MTH643 Final Term Lectures 8-15 by Yasir

Mth-643 Final term lect8to15(yasir)

Matrix Algebra:

clear all clc Rowvector=[1,2,3] Columnvector=[1;2;3] threeby3matrix=[1,2,3;3,4,5;7,6,8] transpose(threeby3matrix)

%Built in Functions

A=pascal(3) %symmetric matix A^T=A B=magic(3) %antisymmetric F=ones(3) %matrix of ones G=eye(3) %Identiy matrix H=zeros(2) % Zero matrix

%diagonal matrix from a vector

v=[2,3,4]; D=diag(v) E=diag(A) % diagonal enteries elehda sy vector mn store

%add scalar quantity to each element of

vector

J=[2,4,5;6,7,1;1,1,1]; K=3+J

%Matrix Multiplication 1st n=2nd m

syms x y T=[1,2;7,5]; X=[x;y] Z=T*X

Matrix Algebra:

%Matrix Multiplication using for Loop

matrix1=[1;2;3]; matrix2=[1,4,5]; m=3; n=3; for i=1:m for j=1:n c(i,j)=matrix1(i,:)*matrix2(:,j)

%Matrix Multiplication Simple method

A=[1;2;3]; %follow multiplication rule B=[1,2,3]; C=A*B

% Multiply Individual Entries

A=[1,2,3];B=[4,3,2]; C=A.*B

%divide Idividual entries

A=[2,4,6]; B=[6,8,6]; C=A./B %A oper B nechy D=A.\B %B oper A nechy

%Eigen Values

A=[1,2,3;2,3,5;4,3,2]; eig(A); [V,D]=eig(A)

%Reduced Row Echelon Form

A=[1,2,3;1,3,4;2,5,7]; B=rref(A)

System of Linear Equations:

%x+2y+4z=1,%3x+2y+4z=2,%2x+4y+5z=5

A=[1,2,4;3,2,4;2,4,5]; X=[x;y;z]; B=[1;2;5]; X=inv(A)*B; %1st method X=vpa(A\B); %2nd method X=mldivide(A,B); %3rd method eqn1='x+2*y+4*z=1'; %4th method eqn2='3*x+2*y+4*z=2'; eqn3='2*x+4*y+5*z=5'; sol=solve(eqn1,eqn2,eqn3) X=double([sol.x,sol.y,sol.z])'

Matrix Equation of Sylvester Type:

AX+BX=C

A=[1,2,4;5,6,7;2,3,5]; B=[1,3,4;5,2,3;1,4,6]; C=[1,3,9;3,6,8;2,3,5]; X=sylvester(A,B,C)

For Loop: Example(1)

%we want to add each element of vector v=1:10; sum=0; for i=1:length(v) sum=sum+i

fprintf('sum of all enteries of vector= %d\n',sum)

For Loop: Example(2)

%we want to add 1st 30 odd numbers v=1:60; sum=0; for i=1:2:length(v) sum=sum+i

fprintf('sum of 1st 30 odd enteries of vector= %d\n',sum)

For Loop: Example(3)

%we want to add 1st 30 even numbers v=1:60; sum=0; for i=1:length(v) if(rem(i,2)==0) sum=sum+i;

fprintf('sum of 1st 30 even enteries of vector= %d\n',sum)

For Loop: Example(4)

%Calculate factorial num=input(' Enter a number= '); fact=1; while num<0 disp(' Enter a positive Number') num=input(' Enter a number= ');

if num>0 for i=1:num fact=fact*i; fprintf(' The factorial of %d is= %d\n',num,fact)

Nested For Loop: Example(5)

%print a patteren rows=input(' How many rows do you want to print: '); for R=1:rows for S=1:R fprintf('*')

fprintf('\n')

Differentiation and Integration:

%y=x^3

f=x^3; df=diff(f,x,2); %differentiation i=int(f,0,2); %Integration %sin(2x) g=sin(2*x); dg=diff(g,x); integ=int(g,x,0,pi/2); %x^2y^3z^2 h=x^2*y^3*z^2; dh=diff(h,z); Ih=int(int(int(h,0,pi),0,pi/2),0,1)

Solution Of ODES: Example(1)

%dy/dt=ty %y(0)=1 syms y(t) ode=diff(y,t)==t*y; cond=y(0)==1; soly(t)=dsolve(ode,cond)

Solution Of ODES: Example(2)

%dy/dt+4y(t)=exp(-t) %y(0)=1 syms y(t) ode=diff(y,t)+4*y(t)== exp(-t); cond=y(0)==1; soly(t)=dsolve(ode,cond)

Solution Of ODES: Example(3)

