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/* -- translated by f2c (version 20240504).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
/* Table of constant values */
static doublereal c_b3 = .66666666666666663;
static integer c__1 = 1;
static doublereal c_b37 = 0.;
static doublereal c_b38 = 1.;
static logical c_true = TRUE_;
static doublereal c_b64 = -1.;
/* \BeginDoc
\Name: dneupd
\Description:
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;
(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 DNAUPD . DNAUPD 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 DNAUPD 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 DNAUPD .
\Usage:
call dneupd
( RVEC, HOWMNY, SELECT, DR, DI, Z, LDZ, SIGMAR, SIGMAI, WORKEV, BMAT,
N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR, WORKD, WORKL,
LWORKL, INFO )
\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 the 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 (DR(j), DI(j)), SELECT(j) must be set to .TRUE..
If HOWMNY = 'A' or 'P', SELECT is used as internal workspace.
DR Double precision array of dimension NEV+1. (OUTPUT)
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
DNAUPD . A further computation must be performed by the user
to transform the Ritz values computed for OP by DNAUPD to those
of the original system A*z = lambda*B*z. See remark 3 below.
DI Double precision array of dimension NEV+1. (OUTPUT)
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 Double precision N by NEV+1 array if RVEC = .TRUE. and HOWMNY = 'A'. (OUTPUT)
On exit, if RVEC = .TRUE. and HOWMNY = 'A', then the columns of
Z represent 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 DNAUPD . 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 >= max( 1, N ). In any case, LDZ >= 1.
SIGMAR Double precision (INPUT)
If IPARAM(7) = 3 or 4, represents the real part of the shift.
Not referenced if IPARAM(7) = 1 or 2.
SIGMAI Double precision (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.
WORKEV Double precision work array of dimension 3*NCV. (WORKSPACE)
**** The remaining arguments MUST be the same as for the ****
**** call to DNAUPD that was just completed. ****
NOTE: The remaining arguments
BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR,
WORKD, WORKL, LWORKL, INFO
must be passed directly to DNEUPD following the last call
to DNAUPD . These arguments MUST NOT BE MODIFIED between
the the last call to DNAUPD and the call to DNEUPD .
Three of these parameters (V, WORKL, INFO) are also output parameters:
V Double precision N by NCV array. (INPUT/OUTPUT)
Upon INPUT: the NCV columns of V contain the Arnoldi basis
vectors for OP as constructed by DNAUPD .
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 Double precision work array of length LWORKL. (OUTPUT/WORKSPACE)
WORKL(1:ncv*ncv+3*ncv) contains information obtained in
dnaupd . They are not changed by dneupd .
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.
Note: IPNTR(9:13) contains the pointer into WORKL for addresses
of the above information computed by dneupd .
-------------------------------------------------------------
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
dneupd 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 dlahqr
could not be reordered by LAPACK routine dtrsen .
Re-enter subroutine dneupd 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-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 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: DNAUPD did not find any eigenvalues to sufficient
accuracy.
= -15: DNEUPD got a different count of the number of converged
Ritz values than DNAUPD got. This indicates the user
probably made an error in passing data from DNAUPD to
DNEUPD or that the data was modified before entering
DNEUPD
\BeginLib
\References:
1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
pp 357-385.
2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly
Restarted Arnoldi Iteration", Rice University Technical Report
TR95-13, Department of Computational and Applied Mathematics.
3. B.N. Parlett & Y. Saad, "Complex Shift and Invert Strategies for
Real Matrices", Linear Algebra and its Applications, vol 88/89,
pp 575-595, (1987).
\Routines called:
ivout ARPACK utility routine that prints integers.
dmout ARPACK utility routine that prints matrices
dvout ARPACK utility routine that prints vectors.
dgeqr2 LAPACK routine that computes the QR factorization of
a matrix.
dlacpy LAPACK matrix copy routine.
dlahqr LAPACK routine to compute the real Schur form of an
upper Hessenberg matrix.
dlamch LAPACK routine that determines machine constants.
dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
dlaset LAPACK matrix initialization routine.
dorm2r LAPACK routine that applies an orthogonal matrix in
factored form.
dtrevc LAPACK routine to compute the eigenvectors of a matrix
in upper quasi-triangular form.
dtrsen LAPACK routine that re-orders the Schur form.
dtrmm Level 3 BLAS matrix times an upper triangular matrix.
dger Level 2 BLAS rank one update to a matrix.
dcopy Level 1 BLAS that copies one vector to another .
ddot Level 1 BLAS that computes the scalar product of two vectors.
dnrm2 Level 1 BLAS that computes the norm of a vector.
dscal Level 1 BLAS that scales a vector.
