Graduate · Contraction estimates · 509 of 650
Contraction estimates · Contraction slope q=-0.5; Constant b=1.5
Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.5; Constant b=1.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Iterate T(x)=(-0.5)x+(1.5) on the complete real line, starting with x₀=3. Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point.
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A starting point
Solve x*=qx*+b. Then xₙ−x*=qⁿ(3−x*), whose absolute value supplies the exact error.
Work through the reasoning
Step 1
Identify the model and target
Iterate T(x)=(-0.5)x+(1.5) on the complete real line, starting with x₀=3. Use n=7 iterations. The governing relation is T(x)=qx+b; x*=b/(1−q); |xₙ−x*|=|q|ⁿ|3−x*|. Solve x*=qx*+b. Then xₙ−x*=qⁿ(3−x*), whose absolute value supplies the exact error.
Step 2
Substitute and calculate
Solve x*=(-0.5)x*+(1.5) to get x*=1.5/(1−(-0.5))=1. Then xₙ=x*+(3−x*)(-0.5)^7=0.984375, and the absolute error is 0.015625.
Step 3
Check the mathematical meaning
The Lipschitz constant is |q|=0.5<1. Directly, T(x*)−x*=0 and the error is |-0.5|⁷|3−1|=0.015625. Iteration: 7; Current iterate: 0.984; Unique fixed point: 1; Exact absolute error: 0.016. Decimal values are rounded, so use unrounded intermediate values.
The answer
Solve x*=(-0.5)x*+(1.5) to get x*=1.5/(1−(-0.5))=1. Then xₙ=x*+(3−x*)(-0.5)^7=0.984375, and the absolute error is 0.015625. The Lipschitz constant is |q|=0.5<1. Directly, T(x*)−x*=0 and the error is |-0.5|⁷|3−1|=0.015625. Animation check: Iteration: 7; Current iterate: 0.984; Unique fixed point: 1; Exact absolute error: 0.016. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
A contraction controls all pairs of inputs, not just the sampled orbit. For this affine map, the Lipschitz constant is |q|<1 on complete real space, yielding a unique fixed point. The explicit error identity lets the animation illustrate the theorem without replacing its hypotheses with empirical convergence. Iterate T(x)=(-0.5)x+(1.5) on the complete real line, starting with x₀=3. Use n=7 iterations. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
T(x)=qx+b; x*=b/(1−q); |xₙ−x*|=|q|ⁿ|3−x*|
03 · Reflect and transfer
Explain what changes and why.
Compare a positive and negative slope with the same absolute value. Which convergence feature changes and which error bound stays identical?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.