%-------------------------------------------------------------
%Andrew Barnard
%MA 4610: Numerical Linear Algebra
%Homework #2: Row Reduction and Solving Linear Systems
%01/27/2003
%
%RowReduction solves a linear system A.x=b using Gauss
%elimination without pivoting.  The Matrix A and the vector
%b must be fed into the function.  The ouput arguments are as
%follows:
%           1. Matlab built in solver solution
%           2. My Algorithm (Gauss no pivot) solution
%           3. Percent difference between solutions
%
%[MatlabSol,MyAlgorithmSol,PercentDiff]=RowReduction(A,b)
%
%Also the time it takes to do these computations will be 
%output into the command window.  The first time is the 
%matlab solver, the second time is my algorithm time.
%-------------------------------------------------------------

format long g

%-------------------------------------------------------------
%Checks matrix-vector size matching and
%Solves the Linear System with Matlab's Built in Command
%and times the activity
%-------------------------------------------------------------

[r,c]=size(A);
[r1,c1]=size(b);
if c~=r1
    disp('Matrix Vector Multiplication Not of Correct Size')
    break
else
end  
tic;
MatlabSolution=A\b;
toc;
MatTime=toc;

%-------------------------------------------------------------
%My Algorithm for solving a linear system by Gauss Elimination
%without pivitong (solves system and time activity)
%-------------------------------------------------------------

tic;
N=[A,b];
for i=1:1:r
    N(i,:)=N(i,:)./N(i,i);
    for j=i+1:1:r
        N(j,:)=N(j,:)-N(j,i).*N(i,:);
end
end
for i=r:-1:2
    for j=i-1:-1:1
        N(j,:)=N(j,:)-N(j,i).*N(i,:);
    end
end
toc;
MyTime=toc;
MyAlgorithmSolution=N(:,r+1);

%------------------------------------------------------------
%Calculates Percent Differences between my solution and 
%Matlab's built in solution technique
%------------------------------------------------------------

PercentDifference=abs((MatlabSolution-MyAlgorithmSolution)./MatlabSolution).*100;




% Andrew Barnard
% MA 4610: Numerical Linear Algebra
% Homework #3
% Timing Linear System Solvers

clear 
clc
format short g

% creates random square matrices of size n and uses my 
% row reduction program to solve the system and time the 
% activity for both matlab and my solution technique

p=11;
n=1.8;
j=1;
for i=2:1:p
    A=randn(round(n^i));
    b=randn(round(n^i),1);
    [Mat,My,Diff,MatT,MyT]=RowReduction(A,b);
    MatTime(j,1)=MatT;
    MyTime(j,1)=MyT;
    Point(j,1)=n^i;
    j=j+1;
end

% displays the sizes of matrices computed and the time info

[round(Point),MyTime,MatTime]

% finds and fits a power function to the raw data

MySlope=polyfit(log10(Point(6:p-1,1)),log10(MyTime(6:p-1,1)),1)
MatSlope=polyfit(log10(Point(7:p-1,1)),log10(MatTime(7:p-1,1)),1)
N=[10:1:50000];
MyFit=10^MySlope(2)*N.^MySlope(1);
MatFit=10^MatSlope(2)*N.^MatSlope(1);

% plots the data for my solver, matlab solver, and fit functions

figure(1)
loglog(Point,MyTime,'o-',Point,MatTime,'o-',N,MyFit,':',N,MatFit,':')
title('Time to comupute solutions to a size n by n linear system')
xlabel('Size of system (n)')
ylabel('Time to compute solution (sec)')
legend('My Algorithm','Matlab Solution','My Fit','Matlab Fit')
axis([1,10e4,10e-3,10e2])

% solves the interpolated power function to find the size matrices
% that will take 2 minutes to solve using my solution method and 
% matlabs built in solver

MySize=(120/(10^MySlope(2)))^(1/MySlope(1))
MatSize=(30/(10^MatSlope(2)))^(1/MatSlope(1))

% uses my solver to solve the size matrix that should take 2 minutes

A=randn(round(MySize));
b=randn(round(MySize),1);
[Mat,My,Diff,MatT,MyT]=RowReduction(A,b);
MyTime2min=MyT

% uses matlat solver to solve the size matrix that should take 2 minutes

A=randn(round(MatSize));
b=randn(round(MatSize),1);
tic
MatSol=A\b;
toc
MatTime2Min=toc