ZNEUPD(3) MathKeisan ARPACK routine ZNEUPD(3)
NAME
ZNEUPD - Postprocessing routine for large-scale complex eigenvalue
calculation.
SYNOPSIS
SUBROUTINE ZNEUPD(RVEC, HOWMNY, SELECT, D, Z, LDZ, WORKEV, SIGMA, BMAT,
N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR,
WORKD, WORKL, LWORKL, INFO )
LOGICAL RVEC
LOGICAL SELECT(NCV)
INTEGER N, NEV, NCV, LDZ, LDV, LWORKL, INFO
INTEGER IPARAM(11), IPNTR(14)
DOUBLE PRECISION TOL
DOUBLE PRECISION RWORK(NCV)
COMPLEX*16 SIGMA
COMPLEX*16 D(NEV), RESID(N), WORKD(3*N), WORKL(LWORK),
Z(N, NEV), V(N, NCV), WORKEV(2*NCV)
CHARACTER BMAT*1, WHICH*2, HOWMNY*1
PURPOSE
ZNEUPD returns the converged approximations to eigenvalues
of A*z = lambda*B*z and (optionally):
(1) The corresponding approximate eigenvectors;
(2) An orthonormal basis for the associated approximate
invariant subspace;
(3) Both.
There is negligible additional cost to obtain eigenvectors. An orthonormal
basis is always computed. There is an additional storage cost of n*nev
if both are requested (in this case a separate array Z must be supplied).
The approximate eigenvalues and eigenvectors of A*z = lambda*B*z
are derived from approximate eigenvalues and eigenvectors of
of the linear operator OP prescribed by the MODE selection in the
call to ZNAUPD. ZNAUPD must be called before this routine is called.
These approximate eigenvalues and vectors are commonly called Ritz
values and Ritz vectors respectively. They are referred to as such
in the comments that follow. The computed orthonormal basis for the
invariant subspace corresponding to these Ritz values is referred to as a
Schur basis.
The definition of OP as well as other terms and the relation of computed
Ritz values and vectors of OP with respect to the given problem
A*z = lambda*B*z may be found in the header of ZNAUPD. For a brief
description, see definitions of IPARAM(7), MODE and WHICH in the
documentation of ZNAUPD.
ARGUMENTS
RVEC LOGICAL (INPUT)
Specifies whether a basis for the invariant subspace corresponding
to the converged Ritz value approximations for the eigenproblem
A*z = lambda*B*z is computed.
RVEC = .FALSE. Compute Ritz values only.
RVEC = .TRUE. Compute Ritz vectors or Schur vectors.
See Remarks below.
HOWMNY Character*1 (INPUT)
Specifies the form of the basis for the invariant subspace
corresponding to the converged Ritz values that is to be computed.
= 'A': Compute NEV Ritz vectors;
= 'P': Compute NEV Schur vectors;
= 'S': compute some of the Ritz vectors, specified
by the logical array SELECT.
SELECT Logical array of dimension NCV. (INPUT)
If HOWMNY = 'S', SELECT specifies the Ritz vectors to be
computed. To select the Ritz vector corresponding to a
Ritz value D(j), SELECT(j) must be set to .TRUE..
If HOWMNY = 'A' or 'P', SELECT need not be initialized
but it is used as internal workspace.
D Complex*16 array of dimension NEV+1. (OUTPUT)
On exit, D contains the Ritz approximations
to the eigenvalues lambda for A*z = lambda*B*z.
Z Complex*16 N by NEV array (OUTPUT)
On exit, if RVEC = .TRUE. and HOWMNY = 'A', then the columns of
Z represents approximate eigenvectors (Ritz vectors) corresponding
to the NCONV=IPARAM(5) Ritz values for eigensystem
A*z = lambda*B*z.
If RVEC = .FALSE. or HOWMNY = 'P', then Z is NOT REFERENCED.
NOTE: If if RVEC = .TRUE. and a Schur basis is not required,
the array Z may be set equal to first NEV+1 columns of the Arnoldi
basis array V computed by ZNAUPD. In this case the Arnoldi basis
will be destroyed and overwritten with the eigenvector basis.
