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c\BeginDoc
c
c\Name: igraphdneupd
c
c\Description:
c
c This subroutine returns the converged approximations to eigenvalues
c of A*z = lambda*B*z and (optionally):
c
c (1) The corresponding approximate eigenvectors;
c
c (2) An orthonormal basis for the associated approximate
c invariant subspace;
c
c (3) Both.
c
c There is negligible additional cost to obtain eigenvectors. An orthonormal
c basis is always computed. There is an additional storage cost of n*nev
c if both are requested (in this case a separate array Z must be supplied).
c
c The approximate eigenvalues and eigenvectors of A*z = lambda*B*z
c are derived from approximate eigenvalues and eigenvectors of
c of the linear operator OP prescribed by the MODE selection in the
c call to DNAUPD. DNAUPD must be called before this routine is called.
c These approximate eigenvalues and vectors are commonly called Ritz
c values and Ritz vectors respectively. They are referred to as such
c in the comments that follow. The computed orthonormal basis for the
c invariant subspace corresponding to these Ritz values is referred to as a
c Schur basis.
c
c See documentation in the header of the subroutine DNAUPD for
c definition of OP as well as other terms and the relation of computed
c Ritz values and Ritz vectors of OP with respect to the given problem
c A*z = lambda*B*z. For a brief description, see definitions of
c IPARAM(7), MODE and WHICH in the documentation of DNAUPD.
c
c\Usage:
c call igraphdneupd
c ( RVEC, HOWMNY, SELECT, DR, DI, Z, LDZ, SIGMAR, SIGMAI, WORKEV, BMAT,
c N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR, WORKD, WORKL,
c LWORKL, INFO )
c
c\Arguments:
c RVEC LOGICAL (INPUT)
c Specifies whether a basis for the invariant subspace corresponding
c to the converged Ritz value approximations for the eigenproblem
c A*z = lambda*B*z is computed.
c
c RVEC = .FALSE. Compute Ritz values only.
c
c RVEC = .TRUE. Compute the Ritz vectors or Schur vectors.
c See Remarks below.
c
c HOWMNY Character*1 (INPUT)
c Specifies the form of the basis for the invariant subspace
c corresponding to the converged Ritz values that is to be computed.
c
c = 'A': Compute NEV Ritz vectors;
c = 'P': Compute NEV Schur vectors;
c = 'S': compute some of the Ritz vectors, specified
c by the logical array SELECT.
c
c SELECT Logical array of dimension NCV. (INPUT)
c If HOWMNY = 'S', SELECT specifies the Ritz vectors to be
c computed. To select the Ritz vector corresponding to a
c Ritz value (DR(j), DI(j)), SELECT(j) must be set to .TRUE..
c If HOWMNY = 'A' or 'P', SELECT is used as internal workspace.
c
c DR Double precision array of dimension NEV+1. (OUTPUT)
c If IPARAM(7) = 1,2 or 3 and SIGMAI=0.0 then on exit: DR contains
c the real part of the Ritz approximations to the eigenvalues of
c A*z = lambda*B*z.
c If IPARAM(7) = 3, 4 and SIGMAI is not equal to zero, then on exit:
c DR contains the real part of the Ritz values of OP computed by
c DNAUPD. A further computation must be performed by the user
c to transform the Ritz values computed for OP by DNAUPD to those
c of the original system A*z = lambda*B*z. See remark 3 below.
c
c DI Double precision array of dimension NEV+1. (OUTPUT)
c On exit, DI contains the imaginary part of the Ritz value
c approximations to the eigenvalues of A*z = lambda*B*z associated
c with DR.
c
c NOTE: When Ritz values are complex, they will come in complex
c conjugate pairs. If eigenvectors are requested, the
c corresponding Ritz vectors will also come in conjugate
c pairs and the real and imaginary parts of these are
c represented in two consecutive columns of the array Z
c (see below).
c
c Z Double precision N by NEV+1 array if RVEC = .TRUE. and HOWMNY = 'A'. (OUTPUT)
c On exit, if RVEC = .TRUE. and HOWMNY = 'A', then the columns of
c Z represent approximate eigenvectors (Ritz vectors) corresponding
c to the NCONV=IPARAM(5) Ritz values for eigensystem
c A*z = lambda*B*z.
c
c The complex Ritz vector associated with the Ritz value
c with positive imaginary part is stored in two consecutive
c columns. The first column holds the real part of the Ritz
c vector and the igraphsecond column holds the imaginary part. The
c Ritz vector associated with the Ritz value with negative
c imaginary part is simply the complex conjugate of the Ritz vector
c associated with the positive imaginary part.
