GenMinRes Class

GenMinRes Class

Linear system solver using the restarted Generalized Minimum Residual (GMRES) method.

Inheritance Hierarchy

SystemObject
Imsl.MathGenMinRes

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class GenMinRes

<SerializableAttribute>
Public Class GenMinRes

[SerializableAttribute]
public ref class GenMinRes

[<SerializableAttribute>]
type GenMinRes =  class end

The GenMinRes type exposes the following members.

Constructors

	Name	Description
	GenMinRes	GMRES linear system solver constructor.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetGuess	Returns the initial guess of the solution.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetVectorProducts	Returns the user-supplied functions for the inner product and, optionally, the norm used in the Gram-Schmidt implementations.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetGuess	Sets the initial guess of the solution.
	SetVectorProducts	Sets the user-supplied functions for the inner product and, optionally, the norm to be used in the Gram-Schmidt implementations.
	Solve	Generate an approximate solution to using the Generalized Residual Method.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	Iterations	The actual number of GMRES iterations used.
	MaxIterations	The maximum number of iterations allowed.
	MaxKrylovDim	The maximum Krylov subspace dimension.
	Method	The implementation method to be used.
	PreconditionerSolves	The total number of GMRES right preconditioner solves.
	Products	The total number of GMRES matrix-vector products used.
	RelativeError	The stopping tolerance.
	ResidualNorm	The final residual norm, ${\Vert b-Ax \Vert}_2$ .
	ResidualUpdating	The residual updating method to be used.

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Remarks

GenMinRes implements restarted GMRES to generate an approximate solution to . It is based on GMRES by Homer Walker (1988).

The GMRES method begins with an approximate solution and an initial residual . At iteration m, a correction is determined in the Krylov subspace

${\kappa}_m (\nu)=\rm{span}(\nu,A\nu,\ldots,A^{m-1}\nu )$

$\nu = r_0$ which solves the least squares problem

$\min_{z \in \kappa_m(r_0)}{\Vert b-A(x_0 + z) \Vert}_2$

Then at iteration m,

There are four distinct GMRES implementations, selectable through property Method. The first Gram-Schmidt implementation is essentially the original algorithm by Saad and Schultz (1986). The second Gram-Schmidt implementation, developed by Homer Walker and Lou Zhou, is simpler than the first implementation. The least squares problem is constructed in upper-triangular form and the residual vector updating at the end of a GMRES cycle is cheaper. The first Householder implementation is algorithm 2.2 of Walker (1988), but with more efficient correction accumulation at the end of each GMRES cycle. The second Householder implementation is algorithm 3.1 of Walker (1988). The products of Householder transformations are expanded as sums, allowing most work to be formulated as large scale matrix-vector operations.

The Gram-Schmidt implementations are less expensive than the Householder, the latter requiring about twice as many computations beyond the coefficient matrix/vector products. However, the Householder implementations may be more reliable near the limits of residual reduction. See Walker (1988) for details. Issues such as the cost of coefficient matrix/vector products, availability of effective preconditioners, and features of particular computing environments may serve to mitigate the extra expense of the Householder implementations.

Reference

Imsl.Math Namespace

Other Resources

Simple Example

Simple Example with User Supplied Inner Product

Sparse Example

Sparse Example with Preconditioning

Second Householder Implementation Example

Second Householder Implementation Example with Preconditioning