C++ for Scientists - Technische Universität Dresden

14.1. COMPUTING AN EIGENFUNCTION OF THE POISSON EQUATION 267
glas::random( f, seed ) ;
v type x( 10 ) ;
x = 0.0 ;
// Richardson iteration
double res nrm = athens::richardson( poisson scaled(), 0.5∗f, x, 1.e−4, 1000 ) ;
{
glas::dense vector r( size(x) ) ;
athens::poisson 1d( x, r ) ;
std::cout ≪ ”res nrm = ” ≪ norm 2( f −r ) ≪ std::endl ;
std::cout ≪ ”f = ” ≪ f ≪ std::endl ;
std::cout ≪ ”x = ” ≪ x ≪ std::endl ;
}
return 0 ;
}
We multiply right-hand side and matrix vector product by 0.5 to make sure the Richardson
method converges.
The output looks like
res_nrm = 0.000195164
f = (10)[0.0484811,0.822283,0.102721,0.436631,0.46112,0.0475317,0.864644,0.0772845,0.920099,0.105434]
x = (10)[1.85463,3.66081,4.64473,5.52601,5.97071,5.9544,5.8906,4.96226,3.95668,2.03105]
Note that the Richardson method converges very slowly. For the Poisson equation, there exist
much faster methods.
14.1.3 LAPACK tridiagonal solver
The LAPACK [?] software package contains routines for solving linear systems with a symmetric
positive definite tridiagonal matrix. This package is written in FORTRAN 77. The
corresponding functions are
• Cholesky factorization: A = LL T by
SUBROUTINE DPTTRF( N, D, E, INFO )
• Linear solve: Ax = b using LL T x = b by
SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO )
In order to solve Au = f, first A is factorized by the Cholesky factorization into A = LDL T
where L is a matrix consisting of a main diagonal of ones and a diagonal below the main diagonal
and D is a diagonal matrix. Once the factorization is performed, the solution is computed as
u = L −T D(L −1 f). Note the inversions of L and L T are not computed explicitly. For example
L −1 f is computed as a linear solve with L. Linear solves for triangular matrices are easy to
program. This is what DPTTRS does for us.
A C++ interface to DPTTRF and DPTTRS is available from the BoostSandbox.Bindings. For
our application, we can solve a linear system as follows.