2D1240 Numerical Methods II / André Jaun, NADA, KTH NADA

Transcription

1 A NADA D4 Numerical Methods II / André Jaun, NADA, KTH 8. PRECISION INTEGRATION, NON-LINEAR EQUATIONS 8.. Remember: what we saw during the last lesson Numerical error propagation, convergence and Richardson etrapolation Differentiation using nite differences Integration Newton-Cotes quadrature rules (mid-point, trapezoidal, Simpson) 8.. Overview: what you will learn today Integration of periodic s, cubic Hermite polynomials and splines Gauss & Gauss-Konrod quadrature Methods for non-linear polynomial equations, bisection, secant and Newton-Raphson

2 I=quad(inline('sqrt(.5+*ep(-).*sin(*.^))'),,,E-7); Recursion level limit in quad reached 48 times. Singularity likely. Warning: Function dened in a separate le fq.m I=[I I] = T (h) f (b) f (a) I + f (b) f (a) h h 4 ::: 7 8. More on Newton-Cotes quadrature (NAM 5., H 8.3.) Error estimates are obtained eperimentally by comparing successive approimations. Using the highest resolution estimates from the previous eample: err= abs( ) = e-9 err with the nal result R q + ep( )sin( )d = : ± 5 9. The mid-point and trapezoidal rules both integrate eactly linear polynomials. The Simpson rule integrates eactly cubic polynomials. In Matlab use recursive Newton-Cotes quadratures with quad() or quadl() I = Periodic s are best integrated with a trapezoidal rule without etrapolation. It can be shown that the periodicity cancels out the errors for all powers resulting in an asymtotic convergence that is more rapid than any power of h. Higher order rules can in principle be dened using Romberg s method. Runge oscillations however yield negative weights w i and large cancellation errors.

3 Z b Z Z Z d = ; w + w = NX Z 8.3 Gauss quadrature (NAM 5.3, H 8.3.3) Choose N = 4 coefcients ( ;w ); ( ;w ) for a quadrature f()d ß G w f( ) + w f( ) so as to integrate eactly polynomials up to N = 3rd order (cubic) I = w + w = d = ; + w = Z w d = 3 ; w 3 + w 3 = Z 3 d = : This requires solving a system N = 4 of non-linear equation; the solution for the nodes i coincide with the zeros of Legre polynomials and are generally tabulated: here = = p ; w 3, = ; leads to a quadrature A linear transformation can be used to map any interval [a; b] to [ ; ] I ß f( :57735) + f(:57735) = b a f(t)dt f b a + b + a ß b a d a b i + b + a w i f a i=

4 Z b f(t)dt ß G N b a NX b a Ig=gauss(N,F,a,b) Gauss quadrature of order N, integrand F, interval [a,b] N switch, g=[ ]; wg=[ ]; case b + a 3, g=[ ]; wg=[ ]; case 4, g=[ ]; wg=[ ]; case 5, g=[ ]; case ]; wg=[ , g=[ ]; case ]; wg=[ , g=[ ]; case ]; wg=[ out=nan; return otherwise g]; wg=[fliplr(wg(n:nh)) wg]; g=[-fliplr(g(n:nh)) abm=(a+b)/; tg=hh*g+abm; Change interval to [-;] hh=(b-a)/; g=[]; for k=:5, g=[g gauss(k,inline('ep(-.^)'),,)]; ; g = g Gauss quadrature using N nodes to integrate eactly polynomials of order N w i f(t i ); t i = i + With Matlab the nodes and weights can be tabulated as a i= nh=floor((n+)/); n=nh-floor(n/)+; Complete nodes and weights fg=feval(f,tg); Ig=hh*sum(wg.*fg); Result of Gauss quadrature Runge oscillations do not appear for large N because i are not evenly spaced: the weights w i remain all positive, preventing cancellation errors. Unfortunately previous evaluations cannot be re-used when increasing N! N +.

5 7X = K b a b + a b + a Gauss-Konrod uses 7 Gauss nodes and 8 Konrod nodes for two quadratures w i f(t i ); t i = 7 = b a G i + i= 7 v i f( i ) 5 = b a K i= 8X i= A ; i = b a K where 5 reuses the f(t values ) i G from 7 to integrate eactly polynomials only up to order (instead of 3 with 5 Gauss points), but provides an error estimate u i f(t i ) + ο i + d ; d = jk 5 G 7 j j 5 qd=jk Error < With Matlab divide the quadrature into 3 intervals.4. Gauss Konrod quadrature G 7 K 5 Gauss Konrod =[.8.5 ]; Ig=[]; Ik=[]; Ek=[]; for i=:length()-, =gauss(7,@fq,(i),(i+)); Ig=[Ig Igi]; Igi Ik=[Ik Iki]; [Iki,e]=g7k5(@fq,(i),(i+)); e]; Ek=[Ek G7=sum(Ig), K5= sum([ik;ek],) = G7 f()=[.5+ep( )sin( )]

