7. Numerical evaluation of definite integrals

Transcription

1 7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh - Adptilit Hn Hldíková

2 Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re clled qudrture formuls. We pproximte the vlue of n indefinite integrl I(f = f(xdx, where f is continuous on the intervl,. If, moreover, f is smooth (its first derivtive is continuous, then geometricl interprettion of I(f is the re etween the grph of f nd x-xis on the intervl,. Numericl methods re used if the function is given y tlet, the integrl cnnot e evluted directly, c the evlution is too complicted. Sustitute the given function y φ so tht the integrl of I(φ = φ(xdx cn e evluted more esily. Let I(f e the exct vlue nd I(φ its pproximtion, i.e. I(f I(φ. Numericl evlution of definite intgrl is stle (unlike numericl derivtive. If φ is n pproximtion of the function f on ;, then I(φ is n pproximtion of I(f, ecuse: f(xdx ϵ {}}{ φ(xdx f(x φ(x dx ( sup( f(x φ(x ( ϵ, Newton-Cotes qudrture formuls The given intervl, will e divided into n suintervls, +1, = x 0 < x 1 < x < < x n 1 < x n =. We tke equidistnt prtition, where +1 = h = n. The integrl is prtitioned: I = n 1 f(xdx = k=0 ( f(xdx. On the intervl, +N+1 sustitute the function f y the Lgrnge polynomil N-th degree nd evlute the integrl of it. Such formul is clled sic Newton-Cotes formul - (sic NC formul. The sum for the integrl, is composed qudrture formul. We cll the formul qudrture formul of the order n if it pproximtes exctly polynomil of degree n (for hiher degrees then n we get nonzero error. Newton-Cotes qudrture formul - rectngle rule ( On the intervl, +1 the function f will e ( sustituted y constnt function L 0 (x = f xk+1 + going through the points [ +1+, f xk+1 + ]. We get qudrture formul of order 0. f(xdx ( ( xk+1 + xk+1 + L 0 (xdx = (+1 f = h.f = I Rec (f. ( For f xk+1 + > 0 the vlue I Rec (f is the re of the rectngle with sides

3 ( +1 f xk+1 +. See Fig. 7.1, where + 1 = +1+. f(x f( +1 f( +1/ f(x f( +1/ +1 x Fig. 7.1 Rectngulr rule. Composed formul: n 1 f(xdx I Rec = h f i=0 ( xi+1 + x i Remrk: Let f hve on, continuous second derivtive, then: I I Rec = 4 f (ξh, where ξ (,. f If M = mx x, (x, then I I Od 4 M h. Newton-Cotes qudrture formul - trpezoid rule On, +1 the function f is sustituted y liner polynomil L 1 (x going through the points [, f( ] nd [+1, f(+1 ]. We get qudrture formul of order 1 f(xdx L 1 (xdx = (+1 (f ( + f (+1 = = h (f ( + f (+1 = I T rp. For f(+1 > 0 f( > 0 is I T rp (f the re of the trpezoid with the sides of the lengths f(+1 nd f( nd the height is +1. Geometricl interprettion is in the Fig

4 f(x f( +1 f(x f( +1 x Fig. 7. Trpezoid rule. The composed formul: f(xdx I T rp = h [f(x 0 + f(x 1 + f(x f(x n 1 + f(x n ] = [ ] n 1 1 = h f(x 0 + f ( + 1 f(x n. Remrk: If on, the function f hs continuous second derivtive, then: f If M = mx x, (x, then i=1 I I T rp = 1 f (ξh, kde ξ (,. I I Lich 1 M h. Newton-Cotes qudrture formul - Simpson s rule On, + the function f will e sustituted y qudrtic polynomil L (xgoing through the points [, f( ], [+1, f(+1 ] nd [+, f(+ ]. We get the qudrture formul of order. xk+ f(xdx xk+ L (xdx = (+ 6 (f ( + 4f (+1 + f (+ = = h 3 (f ( + 4f (+1 + (+ = I Simp. The geometricl interprettion is on the Fig

5 f(x f( + f( f( +1 f(x +1 + x Fig. 7.3 Simpson s rule. The composed formul for n even: f(xdx I Simp = h 3 [f(x 0 + 4f(x 1 + f(x + 4f(x 3 + f(x f(x n + 4f(x n 1 + f(x n ]. Remrk: If on, the function f hs continuous forth derivtive, then: If M 4 = mx x, f (4 (x, then I I Simps = 90 f (4 (ξh 4, where ξ (,. I I Simps 90 M 4h 4. Exmple 1: Evlute the pproximte vlue 1 1 ex dx for n = 4 using the foregoing rules, estimte the error nd compre it to the exct vlue. h = 1 ( 1 4 = 0, 5 x 0 = 1; x 1 = 1 + 0, 5 = 0, 5; x = 0; x 3 = 0, 5; x 4 = 1. f(x = f (1 (x = f ( (x = f (4 (x = e x, e x < 3, x 1, 1 Exct vlue: (1 Rectngle rule: I = 1 1 e x dx = [e x ] 1 1 = e1 e 1 =, ] I Rec = 0, 5 [e 1+( 0,5 + e 0,5+0 + e 0+0,5 + e 0,5+1 =, Error estimtion: I I Od 4 M h 4 3 0, 5 = 0, 065 I Rec I = 0,

6 ( Trpezoid rule: I T rp = 0, 5 [ e 1 + e 0,5 + e 0 + e 0,5 + e 1] =, Error estimtion: I I Lich 1 M h 1 3 0, 5 = 0, 15 I T rp I = 0, (3 Simpson s rule: I Simps = 0, 5 3 [ e 1 + 4e 0,5 + e 0 + 4e 0,5 + e 1] =, Error estimtion: I I Simps 90 M 4h , 54 = 0, 0083 I Simps I = 0, MATLAB trpz(@ fce,, - integrl y trpezoid rule of fce on,, qud(@ fce,, - integr l y Simpson s rule of fce on,, dlqud(@ fce,xmin,xmx,ymin,ymx - dvojný integrál, triplequd(@ fce,xmin,xmx,ymin,ymx,zmin,zmx - trojný integrál. Evlute 0 x dx We cn insert function using the commnd inline: >> F = inline( (x. >> Q = qud(f,0,; We cn define function: >> Q = qud(@ mojefunkce,0,; where mojefunkce.m is n M-file: function y = mojefunkce(x y = x. ; Symolic toolox: >> syms x >> int(x,0, ns = 4 For finding the ntiderivtive for function use int with one prmeter: int(expression. >> syms x >> int(x -5 ns = 1/3*x 3-5*x 6