\Remarks
1. Currently only HOWMNY = 'A' and 'P' are implemented.
Let trans(X) denote the transpose of X.
2. Schur vectors are an orthogonal representation for the basis of
Ritz vectors. Thus, their numerical properties are often superior.
If RVEC = .TRUE. then the relationship
A * V(:,1:IPARAM(5)) = V(:,1:IPARAM(5)) * T, and
trans(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 DNAUPD for OP to those of A*z = lambda*B*z.
Set RVEC = .true. and HOWMNY = 'A', and
compute
trans(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
trans(Z(:,I)) * A * Z(:,I) + trans(Z(:,I+1)) * A * Z(:,I+1),
trans(Z(:,I)) * A * Z(:,I+1) - trans(Z(:,I+1)) * A * Z(:,I),
respectively.
Another possibility is to set RVEC = .true. and HOWMNY = 'P' and
compute trans(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.
\Authors
Danny Sorensen Phuong Vu
Richard Lehoucq CRPC / Rice University
Chao Yang Houston, Texas
Dept. of Computational &
Applied Mathematics
Rice University
Houston, Texas
\SCCS Information: @(#)
FILE: neupd.F SID: 2.7 DATE OF SID: 09/20/00 RELEASE: 2
\EndLib
-----------------------------------------------------------------------
Subroutine */ int igraphdneupd_(logical *rvec, char *howmny, logical *select,
doublereal *dr, doublereal *di, doublereal *z__, integer *ldz,
doublereal *sigmar, doublereal *sigmai, doublereal *workev, char *
bmat, integer *n, char *which, integer *nev, doublereal *tol,
doublereal *resid, integer *ncv, doublereal *v, integer *ldv, integer
*iparam, integer *ipntr, doublereal *workd, doublereal *workl,
integer *lworkl, integer *info)
{
/* System generated locals */
integer v_dim1, v_offset, z_dim1, z_offset, i__1;
doublereal d__1, d__2;
/* Builtin functions */
double pow_dd(doublereal *, doublereal *);
integer s_cmp(char *, char *, ftnlen, ftnlen);
/* Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen);
/* Local variables */
integer j, k, ih, jj, np;
doublereal vl[1] /* was [1][1] */;
integer ibd, ldh, ldq, iri;
doublereal sep;
integer irr, wri, wrr;
extern /* Subroutine */ int igraphdger_(integer *, integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
integer *);
integer mode;
doublereal eps23;
integer ierr;
doublereal temp;
integer iwev;
char type__[6];
extern doublereal igraphdnrm2_(integer *, doublereal *, integer *);
doublereal temp1;
extern /* Subroutine */ int igraphdscal_(integer *, doublereal *, doublereal *,
integer *);
integer ihbds, iconj;
extern /* Subroutine */ int igraphdgemv_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *);
doublereal conds;
logical reord;
extern /* Subroutine */ int igraphdcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
integer nconv;
extern /* Subroutine */ int igraphdtrmm_(char *, char *, char *, char *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *), igraphdmout_(
integer *, integer *, integer *, doublereal *, integer *, integer
*, char *, ftnlen);
integer iwork[1];
doublereal rnorm;
integer ritzi;
extern /* Subroutine */ int igraphdvout_(integer *, integer *, doublereal *,
integer *, char *, ftnlen), igraphivout_(integer *, integer *, integer *
, integer *, char *, ftnlen);
integer ritzr;
extern /* Subroutine */ int igraphdgeqr2_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
extern doublereal igraphdlapy2_(doublereal *, doublereal *);
integer nconv2;
extern /* Subroutine */ int igraphdorm2r_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *);
extern doublereal igraphdlamch_(char *);
integer iheigi, iheigr;
extern /* Subroutine */ int igraphdlahqr_(logical *, logical *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *, integer *, doublereal *, integer *,
integer *), igraphdlacpy_(char *, integer *, integer *, doublereal *,
integer *, doublereal *, integer *), igraphdlaset_(char *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