c
c If RVEC = .FALSE. or HOWMNY = 'P', then Z is not referenced.
c
c NOTE: If if RVEC = .TRUE. and a Schur basis is not required,
c the array Z may be set equal to first NEV+1 columns of the Arnoldi
c basis array V computed by DNAUPD. In this case the Arnoldi basis
c will be destroyed and overwritten with the eigenvector basis.
c
c LDZ Integer. (INPUT)
c The leading dimension of the array Z. If Ritz vectors are
c desired, then LDZ >= max( 1, N ). In any case, LDZ >= 1.
c
c SIGMAR Double precision (INPUT)
c If IPARAM(7) = 3 or 4, represents the real part of the shift.
c Not referenced if IPARAM(7) = 1 or 2.
c
c SIGMAI Double precision (INPUT)
c If IPARAM(7) = 3 or 4, represents the imaginary part of the shift.
c Not referenced if IPARAM(7) = 1 or 2. See remark 3 below.
c
c WORKEV Double precision work array of dimension 3*NCV. (WORKSPACE)
c
c **** The remaining arguments MUST be the same as for the ****
c **** call to DNAUPD that was just completed. ****
c
c NOTE: The remaining arguments
c
c BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR,
c WORKD, WORKL, LWORKL, INFO
c
c must be passed directly to DNEUPD following the last call
c to DNAUPD. These arguments MUST NOT BE MODIFIED between
c the the last call to DNAUPD and the call to DNEUPD.
c
c Three of these parameters (V, WORKL, INFO) are also output parameters:
c
c V Double precision N by NCV array. (INPUT/OUTPUT)
c
c Upon INPUT: the NCV columns of V contain the Arnoldi basis
c vectors for OP as constructed by DNAUPD .
c
c Upon OUTPUT: If RVEC = .TRUE. the first NCONV=IPARAM(5) columns
c contain approximate Schur vectors that span the
c desired invariant subspace. See Remark 2 below.
c
c NOTE: If the array Z has been set equal to first NEV+1 columns
c of the array V and RVEC=.TRUE. and HOWMNY= 'A', then the
c Arnoldi basis held by V has been overwritten by the desired
c Ritz vectors. If a separate array Z has been passed then
c the first NCONV=IPARAM(5) columns of V will contain approximate
c Schur vectors that span the desired invariant subspace.
c
c WORKL Double precision work array of length LWORKL. (OUTPUT/WORKSPACE)
c WORKL(1:ncv*ncv+3*ncv) contains information obtained in
c igraphdnaupd. They are not changed by igraphdneupd.
c WORKL(ncv*ncv+3*ncv+1:3*ncv*ncv+6*ncv) holds the
c real and imaginary part of the untransformed Ritz values,
c the upper quasi-triangular matrix for H, and the
c associated matrix representation of the invariant subspace for H.
c
c Note: IPNTR(9:13) contains the pointer into WORKL for addresses
c of the above information computed by igraphdneupd.
c -------------------------------------------------------------
c IPNTR(9): pointer to the real part of the NCV RITZ values of the
c original system.
c IPNTR(10): pointer to the imaginary part of the NCV RITZ values of
c the original system.
c IPNTR(11): pointer to the NCV corresponding error bounds.
c IPNTR(12): pointer to the NCV by NCV upper quasi-triangular
c Schur matrix for H.
c IPNTR(13): pointer to the NCV by NCV matrix of eigenvectors
c of the upper Hessenberg matrix H. Only referenced by
c igraphdneupd if RVEC = .TRUE. See Remark 2 below.
c -------------------------------------------------------------
c
c INFO Integer. (OUTPUT)
c Error flag on output.
c
c = 0: Normal exit.
c
c = 1: The Schur form computed by LAPACK routine dlahqr
c could not be reordered by LAPACK routine dtrsen.
c Re-enter subroutine igraphdneupd with IPARAM(5)=NCV and
c increase the size of the arrays DR and DI to have
c dimension at least dimension NCV and allocate at least NCV
c columns for Z. NOTE: Not necessary if Z and V share
c the same space. Please notify the authors if this error
c occurs.
c
c = -1: N must be positive.
c = -2: NEV must be positive.
c = -3: NCV-NEV >= 2 and less than or equal to N.
c = -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'
c = -6: BMAT must be one of 'I' or 'G'.
c = -7: Length of private work WORKL array is not sufficient.
c = -8: Error return from calculation of a real Schur form.
c Informational error from LAPACK routine dlahqr.
c = -9: Error return from calculation of eigenvectors.
c Informational error from LAPACK routine dtrevc.
c = -10: IPARAM(7) must be 1,2,3,4.
