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Name: BAGARAGAZA Romuald

Student number: M2014028

Assagnment 4

Numerical Analysis

P203, EXERCISE SET 4.4, 2, f

QUESTION

Use the Composite Trapezoidal rule with the indicated values of n to

approximate the following integrals.

, n=8

Solution

clear

S=0;

a=1;

b=3;

n=32

h=(b-a)/n;

for i=1:n-1

x(i)=a+i*h;

end

S=f(a)+f(b);

for i=1:n/2-1

S=S+2*f(x(2*i))+4*f(x(2*i-1));

end

S=S+4*f(x(n-1));

S=h/3*S

TRUEVALUE=(log(13)-log(5))/2

S-TRUEVALUE

f.m

function ff=f(x)

ff=x/(x*x+4);

n= 4

S = 0.47772983114447

TRUEVALUE = 0.47775572251372

ans = -2.589136925290614e-005

n = 8

S = 0.47775464628479

TRUEVALUE = 0.47775572251372

ans = -1.076228928031942e-006

n = 16

S = 0.47775566285229

2

TRUEVALUE = 0.47775572251372

ans = -5.966143212798869e-008

n = 32

S = 0.47775571889878

TRUEVALUE = 0.47775572251372

ans = -3.614939236840087e-009

P281, EXERCISE SET 5.4 10 (b)

QUESTION

Use the Modified Euler method to approximate the solution to each of the

following initial-value problems, and compare the results to the actual

values.

Using the Runge-Kutta method of order four.

SOLUTION

a=2;

b=3;

N=2;

h=(b-a)/N;

t=a;

w(1)=1;

for i=1:N+1

t(i)=a+(i-1)*h;

end

for i=1:N

K1=h*f(t(i),w(i));

K2=h*f(t(i)+h/2,w(i)+K1/2);

K3=h*f(t(i)+h/2,w(i)+K2/2);

K4=h*f(t(i)+h,w(i)+K3);

w(i+1)=w(i)+(K1+2*K2+2*K3+K4)/6;

end

for i=1:N+1

ww(i)=t(i)+1/(1-t(i));

end

for i=1:N+1

3

error(i)=w(i)-ww(i);

end

w

ww

error

f.m

function ff=f(t,y)

ff=1+(t-y)^2;