6 [Igk,err]=g7k5(F,a,b); Gauss-Kronrod with 5-points, integrand F, interval [a,b] 7 Gauss points and weights ]; g=[ ]; wg=[ g]; wg=[fliplr(wg) wg]; g=[-fliplr(g) 8 Kronrod points and new weights ]; k=[ ]; u=[ ]; v=[ k]; u=[fliplr(u) u]; v=[fliplr(v) v]; k=[-fliplr(k) Quadrature in standard interval [-,] Error estimate err=*d*sqrt(d/abs(k5)); d=abs(k5-g7); Return approimation and error estimate err=ma([eps*abs(k5) err]); Igk=K5; - trapezoidal Simpson (Romberg) & G. G 7 K 5 In Matlab the Gauss-Konrod nodes and weights can be tabulated as hh=(b-a)/; abm=(a+b)/; tg=hh*g+abm; fg=feval(f,tg); G7 = hh*sum(wg.*fg); tk=hh*k+abm; fk=feval(f,tk); K5= hh*(sum(u.*fg)+sum(v.*fk)); Peer Teaching ( minutes to think, eplain to your neighbour and vote) Choosing a quadrature. Which method / rule should you use to integrate numerically with the highest precision possible a) A cubic (Hermite or Bézier) polynomial c) R ß ( sin )ecos d ß b) A square root d) + R ( + sin )e p 3 d

7 Z ß= Z ß= Z cos() p p = t p + p = p sin Z p ß= Z ß= Z 5 Z p ß= Z ß= d 4=X 3 =3 q + 6; Error < Tricks and other quadratures (NAM , H 8.4) Singularities: change variable, integrate analytically or use the lowest order possible cos(t ) = tdt = cos(t )dt t ψ p! p d sin p d + z } p ß analytically= z } behaved well Improper integrals: transform with =-logt, = t t or nd an upper bound for the tail d d d X q + 6 = Z X q Z q z } R X Error< d = 4 +

8 f=fqd(,y) f=sqrt(3+sin(3**y)+cos(3*y)).*ep(-(/pi).^); I=qd(Q); in D, Q the renement factor Quadrature =(a:d:b)'; d=(b-a)/n; c=.^/pi-+pi/4; F=[]; G=[]; Y=[]; i=:n+ for y=c(i):dy:d(i); dy=(d(i)-c(i))/ny; f=fqd((i),y); g=dy*(sum(f)-.5*(f()+f(ny+)));trapezoidal-y y]; F=[F; f]; G=[G; g]; Y=[Y; Double integrals are treated one dimension after the other Z Z f(; y)ddy = Z b d() "Z f(; y)dy # c() d D a z } g() n=*q; ny=5*q; a=; b=pi; Domain a,b Domain c() d=/pi+ep(-/pi)+ep(-(-).^); d() Trapezoidal- I=d*(sum(G)-.5*(G()+G(n+))); return I=qd(); I=qd(); I=[I I I+(I-I)/3] = Simple resolution I Double resolution Romberg / Simpson D f(,y) 3.5 y.5 For rectangular integration domains D, youcanalsousedblquad().

9 Problem: nd a solution of the non-linear f() = equation f() changes sign unless is a double root, e.g. ( ) = ffl g7=roots([ ])' = g7 8.5 Non-linear equations (NAM , H 5.5) ffl f () changes sign if is a double root, e.g. = Polynomial equations are solved using Matlab s command root() The zeros of P 7 () = ( )=6, the Legre polynomial used in the Gauss-Konrod G 7 K 5 quadrature, are Bisection method: divide the interval succesively by two =bisection(f,a,b,tol) fa=feval(f,a); abs(b-a)>tol while f=feval(f,); =a+(b-a)/; 8 Interval bisection 6 sign(fa)==sign(f) if fa=f; a=m; else fb=f; b=m; f()= 4sin() 4 3 return bisection(inline('.^-4*sin()'),.,.5,e-6) = ( iterations) ans Robust, but very Error < log slow with b a tol 4

10 Newton Raphson method: = n f( n) f ( n+ ) n =newton(f,fp,,tol) a=+*tol; Secant method: = n ( n n )f( n ) f( n+ ) f( n ) n =secant(f,a,b,tol) fb=feval(f,b); fa=feval(f,a); 8 Secant method 6 abs(b-a)>tol while f=feval(f,); =b-fb*(b-a)/(fb-fa); fa=fb; a=b; fb=f; b=; return secant(inline('.^-4*sin()'),.5,.5,e-6) = (6 iterations) ans Fast superlinear convergence, but requires a good initial guess. Doesn t f need (). f()= 4sin() Newton method abs(-a)>tol while a=; 8 6 fp=feval(fp,) f=feval(f,); =-f/fp; return newton(inline('.^-4*sin()'),... inline('*-4*cos()'),.3,e-6) f()= 4sin() 4 3 ans = Superfast quadratic convergence, but requires a good initial guess and f ()

11 = f( n) ( n ) + ff n + (ff n ) f (n ) f ( n ) + ::: f f ( n ) ( n ) + ::: f Convergence rate p dened with j n+ ffj ß Cj n ffj p Denote the solution = ff and epand with Taylor around n = f( n ) + (ff n )f ( n ) + (ff n) f(ff) f ( n ) + ::: Divide f by ( n ) 6= and substitute the solution f(ff) = n f( n) f ( n ) ff = (ff n) Newton: n+ which yields a quadratic convergence near a single root z } j n+ ffj ß Cj n ffj ; f ( n ) C lim = n! f (ff) = ( n ) f (ff) f The convergence gets down to linear near a double f root (ff) =.

12 =iter(f,,tol) a=+*tol; return iter(inline('*sqrt(sin())'),.5,e-6) Fied point iteration method: rewrite f() = inaform n+ = G( n ) f() = 4 sin = ) n+ = p sin n 3.5 Fied point iteration f()= abs(-a)>tol while a=; =feval(f,); =[sin()] /.5 f()=[sin()] / (5 iterations) Converges linearly if jg ()j <, but is of little practical interest. ans =