Math 2.0

Approximations and expansions Interpolation Theorem

Interpolation error formula

Builds on Polynomial interpolationDerivativeRolle's theorem

Statement

Let n≥0n \ge 0, let a<ba < b, and let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be n+1n + 1 times continuously differentiable. Let x0,…,xn∈[a,b]x_0, \dots, x_n \in [a, b] be pairwise distinct, and let pp be the interpolation polynomial of degree at most nn with p(xi)=f(xi)p(x_i) = f(x_i) for all ii. Then for every x∈[a,b]x \in [a, b] there is ξ\xi in the smallest closed interval that contains x,x0,…,xnx, x_0, \dots, x_n such that f(x)−p(x)=f(n+1)(ξ)(n+1)!∏i=0n(x−xi).f(x) - p(x) = \frac{f^{(n+1)}(\xi)}{(n + 1)!} \prod_{i=0}^{n} (x - x_i).

Remarks

If xx is a node, both sides are 00. Otherwise set ω(t)=∏i(t−xi)\omega(t) = \prod_i (t - x_i) and choose the constant cc so that g(t)=f(t)−p(t)−c ω(t)g(t) = f(t) - p(t) - c \, \omega(t) vanishes at t=xt = x. Then gg has n+2n + 2 distinct zeros, and n+1n + 1 applications of Rolle's theorem give a zero ξ\xi of the derivative g(n+1)=f(n+1)−c (n+1)!g^{(n+1)} = f^{(n+1)} - c \, (n + 1)!. This gives c=f(n+1)(ξ)/(n+1)!c = f^{(n+1)}(\xi)/(n + 1)!.

Example: take f=exp⁡f = \exp on [0,1][0, 1] with the nodes 00 and 11. At x=1/2x = 1/2 the error of the linear interpolant is eξ2⋅12⋅(−12)\frac{e^{\xi}}{2} \cdot \frac{1}{2} \cdot \left(-\frac{1}{2}\right) for some ξ∈[0,1]\xi \in [0, 1]. So its absolute value is at most e/8e/8.

For n=0n = 0 the formula is the mean value theorem. It is the analogue of Taylor's theorem with the Lagrange remainder, with n+1n + 1 separate points in place of one point of high contact.

Going deeper

For a student

Taking the maximum over [a,b][a, b] gives ∥f−p∥∞≤∥f(n+1)∥∞(n+1)!max⁡x∈[a,b]∣ω(x)∣\norm{f - p}_\infty \le \frac{\norm{f^{(n+1)}}_\infty}{(n + 1)!} \max_{x \in [a, b]} \abs{\omega(x)}. The first factor depends only on ff, the second only on the nodes. Chebyshev nodes make the second factor as small as possible on an interval. When the derivatives of ff grow fast, the bound can fail to tend to 00, which is what happens in Runge's phenomenon. In terms of divided differences, for xx not a node the error equals f[x0,…,xn,x] ω(x)f[x_0, \dots, x_n, x] \, \omega(x).

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Derivative, concept, no evidence levelDerivativeconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPolynomial interpolation, concept, no evidence levelPolynomial interpolat…conceptSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementInterpolation error formula, statement, no evidence levelInterpolation error f…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Derivative concept
  2. Limit point of a set definition
  3. Local maximum definition
  4. Polynomial interpolation concept
  5. Sequence definition
  6. Limit of a function definition
  7. Local minimum definition
  8. Real sequence definition
  9. Subsequence definition
  10. Continuity at a point definition
  11. Derivative definition
  12. Bounded sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Continuous function definition
  16. Fermat's theorem on stationary points statement
  17. Convergent sequence definition
  18. Monotone convergence theorem for sequences statement
  19. Every sequence has a monotone subsequence statement
  20. Bolzano–Weierstrass theorem statement
  21. Boundedness theorem statement
  22. Extreme value theorem statement
  23. Rolle's theorem statement
  24. Interpolation error formula statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_interpolation
About this page
TagHASA2XSG66
URIm2://HASA2XSG66
Kindstatement, theorem
WritingW0 stub
MSC41A05, 65D05, 41A80
StatusThis copy of the site has no build data. Run m2 status HASA2XSG66.
AboutPolynomial interpolation
UsesPolynomial interpolation, Derivative, Rolle's theorem

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