c = -11: IPARAM(7) = 1 and BMAT = 'G' are incompatible.
c = -12: HOWMNY = 'S' not yet implemented
c = -13: HOWMNY must be one of 'A' or 'P' if RVEC = .true.
c = -14: DNAUPD did not find any eigenvalues to sufficient
c accuracy.
c
c\BeginLib
c
c\References:
c 1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
c a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
c pp 357-385.
c 2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly
c Restarted Arnoldi Iteration", Rice University Technical Report
c TR95-13, Department of Computational and Applied Mathematics.
c 3. B.N. Parlett & Y. Saad, "Complex Shift and Invert Strategies for
c Real Matrices", Linear Algebra and its Applications, vol 88/89,
c pp 575-595, (1987).
c
c\Routines called:
c igraphivout ARPACK utility routine that prints integers.
c igraphdmout ARPACK utility routine that prints matrices
c igraphdvout ARPACK utility routine that prints vectors.
c dgeqr2 LAPACK routine that computes the QR factorization of
c a matrix.
c dlacpy LAPACK matrix copy routine.
c dlahqr LAPACK routine to compute the real Schur form of an
c upper Hessenberg matrix.
c dlamch LAPACK routine that determines machine constants.
c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c dlaset LAPACK matrix initialization routine.
c dorm2r LAPACK routine that applies an orthogonal matrix in
c factored form.
c dtrevc LAPACK routine to compute the eigenvectors of a matrix
c in upper quasi-triangular form.
c dtrsen LAPACK routine that re-orders the Schur form.
c dtrmm Level 3 BLAS matrix times an upper triangular matrix.
c dger Level 2 BLAS rank one update to a matrix.
c dcopy Level 1 BLAS that copies one vector to another .
c ddot Level 1 BLAS that computes the scalar product of two vectors.
c dnrm2 Level 1 BLAS that computes the norm of a vector.
c dscal Level 1 BLAS that scales a vector.
c
c\Remarks
c
c 1. Currently only HOWMNY = 'A' and 'P' are implemented.
c
c Let X' denote the transpose of X.
c
c 2. Schur vectors are an orthogonal representation for the basis of
c Ritz vectors. Thus, their numerical properties are often superior.
c If RVEC = .TRUE. then the relationship
c A * V(:,1:IPARAM(5)) = V(:,1:IPARAM(5)) * T, and
c V(:,1:IPARAM(5))' * V(:,1:IPARAM(5)) = I are approximately satisfied.
c Here T is the leading submatrix of order IPARAM(5) of the real
c upper quasi-triangular matrix stored workl(ipntr(12)). That is,
c T is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
c each 2-by-2 diagonal block has its diagonal elements equal and its
c off-diagonal elements of opposite sign. Corresponding to each 2-by-2
c diagonal block is a complex conjugate pair of Ritz values. The real
c Ritz values are stored on the diagonal of T.
c
c 3. If IPARAM(7) = 3 or 4 and SIGMAI is not equal zero, then the user must
c form the IPARAM(5) Rayleigh quotients in order to transform the Ritz
c values computed by DNAUPD for OP to those of A*z = lambda*B*z.
c Set RVEC = .true. and HOWMNY = 'A', and
c compute
c Z(:,I)' * A * Z(:,I) if DI(I) = 0.
c If DI(I) is not equal to zero and DI(I+1) = - D(I),
c then the desired real and imaginary parts of the Ritz value are
c Z(:,I)' * A * Z(:,I) + Z(:,I+1)' * A * Z(:,I+1),
c Z(:,I)' * A * Z(:,I+1) - Z(:,I+1)' * A * Z(:,I), respectively.
c Another possibility is to set RVEC = .true. and HOWMNY = 'P' and
c compute V(:,1:IPARAM(5))' * A * V(:,1:IPARAM(5)) and then an upper
c quasi-triangular matrix of order IPARAM(5) is computed. See remark
c 2 above.
