Romberg Integration. MATH 375 Numerical Analysis. J. Robert Buchanan. Spring Department of Mathematics

Transcription

1 Romberg Integration MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Spring 019

2 Objectives and Background In this lesson we will learn to obtain high accuracy approximations to definite integrals using the Composite Trapezoidal Rule. b f (x) dx = h n 1 f (a) + f (x j ) + f (b) (b a)f (µ) 1 a where h = b a n j=1 and x j = a + jh for j = 0, 1,..., n. h

3 Objectives and Background In this lesson we will learn to obtain high accuracy approximations to definite integrals using the Composite Trapezoidal Rule. b f (x) dx = h n 1 f (a) + f (x j ) + f (b) (b a)f (µ) 1 a where h = b a n j=1 and x j = a + jh for j = 0, 1,..., n. We will use Richardson s Extrapolation to develop O(h m ) approximations to the definite integral. h

4 Alternative Truncation Error If f C [a, b] then we may write the truncation error of the Composite Trapezoidal Rule as b f (x) dx = h n 1 f (a) + f (x j ) + f (b) a where the coefficients K i are constants. j=1 + K 1 h + K h 4 + K 3 h 6 + The truncation error contains only even powers of h and we have seen Richardson s extrapolation be very effective in this situation. Extrapolation applied to this quadrature formula is called Romberg integration.

5 Outline Given b a f (x) dx, use the Composite Trapezoidal rule with n = 1,, 4, 8,..., k,... to form approximations labeled respectively R 1,1, R,1, R 3,1, R 4,1,... R k+1,1,... Since these are O(h ) approximations we can combine them using Richardson s extrapolation to form O(h 4 ) approximations.

6 Outline Given b a f (x) dx, use the Composite Trapezoidal rule with n = 1,, 4, 8,..., k,... to form approximations labeled respectively R 1,1, R,1, R 3,1, R 4,1,... R k+1,1,... Since these are O(h ) approximations we can combine them using Richardson s extrapolation to form O(h 4 ) approximations. For example, for k =, 3,.... R k, = R k, ( Rk,1 R k 1,1 ),

7 Test Case Throughout this presentation we will use the following definite integral to evaluate the performance of the various approximations. π 0 x cos 3x dx = 1 9 cos 3x x sin 3x π =

8 Example (1 of 5) Approximate the definite integral using the Composite Trapezoidal Rule. k R k+1,1 Abs. Err

9 Example ( of 5) Now form the O(h 4 ) approximations using the formula Note that R k, = R k, ( Rk,1 R k 1,1 ) k R k+1,1 R k+1, Abs. Err R, = R, (R,1 R 1,1 ) = (.4674 ( )) 3 =

10 Example (3 of 5) Now form the O(h 6 ) approximations using the formula R k,3 = R k, ( Rk, R k 1, ) k R k+1,1 R k+1, R k+1,3 Abs. Err Note that R 4,3 = R 4, (R 4, R 3, ) = ( ) 15 =

11 Example (4 of 5) Now form the O(h 8 ) approximations using the formula R k,4 = R k, ( Rk,3 R k 1,3 ) k R k+1,1 R k+1, R k+1,3 R k+1,4 Abs. Err Note that R 4,4 = R 4, (R 4,3 R 3,3 ) = ( ) 63 =

12 Example (5 of 5) Now form the O(h 10 ) approximations using the formula R k,5 = R k,4 + 1 ( ) Rk,4 R k 1,4 55 k R k+1,1 R k+1, R k+1,3 R k+1,4 R k+1,5 Abs. Err Note that R 5,5 = R 5, (R 5,4 R 4,4 ) = ( 0.19 ( )) 55 =

13 Remarks Since n increases by a factor of each time, R k+1,1 uses all the function evaluations of R k,1 and adds the function evaluations half-way between the previous nodes. The old function evaluations do not need to be re-computed, only the k new values must be computed.

14 Remarks Since n increases by a factor of each time, R k+1,1 uses all the function evaluations of R k,1 and adds the function evaluations half-way between the previous nodes. The old function evaluations do not need to be re-computed, only the k new values must be computed. We can express the Composite Trapezoidal rule in a recursive manner.

15 Recursion Define h k = b a, then k 1 R 1,1 = b a = h 1 [f (a) + f (b)] [f (a) + f (b)]

16 Recursion Define h k = b a, then k 1 R 1,1 = b a = h 1 R,1 = b a 4 [f (a) + f (b)] [f (a) + f (b)] [ f (a) + f (b) + f = h [f (a) + f (b) + f (a + h )] = 1 [R 1,1 + h 1 f (a + h )] ( a + b a )]

17 Recursion Define h k = b a, then k 1 R 1,1 = b a = h 1 R,1 = b a 4 [f (a) + f (b)] [f (a) + f (b)] [ f (a) + f (b) + f = h [f (a) + f (b) + f (a + h )] = 1 [R 1,1 + h 1 f (a + h )] ( a + b a )] R 3,1 = 1 {R,1 + h [f (a + h 3 ) + f (a + 3h 3 )]}. R k,1 = 1 k R k 1,1 + h k 1 f (a + (i 1)h k ) i=1

18 Romberg Table Once we have calculated R k,1 for k = 1,,..., n we use the extrapolation formula below to produce the O(h j k ) approximation. R k,j = R k,j 1 + for k = j, j + 1, ( ) Rk,j 1 4 j 1 R k 1,j 1 1 k O(hk ) O(h4 k ) O(h6 k ) O(h8 k ) O(hn k ) 1 R 1,1 R,1 R, 3 R 3,1 R 3, R 3,3 4 R 4,1 R 4, R 4,3 R 4, n R n,1 R n, R n,3 R n,4 R n,n

19 Romberg Table Once we have calculated R k,1 for k = 1,,..., n we use the extrapolation formula below to produce the O(h j k ) approximation. R k,j = R k,j 1 + for k = j, j + 1, ( ) Rk,j 1 4 j 1 R k 1,j 1 1 k O(hk ) O(h4 k ) O(h6 k ) O(h8 k ) O(hn k ) 1 R 1,1 R,1 R, 3 R 3,1 R 3, R 3,3 4 R 4,1 R 4, R 4,3 R 4, n R n,1 R n, R n,3 R n,4 R n,n Remark: calculate the table one row at a time.

20 Example Add another row to our previously calculated table. k R k+1,1 R k+1, R k+1,3 R k+1,4 R k+1,5 R k+1,

21 Example Add another row to our previously calculated table. k R k+1,1 R k+1, R k+1,3 R k+1,4 R k+1,5 R k+1,

22 Example Add another row to our previously calculated table. k R k+1,1 R k+1, R k+1,3 R k+1,4 R k+1,5 R k+1, Remark: Romberg integration is efficient since only the entries in the R k,1 column are calculated by evaluating f (x). All others are found by weighted averaging.

23 Romberg Algorithm INPUT Endpoints a, b and integer n. STEP 1 Set h = b a; set R 1,1 = h (f (a) + f (b)). STEP OUTPUT R 1,1. STEP 3 For i =, 3,..., n do STEPS 4 8. STEP 4 Set R,1 = 1 i R 1,1 + h f (a + (k 0.5)h). STEP 5 For j =, 3,..., i set k=1 R,j = R,j 1 + R,j 1 R 1,j 1 4 j 1. 1 STEP 6 For j = 1,,..., i OUTPUT R,j. STEP 7 Set h = h. STEP 8 For j = 1,,..., i set R 1,j = R,j. STEP 9 STOP.

24 Homework Read Section 4.5. Exercises: 1ad, 3ad, 5ad, 18, 19