3 Numerical integration Numerical quadrature Given the continuous

3. Numerical integration (Numerical quadrature). • Given the continuous function f(x) on [a, b], approximate • Newton-Cotes Formulas: For the given abscissas, approximate the integral I(f) by the integral of interpolating formula with degree n, I(pn). Formulas that use end points a, and b as data points are called closed formulas. Those not use the end points called open (semi-open) formulas.

• Newton-Cotes formula for with equally spaced abscissas. Rectangle rule Mid point rule. Trapezoidal rule Simpson’s 3/8 rule Bode’s rule • Rectangle and Mid point rules are the (semi-)open formulas. Definition: Degree of Precision (or Accuracy) of a quadrature rule In(f) is the positive integer D, if I(xk) = In(xk) for the degree k · D, and I(xk) ¹ In(xk) for the degree k = D + 1.

• Closed Newton-Cotes formula of degree D=n=8. Weights wi contain some negative coefficients. This thorem suggests that the higher order Newton-Cotes formula wouldn’t be useful for practical numerical computations. Piecewise ! composite rules. Optimize data points ! A family of Gauss formulas.

Theorem: (The error associated with Newton-Cotes formulas. ) For Newton-Cotes formula with n+1 abscissas (open or closed) a) For even n, and f(x)2 C(n+2)(a, b), 9 x 2(a, b) such that, b) For odd n, and f(x)2 C(n+1)(a, b), 9 x 2(a, b) such that, • n=even cases are generally better in the degree of precision. • Constants c, c’ depend on n and type of formula open or closed. • For a given n, c of the closed formulas are typically smaller than the open formula. The closed formulas are more used in practice. • If the function has a singularity at the end point, open formulas can be useful.

Theorem: (Weighted Mean-Value Theorem for Integrals. ) Exc. 3 -1) Prove this. The above theorem is used to determine the error of a Newton_Cotes formula. Ex) Trapezoidal formula.

Exc. 3 -2) Verify Boole’s rule using an algebraic computing software. Exc. 3 -3) Derive the error term for the Simplson’s rule using the interpolation error formula, Exc 3 -4) Derive the error term for the mid-point rule and Simpson’s rule. Exc 3 -5) Derive the error associated with Newton-Cotes formulas.

• Extended (composite, compound) formula. Trapezoidal formula Simpson formula

For the composite trapezoidal rule, Theorem: (Euler-Maclaurin Sum Formula). If the f(x) has odd derivatives that are equal at the end points of interval [a, b], such as a periodic function on [a, b], the composite trapezoidal rule becomes more accurate. (Also extended mid-point rule. )

• Romberg integration. Extrapolation applied to the composite trapezoidal rule. Euler-Maclaurin summation formula, Level of extrapolation Step size Romberg approximations is written Rk, j , Exc 3 -6) Using Romberg integration, calculate the definite integral and estimate the error up to R 4, 4.

• Gaussian quadratures. Approximating the integral in the form, optimize the location of data points and the associated weights, in the way that the integrals of polynomials have exact value, (From 2 n parameters wi and xi , a quadrature formula with the degree of precision 2 n-1 can be constructed at best. )

Exc 3 -7) Prove the above theorem.

Exc 3 -8) Prove the above theorem. hint) show the following facts.

Exc 3 -9) Try some of the above. More topics for the numerical integration. • Higher precision integration formulas. ex) IMT type formula, (DE formula. ) • Integration of improper integrals. (i. e. infinite integral region, or discontinuity of integrand. ) • Integration of multivariate functions. ex) Monte-Carlo. Multivariate Gauss formulas.