Page 78 - Applied Numerical Methods Using MATLAB

PROBLEMS 67
of the difference between the two results? How do you compare
the running times of the two methods?
(cf) Note that Jkb(K,beta) computes J k (β) of order k = 1:K, while the inte-
gration does for only k = K.

function [J,JJ] = Jkb(K,beta) %the 1st kind of kth-order Bessel ftn
tmpk = ones(size(beta));
for k = 0:K
tmp = tmpk; JJ(k + 1,:) = tmp;
for m = 1:100
tmp = ?????????????????????;
JJ(k + 1,:) = JJ(k + 1,:)+ tmp;
if norm(tmp)<.001, break; end
end
tmpk = tmpk.*beta/2/(k + 1);
end
J = JJ(K+1,:);
%nm1p21b: Bessel_ftn
clear, clf
beta = 0:.05:15; K = 15;
tic
for i = 1:length(beta) %Integration
J151(i) = quad(’Bessel_integrand’,0,pi,[],0,beta(i),K)/pi;
end
toc
tic, J152 = Jkb(K,beta); toc %Recursive Computation
discrepancy = norm(J151-J152)

1.22 Find the four routines in Chapter 5 and 7, which are fabricated in a nested
(recursive calling) structure.
(cf) Don’t those algorithms, which are the souls of the routines, seem to have been
born to be in a nested structure?
1.23 Avoiding Runtime Error in Case of Deficient/Nonadmissible Input Argu-
ments
(a) Consider the MATLAB routine “rotation_r(x,M)”, which you made
in Problem 1.15(h). Does it work somehow when the user gives a
negative integer as the second input argument M ? If not, add a statement
so that it performs the rotation left by −M samples for M < 0, say,
making
rotate_r([1234 5],-2) = [34512]

(b) Consider the routine ‘trpzds(f,a,b,N)’ in Section 5.6, which com-
putes the integral of function f over [a, b] by dividing the integration
interval into N sections and applying the trapezoidal rule. If the user
tries to use it without the fourth input argument N, will it work? If not,
make it work with N = 1000 by default even without the fourth input
argument N.