%d^2y/dt^2=cos(2t)-y %y(0)=1 %y'(0)=0 syms y(t) Dy=diff(y); ode=diff(y,t,2)==cos(2*t)-y; cond1=y(0)==1; cond2=Dy(0)==0; cond=[cond1 cond2]; soly(t)=dsolve(ode,cond) simplesol=simplify(soly(t))

Solution Of ODES: Example(4)

%d^3u/dx^3=u %u(0)=1 %u'(0)=-1 %u''(0)=pi syms u(x) Du=diff(u); D2u=diff(u,2); ode=diff(u,x,3)==u; cond1=u(0)==1; cond2=Du(0)==-1; cond3=D2u(0)==pi; cond=[cond1 cond2 cond3]; solu(x)=dsolve(ode,cond) simplesolu(x)=simplify(solu(x))

System Of ODES: Example(1)

%du/dt=3u+4v %dv/dt=-4u+3v syms u(t) v(t) ode1=diff(u,t)==3*u+4*v; ode2=diff(v,t)==-4*u+3*v; ode=[ode1 ode2]; sol=dsolve(ode); sol.u; sol.v; %OR [usol vsol]=dsolve(ode)

Ode45:Example(1)

%F=3exp(-t) %tspan=[0,5] %x0=0 %F=3exp(-t) %tspan=[0,5] %y0=0 syms t

tspan=[0,5]; f0=0; [t,F]=ode45(F,tspan,f0) plot(t,F)

Ode45:Example(2)

%y'=2t %tspan=[0,5] %y0=0

tspan=[0,5]; y0=0; [t y]=ode45(yprime,tspan,y0) plot(t,y) xlabel('t-axis');

title(' Graph of yprime=2t')

Ode45:Example(3)

%u'=2u+v+5w+exp(-2t) %v'=-3u-2v-8w+2exp(-2t)-cos(3t) %w'=3u+3v+2w+cos(3t) %tspan=[0,5] %u(0)=1 %v(0)=-1 %w(0)=0 clear all clc close all syms u v w t uprime=2*u+v+5*w+exp(-2*t); vprime=-3*u-2*v-8*w+2*exp(-2*t)-cos(3*t); wprime=3*u+3*v+2*w+cos(3*t); tspan=[0,5]; u0=1; v0=-1; w0=0; cond=[u0,v0,w0];

3*x(1)-2*x(2)-8*x(3)+2*exp(-2*t)- cos(3*t);3*x(1)+3*x(2)+2*x(3)+cos(3*t)] [t,x]=ode45(ode,tspan,cond) plot(t,x) xlabel('X-axis')

title(' Plot of Solution Curves')

Solution of differential equation using · numerical method: · Euler’s Method:

%Euler Method %y(x+h)=y(x)+hy'(x) %y(x+h)=y(x)+hf(x,y) %f(x,y)=x/y %Euler Method %y(x+h)=y(x)+hf(x,y) %f(x,y)=x/y %a=0 %b=2 %y0=1 %x0=0 %n=10 clear all clc

a=0; b=2; y0=1; x0=0; n=10; h=(b-a)/n; x(1)=x0; y(1)=y0; for j=1:n y(j+1)=y(j)+h*f(x(j),y(j)); x(j+1)=x(j)+h;

plot(x,y) hold on syms Y(X) ode=diff(Y,X)==X/Y; cond=Y(0)==1; S=dsolve(ode,cond) F=matlabFunction(S); fplot(F,[0,2],'r',2) hold off

Trapezoidal Rule:

𝑏

𝑎 ∆𝑥 𝑛−1

𝑗=1 %Trapezoidal Rule %find area of curve x^2 %a=0,%b=2, %n=3

a=0; b=2; n=3; h=(b-a)/n; sum=0; x(1)=1; for i=1:n-1 x(i)=a+i*h; y(i)=f(x(i)); sum=sum+y(i);

Area=h/2*[f(a)+f(b)+2*sum] %find area of curve exp(x^2)

a=0; b=2; n=20; h=(b-a)/n; sum=0; x(1)=1; for i=1:n-1 x(i)=a+i*h; y(i)=f(x(i)); sum=sum+y(i);

Area=h/2*[f(a)+f(b)+2*sum]

Simpson’s 1/3 Rule:

𝑏

𝑎 ℎ 𝑛 2−1

𝑗=1 𝑛

𝑗=1 %Simpson1by3rule %y=x^3 %a=0, %b=2, %n=10, clear all clc

a=0; b=2; n=10; % n must even h=(b-a)/n; if (rem(n,2)~=0) fprintf('\n Enter Valid n'); n=input('\n Enter n as Even');

Simpson’s 3/8 Rule:

𝑏

𝑎 3ℎ 𝑛 3−1

𝑗=1

𝑛−1

%Simpson 3/8 rule %exp(2x)sin(6x) clear all clc

a=0; b=4; n=12; % n must be multiple of 3 if (rem(n,3)~=0)