ConjugateGradient Class

Solves a real symmetric definite linear system using the conjugate gradient method with optional preconditioning.

For a list of all members of this type, see ConjugateGradient Members.

System.Object
Imsl.Math.ConjugateGradient

public class ConjugateGradient

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

Class ConjugateGradient solves the symmetric positive or negative definite linear system using the conjugate gradient method with optional preconditioning. This method is described in detail by Golub and Van Loan (1983, Chapter 10), and in Hageman and Young (1981, Chapter 7).

The preconditioning matrix M is a matrix that approximates A, and for which the linear system Mz=r is easy to solve. These two properties are in conflict; balancing them is a topic of current research. If no preconditioning matrix is specified, is set to the identity, i.e. .

The number of iterations needed depends on the matrix and the error tolerance. As a rough guide,

${\rm{itmax}}=\sqrt{n}\;\mbox{for}\; n\gg 1,$

where n is the order of matrix A. See the references for details.

Let M be the preconditioning matrix, let b,p,r,x and z be vectors and let $\tau$ be the desired relative error. Then the algorithm used is as follows:

$\lambda = -1$
for $k=1,\ldots,\mbox{{\tt{itmax}}}$

$z_k = M^{-1}r_k$
if then

$\beta_k = 1$

else

$\beta_k=(z_k^Tr_k)/(z_{k-1}^Tr_{k-1})$
$p_k=z_k+\beta_kp_{k-1}$

$\mbox{endif}$
$\alpha_k=(r_k^Tz_k)/(p_k^TAp_k)$
$x_k=x_{k-1}+\alpha_kp_k$
$r_{k+1}=r_k-\alpha_kAp_k$
if $(\|Ap_k\|_2 \leq \tau (1-\lambda)\|x_k\|_2)$ then

recompute $\lambda$
if $(\|Ap_k\|_2 \leq \tau(1-\lambda)\|x_k\|_2)$ exit

endif

endfor

Here, $\lambda$ is an estimate of $\lambda_{\max}(\Gamma)$ , the largest eigenvalue of the iteration matrix $\Gamma=I-M^{-1}A$ . The stopping criterion is based on the result (Hageman and Young 1981, pp. 148-151)

$\frac{\|x_k-x\|_M}{\|x\|_M} \leq \left(\frac{1}{1-\lambda_{\max}(\Gamma)}\right)\left(\frac{\|z_k\|_M}{\|x_k\|_M}\right),$

where

$\|x\|_M^2=x^TMx\,.$

It is also known that

$\lambda_{\max}(T_1) \leq \lambda_{\max}(T_2) \leq \ldots \leq \lambda_{\max}(\Gamma) \lt 1,$

where the

are the symmetric, tridiagonal matrices

$T_l = \left[ \begin{array}{ccccc} \mu_1 & \omega_2 & & & \\ \omega_2 & \mu_2 & \omega_3 & & \\ & \omega_3 & \mu_3 & \raisebox{-1ex}{$\ddots$} & \\ & & \ddots & \ddots & \omega_l \\ & & & \omega_l & \mu_l \end{array} \right]$

with $\mu_1=1-1/\alpha_1$ and, for $k=2,\ldots,l$ ,

$\mu_k=1-\beta_k/\alpha_{k-1}-1/\alpha_k \quad \mbox{and} \quad \omega_k=\sqrt{\beta_k}/\alpha_{k-1}.$

Usually, the eigenvalue computation is needed for only a few of the iterations.

Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

ConjugateGradient Class

Thread Safety

Remarks

Requirements

See Also