# umfpack

solve sparse linear system

## Calling Sequence

    x = umfpack(A,"\",b)
                x = umfpack(b,"/",A)

## Arguments

A

  a sparse (real or complex) square matrix n x n

b

  in the first case, a column vector (n x 1) or a n x m matrix ; in the second case, a row vector (1 x n) or a m x n matrix

x

  in the first case , a column vector (n x 1) or a n x m matrix ; in the second case, a row vector (1 x n) or a m x n matrix

2d arg

  string specifier "\" or "/"

## Description

This function is intended to work like the classic operators \ and / x = A\b and x = b/A) i.e. it solves a linear system Ax = b or xA = b with a sparse square (says n x n) real or complex matrix and with a compatible rhs b : n x m in the first case and m x n in the second.

## Details

First an LU factorization of the matrix is computed (`P R^(-1) A Q = LU` where P and Q are permutation matrices, R is a diagonal matrix (row scaling), L a lower triangular matrix with a diagonal of 1, and U an upper triangular matrix) then a first solution is computed with forward/backward subtitutions ; finaly the solution is improved by iterative refinement.

## Examples

    // this is the small linear test system from UMFPACK
    // whom solution must be [1;2;3;4;5]
    A = sparse( [ 2  3  0  0  0;
                  3  0  4  0  6;
                  0 -1 -3  2  0;
                  0  0  1  0  0;
                  0  4  2  0  1] );
    b = [8 ; 45; -3; 3; 19];
    x = umfpack(A,"\",b)

    // test the other form x A = b
    b = [8  20  13  6  17];
    x = umfpack(b,"/",A)   // solution must be [1 2 3 4 5]

    // test multiple rhs
    b = rand(5,3);
    x = umfpack(A,"\",b)
    norm(A*x - b)

    // test multiple rhs for x A = b
    b = rand(3,5);
    x = umfpack(b,"/",A)
    norm(x*A - b)

    // solve a complex system
    A = sparse( [ 2+%i  3+2*%i  0      0    0;
                  3-%i  0       4+%i   0    6-3*%i;
                  0    -1+%i   -3+6*%i 2-%i 0;
                  0     0       1-5*%i 0    0;
                  0     4       2-%i   0    1] );
    b = [ 3+13*%i ; 58+32*%i ; -19+13*%i ; 18-12*%i ; 22+16*%i ];
    x = umfpack(A,"\",b)  // x must be [1+i; 2+2i; 3+3i; 4 + 4i; 5+5i]

    // A benchmark of several linear solvers

    [A,descr,ref,mtype] = ReadHBSparse(SCI+"/modules/umfpack/examples/bcsstk24.rsa");

    b = 0*ones(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

- umf_lufact
- umf_lusolve
- umf_ludel
- umf_luinfo
- umf_luget
- linsolve
- backslash