Chapters 2 – 14 Covered
w w w w
SOLUTIONS
, SOLUTION MANUAL
w
NUMERICAL AND ANALYTICAL METHODS WITH
w w w w
MATLAB
Table of Contents
w w
Page
Chapter 2
w 1
Chapter 3
w 46
Chapter 4
w 58
Chapter 5
w 98
Chapter 6
w 107
Chapter 7
w 176
Chapter 8
w 180
Chapter 9
w 188
Chapter 10
w 214
Chapter 11
w 271
Chapter 12
w 303
Chapter 13
w 309
Chapter 14
w 339
, CHAPTER 2 w
P2.1. Taylor series expansion of f (x) about x = 0 is:
w w w w w w w w w w w w
f ' ' (0) 2 f ' ' ' (0) 3 f 1V 4
f (x) f (0) f ' (0) x x x x ...
w ww w w www w w
w w w w w w w w w w w w w w w
2! 3! 4!
For f (x) cos (x) ,w w w w w w w f (0) 1,
w w w
f (x) sin(x), f ' (0) 0,
w w w w w w w w w w w
f ' '( x) cos( x), f ' '(0) 1,
w ww w w w w w w ww w w w
f ' ' ' (x) sin(x), f ' ' ' (0) 0,
w www w w w w w w w www w w
f 1V (x) cos(x), f 1V (0) 1
w w w w w w w w w w w w
We can see that
w w w
x 2 x4 x6 8
cos(x) 1 x ...
w w w
w w w w w
2! 4! 6! 8!
w w w w
and that w
x2
term (k) term (k 1)
w
2 k (2 k 1)
w w w w w w w w
w w w w
The following program evaluates cos(x) by both an arithmetic statement and by the above
w w w w w w w w w w w w w w
series for -π ≤ x ≤ π in step of 0.1 .
w w w w w w w w w w w w w
% cosf.m
w
% This program evaluates cos(x) by both arithmetic statement and by
w w w w w w w w w w
% series for -π ≤ x ≤ π in steps of 0.1 π
w w w w w w w w w w w w
clear; clc; w
xi=-
pi;dx=0.1*pi; for j=1
w w w
:21
x(j)=xi+(j-1)*dx;
cos_arith(j)= cos(x(j)); w
, sum=1.0;term=1.0; fo w w
r k=1:50
w
den=2*k*(2*k-1);
term=-
term*x(j)^2/den; sum=su w
m+term; test=abs(sum*1.
w
0e-
6); if abs(term) <= test
w w w w
;
break;
end
end cos_ser(j)=su
w
m; nterms(j)=k;
w
end fo=fopen('output.dat','w'
w
);
fprintf(fo,'x cos(x) cos (x)
w
terms in \n'); fprintf(fo,'
w w w by arith stm
w w
by series
w
the series \n'); fprintf(fo,'=========================
w w w
============================\n');
for j=1:21
w
fprintf(fo,'%10.5f %10.5f %10.5f
%3i\n',... x(j),cos_arith(j),cos_se
w w
r(j),nterms(j));
fprintf(fo,' \n');
end fclose(fo
w
);
plot(x,cos_arith),xlabel('x'),ylabel('cos(x)'), title('cos(x w
) vs. x'),grid;
w w
w w w w
SOLUTIONS
, SOLUTION MANUAL
w
NUMERICAL AND ANALYTICAL METHODS WITH
w w w w
MATLAB
Table of Contents
w w
Page
Chapter 2
w 1
Chapter 3
w 46
Chapter 4
w 58
Chapter 5
w 98
Chapter 6
w 107
Chapter 7
w 176
Chapter 8
w 180
Chapter 9
w 188
Chapter 10
w 214
Chapter 11
w 271
Chapter 12
w 303
Chapter 13
w 309
Chapter 14
w 339
, CHAPTER 2 w
P2.1. Taylor series expansion of f (x) about x = 0 is:
w w w w w w w w w w w w
f ' ' (0) 2 f ' ' ' (0) 3 f 1V 4
f (x) f (0) f ' (0) x x x x ...
w ww w w www w w
w w w w w w w w w w w w w w w
2! 3! 4!
For f (x) cos (x) ,w w w w w w w f (0) 1,
w w w
f (x) sin(x), f ' (0) 0,
w w w w w w w w w w w
f ' '( x) cos( x), f ' '(0) 1,
w ww w w w w w w ww w w w
f ' ' ' (x) sin(x), f ' ' ' (0) 0,
w www w w w w w w w www w w
f 1V (x) cos(x), f 1V (0) 1
w w w w w w w w w w w w
We can see that
w w w
x 2 x4 x6 8
cos(x) 1 x ...
w w w
w w w w w
2! 4! 6! 8!
w w w w
and that w
x2
term (k) term (k 1)
w
2 k (2 k 1)
w w w w w w w w
w w w w
The following program evaluates cos(x) by both an arithmetic statement and by the above
w w w w w w w w w w w w w w
series for -π ≤ x ≤ π in step of 0.1 .
w w w w w w w w w w w w w
% cosf.m
w
% This program evaluates cos(x) by both arithmetic statement and by
w w w w w w w w w w
% series for -π ≤ x ≤ π in steps of 0.1 π
w w w w w w w w w w w w
clear; clc; w
xi=-
pi;dx=0.1*pi; for j=1
w w w
:21
x(j)=xi+(j-1)*dx;
cos_arith(j)= cos(x(j)); w
, sum=1.0;term=1.0; fo w w
r k=1:50
w
den=2*k*(2*k-1);
term=-
term*x(j)^2/den; sum=su w
m+term; test=abs(sum*1.
w
0e-
6); if abs(term) <= test
w w w w
;
break;
end
end cos_ser(j)=su
w
m; nterms(j)=k;
w
end fo=fopen('output.dat','w'
w
);
fprintf(fo,'x cos(x) cos (x)
w
terms in \n'); fprintf(fo,'
w w w by arith stm
w w
by series
w
the series \n'); fprintf(fo,'=========================
w w w
============================\n');
for j=1:21
w
fprintf(fo,'%10.5f %10.5f %10.5f
%3i\n',... x(j),cos_arith(j),cos_se
w w
r(j),nterms(j));
fprintf(fo,' \n');
end fclose(fo
w
);
plot(x,cos_arith),xlabel('x'),ylabel('cos(x)'), title('cos(x w
) vs. x'),grid;
w w
