C++ for Scientists - Technische Universität Dresden

14.1. COMPUTING AN EIGENFUNCTION OF THE POISSON EQUATION 265
} // namespace athens
#endif
where we assume that the types X and Y are models of the concept glas::DenseVectorCollection.
14.1.2 Richardson iteration
Richardson iteration is an iterative method for the solution of the linear system
Bu = g
that starts from an initial guess u0 and computes ui = ui−1 + ri−1 at iteration i, where ri−1 is
the residual g − Bui−1. It works as follows: The method converges when the eigenvalues of B
1
2
3
4
5
1. For i = 1, . . . , max it:
1.1. Compute residual ri−1 = g − Bui−1
1.2. If �ri−1�2 ≤ τ: return
1.3. Compute new solution ui = ui−1 + ri−1
lie between 0 and 2.
The eigenvalues of the Poisson matrix A are λj = 2(1 − cos(πj/(n + 1))) for j = 1, . . . , n. The
eigenvalues are thus bounded by 0 < λj < 4. We therefore first multiply Au = f by 0.5 into
(0.5A)u = (0.5f)
Note that the solution u does not change. Define B = 0.5A and g = 0.5f, then Bu = g and the
eigenvalues of B lie in (0, 2). For such matrix, we can use the Richardson iteration method.
We develop the following function
template
double richardson( Op const& op, G const& g, U& u, double const& tol, int max it ) ;
where op is a BinaryFunction op(x,y) that computes y = Bx for a given input argument x, and
where u is an initial estimate of the solution on input and the computed solution on output.
The vector g is the right-hand side of the system. The return value of richardson is the residual
norm. This allows us to check how accurate the solution is without having to compute the
residual explicitly. The parameter tol corresponds to the tolerance τ.
First, we set conceptual conditions on all arguments.
• U is a model of concept glas::DenseVectorCollection, i.e. we assume that a dense vector
from GLAS is used.
• Op is a model of BinaryFunction, i.e. the following are valid expressions for op of type Op:
– op(x,y) where x and y are instances of type X where X is a model of the concept
glas::DenseVectorCollection.