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Numerical Integration (Quadrature)

Numerical Integration (Quadrature) Sachin ShanbhagDept. Scientific Computing(based on material borrowed from Dennis Duke, Samir Al-Amer,David Kofke, Holistic Numerical Methods Institute) Numerical IntegrationWhy do we need it? many integrals cannot be evaluated analytically even if you can, you might need to check your answer even if you can, Numerical evaluation of the answer can be bothersomeExamples:00(1)2cosh21kkdxxxk!" "=#=+$% ! e"x2dxab# Error functionAn example of an integral that needs checking:Possible Issuesthe integrand is some sort of table of numbers regularly spaced irregularly spaced contaminated with noise (experimental data)the integrand is computable everywhere in the range of Integration ,but there may be infinite range of Integration local discontinuitiesconsiderations time to compute the integral estimate of the error due to- truncation- round-off- noise in tabulated values In the differential limit, an integral is equivalent to a summationoperation: Approximate methods for determining integrals are mostly based onidea of area between integrand and as Riemann sum!

• Gauss Quadrature Like Newton-Cotes, but instead of a regular grid, choose a set that lets you get higher order accuracy • Monte Carlo Integration Use randomly selected grid points. Useful for higher dimensional integrals (d>4) Newton-Cotes Methods • In Newton-Cotes Methods, the function is approximated by a polynomial of order n

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