c
c\Authors
c Danny Sorensen Phuong Vu
c Richard Lehoucq CRPC / Rice University
c Chao Yang Houston, Texas
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\SCCS Information: @(#)
c FILE: neupd.F SID: 2.5 DATE OF SID: 7/31/96 RELEASE: 2
c
c\EndLib
c
c-----------------------------------------------------------------------
subroutine igraphdneupd (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)
c
c %----------------------------------------------------%
c | Include files for debugging and timing information |
c %----------------------------------------------------%
c
include 'debug.h'
include 'stat.h'
c
c %------------------%
c | Scalar Arguments |
c %------------------%
c
character bmat, howmny, which*2
logical rvec
integer info, ldz, ldv, lworkl, n, ncv, nev
Double precision
& sigmar, sigmai, tol
c
c %-----------------%
c | Array Arguments |
c %-----------------%
c
integer iparam(11), ipntr(14)
logical select(ncv)
Double precision
& dr(nev+1), di(nev+1), resid(n), v(ldv,ncv), z(ldz,*),
& workd(3*n), workl(lworkl), workev(3*ncv)
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& one, zero
parameter (one = 1.0D+0, zero = 0.0D+0)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character type*6
integer bounds, ierr, ih, ihbds, iheigr, iheigi, iconj, nconv,
& invsub, iuptri, iwev, iwork(1), j, k, ktrord,
& ldh, ldq, mode, msglvl, outncv, ritzr, ritzi, wri, wrr,
& irr, iri, ibd
logical reord
Double precision
& conds, rnorm, sep, temp, thres, vl(1,1), temp1, eps23
c
c %----------------------%
c | External Subroutines |
c %----------------------%
c
external dcopy, dger, dgeqr2, dlacpy, dlahqr, dlaset,
& igraphdmout, dorm2r, dtrevc, dtrmm, dtrsen, dscal,
& igraphdvout, igraphivout
c
c %--------------------%
c | External Functions |
c %--------------------%
c
Double precision
& dlapy2, dnrm2, dlamch, ddot
external dlapy2, dnrm2, dlamch, ddot
c
c %---------------------%
c | Intrinsic Functions |
c %---------------------%
c
intrinsic abs, min, sqrt
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
c %------------------------%
c | Set default parameters |
c %------------------------%
c
msglvl = mneupd
mode = iparam(7)
nconv = iparam(5)
info = 0
c
c %---------------------------------%
c | Get machine dependent constant. |
c %---------------------------------%
c
eps23 = dlamch('Epsilon-Machine')
eps23 = eps23**(2.0D+0 / 3.0D+0)
c
c %--------------%
c | Quick return |
c %--------------%
c
ierr = 0
c
if (nconv .le. 0) then
ierr = -14
else if (n .le. 0) then
ierr = -1
else if (nev .le. 0) then
ierr = -2
else if (ncv .le. nev+1 .or. ncv .gt. n) then
ierr = -3
else if (which .ne. 'LM' .and.
& which .ne. 'SM' .and.
& which .ne. 'LR' .and.
& which .ne. 'SR' .and.
& which .ne. 'LI' .and.
& which .ne. 'SI') then
ierr = -5
else if (bmat .ne. 'I' .and. bmat .ne. 'G') then
ierr = -6
else if (lworkl .lt. 3*ncv**2 + 6*ncv) then
ierr = -7
else if ( (howmny .ne. 'A' .and.
& howmny .ne. 'P' .and.
& howmny .ne. 'S') .and. rvec ) then
ierr = -13
else if (howmny .eq. 'S' ) then
ierr = -12
end if
c
if (mode .eq. 1 .or. mode .eq. 2) then
type = 'REGULR'
else if (mode .eq. 3 .and. sigmai .eq. zero) then
type = 'SHIFTI'
else if (mode .eq. 3 ) then
type = 'REALPT'
else if (mode .eq. 4 ) then
type = 'IMAGPT'
else
ierr = -10
end if
if (mode .eq. 1 .and. bmat .eq. 'G') ierr = -11
c
c %------------%
c | Error Exit |
c %------------%
c
if (ierr .ne. 0) then
info = ierr
go to 9000
end if
c
c %--------------------------------------------------------%
c | Pointer into WORKL for address of H, RITZ, BOUNDS, Q |
c | etc... and the remaining workspace. |
c | Also update pointer to be used on output. |
c | Memory is laid out as follows: |
c | workl(1:ncv*ncv) := generated Hessenberg matrix |
c | workl(ncv*ncv+1:ncv*ncv+2*ncv) := real and imaginary |
c | parts of ritz values |
c | workl(ncv*ncv+2*ncv+1:ncv*ncv+3*ncv) := error bounds |
c %--------------------------------------------------------%
c
