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.

• Rectangle and Mid point rules are the (semi-)open formulas.

Rectangle rule

Mid point rule.

Trapezoidal rule

Simpson’s rule

Simpson’s 3/8 rule

Bode’s rule

• Newton-Cotes formula for with equally spaced abscissas .

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) In(xk) for the degree k = D + 1.

Weights wi contain some negative coefficients.

• Closed Newton-Cotes formula of degree D=n=8.

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.

• 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: (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 2(a,b) such that,

b) For odd n, and f(x)2 C(n+1)(a,b), 9 2(a,b) such that,

The above theorem is used to determine the error of a Newton_Cotes formula.

Ex) Trapezoidal formula.

Theorem: (Weighted Mean-Value Theorem for Integrals.)

Exc. 3-1) Prove this.

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.

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,

Trapezoidal formula

• Extended (composite, compound) 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,

Romberg approximations is written Rk,j , Level of extrapolation

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Exc 3-6) Using Romberg integration, calculate the definite integral

and estimate the error up to R4,4 .

• 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 2n-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.

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.

Exc 3-9) Try some of the above.