function x=nma_BackSub(U,v)
%function x=nma_BackSub(U,v)
%
%Backward substitution that solves the lower triangular system
%Ux=v for x
%
%
%INPUT:
%  U:  an nxn U, upper tirangular square matrix
%  v:  vector of size n
%
%OUTPUT:
%  x: the solution to U*x=v

% By Nasser M. Abbasi
% HW 4, MATH 501, CSUF
%
% Backsubstition matlab function.
% Copyright (C) 2007  Nasser Abbasi
% 
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
% 
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
% 
% You should have received a copy of the GNU General Public License
% along with this program; if not, write to the Free Software
% Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

    if nargin ~= 2
       error 'Only two inputs are required'
    end

    if ~(isnumeric(U)&isnumeric(v))
        error 'input must be numeric'
    end

    [nRow,nCol]=size(v);
    if nRow>1 & nCol>1
        error 'v must be a vector not a matrix'
    end
    
    [nRow,nCol]=size(U);
    if nRow ~= nCol
        error 'Matrix U must be square'
    end

    if length(v) ~= nRow
       error 'v length does not match U matrix dimension'        
    end
    
    x=zeros(nRow,1);
    x(nRow)=v(nRow)/U(nRow,nRow);
    
    v=v(:);
    
    for n=(nRow-1):-1:1
        x(n)=(v(n)-(U(n,n+1:end)*x(n+1:end))) / U(n,n);
    end
   
end