This document discusses several numerical computing techniques: 1) Gauss elimination for solving systems of linear equations. 2) LU decomposition to solve Ax=b. 3) Gram-Schmidt process for orthogonalizing vectors. 4) QR decomposition to decompose a matrix as a product of an orthogonal and upper triangular matrix. 5) Cholesky decomposition to decompose a symmetric positive-definite matrix into the product of a lower triangular matrix and its transpose. 6) Shooting method using secant method to solve initial value problems.

1. SCIENTIFIC COMPUTING
Programming language used – MATLAB
Saurabh S. Ramteke

2. Contents
Gauss Elimination......................................................................................................................... 2
LU decomp...................................................................................................................................3
Gram Schmidt.............................................................................................................................. 4
QR............................................................................................................................................... 5
Choleski.......................................................................................................................................6
Shooting using secant................................................................................................................... 7

3. Gauss Elimination
%% Double for loop
clear all; clc;
%% Equatiions AX = b
A = [12 -2 3; 1 10 -2; 3 1 15];
b = [18 16 52];
%% Initilization
n = length(A);
%% Gauss
for i = 1 : (n-1)
for j = n:-1 : i+1
m = A(j,i) / A(i,i);
A(j,:) = A(j,:) - m * A(i,:);
b(j) = b(j) - m * b(i);
end
end
%% Back - Substitution
x = zeros(n,1);
x(n) = b(n)/A(n,n);
for j = n-1:-1:1
sum = 0;
for i = n:-1:j+1
sum = sum + A(j,i)*x(i);
end
x(j) = (b(j)- sum)/A(j,j);
end

4. LU decomp
clc;
clear all;
%% Initialization
A = [1 2 3;
4 5 6;
7 8 9];
b = [1 2 3]';
%% Computation
[L,U] = lu(A);
% Ax = b => LUx=b => Ux = z
z = Lb;
x = Uz;

5. Gram Schmidt
V = [1 1 1 1;
1 2 4 5;
1 -3 -4 -2];
n = 3;
sum = 0;
for i = 1: n
A = V(:,i);
for j = 1: n
d = V(:,j) .* V(:,j);
n = V(:,j) .* A;
sum = sum + (n' / d') .* V(:,j);
end
B = A - sum;
q = norm(B);
q1 = B/q;
Q(:,i) = q1;
end

6. QR
A = [1 2 0;
0 1 1;
1 0 1];
n = 3;
for j =1 : n
v = A(:,j);
for i=1 : j-1
R(i,j) = Q(:,i)' * A(:,j);
v = v - R(i,j)*Q(:,i);
end
R(j,j) = norm(v);
Q(:,j) = v/R(j,j);
end

7. Choleski
clc;
clear all;
%% Initialization
A = [4 -1 -1;
-1 4 -3;
-1 -3 5];
b = [3 -0.5 0]';
%% Computation -
U = chol(A); % upper traiagular matrix
z = U'b; % U(U'X) = b => U'z = b
x = Uz; % Ux = z

8. Shooting using secant
clc;
clear all;
%% Initiliazation
h = 1/3;
a0 = 1.2; a1 = 1.5;
y_soln = 5;
% fn - y'' = 6y^2 - x
%% IVP 1
ys1 = 1;
zs1 = 1.2;
x1 = 0;
for i = 0:2
y1 = ys1 + h*zs1;
z1 = zs1 + h*(6*(ys1^2) - x1);
ys1 = y1;
zs1 = z1;
x1 = x1 + h;
end
%% IVP 2
ys2 = 1;
zs2 = 1.5;
x2 = 0;
for i = 0:2
y2 = ys2 + h*zs2;
z2 = zs2 + h*(6*(ys2^2) - x2);
ys2 = y2;
zs2 = z2;
x2 = x2 + h;
end
%% ph
phi_a0 = abs( y_soln - y1) ;
phi_a1 = abs( y_soln - y2) ;
%% a2
a2 = a1 - (a1 - a0)*phi_a1/(phi_a1 - phi_a0);