c %-----------------------------------------------------------%
c | The following is used and set by DNEUPD. |
c | workl(ncv*ncv+3*ncv+1:ncv*ncv+4*ncv) := The untransformed |
c | real part of the Ritz values. |
c | workl(ncv*ncv+4*ncv+1:ncv*ncv+5*ncv) := The untransformed |
c | imaginary part of the Ritz values. |
c | workl(ncv*ncv+5*ncv+1:ncv*ncv+6*ncv) := The untransformed |
c | error bounds of the Ritz values |
c | workl(ncv*ncv+6*ncv+1:2*ncv*ncv+6*ncv) := Holds the upper |
c | quasi-triangular matrix for H |
c | workl(2*ncv*ncv+6*ncv+1: 3*ncv*ncv+6*ncv) := Holds the |
c | associated matrix representation of the invariant |
c | subspace for H. |
c | GRAND total of NCV * ( 3 * NCV + 6 ) locations. |
c %-----------------------------------------------------------%
c
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
c
c %-----------------------------------------%
c | irr points to the REAL part of the Ritz |
c | values computed by _neigh before |
c | exiting _naup2. |
c | iri points to the IMAGINARY part of the |
c | Ritz values computed by _neigh |
c | before exiting _naup2. |
c | ibd points to the Ritz estimates |
c | computed by _neigh before exiting |
c | _naup2. |
c %-----------------------------------------%
c
irr = ipntr(14)+ncv*ncv
iri = irr+ncv
ibd = iri+ncv
c
c %------------------------------------%
c | RNORM is B-norm of the RESID(1:N). |
c %------------------------------------%
c
rnorm = workl(ih+2)
workl(ih+2) = zero
c
if (rvec) then
c
c %-------------------------------------------%
c | Get converged Ritz value on the boundary. |
c | Note: converged Ritz values have been |
c | placed in the first NCONV locations in |
c | workl(ritzr) and workl(ritzi). They have |
c | been sorted (in _naup2) according to the |
c | WHICH selection criterion. |
c %-------------------------------------------%
c
if (which .eq. 'LM' .or. which .eq. 'SM') then
thres = dlapy2( workl(ritzr), workl(ritzi) )
else if (which .eq. 'LR' .or. which .eq. 'SR') then
thres = workl(ritzr)
else if (which .eq. 'LI' .or. which .eq. 'SI') then
thres = abs( workl(ritzi) )
end if
c
if (msglvl .gt. 2) then
call igraphdvout(logfil, 1, thres, ndigit,
& '_neupd: Threshold eigenvalue used for re-ordering')
end if
c
c %----------------------------------------------------------%
c | Check to see if all converged Ritz values appear at the |
c | top of the upper quasi-triangular matrix computed by |
c | _neigh in _naup2. This is done in the following way: |
c | |
c | 1) For each Ritz value obtained from _neigh, compare it |
c | with the threshold Ritz value computed above to |
c | determine whether it is a wanted one. |
c | |
c | 2) If it is wanted, then check the corresponding Ritz |
c | estimate to see if it has converged. If it has, set |
c | correponding entry in the logical array SELECT to |
c | .TRUE.. |
c | |
c | If SELECT(j) = .TRUE. and j > NCONV, then there is a |
c | converged Ritz value that does not appear at the top of |
c | the upper quasi-triangular matrix computed by _neigh in |
c | _naup2. Reordering is needed. |
c %----------------------------------------------------------%
c
reord = .false.
ktrord = 0
do 10 j = 0, ncv-1
select(j+1) = .false.
if (which .eq. 'LM') then
if (dlapy2(workl(irr+j), workl(iri+j))
& .ge. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
else if (which .eq. 'SM') then
if (dlapy2(workl(irr+j), workl(iri+j))
& .le. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
else if (which .eq. 'LR') then
if (workl(irr+j) .ge. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
else if (which .eq. 'SR') then
if (workl(irr+j) .le. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
else if (which .eq. 'LI') then
if (abs(workl(iri+j)) .ge. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
else if (which .eq. 'SI') then
if (abs(workl(iri+j)) .le. thres) then
temp1 = max( eps23,
& dlapy2( workl(irr+j), workl(iri+j) ) )
if (workl(ibd+j) .le. tol*temp1)
& select(j+1) = .true.