integer *);
integer logfil=6, ndigit=-3;
extern /* Subroutine */ int igraphdngets_(integer *, char *, integer *, integer
*, doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
integer ishift;
extern /* Subroutine */ int igraphdtrevc_(char *, char *, logical *, integer *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
integer *, integer *, integer *, doublereal *, integer *);
integer mneupd=0, bounds;
extern /* Subroutine */ int igraphdtrsen_(char *, char *, logical *, integer *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
integer *, integer *, integer *, integer *);
integer msglvl, invsub, numcnv, iuptri, outncv;
/* %----------------------------------------------------%
| Include files for debugging and timing information |
%----------------------------------------------------%
%------------------%
| Scalar Arguments |
%------------------%
%-----------------%
| Array Arguments |
%-----------------%
%------------%
| Parameters |
%------------%
%---------------%
| Local Scalars |
%---------------%
%----------------------%
| External Subroutines |
%----------------------%
%--------------------%
| External Functions |
%--------------------%
%---------------------%
| Intrinsic Functions |
%---------------------%
%-----------------------%
| Executable Statements |
%-----------------------%
%------------------------%
| Set default parameters |
%------------------------%
Parameter adjustments */
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--workd;
--resid;
--di;
--dr;
--workev;
--select;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
--iparam;
--ipntr;
--workl;
/* Function Body */
msglvl = mneupd;
mode = iparam[7];
nconv = iparam[5];
*info = 0;
/* %---------------------------------%
| Get machine dependent constant. |
%---------------------------------% */
eps23 = igraphdlamch_("Epsilon-Machine");
eps23 = pow_dd(&eps23, &c_b3);
/* %--------------%
| Quick return |
%--------------% */
ierr = 0;
if (nconv <= 0) {
ierr = -14;
} else if (*n <= 0) {
ierr = -1;
} else if (*nev <= 0) {
ierr = -2;
} else if (*ncv <= *nev + 1 || *ncv > *n) {
ierr = -3;
} else if (s_cmp(which, "LM", (ftnlen)2, (ftnlen)2) != 0 && s_cmp(which,
"SM", (ftnlen)2, (ftnlen)2) != 0 && s_cmp(which, "LR", (ftnlen)2,
(ftnlen)2) != 0 && s_cmp(which, "SR", (ftnlen)2, (ftnlen)2) != 0
&& s_cmp(which, "LI", (ftnlen)2, (ftnlen)2) != 0 && s_cmp(which,
"SI", (ftnlen)2, (ftnlen)2) != 0) {
ierr = -5;
} else if (*(unsigned char *)bmat != 'I' && *(unsigned char *)bmat != 'G')
{
ierr = -6;
} else /* if(complicated condition) */ {
/* Computing 2nd power */
i__1 = *ncv;
if (*lworkl < i__1 * i__1 * 3 + *ncv * 6) {
ierr = -7;
} else if (*(unsigned char *)howmny != 'A' && *(unsigned char *)
howmny != 'P' && *(unsigned char *)howmny != 'S' && *rvec) {
ierr = -13;
} else if (*(unsigned char *)howmny == 'S') {
ierr = -12;
}
}
if (mode == 1 || mode == 2) {
s_copy(type__, "REGULR", (ftnlen)6, (ftnlen)6);
} else if (mode == 3 && *sigmai == 0.) {
s_copy(type__, "SHIFTI", (ftnlen)6, (ftnlen)6);
} else if (mode == 3) {
s_copy(type__, "REALPT", (ftnlen)6, (ftnlen)6);
} else if (mode == 4) {
s_copy(type__, "IMAGPT", (ftnlen)6, (ftnlen)6);
} else {
ierr = -10;
}
if (mode == 1 && *(unsigned char *)bmat == 'G') {
ierr = -11;
}
/* %------------%
| Error Exit |
%------------% */
if (ierr != 0) {
*info = ierr;
goto L9000;
}
/* %--------------------------------------------------------%
| Pointer into WORKL for address of H, RITZ, BOUNDS, Q |
| etc... and the remaining workspace. |
| Also update pointer to be used on output. |
| Memory is laid out as follows: |
| workl(1:ncv*ncv) := generated Hessenberg matrix |
| workl(ncv*ncv+1:ncv*ncv+2*ncv) := real and imaginary |
| parts of ritz values |
