% Class 14
% Coded by NFR on 10.10.19
% Systems of Linear Equations
%
% General Example of solution form
%
% 3x1 + 4x2 + 2x3 = 10
% 4x1 + 5x2 + 3x3 = 7
% 2x1 + 2x2 + 4x4 = 12
%
% Put coefficients into an a matrix
A = [3 4 2; 4 5 3; 2 2 4];
% Put the rhs solutions in a vector
b = [10; 7; 12];
% You can solve with inverse matrix
tic
xvec = inv(A)*b
toc
% Alternatively you can use numerical method
tic
xvec = A\b % <-- -where does this come from?
toc
% Example 8.3
A = [0 -6 5; 0 2 7;4 -3 7];
b = [50; -30; -50];
xvec2 = A\b
% To test if a system of eqns. is ill-defined
% look at the determinant of the A matrix
test = det(A)
% Example of using Crammer's method 
% for solving...
A = [10 2 -1; -3 -5 2; 1 1 6];
b = [27; -61.5; -21.5];
% Solve for x1
Amod = [b A(:,2:3)];
x1 = det(Amod)/det(A)
% Solve for x2
Amod = [A(:,1) b A(:,3)];
x2 = det(Amod)/det(A)
% Solve for x3
Amod = [A(:,1:2) b];
x3 = det(Amod)/det(A)
% Double check with \ command
xvec = A\b
% Use Gauss Elimination (w/ no pivot)
% to solve the system of equations from 
% cramer rule example
%
xvec = GaussNaive(A,b);
%
% Example of a sparse or banded matrix
% Heated rod (see printed work)
e = [1 1 1];
f = [-2.04 -2.04 -2.04 -2.04];
g = [ 1 1 1];
b = [-40.8 -.8 -.8 -200.8];
% Function for this is custom, tridiag
Tvec = tridiag(e,f,g,b)
% Plot the answer
Xvec = linspace(0,5,6);
Tvec_full = [40 Tvec' 200];
plot(Xvec,Tvec_full)
xlabel('X position (m)')
ylabel('Temperature (deg C)')