end if
end if
if (j+1 .gt. nconv ) reord = ( select(j+1) .or. reord )
if (select(j+1)) ktrord = ktrord + 1
10 continue
c
if (msglvl .gt. 2) then
call igraphivout(logfil, 1, ktrord, ndigit,
& '_neupd: Number of specified eigenvalues')
call igraphivout(logfil, 1, nconv, ndigit,
& '_neupd: Number of "converged" eigenvalues')
end if
c
c %-----------------------------------------------------------%
c | Call LAPACK routine dlahqr to compute the real Schur form |
c | of the upper Hessenberg matrix returned by DNAUPD. |
c | Make a copy of the upper Hessenberg matrix. |
c | Initialize the Schur vector matrix Q to the identity. |
c %-----------------------------------------------------------%
c
call dcopy (ldh*ncv, workl(ih), 1, workl(iuptri), 1)
call dlaset ('All', ncv, ncv, zero, one, workl(invsub), ldq)
call dlahqr (.true., .true., ncv, 1, ncv, workl(iuptri), ldh,
& workl(iheigr), workl(iheigi), 1, ncv,
& workl(invsub), ldq, ierr)
call dcopy (ncv, workl(invsub+ncv-1), ldq, workl(ihbds), 1)
c
if (ierr .ne. 0) then
info = -8
go to 9000
end if
c
if (msglvl .gt. 1) then
call igraphdvout (logfil, ncv, workl(iheigr), ndigit,
& '_neupd: Real part of the eigenvalues of H')
call igraphdvout (logfil, ncv, workl(iheigi), ndigit,
& '_neupd: Imaginary part of the Eigenvalues of H')
call igraphdvout (logfil, ncv, workl(ihbds), ndigit,
& '_neupd: Last row of the Schur vector matrix')
if (msglvl .gt. 3) then
call igraphdmout (logfil, ncv, ncv, workl(iuptri), ldh,
& ndigit,
& '_neupd: The upper quasi-triangular matrix ')
end if
end if
c
if (reord) then
c
c %-----------------------------------------------------%
c | Reorder the computed upper quasi-triangular matrix. |
c %-----------------------------------------------------%
c
call dtrsen ('None', 'V', select, ncv, workl(iuptri), ldh,
& workl(invsub), ldq, workl(iheigr), workl(iheigi),
& nconv, conds, sep, workl(ihbds), ncv, iwork, 1, ierr)
c
if (ierr .eq. 1) then
info = 1
go to 9000
end if
c
if (msglvl .gt. 2) then
call igraphdvout (logfil, ncv, workl(iheigr), ndigit,
& '_neupd: Real part of the eigenvalues of H--reordered')
call igraphdvout (logfil, ncv, workl(iheigi), ndigit,
& '_neupd: Imag part of the eigenvalues of H--reordered')
if (msglvl .gt. 3) then
call igraphdmout (logfil, ncv, ncv, workl(iuptri),
& ldq, ndigit,
& '_neupd: Quasi-triangular matrix after re-ordering')
end if
end if
c
end if
c
c %---------------------------------------%
c | Copy the last row of the Schur vector |
c | into workl(ihbds). This will be used |
c | to compute the Ritz estimates of |
c | converged Ritz values. |
c %---------------------------------------%
c
call dcopy(ncv, workl(invsub+ncv-1), ldq, workl(ihbds), 1)
c
c %----------------------------------------------------%
c | Place the computed eigenvalues of H into DR and DI |
c | if a spectral transformation was not used. |
c %----------------------------------------------------%
c
if (type .eq. 'REGULR') then
call dcopy (nconv, workl(iheigr), 1, dr, 1)
call dcopy (nconv, workl(iheigi), 1, di, 1)
end if
c
c %----------------------------------------------------------%
c | Compute the QR factorization of the matrix representing |
c | the wanted invariant subspace located in the first NCONV |
c | columns of workl(invsub,ldq). |
c %----------------------------------------------------------%
c
call dgeqr2 (ncv, nconv, workl(invsub), ldq, workev,
& workev(ncv+1), ierr)
c
c %---------------------------------------------------------%
c | * Postmultiply V by Q using dorm2r. |
c | * Copy the first NCONV columns of VQ into Z. |
c | * Postmultiply Z by R. |
c | The N by NCONV matrix Z is now a matrix representation |
c | of the approximate invariant subspace associated with |
c | the Ritz values in workl(iheigr) and workl(iheigi) |
c | The first NCONV columns of V are now approximate Schur |
c | vectors associated with the real upper quasi-triangular |
c | matrix of order NCONV in workl(iuptri) |
c %---------------------------------------------------------%
c
call dorm2r ('Right', 'Notranspose', n, ncv, nconv,
& workl(invsub), ldq, workev, v, ldv, workd(n+1), ierr)
call dlacpy ('All', n, nconv, v, ldv, z, ldz)
c
do 20 j=1, nconv
c
c %---------------------------------------------------%
c | Perform both a column and row scaling if the |
c | diagonal element of workl(invsub,ldq) is negative |
c | I'm lazy and don't take advantage of the upper |
c | quasi-triangular form of workl(iuptri,ldq) |
c | Note that since Q is orthogonal, R is a diagonal |
c | matrix consisting of plus or minus ones |
c %---------------------------------------------------%
c
if (workl(invsub+(j-1)*ldq+j-1) .lt. zero) then
call dscal (nconv, -one, workl(iuptri+j-1), ldq)
call dscal (nconv, -one, workl(iuptri+(j-1)*ldq), 1)
end if
c
20 continue
c
if (howmny .eq. 'A') then
c
c %--------------------------------------------%
c | Compute the NCONV wanted eigenvectors of T |
c | located in workl(iuptri,ldq). |
c %--------------------------------------------%
c
do 30 j=1, ncv
if (j .le. nconv) then
select(j) = .true.
