umfpack

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 substitutions ; 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/demos/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 — lu factorization of a sparse matrix
• umf_lusolve — solve a linear sparse system given the LU factors
• umf_ludel — utility function used with umf_lufact
• umf_luinfo — get information on LU factors
• umf_luget — retrieve lu factors at the Scilab level
• linsolve — solveur d'équation linéaire
• backslash — (\) division matricielle à gauche : solution exacte ou selon moindres carrés