Math 2.0

Numerical analysis Definition

Finite difference coefficients

Builds on Numerical differentiationVandermonde matrixTaylor's theorem

Definition

Let m≥0m \ge 0 and N≥m+1N \ge m + 1 be integers, and let s1,…,sNs_1, \ldots, s_N be distinct real numbers, the stencil. The finite difference coefficients for the mm-th derivative on this stencil are the real numbers a1,…,aNa_1, \ldots, a_N that solve ∑j=1Naj sj i={m!if i=m,0if i≠m,i=0,1,…,N−1.\sum_{j=1}^{N} a_j \, s_j^{\,i} = \begin{cases} m! & \text{if } i = m, \\ 0 & \text{if } i \ne m, \end{cases} \qquad i = 0, 1, \ldots, N - 1. The associated formula is f(m)(x)≈h−m∑j=1Najf(x+sjh)f^{(m)}(x) \approx h^{-m} \sum_{j=1}^{N} a_j f(x + s_j h) for a step h>0h > 0.

Remarks

The matrix of the system is a Vandermonde matrix in the distinct numbers sjs_j, so the coefficients exist and are unique. Expanding f(x+sjh)f(x + s_j h) by Taylor's theorem up to order N−1N - 1 shows that for f∈CNf \in C^N the formula has error O(hN−m)O(h^{N - m}); for symmetric stencils the order is often one higher. Example: m=1m = 1 and the forward stencil 0,1,20, 1, 2 give a=(−32,2,−12)a = (-\tfrac32, 2, -\tfrac12). Check: ∑aj=0\sum a_j = 0, ∑ajsj=2−1=1=1!\sum a_j s_j = 2 - 1 = 1 = 1!, and ∑ajsj2=2−2=0\sum a_j s_j^2 = 2 - 2 = 0. So f′(x)≈−3f(x)+4f(x+h)−f(x+2h)2hf'(x) \approx \frac{-3 f(x) + 4 f(x+h) - f(x+2h)}{2h}, with error O(h2)O(h^2). Non-example: for m=2m = 2 the weights (1,−2,1)(1, -2, 1) on the stencil 0,1,30, 1, 3 are not the coefficients, since ∑ajsj=−2+3=1≠0\sum a_j s_j = -2 + 3 = 1 \ne 0; the correct ones there are (23,−1,13)(\tfrac23, -1, \tfrac13).

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionNumerical differentiation, concept, no evidence levelNumerical differentia…conceptVandermonde matrix, concept, no evidence levelVandermonde matrixconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionTaylor's theorem, statement, no evidence levelTaylor's theoremstatementFinite difference coefficients, definition, no evidence levelFinite difference coe…definition
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  1. Limit point of a set definition
  2. Numerical differentiation concept
  3. Vandermonde matrix concept
  4. Limit of a function definition
  5. Derivative definition
  6. Differentiable function definition
  7. Derivative function definition
  8. Higher derivative definition
  9. Taylor polynomial definition
  10. Taylor's theorem statement
  11. Finite difference coefficients definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_difference_coefficient
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Kinddefinition
WritingW0 stub
MSC65D25, 65N06
DefinesFinite difference coefficients
UsesNumerical differentiation, Vandermonde matrix, Taylor's theorem

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