| workl(ncv*ncv+2*ncv+1:ncv*ncv+3*ncv) := error bounds |
%--------------------------------------------------------%
%-----------------------------------------------------------%
| The following is used and set by DNEUPD . |
| workl(ncv*ncv+3*ncv+1:ncv*ncv+4*ncv) := The untransformed |
| real part of the Ritz values. |
| workl(ncv*ncv+4*ncv+1:ncv*ncv+5*ncv) := The untransformed |
| imaginary part of the Ritz values. |
| workl(ncv*ncv+5*ncv+1:ncv*ncv+6*ncv) := The untransformed |
| error bounds of the Ritz values |
| workl(ncv*ncv+6*ncv+1:2*ncv*ncv+6*ncv) := Holds the upper |
| quasi-triangular matrix for H |
| workl(2*ncv*ncv+6*ncv+1: 3*ncv*ncv+6*ncv) := Holds the |
| associated matrix representation of the invariant |
| subspace for H. |
| GRAND total of NCV * ( 3 * NCV + 6 ) locations. |
%-----------------------------------------------------------% */
ih = ipntr[5];
ritzr = ipntr[6];
ritzi = ipntr[7];
bounds = ipntr[8];
ldh = *ncv;
ldq = *ncv;
iheigr = bounds + ldh;
iheigi = iheigr + ldh;
ihbds = iheigi + ldh;
iuptri = ihbds + ldh;
invsub = iuptri + ldh * *ncv;
ipntr[9] = iheigr;
ipntr[10] = iheigi;
ipntr[11] = ihbds;
ipntr[12] = iuptri;
ipntr[13] = invsub;
wrr = 1;
wri = *ncv + 1;
iwev = wri + *ncv;
/* %-----------------------------------------%
| irr points to the REAL part of the Ritz |
| values computed by _neigh before |
| exiting _naup2. |
| iri points to the IMAGINARY part of the |
| Ritz values computed by _neigh |
| before exiting _naup2. |
| ibd points to the Ritz estimates |
| computed by _neigh before exiting |
| _naup2. |
%-----------------------------------------% */
irr = ipntr[14] + *ncv * *ncv;
iri = irr + *ncv;
ibd = iri + *ncv;
/* %------------------------------------%
| RNORM is B-norm of the RESID(1:N). |
%------------------------------------% */
rnorm = workl[ih + 2];
workl[ih + 2] = 0.;
if (msglvl > 2) {
igraphdvout_(&logfil, ncv, &workl[irr], &ndigit, "_neupd: Real part of Rit"
"z values passed in from _NAUPD.", (ftnlen)55);
igraphdvout_(&logfil, ncv, &workl[iri], &ndigit, "_neupd: Imag part of Rit"
"z values passed in from _NAUPD.", (ftnlen)55);
igraphdvout_(&logfil, ncv, &workl[ibd], &ndigit, "_neupd: Ritz estimates p"
"assed in from _NAUPD.", (ftnlen)45);
}
if (*rvec) {
reord = FALSE_;
/* %---------------------------------------------------%
| Use the temporary bounds array to store indices |
| These will be used to mark the select array later |
%---------------------------------------------------% */
i__1 = *ncv;
for (j = 1; j <= i__1; ++j) {
workl[bounds + j - 1] = (doublereal) j;
select[j] = FALSE_;
/* L10: */
}
/* %-------------------------------------%
| Select the wanted Ritz values. |
| Sort the Ritz values so that the |
| wanted ones appear at the tailing |
| NEV positions of workl(irr) and |
| workl(iri). Move the corresponding |
| error estimates in workl(bound) |
| accordingly. |
%-------------------------------------% */
np = *ncv - *nev;
ishift = 0;
igraphdngets_(&ishift, which, nev, &np, &workl[irr], &workl[iri], &workl[
bounds], &workl[1], &workl[np + 1]);
if (msglvl > 2) {
igraphdvout_(&logfil, ncv, &workl[irr], &ndigit, "_neupd: Real part of"
" Ritz values after calling _NGETS.", (ftnlen)54);
igraphdvout_(&logfil, ncv, &workl[iri], &ndigit, "_neupd: Imag part of"
" Ritz values after calling _NGETS.", (ftnlen)54);
igraphdvout_(&logfil, ncv, &workl[bounds], &ndigit, "_neupd: Ritz valu"
"e indices after calling _NGETS.", (ftnlen)48);
}
/* %-----------------------------------------------------%
| Record indices of the converged wanted Ritz values |
| Mark the select array for possible reordering |
%-----------------------------------------------------% */
numcnv = 0;
i__1 = *ncv;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
