Solutions Manual for Numerical and Analytical Methods with MATLAB 1st Edition by Bober

Chapters 2 – 14 Covered
ks ks ks ks




SOLUTIONS

, SOLUTION MANUAL ks




NUMERICALANDANALYTICALMETHODS WITH
ks ks ks ks




MATLAB

Table of Contents
ks ks




Page

Chapter 2
ks 1

Chapter 3
ks 46

Chapter 4
ks 58

Chapter 5
ks 98

Chapter 6
ks 107

Chapter 7
ks 176

Chapter 8
ks 180

Chapter 9
ks 188

Chapter 10
ks 214

Chapter 11
ks 271

Chapter 12
ks 303

Chapter 13
ks 309

Chapter 14
ks 339




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, CHAPTER 2 ks




P2.1. Taylor series expansion of
ks ks ks ks k s f (x) about x = 0 is:
ks ks k s ks ks ks ks




f 1V 4
f (x) = f (0) + f '(0) x+ f' '(0) x2 + f' ''(0) x3 + x +... ks ks ks ks ks ks ks ks


ks ks ks k s ks ks k s ks ks ks ks k s ks ks ks



2! 3! 4! ks ks




For f ( x) =cos(x) , ks ks ks ks ks ks ks f (0) = 1,
ks ks ks




f ( x) = − sin(x),
ks ks ks ks ks ks k s f '(0) =0,
ks ks ks ks




f ''( x) = −cos(x),
ks ks ks ks ks k s ks k s f ''(0)=−1,
ks ks ks ks ks ks




f '''(x) = +sin(x),
ks ks ks ks ks ks k s ks ks k s f '''(0) =0,
ks ks ks ks ks ks




f1V(x) = + cos(x),
ks ks ks ks ks ks ks k s f1V(0) =1
ks ks ks ks




We can see that ks ks ks




x2 x4 x6 8
cos( x) =1− − +x
k s ks




6! 8! − + − +...
ks ks ks k s k s




4! ks ks ks




+
k s

ks




2!

and that ks




x2
term (k) = −term (k −1) 
ks




2k(2 k −1)
ks ks ks ks ks ks ks ks



ks ks ks ks




The following program evaluates cos(x) by both an arithmetic statement and by t he above
ks ks ks k s ks ks ks ks ks ks ks ks ks ks




series for -π ≤ x ≤ π in step of 0.1  .
ks ks ks ks ks ks ks ks ks ks ks ks ks




% cosf.m
ks




% This program evaluates cos(x) by both arithmetic statement and by
ks ks ks ks ks ks ks ks ks ks




% series for
ks ks ks -π ≤ x ≤ π ks ks ks ks k s in steps of 0.1 ks ks ks ks π

clear; clc; ks




xi=-pi;dx=0.1*pi; s
k




ks for j=1:21ks




x(j)=xi+(j-1)*dx;

cos_arith(j)= cos(x(j)); ks




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, sum=1.0;term=1.0; s
k




ks for k=1:50
ks




den=2*k*(2*k-1);

term=-term*x(j)^2/den;

ks sum=sum+term;

ks test=abs(sum*1.0e-6);

ks if abs(term) <= test;
ks ks ks




break;

end

end

ks cos_ser(j)=sum;

ks nterms(j)=k;

end

ks fo=fopen('output.dat','w');

fprintf(fo,'x cos(x) cos (x) ks terms in ks k s \n');

ks fprintf(fo,' by arith stm
ks ks by series
ks the series
ks k s \n');

ks fprintf(fo,'===================================================== \n');

for j=1:21
ks




fprintf(fo,'%10.5f %10.5f %10.5f %3i\n',...
s
k




ks x(j),cos_arith(j),cos_ser(j),nterms(j));

fprintf(fo,' \n');

end

ks fclose(fo);

plot(x,cos_arith),xlabel('x'),ylabel('cos(x)'),

ks title('cos(x) vs. x'),grid; ks ks




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