psfun_d_saiarnoldi Interface

interface

This interface contains the Shift-and-Invert method based on Arnoldi orthogonalization.

public module subroutine psfun_d_saiarnoldi(fun, a, kryl, prec, tau, desc_a, initmax, initrace, inistop, y, x, eps, info, itmax, itrace, istop, iter, err, res)

Arguments

Type	Intent	Optional	Attributes		Name
type(psfun_d_serial),	intent(inout)			::	fun	Function object
type(psb_dspmat_type),	intent(in)			::	a	Distribute sparse matrix
character(len=*),	intent(in)			::	kryl	Krylov method for the solution of inner systems
type(amg_dprec_type),	intent(inout)			::	prec	Preconditioner for the inner method
real(kind=psb_dpk_),	intent(in)			::	tau	Shift parameter of the method
type(psb_desc_type),	intent(in)			::	desc_a	Descriptor for the sparse matrix
integer(kind=psb_ipk_),	intent(in)			::	initmax	Maximum number of iteration (inner method)
integer(kind=psb_ipk_),	intent(in)			::	initrace	Trace for logoutput (inner method)
integer(kind=psb_ipk_),	intent(in)			::	inistop	Stop criterion (inner method)
type(psb_d_vect_type),	intent(inout)			::	y	Output vector
type(psb_d_vect_type),	intent(inout)			::	x	Input vector
real(kind=psb_dpk_),	intent(in)			::	eps	Requested tolerance
integer(kind=psb_ipk_),	intent(out)			::	info	Output flag
integer(kind=psb_ipk_),	intent(in),	optional		::	itmax	Maximum number of iteration
integer(kind=psb_ipk_),	intent(in),	optional		::	itrace	Trace for logoutput
integer(kind=psb_ipk_),	intent(in),	optional		::	istop	Stop criterion
integer(kind=psb_ipk_),	intent(out),	optional		::	iter	Number of iteration
real(kind=psb_dpk_),	intent(out),	optional		::	err	Last estimate error
real(kind=psb_dpk_),	intent(out),	optional	allocatable	::	res(:)	Vector of the residuals

Description

Shift-and-invert method based on the Arnoldi orthogonalization procedure the method builds a basis $V_k$ for the shifted Krylov subspace

$$\begin{equation*} \mathcal{K}_( (A + \tau I)^{-1},x ) = \{ x,(A + \tau I)^{-1}x,\ldots,(A + \tau I)^{k-1}x\}, \end{equation*}$$

and approximate $y = f(\alpha A)x \approx \beta_1 f((H_k^{-1} - \tau I)^{-1}) e_1$,for $\beta_1 = \|x\|_2$, $e_1$ the first vector of the canonical base of $\mathbb{R}^k$, and $H_k$ the Hessemberg matrix given by the projection of $A$ into the subspace $\mathcal{K}_( (A + \tau I)^{-1},x )$.

To march the algorithm one needs to solve at each step a shifted linear system for which we employ the routines from psfun_krylov_mod and the preconditioners from AMG4PSBLAS.