d__1 = eps23, d__2 = igraphdlapy2_(&workl[irr + *ncv - j], &workl[iri +
*ncv - j]);
temp1 = max(d__1,d__2);
jj = (integer) workl[bounds + *ncv - j];
if (numcnv < nconv && workl[ibd + jj - 1] <= *tol * temp1) {
select[jj] = TRUE_;
++numcnv;
if (jj > nconv) {
reord = TRUE_;
}
}
/* L11: */
}
/* %-----------------------------------------------------------%
| Check the count (numcnv) of converged Ritz values with |
| the number (nconv) reported by dnaupd. If these two |
| are different then there has probably been an error |
| caused by incorrect passing of the dnaupd data. |
%-----------------------------------------------------------% */
if (msglvl > 2) {
igraphivout_(&logfil, &c__1, &numcnv, &ndigit, "_neupd: Number of spec"
"ified eigenvalues", (ftnlen)39);
igraphivout_(&logfil, &c__1, &nconv, &ndigit, "_neupd: Number of \"con"
"verged\" eigenvalues", (ftnlen)41);
}
if (numcnv != nconv) {
*info = -15;
goto L9000;
}
/* %-----------------------------------------------------------%
| Call LAPACK routine dlahqr to compute the real Schur form |
| of the upper Hessenberg matrix returned by DNAUPD . |
| Make a copy of the upper Hessenberg matrix. |
| Initialize the Schur vector matrix Q to the identity. |
%-----------------------------------------------------------% */
i__1 = ldh * *ncv;
igraphdcopy_(&i__1, &workl[ih], &c__1, &workl[iuptri], &c__1);
igraphdlaset_("All", ncv, ncv, &c_b37, &c_b38, &workl[invsub], &ldq);
igraphdlahqr_(&c_true, &c_true, ncv, &c__1, ncv, &workl[iuptri], &ldh, &
workl[iheigr], &workl[iheigi], &c__1, ncv, &workl[invsub], &
ldq, &ierr);
igraphdcopy_(ncv, &workl[invsub + *ncv - 1], &ldq, &workl[ihbds], &c__1);
if (ierr != 0) {
*info = -8;
goto L9000;
}
if (msglvl > 1) {
igraphdvout_(&logfil, ncv, &workl[iheigr], &ndigit, "_neupd: Real part"
" of the eigenvalues of H", (ftnlen)41);
igraphdvout_(&logfil, ncv, &workl[iheigi], &ndigit, "_neupd: Imaginary"
" part of the Eigenvalues of H", (ftnlen)46);
igraphdvout_(&logfil, ncv, &workl[ihbds], &ndigit, "_neupd: Last row o"
"f the Schur vector matrix", (ftnlen)43);
if (msglvl > 3) {
igraphdmout_(&logfil, ncv, ncv, &workl[iuptri], &ldh, &ndigit,
"_neupd: The upper quasi-triangular matrix ", (ftnlen)
42);
}
}
if (reord) {
/* %-----------------------------------------------------%
| Reorder the computed upper quasi-triangular matrix. |
%-----------------------------------------------------% */
igraphdtrsen_("None", "V", &select[1], ncv, &workl[iuptri], &ldh, &
workl[invsub], &ldq, &workl[iheigr], &workl[iheigi], &
nconv2, &conds, &sep, &workl[ihbds], ncv, iwork, &c__1, &
ierr);
if (nconv2 < nconv) {
nconv = nconv2;
}
if (ierr == 1) {
*info = 1;
goto L9000;
}
if (msglvl > 2) {
igraphdvout_(&logfil, ncv, &workl[iheigr], &ndigit, "_neupd: Real "
"part of the eigenvalues of H--reordered", (ftnlen)52);
igraphdvout_(&logfil, ncv, &workl[iheigi], &ndigit, "_neupd: Imag "
"part of the eigenvalues of H--reordered", (ftnlen)52);
if (msglvl > 3) {
igraphdmout_(&logfil, ncv, ncv, &workl[iuptri], &ldq, &ndigit,
"_neupd: Quasi-triangular matrix after re-orderi"
"ng", (ftnlen)49);
}
}
}
/* %---------------------------------------%
| Copy the last row of the Schur vector |
| into workl(ihbds). This will be used |
| to compute the Ritz estimates of |
| converged Ritz values. |
%---------------------------------------% */
igraphdcopy_(ncv, &workl[invsub + *ncv - 1], &ldq, &workl[ihbds], &c__1);
/* %----------------------------------------------------%
| Place the computed eigenvalues of H into DR and DI |
| if a spectral transformation was not used. |
%----------------------------------------------------% */
if (s_cmp(type__, "REGULR", (ftnlen)6, (ftnlen)6) == 0) {
igraphdcopy_(&nconv, &workl[iheigr], &c__1, &dr[1], &c__1);
