MATH2089 Chap.4 Nonlinear Equations
Nonlinear Equations
Root finding balances global guarantees against local speed. Bisection keeps a continuous sign changing bracket and converges linearly, fixed point iteration depends on a contractive mapping, Newton can converge quadratically near a simple root and the secant method avoids an explicit derivative.
This chapter connects convergence order with stopping rules, residual checks and safeguards for weak derivatives, poor starting values and stagnation. Begin by defining the function whose zero represents the original problem. A root of the wrong rearrangement or a value outside the permitted domain is not a valid answer.
Inspect continuity, sign changes, derivatives and available starting information before choosing an iteration. The method should follow from that evidence rather than from a preference for the fastest local formula. Bisection is built on a continuous bracket with opposite endpoint signs. Every step preserves a smaller valid bracket, so the interval width gives an explicit location error bound.
Its linear convergence is slower than Newton near a good root, but the guarantee is valuable when derivative information is weak or the starting point is uncertain. A sign change does not prove uniqueness, and even-multiplicity roots may not produce a sign change. Fixed-point iteration rewrites the problem as x = g(x).
A fixed point can be algebraically correct yet numerically unsuitable if the mapping is not contractive near the solution. Compare successive iterates, inspect the derivative magnitude and watch for oscillation or divergence. Different rearrangements of the same equation can have different convergence behaviour.
Newton uses a tangent update and can converge quadratically near a simple root when the derivative is informative and the initial value is close enough. A small derivative can create an enormous step, a poor initial value can lead to another root, and repeated roots reduce the expected rate. The secant method estimates the derivative from two recent iterates, avoiding an explicit derivative at the cost of a weaker guarantee.
Stopping needs more than one number. A small step can occur during stagnation even when the residual remains large, while a small residual may not imply a small root error for a very flat function. Combine a step or bracket criterion with the residual and the original domain conditions. Report the starting data, iteration rule, tolerance and final check.
If reliability matters, use a safeguarded hybrid: retain a bracket and accept a faster Newton or secant step only when it stays credible.
What this chapter covers
- 01
Sign changing brackets
- 02
Bisection error bounds
- 03
Fixed point contraction
- 04
Newton iteration
- 05
Secant updates
- 06
Convergence order and stopping
Worked example · free
Take two Newton steps
- +1Differentiate to obtain f prime equal to 2x.
- +1The first update gives x1=2.25.
- +1The second update gives x2 about 2.236111.
- +1Check the residual and compare with the positive square root of 5.
Key terms
- Root bracket
- An interval whose endpoint function values have opposite signs for a continuous function.
- Quadratic convergence
- Local behaviour in which the next error is proportional to the square of the current error.
- Contraction mapping
- A mapping that reduces distances and supports fixed point convergence on an invariant interval.
Nonlinear Equations FAQ
Why can Newton's method fail?
A poor start can enter another attraction basin, a small derivative can create a huge step, and difficult geometry can cause divergence or cycling. Local speed does not give a global guarantee.
What certifies a bisection answer?
Continuity and the retained sign change keep a root inside the current interval. The interval width gives a direct location bound after each halving step.
Is a small iterate change enough to stop?
Not always. A sequence can stagnate far from a root. Pair the step change with a residual and, when available, a bracket or other problem specific check.
Exam move
Make a comparison table for bisection, fixed point, Newton and secant methods with starting data, per step work, order and failure mode. Recompute the Newton example from a different starting value and practise one hybrid decision where an unsafe fast step falls back to bisection. For bisection, record the bracket and its width on every line so the error certificate is automatic.
For fixed-point problems, compare two rearrangements of the same equation and use derivative size to predict which will converge. For Newton and secant iterations, monitor both step size and residual to expose stagnation. Add one weak-derivative case and one repeated-root case to the comparison table.
In a timed problem, state the domain, starting evidence, stopping rule and final residual before giving decimal digits; this prevents fast iteration from being mistaken for a guaranteed solution.
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