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ARPACK++ v1.2 2/20/2000
c++ interface to ARPACK code.
MODULE neupp.h.
Interface to ARPACK subroutines dneupd and sneupd.
ARPACK Authors
Richard Lehoucq
Danny Sorensen
Chao Yang
Dept. of Computational & Applied Mathematics
Rice University
Houston, Texas
*/
#ifndef NEUPP_H
#define NEUPP_H
#include <cstddef>
#include "arch.h"
#include "arpackf.h"
inline void neupp(bool rvec, char HowMny, double dr[],
double di[], double Z[], ARint ldz, double sigmar,
double sigmai, double workv[], char bmat, ARint n,
char* which, ARint nev, double tol, double resid[],
ARint ncv, double V[], ARint ldv, ARint iparam[],
ARint ipntr[], double workd[], double workl[],
ARint lworkl, ARint& info)
/*
c++ version of ARPACK routine dneupd.
This subroutine 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,
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 naupp. naupp 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.
See documentation in the header of the subroutine naupp for
definition of OP as well as other terms and the relation of computed
Ritz values and Ritz vectors of OP with respect to the given problem
A*z = lambda*B*z. For a brief description, see definitions of
iparam[7], MODE and which in the documentation of naupp.
Parameters:
rvec (Input) Specifies whether Ritz vectors corresponding to the
Ritz value approximations to the eigenproblem A*z = lambda*B*z
are computed.
rvec = false: Compute Ritz values only.
rvec = true : Compute the Ritz vectors or Schur vectors.
See Remarks below.
HowMny (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;
dr (Output) Array of dimension nev+1.
If iparam[7] = 1,2 or 3 and sigmai=0.0 then on exit: dr
contains the real part of the Ritz approximations to the
eigenvalues of A*z = lambda*B*z.
If iparam[7] = 3, 4 and sigmai is not equal to zero, then on
exit: dr contains the real part of the Ritz values of OP
computed by naupp. A further computation must be performed by
the user to transform the Ritz values computed for OP by naupp
to those of the original system A*z = lambda*B*z. See remark 3.
di (Output) Array of dimension nev+1.
On exit, di contains the imaginary part of the Ritz value
approximations to the eigenvalues of A*z = lambda*B*z
associated with dr.
NOTE: When Ritz values are complex, they will come in complex
conjugate pairs. If eigenvectors are requested, the
corresponding Ritz vectors will also come in conjugate
pairs and the real and imaginary parts of these are
represented in two consecutive columns of the array Z
(see below).
Z (Output) Array of dimension nev*n if rvec = TRUE and HowMny =
'A'. if rvec = TRUE. and HowMny = 'A', then the contains
approximate eigenvectors (Ritz vectors) corresponding to the
NCONV=iparam[5] Ritz values for eigensystem A*z = lambda*B*z.
The complex Ritz vector associated with the Ritz value
with positive imaginary part is stored in two consecutive
columns. The first column holds the real part of the Ritz
vector and the second column holds the imaginary part. The
Ritz vector associated with the Ritz value with negative
imaginary part is simply the complex conjugate of the Ritz
vector associated with the positive imaginary part.
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 naupp. In this
case the Arnoldi basis will be destroyed and overwritten
with the eigenvector basis.
ldz (Input) Dimension of the vectors contained in Z. This
parameter MUST be set to n.
sigmar (Input) If iparam[7] = 3 or 4, represents the real part of
the shift. Not referenced if iparam[7] = 1 or 2.
sigmai (Input) If iparam[7] = 3 or 4, represents the imaginary part
of the shift. Not referenced if iparam[7] = 1 or 2. See
remark 3 below.
workv (Workspace) Array of dimension 3*ncv.
V (Input/Output) Array of dimension n*ncv+1.
Upon Input: V contains the ncv vectors of the Arnoldi basis
for OP as constructed by naupp.
Upon Output: If rvec = TRUE the first NCONV=iparam[5] columns
contain approximate Schur vectors that span the
desired invariant subspace. See Remark 2 below.
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 (Input / Output) Array of length lworkl+1.
workl[1:ncv*ncv+3*ncv] contains information obtained in
naupp. They are not changed by neupp.
workl[ncv*ncv+3*ncv+1:3*ncv*ncv+6*ncv] holds the real and
imaginary part of the untransformed Ritz values, the upper
quasi-triangular matrix for H, and the associated matrix
representation of the invariant subspace for H.
ipntr (Input / Output) Array of length 14. Pointer to mark the
starting locations in the workl array for matrices/vectors
used by naupp and neupp.
ipntr[9]: pointer to the real part of the ncv RITZ values
of the original system.
ipntr[10]: pointer to the imaginary part of the ncv RITZ
values of the original system.
ipntr[11]: pointer to the ncv corresponding error bounds.
ipntr[12]: pointer to the ncv by ncv upper quasi-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 neupp if rvec = TRUE. See Remark 2 below.
info (Output) Error flag.
