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Fixed-point Iteration
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
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Fixed-point Iteration
What is it?
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
In simple words
- 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.
Important terms
Fixed-point iteration: Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
What you should learn
- State the rule for fixed-point iteration.
- Solve a fixed-point iteration problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
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
How it works
- 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.
Remember
- Fixed-point iteration
- rule
- worked example
- worked example
How the ideas connect
Worked example: For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
Deeper understanding
Go beyond memorising Fixed-point Iteration. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: For x=cos x, start x0=.7 and repeatedly evaluate cosine in radians.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Fixed-point iteration
- rule
- worked example
- Fixed-point iteration: Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
- 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 d
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Fixed-point Iteration: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Fixed-point iteration
- rule
- worked example
- Fixed-point iteration: Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
Important formulas
- Fixed-point iteration
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Fixed-point iteration
rule
worked example
Fixed-point iteration: Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
- Definition
- Rewrite f(x)=0 as x=g(x), choose a start, and compute x(n+1)=g(xn).
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Fixed-point Iteration.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Fixed-point Iteration and say why it fits.
Mistake
Q4. State one common mistake students make about Fixed-point Iteration.
Exam note
Q5. Write a short exam note on Fixed-point Iteration: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.