else
select(j) = .false.
end if
30 continue
c
call dtrevc ('Right', 'Select', select, ncv, workl(iuptri),
& ldq, vl, 1, workl(invsub), ldq, ncv, outncv, workev,
& ierr)
c
if (ierr .ne. 0) then
info = -9
go to 9000
end if
c
c %------------------------------------------------%
c | Scale the returning eigenvectors so that their |
c | Euclidean norms are all one. LAPACK subroutine |
c | dtrevc returns each eigenvector normalized so |
c | that the element of largest magnitude has |
c | magnitude 1; |
c %------------------------------------------------%
c
iconj = 0
do 40 j=1, nconv
c
if ( workl(iheigi+j-1) .eq. zero ) then
c
c %----------------------%
c | real eigenvalue case |
c %----------------------%
c
temp = dnrm2( ncv, workl(invsub+(j-1)*ldq), 1 )
call dscal ( ncv, one / temp,
& workl(invsub+(j-1)*ldq), 1 )
c
else
c
c %-------------------------------------------%
c | Complex conjugate pair case. Note that |
c | since the real and imaginary part of |
c | the eigenvector are stored in consecutive |
c | columns, we further normalize by the |
c | square root of two. |
c %-------------------------------------------%
c
if (iconj .eq. 0) then
temp = dlapy2( dnrm2( ncv, workl(invsub+(j-1)*ldq),
& 1 ), dnrm2( ncv, workl(invsub+j*ldq), 1) )
call dscal ( ncv, one / temp,
& workl(invsub+(j-1)*ldq), 1 )
call dscal ( ncv, one / temp,
& workl(invsub+j*ldq), 1 )
iconj = 1
else
iconj = 0
end if
c
end if
c
40 continue
c
call dgemv('T', ncv, nconv, one, workl(invsub),
& ldq, workl(ihbds), 1, zero, workev, 1)
c
iconj = 0
do 45 j=1, nconv
if (workl(iheigi+j-1) .ne. zero) then
c
c %-------------------------------------------%
c | Complex conjugate pair case. Note that |
c | since the real and imaginary part of |
c | the eigenvector are stored in consecutive |
c %-------------------------------------------%
c
if (iconj .eq. 0) then
workev(j) = dlapy2(workev(j), workev(j+1))
workev(j+1) = workev(j)
iconj = 1
else
iconj = 0
end if
end if
45 continue
c
if (msglvl .gt. 2) then
call dcopy(ncv, workl(invsub+ncv-1), ldq,
& workl(ihbds), 1)
call igraphdvout (logfil, ncv, workl(ihbds), ndigit,
& '_neupd: Last row of the eigenvector matrix for T')
if (msglvl .gt. 3) then
call igraphdmout (logfil, ncv, ncv, workl(invsub),
& ldq, ndigit,
& '_neupd: The eigenvector matrix for T')
end if
end if
c
c %---------------------------------------%
c | Copy Ritz estimates into workl(ihbds) |
c %---------------------------------------%
c
call dcopy(nconv, workev, 1, workl(ihbds), 1)
c
c %---------------------------------------------------------%
c | Compute the QR factorization of the eigenvector matrix |
c | associated with leading portion of T in the first NCONV |
c | columns of workl(invsub,ldq). |
c %---------------------------------------------------------%
c
call dgeqr2 (ncv, nconv, workl(invsub), ldq, workev,
& workev(ncv+1), ierr)
c
c %----------------------------------------------%
c | * Postmultiply Z by Q. |
c | * Postmultiply Z by R. |
c | The N by NCONV matrix Z is now contains the |
c | Ritz vectors associated with the Ritz values |
c | in workl(iheigr) and workl(iheigi). |
c %----------------------------------------------%
c
call dorm2r ('Right', 'Notranspose', n, ncv, nconv,
& workl(invsub), ldq, workev, z, ldz, workd(n+1), ierr)
c
call dtrmm ('Right', 'Upper', 'No transpose', 'Non-unit',
& n, nconv, one, workl(invsub), ldq, z, ldz)
c
end if
c
else
c
c %------------------------------------------------------%
c | An approximate invariant subspace is not needed. |
c | Place the Ritz values computed DNAUPD into DR and DI |
c %------------------------------------------------------%
c
call dcopy (nconv, workl(ritzr), 1, dr, 1)
call dcopy (nconv, workl(ritzi), 1, di, 1)
call dcopy (nconv, workl(ritzr), 1, workl(iheigr), 1)
call dcopy (nconv, workl(ritzi), 1, workl(iheigi), 1)
call dcopy (nconv, workl(bounds), 1, workl(ihbds), 1)
