umfpack — solve sparse linear system
x = umfpack(A,"\",b) x = umfpack(b,"/",A)
a sparse (real or complex) square matrix n x n
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
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
string specifier "\" or "/"
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.
First an LU factorisation 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.
// 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);
printf('\ntime needed to solve the system with umfpack: %.3f\n',toc());
tic();
res = linsolve(A,b);
printf('\ntime needed to solve the system with linsolve: %.3f\n',toc());
tic();
res = A\b;
printf('\ntime needed to solve the system with the backslash operator: %.3f\n',toc());