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NUMERICAL COMPUTATIONS

NUMERICAL COMPUTATIONS. 04/06/2011. MATH/CMPSC 451. Gaussian Quadrature. Trapezoid rule: two points, exact for all the polynomials of degree at most 1. Simpson’s rule: three points, exact for all the polynomials of degree at most 3.

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NUMERICAL COMPUTATIONS

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  1. NUMERICAL COMPUTATIONS 04/06/2011 MATH/CMPSC 451

  2. Gaussian Quadrature Trapezoid rule: two points, exact for all the polynomials of degree at most 1. Simpson’s rule: three points, exact for all the polynomials of degree at most 3.

  3. In general, (n+1) equally distributed points, are exact for polynomial of degree at most n (for n odd) or n+1 (for n even). Question:Can we select the (n+1) points such that, the numerical integration formula is exact for the polynomials of degree higher than (n+1)?

  4. Example: Take Check the numerical integration formula is exact for all polynomial of degree less than or equal to 1. Example: Take Check the numerical integration formula is exact for all polynomial of degree less than or equal to 3.

  5. Theorem (Theorem on Gaussian Quadrature): Suppose is orthogonal to all polynomials of degree in the sense Then the numerical integration formula will be exact for all polynomials of degree Answer: Yes!! we should choose such that they are the zeros of a polynomial, which is orthogonal to all polynomials of degree

  6. Legendre Polynomial

  7. Legendre Polynomial Definition: Recursive formula

  8. Zeros for Legendre Polynomial:

  9. Example.

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