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Graduate · Contraction estimates

Contraction estimates: A small negative fixed point

Contraction estimates: investigate a small negative fixed point with contraction slope q = -0.4; constant b = -0.5.

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Contraction estimates: A small negative fixed point. Iteration: 0. Current iterate: 3. Unique fixed point: -0.357. Exact absolute error: 3.357An exact contraction error estimate-1.730612iterateiteration n → · labeled axes rescale to this model
The domain is the complete metric space R, the map is affine, and |q|≤0.8. The initial iterate is fixed at three. The explicit identity is stronger than a general contraction bound and does not describe every nonlinear iteration.

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Make it your experiment

Change one value. Notice what follows.

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Iteration
0
Current iterate
3
Unique fixed point
-0.357
Exact absolute error
3.357

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Understand what you are seeing

The idea behind the motion.

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. This investigation starts with Contraction slope q = -0.4; Constant b = -0.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

T(x)=qx+b; x*=b/(1−q); |xₙ−x*|=|q|ⁿ|3−x*|

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Verify |q|<1 and solve the fixed-point equation before iterating. The starting case is “A small negative fixed point.”

  2. STEP 2

    Follow the changing quantity

    Starting at x₀=3, follow twelve exact iterates. A negative q reverses the sign of the error on each step.

  3. STEP 3

    Explain and test the result

    Compare the actual absolute error with |q|ⁿ times the initial error. Explain which conclusion would lose justification if completeness or strict contraction were removed.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Contraction slope q = -0.4; Constant b = -0.5. Pause the timeline at 100%. Given iteration = 12, calculate current iterate, unique fixed point, exact absolute error. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Solve x*=(-0.4)x*+(-0.5) to get x*=-0.5/(1−(-0.4))=-0.357143. Then xₙ=x*+(3−x*)(-0.4)^12=-0.357087, and the absolute error is 0.000056. Results: Current iterate: -0.357; Unique fixed point: -0.357; Exact absolute error: 0. Decimal values are rounded; retain the original parameters when checking.

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Graduate · Calculus

Turn a contraction into a certified iterative solver

Check a contraction constant and derive a computable error bound.