犰狳C ++ LU分解

Question

I am using the Armadillo C++ library for solving linear systems of medium/large dimensions (1000-5000 equations).

Since I have to solve different linear systems

AX=b

in which A is always the same and B changes, I would like to LU factorize A only once and reuse the LU factorization with different b. Unfortunately I do not know how to perform this kind of operations in Armadillo.

What I did was just the LU factorization of the A matrix:

arma::mat A;
// ... fill the A matrix ...
arma::mat P,L,U;
arma::lu(L, U, P, A);

But now I would like to use the matrices P, L and U to solve several linear systems with different b vectors.

Could you help me please?

Answer 1

Since A = Pt()*L*U (where equality is only approximate due to rounding errors), solving for x in Pt()*L*U*x = b requires to permute rows of B and performing forward and back substitution:

x = solve(trimatu(U), solve(trimatl(L), P*b) );

Due to the lack of a true triangular solver in armadillo, and a fast way to perform row permutation, this procedure will not be very efficient, with respect to a direct call to the relevant computational LAPACK subroutines.

General advice is to avoid explicit LU decomposition in higher level libraries, like armadillo.

1. if all different b 's are known at the same time, store them as columns in a rectangular matrix B and X = solve(A,B);
2. if the different b 's are known one at a time, then precomputing AINV = Ai(); and x = AINV*b; will be more efficient if the number of different rhs vectors is big enough. See this answer to a similar question .