% Class 14 
% Systems of Linear Equations
% Coded by NFR on 2.28.19
%
% Warm up exercise
clc
close all
F1 = @(x,y) 4*exp(-(0.5*(x-2.7).^2+2*0*(x-2.7).*(y-3.8)+0.5*(y-3.8).^2));
F2 = @(x,y) 4*exp(-(x - 27/10).^2/2 - (y - 19/5).^2/2);
xvec = linspace(0,5,100);
yvec = linspace(0,5,100);
[xmat, ymat] = meshgrid(xvec,yvec);
zmat = F2(xmat,ymat);
contourf(xvec,yvec,zmat)
% Now to solve for the max
F3 = @(V) -F2(V(1), V(2));
A = fminsearch(F3,[3 3]);
max_xlocation = A(1)
max_ylocation = A(2)
%
% System of Equation example (8.3 from book)
% We rearranged on the board to have a 
% coefficient matrix (A) and solution right hand
% vector (b)
A = [0 -6 5; 0 2 7; -4 3 -7];
b = [50; -30; 50];
tic
xsol = inv(A)*b
toc
% Matlab has a highly sophisticated algorith for
% solving systems of linear equations
% left hand division '\'
tic
xsol2 = A\b
toc
% If you want to determine if this is ill-conditioned
% You can find the determinant
D = det(A)
% Demo Cramer's rule (another use of determinant)
A = [10 2 -1; -3 -5 2; 1 1 6];
b = [27; -61.5; -21.5]; %NOTE this must be a column vector
D = det(A); % solve for determinant
% Solve for x1
Anew = [b A(:,2:3)];
x1 = det(Anew)/D
% Solve for x2
Anew = [A(:,1) b A(:,3)];
x2 = det(Anew)/D
% Solve for x3
Anew = [A(:,1:2) b];
x3 = det(Anew)/D
% Double check answer
AnsChec = A\b
% Demo of Gauss Elimination
S1 = GaussNaive(A,b)
S2 = GaussPivot(A,b)
% Example of sparse matrix solution
% From book, heat rod
e = [0 -1 -1 -1];
f = ones(1,4)*2.04;
g = [-1 -1 -1 0];
r = [40.8 0.8 0.8 200.8];
Tvec = Tridiag(e,f,g,r)
Tsol = [40 Tvec 200];
xvec = linspace(0,10,6);
plot(xvec,Tsol)
xlabel('x (m)')
ylabel('Temp (deg C)')
% 
% Example of using LU decomposition to solve
% system of linear equations
% We'll use the problem 9.6 from above
[L,U] = lu(A);
d = L\b;
x = U\d
% Distillation Column Example
% Overall mass balance
A = [0.07 0.18 0.15 0.24;...
    0.04 0.24 0.10 0.65;...
    0.54 0.42 0.54 0.1;...
    0.35 0.16 0.21 0.01];
b = [0.15*70 0.25*70 0.4*70 0.2*70]';
Sol = A\b