assuming guaranteed convergenceIteration flies off to infinity"Is the guess near the root?"Open methods can diverge; start close or safeguard.
Derivative near zeroHuge jump in one Newton step"Is f′ small here?"A flat slope throws the estimate far; switch methods.
Secant slope from one pointDivision by zero or stall"Do I have two distinct points?"The secant needs two distinct latest points.
No stopping criterionIterating past machine precision"Is εa below tolerance?"Stop when the approximate error is small enough.
07

Practice ladder

Level 1 · Direct skill

One Newton step on f(x) = x2 − 2 from x0 = 1.5. Find x1.

Show answer

x1 = 1.5 − (2.25 − 2)/(2·1.5) = 1.5 − 0.25/3 = 1.41667.

Level 2 · Mixed concept

Take a second Newton step from x1 = 1.41667 and find εa.

Show answer

x2 = 1.41667 − (2.00694 − 2)/(2·1.41667) = 1.41422. εa = |(1.41422 − 1.41667)/1.41422| × 100 = 0.173%.

Level 3 · Independent problem

Apply Newton-Raphson to f(x) = cos(x) − x from x0 = 0.75 for one step (the root is near 0.739).

Show answer

f′ = −sin(x) − 1. f(0.75) = cos0.75 − 0.75 = 0.7317 − 0.75 = −0.0183; f′ = −0.6816 − 1 = −1.6816. x1 = 0.75 − (−0.0183)/(−1.6816) = 0.75 − 0.01088 = 0.73912, already near the root.

Transfer task | Real engineering

You will solve an implicit equation thousands of times in a simulation loop. Argue why Newton-Raphson is worth its derivative, and how to keep it safe.

What good work looks like

Its quadratic convergence saves many iterations across thousands of calls, justifying the derivative cost. Safety comes from a good initial guess (often the previous solution) and a bracketed fallback to bisection if a step diverges.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my error is squaring each Newton step."
"Give me three functions; I will write the Newton iteration for each."
"Find the root for me." Iterating the tangent yourself is the skill.
"Why did it diverge?" Reasoning from the slope and guess is the point.

Portfolio task

Solve a real nonlinear equation by Newton-Raphson, tabulate the quadratic fall in error, and compare the iteration count to bisection on the same problem.

Must include: the iteration formula, an error table, and a comparison with bracketing.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the Newton-Raphson formula.

xi+1 = xi − f(xi)/f′(xi).

2. What is quadratic convergence?

The error roughly squares each step, doubling the correct digits.

3. How does the secant method estimate the slope?

From the two latest points: (fi − fi−1)/(xi − xi−1).

4. Name two causes of divergence.

A near-zero derivative and a starting guess far from the root.

5. How are open methods made safe?

By safeguarding with a bracket and falling back to bisection if a step leaves it.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-run three Newton steps from a blank page.
+3 daysSolve three new roots, watching the error squaring.
+7 daysMove from one equation to systems of equations, Module 4.
+30 daysReuse Newton-Raphson inside an optimisation or solver loop.
10

Textbook mapping

This module follows Chapra and Canale, Numerical Methods for Engineers, 7th edition. Use these references to read further.

Topic in this moduleWhere to read more
Fixed-point iterationChapra & Canale, Chapter 6
The Newton-Raphson methodChapra & Canale, Chapter 6
The secant method and convergenceChapra & Canale, Chapter 6

Chapter numbers refer to the 7th edition. The open methods are standard, so any recent edition will align closely.