MathematicsChapter 10: Numerical Methods
Fixed-Point Convergence
Convergence near a fixed point is commonly expected when
Learning objectives
- State the rule for fixed-point convergence.
- Solve a fixed-point convergence problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Convergence near a fixed point is commonly expected when
Where / symbols
- Fixed-point convergence
- Convergence near a fixed point is commonly expected when
CONCEPT AND RULE
Convergence near a fixed point is commonly expected when
METHOD
For fixed-point convergence, 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
g'(x)|<1 in the relevant region.|A rearrangement with large |g'| may diverge although it has the same root.
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 convergence
- 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:
- g'(x)|<1 in the relevant region.|A rearrangement with large |g'| may diverge although it has the same root.
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 convergence
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Fixed-Point Convergence
Sticky Notes · Formula and Worked Revision
Core Idea
- Convergence near a fixed point is commonly expected when
Rule / Formula
- Convergence near a fixed point is commonly expected when
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- g'(x)|<1 in the relevant region.|A rearrangement with large |g'| may diverge although it has the same root.
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
- Convergence near a fixed point is commonly expected when
- Worked pattern: g'(x)|<1 in the relevant region.|A rearrangement with large |g'| may diverge although it has the same root.
- Final check: domain, sign, notation, units, and reasonableness.
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