Numerical Integration

Transcription

1 Numerical Integration Sanzheng Qiao Department of Computing and Software McMaster University February, 2014

2 Outline 1 Introduction 2 Rectangle Rule 3 Trapezoid Rule 4 Error Estimates 5 Simpson s Rule 6 Richardson s Extrapolation 7 Adaptive Quadrature 8 2D Quadrature 9 Software Packages

3 Introduction Another (better) term: quadrature Problem: Given a function f(x) on x [a, b], calculate I(f) = b a f(x)dx.

4 Introduction (cont.) Partition a = x 1 < x 2 < < x n+1 = b, and denote h i = x i+1 x i. Then I(f) = I i I i = xi+1 x i f(x)dx Quadrature rule: Approximation of I i Composite quadrature rule: Approximation of I(f) as the sum of I i

5 Rectangle rule We use piecewise constant (degree zero polynomial) to approximate f(x) on each [x i, x i+1 ]. The function is evaluated at the midpoint y i = x i + x i+1, i = 1,..., n, 2 then the rectangle (quadrature) rule is: I i h i f(y i ) x i x i+1

6 Rectangle rule (cont.) The composite rectangle rule is: R(f) = h i f(y i ) A weighted sum of function values. Often the major computation is the evaluation of the function. Thus the complexity is measured by the number of function evaluations. In the rectangle rule Function evaluations: n. Evaluated at midpoints y i.

7 Trapezoid rule We use piecewise linear interpolation (degree one polynomial) to approximate f(x) on each [x i, x i+1 ]. The function is evaluated at the endpoints: I i h i f(x i )+f(x i+1 ) 2 xi xi+1

8 Trapezoid rule (cont.) Composite trapezoid rule: T(f) = h i f(x i )+f(x i+1 ) 2 = h 1 2 f(x 1)+ h 1 + h 2 f(x 2 ) A weighted sum of function values. + h n 1 + h n f(x n )+ h n f(x n+1) Function evaluations: n+1. Evaluated at endpoints x i

9 Error in rectangle rule Taylor expansion f(x) about the midpoint y i = (x i + x i+1 )/2: f(x) = f(y i )+ (x y i ) p f (p) (y i ). p! p=1 Integrate the both sides and note that xi+1 x i (x y i ) p dx = { h p+1 i (p+1)2 p even p 0 odd p

10 Error in rectangle rule (cont.) Then I i = xi+1 x i f(x)dx = h i f(y i ) h3 i f (y i ) h5 i f iv (y i )+ When h i is small, the error I(f) R(f) 1 hi 3 24 f (y i )+ 1 hi f iv (y i ) For equal spacing, h i = h, we have the error in the composite rectangle rule: I(f) R(f) h3 f (y i )+ h5 f iv (y i )

11 Error in trapezoid rule In order to make the error in the trapezoid rule comparable with that in the rectangle rule, we expand f(x) at the midpoint y i. Substituting x = x i and x = x i+1 in the Taylor expansion, we have Thus f(x i ) = f(y i )+ f(x i+1 ) = f(y i )+ ( 1) p hp i 2 p p! f(p) (y i ) p=1 p=1 h p i 2 p p! f(p) (y i ) f(x i )+f(x i+1 ) 2 = f(y i )+ 1 8 h2 i f (y i ) h4 i f iv (y i )+

12 Error in trapezoid rule (cont.) Recall that in the case of rectangle rule, we had I i = xi+1 x i f(x)dx = h i f(y i ) h3 i f (y i ) h5 i f iv (y i )+ Combining the above two equations, we have I i = xi+1 x i f(x)dx = h i f(x i )+f(x i+1 ) h3 i f (y i ) h5 i f iv (y i )+ Then the error in the composite trapezoid rule is I(f) T(f) 1 hi 3 12 f (y i ) 1 hi f iv (y i )+

13 Remarks Compare and I(f) R(f) h3 24 I(f) T(f) 1 12 f (y i )+ h h 3 i f (y i ) f iv (y i ) hi 5 f iv (y i )+ Usually rectangle rule (degree zero approximation) is more accurate than trapezoid rule (degree one approximation). Surprised?

14 Remarks Observe and I(f) R(f) h3 24 I(f) T(f) 1 12 f (y i )+ h h 3 i f (y i ) f iv (y i ) hi 5 f iv (y i )+ Using I(f) R(f) 1 3 (T(f) R(f)), we can estimate the error in R(f) using T(f) and R(f). Similarly, I(f) T(f) 2 3 (R(f) T(f)) can be used to estimate the error in T(f). (But they are approximations, it is possible that R(f) T(f) = 0 whereas I(f) R(f) 0).

15 Remarks Observe I(f) R(f) h3 24 f (y i )+ h f iv (y i ) When each h i is cut in half, I(f) R 1(f) 1 4 (I(f) R(f)). 2 (Why?) Similarly for the trapezoid rule. Doubling the number of panels in either the rectangle rule or the trapezoid rule, it can be expected to roughly quadruple the accuracy. This can be used to estimate the error as well as improving the accuracy. (How?)

16 Example Compute using the trapezoid rule. π 2 0 sin(x)dx m QCTrap(sin, 0.000, 1.571, m) error e e e e-4 where m is the number of points, that is, m 1 is the number of intervals.