igraphdcopy_(&nconv, &workl[iheigi], &c__1, &di[1], &c__1);
}
/* %----------------------------------------------------------%
| Compute the QR factorization of the matrix representing |
| the wanted invariant subspace located in the first NCONV |
| columns of workl(invsub,ldq). |
%----------------------------------------------------------% */
igraphdgeqr2_(ncv, &nconv, &workl[invsub], &ldq, &workev[1], &workev[*ncv +
1], &ierr);
/* %---------------------------------------------------------%
| * Postmultiply V by Q using dorm2r . |
| * Copy the first NCONV columns of VQ into Z. |
| * Postmultiply Z by R. |
| The N by NCONV matrix Z is now a matrix representation |
| of the approximate invariant subspace associated with |
| the Ritz values in workl(iheigr) and workl(iheigi) |
| The first NCONV columns of V are now approximate Schur |
| vectors associated with the real upper quasi-triangular |
| matrix of order NCONV in workl(iuptri) |
%---------------------------------------------------------% */
igraphdorm2r_("Right", "Notranspose", n, ncv, &nconv, &workl[invsub], &ldq,
&workev[1], &v[v_offset], ldv, &workd[*n + 1], &ierr);
igraphdlacpy_("All", n, &nconv, &v[v_offset], ldv, &z__[z_offset], ldz);
i__1 = nconv;
for (j = 1; j <= i__1; ++j) {
/* %---------------------------------------------------%
| Perform both a column and row scaling if the |
| diagonal element of workl(invsub,ldq) is negative |
| I'm lazy and don't take advantage of the upper |
| quasi-triangular form of workl(iuptri,ldq) |
| Note that since Q is orthogonal, R is a diagonal |
| matrix consisting of plus or minus ones |
%---------------------------------------------------% */
if (workl[invsub + (j - 1) * ldq + j - 1] < 0.) {
igraphdscal_(&nconv, &c_b64, &workl[iuptri + j - 1], &ldq);
igraphdscal_(&nconv, &c_b64, &workl[iuptri + (j - 1) * ldq], &c__1);
}
/* L20: */
}
if (*(unsigned char *)howmny == 'A') {
/* %--------------------------------------------%
| Compute the NCONV wanted eigenvectors of T |
| located in workl(iuptri,ldq). |
%--------------------------------------------% */
i__1 = *ncv;
for (j = 1; j <= i__1; ++j) {
if (j <= nconv) {
select[j] = TRUE_;
} else {
select[j] = FALSE_;
}
/* L30: */
}
igraphdtrevc_("Right", "Select", &select[1], ncv, &workl[iuptri], &ldq,
vl, &c__1, &workl[invsub], &ldq, ncv, &outncv, &workev[1],
&ierr);
if (ierr != 0) {
*info = -9;
goto L9000;
}
/* %------------------------------------------------%
| Scale the returning eigenvectors so that their |
| Euclidean norms are all one. LAPACK subroutine |
| dtrevc returns each eigenvector normalized so |
| that the element of largest magnitude has |
| magnitude 1; |
%------------------------------------------------% */
iconj = 0;
i__1 = nconv;
for (j = 1; j <= i__1; ++j) {
if (workl[iheigi + j - 1] == 0.) {
/* %----------------------%
| real eigenvalue case |
%----------------------% */
temp = igraphdnrm2_(ncv, &workl[invsub + (j - 1) * ldq], &c__1);
d__1 = 1. / temp;
igraphdscal_(ncv, &d__1, &workl[invsub + (j - 1) * ldq], &c__1);
} else {
/* %-------------------------------------------%
| Complex conjugate pair case. Note that |
| since the real and imaginary part of |
| the eigenvector are stored in consecutive |
| columns, we further normalize by the |
| square root of two. |
%-------------------------------------------% */
if (iconj == 0) {
d__1 = igraphdnrm2_(ncv, &workl[invsub + (j - 1) * ldq], &
c__1);
d__2 = igraphdnrm2_(ncv, &workl[invsub + j * ldq], &c__1);
temp = igraphdlapy2_(&d__1, &d__2);
d__1 = 1. / temp;
igraphdscal_(ncv, &d__1, &workl[invsub + (j - 1) * ldq], &
c__1);
d__1 = 1. / temp;
igraphdscal_(ncv, &d__1, &workl[invsub + j * ldq], &c__1);
iconj = 1;
} else {
iconj = 0;
}
}
/* L40: */
}
igraphdgemv_("T", ncv, &nconv, &c_b38, &workl[invsub], &ldq, &workl[
