linsolve linear equation solver

## Syntax

    [x0,kerA]=linsolve(A,b [,x0])

## Arguments

A

  a `na x ma` real matrix (possibly sparse)

b

  a `na x 1` vector (same row dimension as `A`)

x0

  a real vector

kerA

  a `ma x k` real matrix

## Description

`linsolve` computes all the solutions to `A*x+b=0`.

`x0` is a particular solution (if any) and `kerA=` nullspace of `A`. Any `x=x0+kerA*w` with arbitrary `w` satisfies `A*x+b=0`.

If compatible `x0` is given on entry, `x0` is returned. If not a compatible `x0`, if any, is returned.

## Examples

    A=rand(5,3)*rand(3,8);
    b=A*ones(8,1);[x,kerA]=linsolve(A,b);A*x+b   //compatible b
    b=ones(5,1);[x,kerA]=linsolve(A,b);A*x+b   //uncompatible b
    A=rand(5,5);[x,kerA]=linsolve(A,b), -inv(A)*b  //x is unique

    // Benchmark with other linear sparse solver:
    [A,descr,ref,mtype] = ReadHBSparse(SCI+"/modules/umfpack/demos/bcsstk24.rsa");

    b = zeros(size(A,1),1);

    tic();
    res = umfpack(A,'\',b);
    mprintf('\ntime needed to solve the system with umfpack: %.3f\n',toc());

    tic();
    res = linsolve(A,b);
    mprintf('\ntime needed to solve the system with linsolve: %.3f\n',toc());

    tic();
    res = A\b;
    mprintf('\ntime needed to solve the system with the backslash operator: %.3f\n',toc());

## See also

- inv
- pinv
- colcomp
- im_inv
- umfpack
- backslash