3. numerical integration (numerical quadrature). given the continuous function f(x) on [a,b],...
Post on 26-Dec-2015
227 Views
Preview:
TRANSCRIPT
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,
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
S
tep
size
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.)
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.
top related