LDZ Integer. (INPUT)
The leading dimension of the array Z. If Ritz vectors are
desired, then LDZ .ge. max( 1, N ) is required.
In any case, LDZ .ge. 1 is required.
SIGMA Complex*16 (INPUT)
If IPARAM(7) = 3 then SIGMA represents the shift.
Not referenced if IPARAM(7) = 1 or 2.
WORKEV Complex*16 work array of dimension 2*NCV. (WORKSPACE)
**** The remaining arguments MUST be the same as for the ****
**** call to ZNAUPD that was just completed. ****
NOTE: The remaining arguments
BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR,
WORKD, WORKL, LWORKL, RWORK, INFO
must be passed directly to ZNEUPD following the last call
to ZNAUPD. These arguments MUST NOT BE MODIFIED between
the the last call to ZNAUPD and the call to ZNEUPD.
Three of these parameters (V, WORKL and INFO) are also output parameters:
V Complex*16 N by NCV array. (INPUT/OUTPUT)
Upon INPUT: the NCV columns of V contain the Arnoldi basis
vectors for OP as constructed by ZNAUPD .
Upon OUTPUT: If RVEC = .TRUE. the first NCONV=IPARAM(5) columns
contain approximate Schur vectors that span the
desired invariant subspace.
NOTE: If the array Z has been set equal to first NEV+1 columns
of the array V and RVEC=.TRUE. and HOWMNY= 'A', then the
Arnoldi basis held by V has been overwritten by the desired
Ritz vectors. If a separate array Z has been passed then
the first NCONV=IPARAM(5) columns of V will contain approximate
Schur vectors that span the desired invariant subspace.
WORKL Double precision work array of length LWORKL. (OUTPUT/WORKSPACE)
WORKL(1:ncv*ncv+2*ncv) contains information obtained in
ZNAUPD. They are not changed by ZNEUPD.
WORKL(ncv*ncv+2*ncv+1:3*ncv*ncv+4*ncv) holds the
untransformed Ritz values, the untransformed error estimates of
the Ritz values, the upper triangular matrix for H, and the
associated matrix representation of the invariant subspace for H.
Note: IPNTR(9:13) contains the pointer into WORKL for addresses
of the above information computed by ZNEUPD.
-------------------------------------------------------------
IPNTR(9): pointer to the NCV RITZ values of the
original system.
IPNTR(10): Not used
IPNTR(11): pointer to the NCV corresponding error estimates.
IPNTR(12): pointer to the NCV by NCV upper triangular
Schur matrix for H.
IPNTR(13): pointer to the NCV by NCV matrix of eigenvectors
of the upper Hessenberg matrix H. Only referenced by
ZNEUPD if RVEC = .TRUE. See Remark 2 below.
-------------------------------------------------------------
INFO Integer. (OUTPUT)
Error flag on output.
= 0: Normal exit.
= 1: The Schur form computed by LAPACK routine csheqr
could not be reordered by LAPACK routine ztrsen.
Re-enter subroutine ZNEUPD with IPARAM(5)=NCV and
increase the size of the array D to have
dimension at least dimension NCV and allocate at least NCV
columns for Z. NOTE: Not necessary if Z and V share
the same space. Please notify the authors if this error
occurs.
= -1: N must be positive.
= -2: NEV must be positive.
= -3: NCV-NEV >= 2 and less than or equal to N.
= -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'
= -6: BMAT must be one of 'I' or 'G'.
= -7: Length of private work WORKL array is not sufficient.
= -8: Error return from LAPACK eigenvalue calculation.
This should never happened.
= -9: Error return from calculation of eigenvectors.
Informational error from LAPACK routine ztrevc.
= -10: IPARAM(7) must be 1,2,3
= -11: IPARAM(7) = 1 and BMAT = 'G' are incompatible.
= -12: HOWMNY = 'S' not yet implemented
= -13: HOWMNY must be one of 'A' or 'P' if RVEC = .true.
= -14: ZNAUPD did not find any eigenvalues to sufficient
accuracy.
MathKeisan ZNEUPD(3)