Math 2.0

Numerical analysis Theorem

Mean value theorem for divided differences

Builds on Divided differencesRolle's theoremDerivativePolynomial interpolation

Statement

Let n≥1n \ge 1, let I⊆ℝI \subseteq \R be an open interval, and let f ⁣:I→ℝf \colon I \to \R be nn times differentiable. Let x0,…,xn∈Ix_0, \dots, x_n \in I be pairwise distinct, and let mm and MM be the smallest and largest of them. Then there is ξ∈(m,M)\xi \in (m, M) such that f[x0,…,xn]=f(n)(ξ)n!,f[x_0, \dots, x_n] = \frac{f^{(n)}(\xi)}{n!}, where f[x0,…,xn]f[x_0, \dots, x_n] is the divided difference.

Remarks

Let pp be the interpolation polynomial of ff at the nodes. Then g=f−pg = f - p has n+1n + 1 distinct zeros in [m,M][m, M]. Applying Rolle's theorem nn times gives a point ξ∈(m,M)\xi \in (m, M) with g(n)(ξ)=0g^{(n)}(\xi) = 0. Since p(n)p^{(n)} is the constant n! f[x0,…,xn]n! \, f[x_0, \dots, x_n], the claim follows.

For n=1n = 1 this is the mean value theorem. Example: for f(x)=x3f(x) = x^3 and the nodes 0,1,20, 1, 2, one has f[0,1]=1f[0, 1] = 1, f[1,2]=7f[1, 2] = 7 and f[0,1,2]=3f[0, 1, 2] = 3. Since f′′(ξ)/2=3ξf''(\xi)/2 = 3\xi, the theorem holds with ξ=1\xi = 1.

Going deeper

For a student

It turns statements about derivatives into statements about divided differences and back. For example, if f(n)≥0f^{(n)} \ge 0 on II, then every nn-th divided difference of ff is nonnegative. Combined with the Newton form of the interpolation polynomial, it gives the interpolation error formula.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

Show the path as a graph
Derivative, concept, no evidence levelDerivativeconceptDivided differences, concept, no evidence levelDivided differencesconceptLimit 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 theoremstatementMean value theorem for divided differences, statement, no evidence levelMean value theorem fo…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. Divided differences concept
  3. Limit point of a set definition
  4. Local maximum definition
  5. Polynomial interpolation concept
  6. Sequence definition
  7. Limit of a function definition
  8. Local minimum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Bounded sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Convergent sequence definition
  19. Monotone convergence theorem for sequences statement
  20. Every sequence has a monotone subsequence statement
  21. Bolzano–Weierstrass theorem statement
  22. Boundedness theorem statement
  23. Extreme value theorem statement
  24. Rolle's theorem statement
  25. Mean value theorem for divided differences statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Mean_value_theorem_(divided_differences)
About this page
TagBQJ9C1CD5X
URIm2://BQJ9C1CD5X
Kindstatement, theorem
WritingW0 stub
MSC65D05, 41A05, 26A24
StatusThis copy of the site has no build data. Run m2 status BQJ9C1CD5X.
AboutDivided differences
UsesDivided differences, Rolle's theorem, Derivative, Polynomial interpolation
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.