3 2 1 b a b a x f x f 3 x f 3 x f 8 h 3 3 a b h Lxdx fxdx Truncation error 3 a

3 2 1 b a b a x f x f 3 x f 3 x f 8 h 3 3 a b h lxdx

This preview shows page 43 - 56 out of 61 pages.

3 2 1 0 b a b a x f x f 3 x f 3 x f 8 h 3 3 a b h ; L(x)dx f(x)dx + + + = - = Truncation error 3 a b h ; f 6480 a b f h 80 3 E 4 5 4 5 t - = - - = - = ) ( ) ( ) ( ) ( ) ( ξ ξ
Image of page 43
Example: Simpson’s Rules Example: Simpson’s Rules Evaluate the integral Simpson’s 1/3-Rule Simpson’s 3/8-Rule dx xe 4 0 x 2 [ ] [ ] % . . . . . ) ( ) ( ) ( ) ( 96 57 926 5216 411 8240 926 5216 411 8240 e 4 e 2 4 0 3 2 4 f 2 f 4 0 f 3 h dx xe I 8 4 4 0 x 2 - = - = = + + = + + = ε [ ] % 71 . 30 926 . 5216 209 . 6819 926 . 5216 209 . 6819 832 . 11923 ) 33933 . 552 ( 3 ) 18922 . 19 ( 3 0 8 ) 4/3 ( 3 ) 4 ( f ) 3 8 ( f 3 ) 3 4 ( f 3 ) 0 ( f 8 h 3 dx xe I 4 0 x 2 - = - = = + + + = + + + = ε
Image of page 44
function I = Simp(f, a, b, n) % integral of f using composite Simpson rule % n must be even h = (b - a)/n; S = feval(f,a); for i = 1 : 2 : n-1 x(i) = a + h*i; S = S + 4*feval(f, x(i)); end for i = 2 : 2 : n-2 x(i) = a + h*i; S = S + 2*feval(f, x(i)); end S = S + feval(f, b); I = h*S/3; Composite Simpson’s Rule Composite Simpson’s Rule
Image of page 45
Simpson’s Rule Simpson’s Rule
Image of page 46
Composite Simpson’s Rule Composite Simpson’s Rule
Image of page 47
» x=0:0.04:4; y=example(x); » x1=0:2:4; y1=example(x1); » c=Lagrange_coef(x1,y1); p1=Lagrange_eval(x,x1,c); » H=plot(x,y,x1,y1, 'r*' ,x,p1, 'r' ); » xlabel('x'); ylabel('y'); title('f(x) = x*exp(2x)'); » set(H,'LineWidth',3,'MarkerSize',12); » x2=0:1:4; y2=example(x2); » c=Lagrange_coef(x2,y2); p2=Lagrange_eval(x,x2,c); » H=plot(x,y,x2,y2, 'r*' ,x,p2, 'r' ); » xlabel( 'x' ); ylabel( 'y' ); title( 'f(x) = x*exp(2x)' ); » set(H, 'LineWidth' ,3, 'MarkerSize' ,12); » » I=Simp( 'example' ,0,4,2) I = 8.2404e+003 » I=Simp( 'example' ,0,4,4) I = 5.6710e+003 » I=Simp( 'example' ,0,4,8) I = 5.2568e+003 » I=Simp( 'example' ,0,4,16) I = 5.2197e+003 » Q=Quad8( 'example' ,0,4) Q = 5.2169e+003 n = 2 n = 4 n = 8 n = 16 MATLAB fun
Image of page 48
Multiple applications of Simpson’s rule Multiple applications of Simpson’s rule with odd number of intervals with odd number of intervals Hybrid Simpson’s 1/3 & 3/8 rules
Image of page 49
Newton-Cotes Closed Newton-Cotes Closed Integration Formulae Integration Formulae ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ' ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ξ ξ ξ ξ ξ 6 7 5 4 3 2 1 0 6 7 4 3 2 1 0 4 5 3 2 1 0 4 5 2 1 0 3 1 0 f h 12096 275 288 x f 19 x f 75 x f 50 x f 50 x f 75 x f 19 a b 5 f h 945 8 90 x f 7 x f 32 x f 12 x f 32 x f 7 a b rule s Boole' 4 f h 80 3 8 x f x f 3 x f 3 x f a b rule 3/8 s Simpson' 3 f h 90 1 6 x f x f 4 x f a b rule 1/3 s Simpson 2 f h 12 1 2 x f x f a b rule l Trapezoida 1 Error Truncation Formula Name n - + + + + + - - + + + + - - + + + - - + + - - + - n a b h - =
Image of page 50
Composite Trapezoidal Rule with Composite Trapezoidal Rule with Unequal Segments Unequal Segments Evaluate the integral h 1 = 2, h 2 = 1, h 3 = 0.5, h 4 = 0.5 dx xe I 4 0 x 2 = [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] % . . . . ) ( ) . ( ) . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( . . 45 14 58 5971 e 4 e 5 3 2 0.5 e 5 3 3e 2 0.5 e 3 e 2 2 1 e 2 0 2 2 4 f 5 3 f 2 h 5 3 f 3 f 2 h 3 f 2 f 2 h 2 f 0 f 2 h dx x f dx x f dx x f dx x f I 8 7 7 6 6 4 4 4 3 2 1 4 5 3 5 3 3 3 2 2 0 - = = + + + + + + + = + + + + + + + = + + + = ε
Image of page 51
Trapezoidal Rule for Unequally Spaced Data Trapezoidal Rule for Unequally Spaced Data
Image of page 52
MATLAB Function: MATLAB Function: trapz trapz » x=[0 1 1.5 2.0 2.5 3.0 3.3 3.6 3.8 3.9 4.0] x = Columns 1 through 7 0 1.0000 1.5000 2.0000 2.5000 3.0000 3.3000 Columns 8 through 11 3.6000 3.8000 3.9000 4.0000 » y=x.*exp(2.*x) y = 1.0e+004 * Columns 1 through 7 0 0.0007 0.0030 0.0109 0.0371 0.1210 0.2426 Columns 8 through 11 0.4822 0.7593 0.9518 1.1924 » integr = trapz(x,y) integr = 5.3651e+003 Z = trapz(x,y)
Image of page 53
Integral of Unevenly-Spaced Data Integral of Unevenly-Spaced Data Trapezoidal rule Could also be evaluated with Simpson’s rule for higher accuracy
Image of page 54
Composite Simpson’s Rule with Composite Simpson’s Rule with Unequal Segments Unequal Segments
Image of page 55
Image of page 56

You've reached the end of your free preview.

Want to read all 61 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes