Solutions Manual for Numerical and Analytical Methods with MATLAB (1st Edition by Bober) – Complete Worked Solutions

Chapters 2 – 14 Covered
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SOLUTIONS

, SOLUTIONMANUAL v




NUMERICALANDANALYTICALMETHODSWITH
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MATLAB

Table of Contents
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Page

Chapter 2 v 1

Chapter 3 v 46

Chapter 4 v 58

Chapter 5 v 98

Chapter 6 v 107

Chapter 7 v 176

Chapter 8 v 180

Chapter 9 v 188

Chapter 10 v 214

Chapter 11 v 271

Chapter 12 v 303

Chapter 13 v 309

Chapter 14 v 339




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, CHAPTER 2 v




P2.1. Taylor series expansion of f (x) about x = 0 is:
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f''(0) 2 f'''(0) 3 f 1V 4
f (x) = f (0)+ f '(0) x+ x + x + x +...
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v v v v v v v v v v v v v



4!
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2! 3! v v




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




f (x) =− sin(x), f '(0)=0,
v v v v v v v v v v v




f ''(x) = −cos(x), f ''(0)=−1,
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f '''(x) = +sin(x), f '''(0)=0,
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f1V(x)=+cos(x), f 1V(0)=1
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We can see that v v v




x x 4 x6 8 2
cos(x) =1− + − + x −+−+...
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v v v v v




2! 4! 6! 8! v v v



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and that v




x2
term(k)=−term(k −1)
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2k (2k −1)
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The following program evaluates cos(x) by both an arithmetic statement and by the above
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v series for -π ≤ x ≤ π in step of 0.1  .
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% cosf.m
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% This program evaluates cos(x) by both arithmetic statement and by
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% series for -π ≤ x ≤ π in steps of 0.1 π
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clear; clc; v




xi=-pi;dx=0.1*pi; v




v for j=1:21
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x(j)=xi+(j-1)*dx;

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




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, sum=1.0;term=1.0; v




for k=1:50
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den=2*k*(2*k-1);

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

sum=sum+term;
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test=abs(sum*1.0e-6);
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if abs(term) <= test;
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break;

end

end

cos_ser(j)=sum;
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nterms(j)=k;
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end

fo=fopen('output.dat','w');
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fprintf(fo,'x cos(x) cos (x) v terms inv v \n');

fprintf(fo,'
v by arith stm
v v by series
v the series
v v \n');

fprintf(fo,'=====================================================\n');
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for j=1:21
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fprintf(fo,'%10.5f %10.5f %10.5f %3i\n',...
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x(j),cos_arith(j),cos_ser(j),nterms(j));
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fprintf(fo,' \n');

end

fclose(fo);
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plot(x,cos_arith),xlabel('x'),ylabel('cos(x)'),

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