ihbds], &c__1, &c_b37, &workev[1], &c__1);
iconj = 0;
i__1 = nconv;
for (j = 1; j <= i__1; ++j) {
if (workl[iheigi + j - 1] != 0.) {
/* %-------------------------------------------%
| Complex conjugate pair case. Note that |
| since the real and imaginary part of |
| the eigenvector are stored in consecutive |
%-------------------------------------------% */
if (iconj == 0) {
workev[j] = igraphdlapy2_(&workev[j], &workev[j + 1]);
workev[j + 1] = workev[j];
iconj = 1;
} else {
iconj = 0;
}
}
/* L45: */
}
if (msglvl > 2) {
igraphdcopy_(ncv, &workl[invsub + *ncv - 1], &ldq, &workl[ihbds], &
c__1);
igraphdvout_(&logfil, ncv, &workl[ihbds], &ndigit, "_neupd: Last r"
"ow of the eigenvector matrix for T", (ftnlen)48);
if (msglvl > 3) {
igraphdmout_(&logfil, ncv, ncv, &workl[invsub], &ldq, &ndigit,
"_neupd: The eigenvector matrix for T", (ftnlen)
36);
}
}
/* %---------------------------------------%
| Copy Ritz estimates into workl(ihbds) |
%---------------------------------------% */
igraphdcopy_(&nconv, &workev[1], &c__1, &workl[ihbds], &c__1);
/* %---------------------------------------------------------%
| Compute the QR factorization of the eigenvector matrix |
| associated with leading portion of T in the first NCONV |
| columns of workl(invsub,ldq). |
%---------------------------------------------------------% */
igraphdgeqr2_(ncv, &nconv, &workl[invsub], &ldq, &workev[1], &workev[*
ncv + 1], &ierr);
/* %----------------------------------------------%
| * Postmultiply Z by Q. |
| * Postmultiply Z by R. |
| The N by NCONV matrix Z is now contains the |
| Ritz vectors associated with the Ritz values |
| in workl(iheigr) and workl(iheigi). |
%----------------------------------------------% */
igraphdorm2r_("Right", "Notranspose", n, ncv, &nconv, &workl[invsub], &
ldq, &workev[1], &z__[z_offset], ldz, &workd[*n + 1], &
ierr);
igraphdtrmm_("Right", "Upper", "No transpose", "Non-unit", n, &nconv, &
c_b38, &workl[invsub], &ldq, &z__[z_offset], ldz);
}
} else {
/* %------------------------------------------------------%
| An approximate invariant subspace is not needed. |
| Place the Ritz values computed DNAUPD into DR and DI |
%------------------------------------------------------% */
igraphdcopy_(&nconv, &workl[ritzr], &c__1, &dr[1], &c__1);
igraphdcopy_(&nconv, &workl[ritzi], &c__1, &di[1], &c__1);
igraphdcopy_(&nconv, &workl[ritzr], &c__1, &workl[iheigr], &c__1);
igraphdcopy_(&nconv, &workl[ritzi], &c__1, &workl[iheigi], &c__1);
igraphdcopy_(&nconv, &workl[bounds], &c__1, &workl[ihbds], &c__1);
}
/* %------------------------------------------------%
| Transform the Ritz values and possibly vectors |
| and corresponding error bounds of OP to those |
| of A*x = lambda*B*x. |
%------------------------------------------------% */
if (s_cmp(type__, "REGULR", (ftnlen)6, (ftnlen)6) == 0) {
if (*rvec) {
igraphdscal_(ncv, &rnorm, &workl[ihbds], &c__1);
}
} else {
/* %---------------------------------------%
| A spectral transformation was used. |
| * Determine the Ritz estimates of the |
| Ritz values in the original system. |
%---------------------------------------% */
if (s_cmp(type__, "SHIFTI", (ftnlen)6, (ftnlen)6) == 0) {
if (*rvec) {
igraphdscal_(ncv, &rnorm, &workl[ihbds], &c__1);
}
i__1 = *ncv;
for (k = 1; k <= i__1; ++k) {
temp = igraphdlapy2_(&workl[iheigr + k - 1], &workl[iheigi + k - 1])
;
workl[ihbds + k - 1] = (d__1 = workl[ihbds + k - 1], abs(d__1)
) / temp / temp;
/* L50: */
}
} else if (s_cmp(type__, "REALPT", (ftnlen)6, (ftnlen)6) == 0) {
i__1 = *ncv;
for (k = 1; k <= i__1; ++k) {
/* L60: */
}
} else if (s_cmp(type__, "IMAGPT", (ftnlen)6, (ftnlen)6) == 0) {
i__1 = *ncv;
for (k = 1; k <= i__1; ++k) {
/* L70: */
}
}
/* %-----------------------------------------------------------%
