04_linear_algebra

Homework #4

Code is known as HilbertHWP2.m The following are my answers:

ans 9.4366e+05

fnormH4 1.5097

fnormH5 1.5809

fnormiH4 1.0342e+04

fnormiH5 3.0416e+05

H4 4x4 double

H5 5x5 double

i 5

iH4 4x4 double

iH5 5x5 double

infnorm_H4 2.0833

infnorm_H5 2.2833

infnorm_iH4 140.0000

infnorm_iH5 630.0000

j 5

n1 4

n2 5

norm1_H4 2.0833

norm1_H5 2.2833

norm1_iH4 140.0000

norm1_iH5 630.0000

norm2_H4 1.5002

norm2_H5 1.5671

norm2_iH4 1.0341e+04

norm2_iH5 3.0414e+05

Problem #2

% HilbertHWP2 creates and modifies 4x4 and 5x5
% Hilbert matrix.  These modifications are the calculations of certain
% norms and their condition numbers of both the regular and inverse Hilbert
% matrices.
n1=4;
n2=5;
%% Hilbert matricies
H4=zeros(n1,n1); %Create first Hilbert matrix
for i=1:n1
    for j=1:n1
        H4(i,j)=1/(i+j-1);
    end
end

H5=zeros(n2,n2); %Create second Hilbert matrix
for i=1:n2
    for j=1:n2
        H5(i,j)=1/(i+j-1);
    end
end

% Calculate the Frobenius Norms and Their Conditions
for i=1:n1
    for j=1:n1
        fnormH4(i,j)=H4(i,j)^2;
    end
end
fnormH4=sqrt(sum(sum(fnormH4)));
cond(H4,'fro');

for i=1:n2
    for j=1:n2
        fnormH5(i,j)=H5(i,j)^2;
    end
end
fnormH5=sqrt(sum(sum(fnormH5)));
cond(H5,'fro');

% Calculate the 1-Norms and Their Conditions
norm1_H4=[];
for j=1:n1
    norm1_H4=max([norm1_H4,sum(H4(:,j))]);
end
cond(H4,1);

norm1_H5=[];
for j=1:n2
    norm1_H5=max([norm1_H5,sum(H5(:,j))]);
end
cond(H5,1);

% Compute 2-Norms and Conditions
norm2_H4=max(sqrt(eig(H4'*H4)));
cond(H4,2);

norm2_H5=max(sqrt(eig(H5'*H5)));
cond(H5,2);

% Compute Infinty Norms and Conditions
infnorm_H4=[];
for i=1:n1
    infnorm_H4=max([infnorm_H4,sum(H4(i,:))]);
end
cond(H4,inf);

infnorm_H5=[];
for i=1:n2
    infnorm_H5=max([infnorm_H5,sum(H5(i,:))]);
end
cond(H5,inf);



  
        
%% Part 2b
% Compute for Inverse Hilbert Matricies  

% Compute Frobenius Norms and Conditions
iH4=inv(H4);
iH5=inv(H5);
for i=1:n1
    for j=1:n1
        fnormiH4(i,j)=iH4(i,j)^2;
    end
end
fnormiH4=sqrt(sum(sum(fnormiH4)));
cond(iH4,'fro');

for i=1:n2
    for j=1:n2
        fnormiH5(i,j)=iH5(i,j)^2;
    end
end
fnormiH5=sqrt(sum(sum(fnormiH5)));
cond(iH5,'fro');

%% Compute 1-Norms and Conditions
norm1_iH4=[];
for j=1:n1
    norm1_iH4=max([norm1_iH4,sum(iH4(:,j))]);
end
cond(iH4,1);

norm1_iH5=[];
for j=1:n2
    norm1_iH5=max([norm1_iH5,sum(iH5(:,j))]);
end
cond(iH5,1);

%% Compute the 2-Norms and Their Conditions
norm2_iH4=max(sqrt(eig(iH4'*iH4)));
cond(iH4,2);

norm2_iH5=max(sqrt(eig(iH5'*iH5)));
cond(iH5,2);

%% Compute Infinty Norms and Conditions
infnorm_iH4=[];
for i=1:n1
    infnorm_iH4=max([infnorm_iH4,sum(iH4(i,:))]);
end
cond(iH4,inf);

infnorm_iH5=[];
for i=1:n2
    infnorm_iH5=max([infnorm_iH5,sum(iH5(i,:))]);
end
cond(iH5,inf);







%{
fprintf('Frobenius norm of 4x4 Hilbert matrix = %.3f\nFrobenius norm of 4x4 Hilbert matrix = %.3f\n',fnormH4, fnormH5)
%}