17 Simpson s rule Recall the rectangle rule R(f) = I(f) 1 hi 3 24 f (y i ) and the trapezoid rule T(f) = I(f) h 3 i f (y i ) hi 5 f iv (y i )+ hi 5 f iv (y i )+ Combining the above two equations (canceling the O(h 3 i ) term), we get a more accurate method (Simpson s rule): S(f) = 2 3 R(f)+ 1 3 T(f) = I(f) hi 5 f iv (y i )+

18 Simpson s rule (cont.) Simpson s rule: I i = 2 3 h if( x i + x i+1 ) h f(x i )+f(x i+1 ) i 2 Composite Simpson s rule: S(f) = Function evaluations: 2n + 1 Error I(f) S(f) = [ 1 6 h i f(x i )+4f( x ] i + x i+1 )+f(x i+1 ) 2 hi 5 f iv (y i )+

19 Remarks Simpson s rule can also be derived by using piecewise quadratic (degree two) approximation. Actually, Simpson s rule is exact for cubic function (one extra order of accuracy), since the error term involves the fourth derivatives. Doubling the number of panels in Simpson s rule can be expected to reduce the error by roughly the factor of 1/16.

20 A general technique: Richardson s extrapolation Idea: Combining two approximations (e.g., R(f) and T(f)) which have similar error terms to achieve a more accurate approximation (e.g., S(f)). Example. Combining S(f) and S 1(f) to obtain an 2 approximation which has error of order hi 7. This gives the Romberg quadrature. Question What are the weights? Answer: S 1 2(f) 1 15 S(f)

21 What is adaptive quadrature? Given a predetermined tolerance ǫ, the algorithm automatically determines the panel sizes so that the computed approximation Q satisfies b Q f(x)dx < ǫ a Software interface: quad(fname, a, b, tol) Why adaptive? The algorithm uses large panel sizes for smooth parts and small panel sizes for the parts where the function changes rapidly. Thus the prescribed accuracy is attained at as small a cost in computing time. (Measured by the number of function evaluations.)

22 Basic idea Compute two approximations (Simpson s rule): one-panel formula P i = h [ i f(x 6 i )+4f(x i + h ] i 2 )+f(x i + h i ) two-panel formula Q i = h i 12 [ f(x i )+4f(x i + h i 4 )+2f(x i + h i 2 ) + 4f(x i + 3h i 4 )+f(x i + h i ) ]

23 Basic idea (cont.) Note From P i to Q i, we need only two function evaluations f(x i + h i 4 ) and f(x i + 3h i 4 ) Q i is the sum of two P s from two subintervals of length h i /2 Compare P i and Q i to obtain an estimate of their accuracy. I i P i = c hi 5 f iv (x i + h i 2 )+ ( ) 5 [ hi I i Q i = c f iv (x 2 i + h i 4 )+f iv (x i + 3h ] i 4 ) +

24 Error estimation Using the approximation we have f iv (x i + h i 4 )+f iv (x i + 3h i 4 ) 2f iv (x i + h i 2 ), I i Q i 2c ( ) 5 hi f iv (x i + h i 2 2 )+ Thus we have a relation between the errors in Q i and P i : Reformulate the above I i Q i 1 2 4(I i P i )+ I i Q i (Q i P i )+ Now the accuracy of Q i is expressed in terms of Q i P i

25 Scheme Bisect each subinterval until Q i P i h i b a ǫ Then b a f(x)dx Q i ǫ b a Q i P i h i = ǫ

26 function [I, err] = AdaptQuad(fname,a,b,tol,maxLev) if maxlev==0 too many levels of recursion, quit; compute one-panel quadrature R1; compute two-panel quadrature R2; use R1 and R2 to estimate error in R2; if the estimated error < tol return R2 and estimated error; else [I1, err1] = AdaptQuad(fname,a,mid,tol/2,maxLev-1); [I2, err2] = AdaptQuad(fname,mid,b,tol/2,maxLev-1); I = I1 + I2; err = err1 + err2;

27 Example Compute π 1 4 = x 2 dx using the adaptive rectangle rule. AdaptQRec( datan,0,1,0.0001,10): estimated error: actual error: Show Adaptive, tol =

28 2D quadrature Consider a 2D integral and let I = b d a g(x) = c d c f(x, y)dydx f(x, y)dy. Applying the composite trapezoid rule to b we get the numerical integration m 1 a g(x)dx g(x i )+g(x i+1 ) h x. 2

29 2D quadrature (cont.) Written in vector form: h x w T x g(x 1 ). g(x m ) where w T x = [1/2, 1,..., 1, 1/2].

30 2D quadrature (cont.) Again, applying the composite trapezoid rule to each g(x i ), we get g(x i ) = In vector form d c f(x i, y)dy n 1 j=1 f(x i, y j )+f(x i, y j+1 ) h y. 2 g(x i ) h y [f(x i, y 1 ),..., f(x i, y n )]w y where w y = [1/2, 1,..., 1, 1/2] T.

31 2D quadrature (cont.) Finally, we have the numerical integration, in matrix-vector form: Q = h x h y w T x Fw y where F = f(x 1, y 1 ) f(x 1, y n ).. f(x m, y 1 ) f(x m, y n ).

32 Example m = n e (x2 +2y 2 )/4 dydx Relative Subintervals Integral Time

33 Software packages IMSL qdag, qdags, twodq, qand MATLAB quad, quad1, dblquad NAG d01ajf, d01daf, d01fcf Octave quad, quadl, trapz

34 Summary Composite quadrature rules: Rectangle rule, trapezoid rule, Simpson s rule Richardson s extrapolation technique: Combining two quadrature rules with similar error terms to achieve a more accurate quadrature rule by canceling the leading error term; Combining one-panel and two-panel results to estimate errors Adaptive quadrature: By using error estimates, determine the panel sizes so that the computed approximation satisfies a predetermined tolerance 2D quadrature: Formulation of the problem

35 References [1 ] George E. Forsyth and Michael A. Malcolm and Cleve B. Moler. Computer Methods for Mathematical Computations. Prentice-Hall, Inc., Ch 5.