= 0 : Normal exit.
= 1 : The Schur form computed by LAPACK routine dlahqr
could not be reordered by LAPACK routine dtrsen.
Re-enter subroutine neupp with iparam[5] = ncv and
increase the size of the arrays DR and DI 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 must satisfy nev+2 <= ncv <= 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 calculation of a real Schur form.
Informational error from LAPACK routine dlahqr.
= -9 : Error return from calculation of eigenvectors.
Informational error from LAPACK routine dtrevc.
= -10: iparam[7] must be 1,2,3,4.
= -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: naupp did not find any eigenvalues to sufficient
accuracy.
NOTE: The following arguments
bmat, n, which, nev, tol, resid, ncv, V, ldv, iparam,
ipntr, workd, workl, lworkl, info
must be passed directly to neupp following the last call
to naupp. These arguments MUST NOT BE MODIFIED between
the the last call to naupp and the call to neupp.
Remarks
1. Currently only HowMny = 'A' and 'P' are implemented.
2. Schur vectors are an orthogonal representation for the basis of
Ritz vectors. Thus, their numerical properties are often superior.
Let X' denote the transpose of X. If rvec = .TRUE. then the
relationship A * V[:,1:iparam[5]] = V[:,1:iparam[5]] * T, and
V[:,1:iparam[5]]' * V[:,1:iparam[5]] = I are approximately satisfied.
Here T is the leading submatrix of order iparam[5] of the real
upper quasi-triangular matrix stored workl[ipntr[12]]. That is,
T is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
each 2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign. Corresponding to each
2-by-2 diagonal block is a complex conjugate pair of Ritz values.
The real Ritz values are stored on the diagonal of T.
3. If iparam[7] = 3 or 4 and sigmai is not equal zero, then the user
must form the iparam[5] Rayleigh quotients in order to transform the
Ritz values computed by naupp for OP to those of A*z = lambda*B*z.
Set rvec = TRUE. and HowMny = 'A', and compute
Z[:,I]' * A * Z[:,I] if di[I] = 0.
If di[I] is not equal to zero and di[I+1] = - D[I],
then the desired real and imaginary parts of the Ritz value are
Z[:,I]' * A * Z[:,I] + Z[:,I+1]' * A * Z[:,I+1],
Z[:,I]' * A * Z[:,I+1] - Z[:,I+1]' * A * Z[:,I], respectively.
Another possibility is to set rvec = .true. and HowMny = 'P' and
compute V[:,1:iparam[5]]' * A * V[:,1:iparam[5]] and then an upper
quasi-triangular matrix of order iparam[5] is computed. See remark
2 above.
*/
{
ARint irvec;
ARlogical* iselect;
double* iZ;
irvec = (ARint) rvec;
iselect = new ARlogical[ncv];
iZ = (Z == NULL) ? &V[1] : Z;
F77NAME(dneupd)(&irvec, &HowMny, iselect, dr, di, iZ, &ldz, &sigmar,
&sigmai, &workv[1], &bmat, &n, which, &nev, &tol,
resid, &ncv, &V[1], &ldv, &iparam[1], &ipntr[1],
&workd[1], &workl[1], &lworkl, &info);
delete[] iselect;
} // neupp (double).
inline void neupp(bool rvec, char HowMny, float dr[],
float di[], float Z[], ARint ldz, float sigmar,
float sigmai, float workv[], char bmat, ARint n,
char* which, ARint nev, float tol, float resid[],
ARint ncv, float V[], ARint ldv, ARint iparam[],
ARint ipntr[], float workd[], float workl[],
ARint lworkl, ARint& info)
/*
c++ version of ARPACK routine sneupd. The only difference between
sneupd and dneupd is that in the former function all vectors have
single precision elements and in the latter all vectors have double
precision elements.
*/
{
ARint irvec;
ARlogical* iselect;
float* iZ;
irvec = (ARint) rvec;
iselect = new ARlogical[ncv];
iZ = (Z == NULL) ? &V[1] : Z;
F77NAME(sneupd)(&irvec, &HowMny, iselect, dr, di, iZ, &ldz, &sigmar,
&sigmai, &workv[1], &bmat, &n, which, &nev, &tol,
resid, &ncv, &V[1], &ldv, &iparam[1], &ipntr[1],
&workd[1], &workl[1], &lworkl, &info );
delete[] iselect;
} // neupp (float).
#endif // NEUPP_H
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