MathematicsChapter 10: Numerical Methods
Fixed-Point Iteration
Rewrite f(x)=0 as x=g(x), choose a start, and compute x_(n+1)=g(x_n).
Learning objectives
- State the rule for fixed-point iteration.
- Solve a fixed-point iteration problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
Mathematical form
x=g(x)
x=cos x
x0=.7 and repeatedly evaluate cosine in rad
Where / symbols
- Fixed-point iteration
- Rewrite f(x)=0 as x=g(x), choose a start, and compute x_(n+1)=g(x_n).
CONCEPT AND RULE
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
METHOD
For fixed-point iteration, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve fixed-point iteration
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Fixed-point iteration
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Fixed-Point Iteration
Sticky Notes · Formula and Worked Revision
Core Idea
- Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
Rule / Formula
- Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
- Worked pattern: For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
- Final check: domain, sign, notation, units, and reasonableness.
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