| * Transform the Ritz values back to the original system. |
| For TYPE = 'SHIFTI' the transformation is |
| lambda = 1/theta + sigma |
| For TYPE = 'REALPT' or 'IMAGPT' the user must from |
| Rayleigh quotients or a projection. See remark 3 above.|
| NOTES: |
| *The Ritz vectors are not affected by the transformation. |
%-----------------------------------------------------------% */
if (s_cmp(type__, "SHIFTI", (ftnlen)6, (ftnlen)6) == 0) {
i__1 = *ncv;
for (k = 1; k <= i__1; ++k) {
temp = igraphdlapy2_(&workl[iheigr + k - 1], &workl[iheigi + k - 1])
;
workl[iheigr + k - 1] = workl[iheigr + k - 1] / temp / temp +
*sigmar;
workl[iheigi + k - 1] = -workl[iheigi + k - 1] / temp / temp
+ *sigmai;
/* L80: */
}
igraphdcopy_(&nconv, &workl[iheigr], &c__1, &dr[1], &c__1);
igraphdcopy_(&nconv, &workl[iheigi], &c__1, &di[1], &c__1);
} else if (s_cmp(type__, "REALPT", (ftnlen)6, (ftnlen)6) == 0 ||
s_cmp(type__, "IMAGPT", (ftnlen)6, (ftnlen)6) == 0) {
igraphdcopy_(&nconv, &workl[iheigr], &c__1, &dr[1], &c__1);
igraphdcopy_(&nconv, &workl[iheigi], &c__1, &di[1], &c__1);
}
}
if (s_cmp(type__, "SHIFTI", (ftnlen)6, (ftnlen)6) == 0 && msglvl > 1) {
igraphdvout_(&logfil, &nconv, &dr[1], &ndigit, "_neupd: Untransformed real"
" part of the Ritz valuess.", (ftnlen)52);
igraphdvout_(&logfil, &nconv, &di[1], &ndigit, "_neupd: Untransformed imag"
" part of the Ritz valuess.", (ftnlen)52);
igraphdvout_(&logfil, &nconv, &workl[ihbds], &ndigit, "_neupd: Ritz estima"
"tes of untransformed Ritz values.", (ftnlen)52);
} else if (s_cmp(type__, "REGULR", (ftnlen)6, (ftnlen)6) == 0 && msglvl >
1) {
igraphdvout_(&logfil, &nconv, &dr[1], &ndigit, "_neupd: Real parts of conv"
"erged Ritz values.", (ftnlen)44);
igraphdvout_(&logfil, &nconv, &di[1], &ndigit, "_neupd: Imag parts of conv"
"erged Ritz values.", (ftnlen)44);
igraphdvout_(&logfil, &nconv, &workl[ihbds], &ndigit, "_neupd: Associated "
"Ritz estimates.", (ftnlen)34);
}
/* %-------------------------------------------------%
| Eigenvector Purification step. Formally perform |
| one of inverse subspace iteration. Only used |
| for MODE = 2. |
%-------------------------------------------------% */
if (*rvec && *(unsigned char *)howmny == 'A' && s_cmp(type__, "SHIFTI", (
ftnlen)6, (ftnlen)6) == 0) {
/* %------------------------------------------------%
| Purify the computed Ritz vectors by adding a |
| little bit of the residual vector: |
| T |
| resid(:)*( e s ) / theta |
| NCV |
| where H s = s theta. Remember that when theta |
| has nonzero imaginary part, the corresponding |
| Ritz vector is stored across two columns of Z. |
%------------------------------------------------% */
iconj = 0;
i__1 = nconv;
for (j = 1; j <= i__1; ++j) {
if (workl[iheigi + j - 1] == 0. && workl[iheigr + j - 1] != 0.) {
workev[j] = workl[invsub + (j - 1) * ldq + *ncv - 1] / workl[
iheigr + j - 1];
} else if (iconj == 0) {
temp = igraphdlapy2_(&workl[iheigr + j - 1], &workl[iheigi + j - 1])
;
if (temp != 0.) {
workev[j] = (workl[invsub + (j - 1) * ldq + *ncv - 1] *
workl[iheigr + j - 1] + workl[invsub + j * ldq + *
ncv - 1] * workl[iheigi + j - 1]) / temp / temp;
workev[j + 1] = (workl[invsub + j * ldq + *ncv - 1] *
workl[iheigr + j - 1] - workl[invsub + (j - 1) *
ldq + *ncv - 1] * workl[iheigi + j - 1]) / temp /
temp;
}
iconj = 1;
} else {
iconj = 0;
}
/* L110: */
}
/* %---------------------------------------%
| Perform a rank one update to Z and |
| purify all the Ritz vectors together. |
%---------------------------------------% */
igraphdger_(n, &nconv, &c_b38, &resid[1], &c__1, &workev[1], &c__1, &z__[
z_offset], ldz);
}
L9000:
return 0;
/* %---------------%
| End of DNEUPD |
%---------------% */
} /* igraphdneupd_ */
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