function [istatus,X,eta,iB,iN,xB] = simplex_method(A,b,c,irule)
%
% Find a basic optimal solution for the linear program
%
%     min:    c*x
%     ST:     Ax=b
%             x>=0,
%
% where A is an (m,n) matrix.
%
%                     Input Parameters:
%
% A - (n,m) constraint matrix
% b - (m,1) POSITIVE vector appearing in the constraint equation above
% c - (1,n) vector giving the coefficients of the objective function
%  
% irule - integer parameter speciying which pivot rule to use:
%     irule = 0 indicates that the smallest coefficient rule should be 
%        used
%     irule = 1 indicates that Bland's rule should be used
% 
%                     Output Parameters:
%
% istatus - integer parameter reporting the condition of the 
%    istatus = 0  indicates normal completeion (i.e., a solution
%       has been found and reported)
%    istatus = 4  indicates the program is infeasible
%    istatus = 16 indicates the program is feasible but our initialization
%       procedure has failed
%    istatus = 32 indicates that the program is unbounded
%
% X  - vector of length n specifying the solution
% eta - the minimum value of the objective function 
% iB - integer vector specifying the m indices of the basic variables
%      after the simplex step
% iN - integer vector specifying the n-m indices of the nonbasic 
%      variables after the simplex step
% xB - vector of length m specifying the values of the basic
%      variables after the simplex step
%


function [istatus,iB,iN,xB] = simplex_step(A,b,c,iB,iN,xB,irule)
%
% Take a single simplex method step for the linear program
%
%     min:    c*x
%     ST:     Ax=b
%             x>=0,
%
% where A is an (m,n) matrix.
%
% That is, given a basic feasible vector vector described by the variables
% iB,iN,XB, 
%   
%                     Input Parameters:
%
% A - (n,m) constraint matrix
% b - (m,1) POSITIVE vector appearing in the constraint equation above
% c - (1,n) vector giving the coefficients of the objective function
% 
% iB - (1,m) integer vector specifying the indices of the basic
%      variables at the beginning of the simplex step
% iN - (1,n-m) integer vector specying the indices of the nonbasic
%      variables at the beginning of the simplex step
% xB - (m,1) vector specifying the values of the basic 
%      variables at the beginning of the simplex step
% 
% irule - integer parameter speciying which pivot rule to use:
%     irule = 0 indicates that the smallest coefficient rule should be 
%        used
%     irule = 1 indicates that Bland's rule should be used
% 
%                     Output Parameters:
%
% istatus - integer parameter reporting the condition of the 
%    istatus = 0  indicates normal simplex method step 
%      completed
%    istatus = 16 indicates the program is unbounded
%    istatus = -1 indicates an optimal feasible vector has been
%      found
%
% iB - integer vector specifying the m indices of the basic variables
%      after the simplex step
% iN - integer vector specifying the n-m indices of the nonbasic 
%      variables after the simplex step
% xB - vector of length m specifying the values of the basic
%      variables after the simplex step
%


function [istatus,iB,iN,xB] = simplex_init(A,b,c)
%
% Attempt to find a basic feasible vector for the linear program
%
%     max:    c*x
%     ST:     Ax=b
%             x>=0,
%
% where A is a (m,n) matrix.
%
%                     Input Parameters:
%
% A - (n,m) constraint matrix
% b - (m,1) vector appearing in the constraint equation above
% c - (1,n) vector giving the coefficients of the objective function
% 
%  
%                     Output Parameters:
%
% istatus - integer parameter reporting the condition of the 
%    istatus = 0  indicates a basic feasible vector was found
%    istatus = 4  indicates that the initialization procedure failed
%    istatus = 16  indicates that the problem is infeasible
%
% iB - integer vector of length m specifying the indices of the basic
%      variables 
% iN - integer vector of length n-m specying the indices of the nonbasic
%      variables 
% xB - vector of length m specifying the values of the basic 
%      variables 
%