end if
c
c %------------------------------------------------%
c | Transform the Ritz values and possibly vectors |
c | and corresponding error bounds of OP to those |
c | of A*x = lambda*B*x. |
c %------------------------------------------------%
c
if (type .eq. 'REGULR') then
c
if (rvec)
& call dscal (ncv, rnorm, workl(ihbds), 1)
c
else
c
c %---------------------------------------%
c | A spectral transformation was used. |
c | * Determine the Ritz estimates of the |
c | Ritz values in the original system. |
c %---------------------------------------%
c
if (type .eq. 'SHIFTI') then
c
if (rvec)
& call dscal (ncv, rnorm, workl(ihbds), 1)
c
do 50 k=1, ncv
temp = dlapy2( workl(iheigr+k-1),
& workl(iheigi+k-1) )
workl(ihbds+k-1) = abs( workl(ihbds+k-1) )
& / temp / temp
50 continue
c
else if (type .eq. 'REALPT') then
c
do 60 k=1, ncv
60 continue
c
else if (type .eq. 'IMAGPT') then
c
do 70 k=1, ncv
70 continue
c
end if
c
c %-----------------------------------------------------------%
c | * Transform the Ritz values back to the original system. |
c | For TYPE = 'SHIFTI' the transformation is |
c | lambda = 1/theta + sigma |
c | For TYPE = 'REALPT' or 'IMAGPT' the user must from |
c | Rayleigh quotients or a projection. See remark 3 above.|
c | NOTES: |
c | *The Ritz vectors are not affected by the transformation. |
c %-----------------------------------------------------------%
c
if (type .eq. 'SHIFTI') then
c
do 80 k=1, ncv
temp = dlapy2( 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
80 continue
c
call dcopy (nconv, workl(iheigr), 1, dr, 1)
call dcopy (nconv, workl(iheigi), 1, di, 1)
c
else if (type .eq. 'REALPT' .or. type .eq. 'IMAGPT') then
c
call dcopy (nconv, workl(iheigr), 1, dr, 1)
call dcopy (nconv, workl(iheigi), 1, di, 1)
c
end if
c
end if
c
if (type .eq. 'SHIFTI' .and. msglvl .gt. 1) then
call igraphdvout (logfil, nconv, dr, ndigit,
& '_neupd: Untransformed real part of the Ritz valuess.')
call igraphdvout (logfil, nconv, di, ndigit,
& '_neupd: Untransformed imag part of the Ritz valuess.')
call igraphdvout (logfil, nconv, workl(ihbds), ndigit,
& '_neupd: Ritz estimates of untransformed Ritz values.')
else if (type .eq. 'REGULR' .and. msglvl .gt. 1) then
call igraphdvout (logfil, nconv, dr, ndigit,
& '_neupd: Real parts of converged Ritz values.')
call igraphdvout (logfil, nconv, di, ndigit,
& '_neupd: Imag parts of converged Ritz values.')
call igraphdvout (logfil, nconv, workl(ihbds), ndigit,
& '_neupd: Associated Ritz estimates.')
end if
c
c %-------------------------------------------------%
c | Eigenvector Purification step. Formally perform |
c | one of inverse subspace iteration. Only used |
c | for MODE = 2. |
c %-------------------------------------------------%
c
if (rvec .and. howmny .eq. 'A' .and. type .eq. 'SHIFTI') then
c
c %------------------------------------------------%
c | Purify the computed Ritz vectors by adding a |
c | little bit of the residual vector: |
c | T |
c | resid(:)*( e s ) / theta |
c | NCV |
c | where H s = s theta. Remember that when theta |
c | has nonzero imaginary part, the corresponding |
c | Ritz vector is stored across two columns of Z. |
c %------------------------------------------------%
c
iconj = 0
do 110 j=1, nconv
if (workl(iheigi+j-1) .eq. zero) then
workev(j) = workl(invsub+(j-1)*ldq+ncv-1) /
& workl(iheigr+j-1)
else if (iconj .eq. 0) then
temp = dlapy2( workl(iheigr+j-1), workl(iheigi+j-1) )
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
end if
110 continue
c
c %---------------------------------------%
c | Perform a rank one update to Z and |
c | purify all the Ritz vectors together. |
c %---------------------------------------%
c
call dger (n, nconv, one, resid, 1, workev, 1, z, ldz)
c
end if
c
9000 continue
c
return
c
c %---------------%
c | End of DNEUPD |
c %---------------%
c
end
|