Problem #3

function [d,u]=chol_tridiag(e,f);
%This function chol.tridiag.m calculates the Cholevsky factorization of a
%tri-diagonal matrix
A=diag(e)+diag(f,1)+diag(f,-1); %initialize matrix to be factorized
[m,n]=size(A);
C=zeros(m,n); %create Cholesky matrix
%do Cholesky factorization
C(1,1)=sqrt(A(1,1));
for j=2:n
    C(1,j)=A(1,j)/C(1,1);
end
for i=2:n
    C(i,i)=sqrt(A(i,i)-sum(C(:,i).^2));
    for j=i+1:n
        C(i,j)=(A(i,j)-sum(C(:,i).*C(:,j)))/(C(i,i));
    end
end
d=diag(C)
u=diag(C,1)
end

Problem #4

function x = solve_tridiag(d,u,b)
%This function solves Cx = b when given inputs of the diagonal (d) and the 
%1 offset from upper diagonal (u) of the Cholesky matrix

d=diag(C); %change the diagonal of the Cholesky matrix to vector form
u=diag(C,1); %put the 1 upper diagonal of the Cholesky matrix into vector form

n=length(d);
b=ones(n,1); %create b matrix of Cx=b
v=zeros(n,1);   
x=zeros(n,1);

x(1)=b(1)/d(1);

for i=2:n  %start forward substitution
    v(i-1)=u(i-1)/d(1);
    d(1)=d(i);
    x(i)=(b(i))/d(1);
end
for j=n-1:-1:1 %start back substitution
   x(j)=x(j)-v(j)*x(j+1);
end
end

Problem #5

%stiffness.m calculates the stiffness matrix for four masses of pre-defined
%spring constants
%give values for spring constants
K1=1000;
K2=1000;
K3=1000;
K4=1000;

K=[K1+K2,-K2,0,0;-K2,K2+K3,-K3,0;0,-K3,K3+K4,-K4;0,0,-K4,K4]; %create stiffness matrix
cond(K) %compute the condition of the stiffness matrix

Results: 29.2841 is the condition of this matrix

Problem #6


%assign values for spring constants
K1=1000;
K2=1000;
K3=1000;
K4=1000;

%assign values for masses
m1=1;
m2=1;
m3=1;
m4=1;

g=9.81;
e=[K1+K2,K2+K3,K3+K4,K4];
f=[-K2,-K3,-K4];
b=g*[m1;m2;m3;m4];
[d,u]=chol_tridiag(e,f);
x=solve_tridiag(d,u,b)

d =

44.7214 38.7298 36.5148 15.8114

u =

-22.3607 -25.8199 -27.3861

x

x =

Problem #7

function [freq, modes] = mass_spring_vibrate(K1,K2,K3,K4)
K1=10;
K2=20;
K3=20;
K4=10;

m1=1;  %input the masses
m2=2;
m3=4; 

M=[m1,0,0;0,m2,0;0,0,m3]; %create mass matrix

K=[K1+K2,-K2,0;-K2,K2+K3,-K3;0,-K3,K3+K4]; %create stiffness matrix

[V,D]=eig(K,M);  %calculate modes values and eigenvalues using eig command
for i=1:length(M)
    freq(i)=sqrt(D(i,i)); %take sqrt of eigenvalues to get frequencies
end
modes=V;
end

ans =

1.6028    3.7959    6.3657

Problem #7

%This computes different modes of a beam given a particular nummber of
%segments and finds both the minimum and maximum loads in this beam
P=10; %force in N
E=76*10^9; %elastic modulus in Pa
I=4*10^-8; %moment of inertia in m^4
L=5; %length in m

%psqrd=P/(E*I);
N=5; %# of segmentsw
dx=L/N; %step size
A=zeros(N-1);
u=2*ones(1,N-1);
v=ones(1,N-2);
A=A+diag(u)+diag(v,-1)+diag(v,1); %create the coefficient matrix

[V,D]=eig(A); %find the eigenvalues and eigenvectors

plot(V(:,end)) %plot mode shapes (end=mode 1, end-1=mode 2, etc.)

P=E*I*(diag(D))./(dx^2); %solve for load vector
max_load=max(P); %find maximum load
min_load=min(P); %find minimum load
number_of_eigenvalues=length(diag(D));

%if the segment length approached zero there would be infinite eigenvalues

of segments | largest load (N) | smallest load (N) | # of eigenvalues |

| ------------ | ---------------- | ----------------- | ---------------- | | 5 | 1.099e4 | 1.1612e3 | 4 | | 6 | 1.6337e4 | 1.1730e3 | 5 | | 10 | 4.7450e4 |1.1903e3 | 9 |

rjg09009/04_linear_algebra

About

Resources

Stars

Watchers

Forks

Releases

Contributors 3

Languages