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|
!> \brief \b DGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
!
! =========== DOCUMENTATION ===========
!
! Definition:
! ===========
!
! SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, &
! M, N, X, LDX, Y, LDY, NRNK, TOL, &
! K, REIG, IMEIG, Z, LDZ, RES, &
! B, LDB, W, LDW, S, LDS, &
! WORK, LWORK, IWORK, LIWORK, INFO )
!.....
! USE, INTRINSIC :: iso_fortran_env, only: real64
! IMPLICIT NONE
! INTEGER, PARAMETER :: WP = real64
!.....
! Scalar arguments
! CHARACTER, INTENT(IN) :: JOBS, JOBZ, JOBR, JOBF
! INTEGER, INTENT(IN) :: WHTSVD, M, N, LDX, LDY, &
! NRNK, LDZ, LDB, LDW, LDS, &
! LWORK, LIWORK
! INTEGER, INTENT(OUT) :: K, INFO
! REAL(KIND=WP), INTENT(IN) :: TOL
! Array arguments
! REAL(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*)
! REAL(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), &
! W(LDW,*), S(LDS,*)
! REAL(KIND=WP), INTENT(OUT) :: REIG(*), IMEIG(*), &
! RES(*)
! REAL(KIND=WP), INTENT(OUT) :: WORK(*)
! INTEGER, INTENT(OUT) :: IWORK(*)
!
!............................................................
!> \par Purpose:
! =============
!> \verbatim
!> DGEDMD computes the Dynamic Mode Decomposition (DMD) for
!> a pair of data snapshot matrices. For the input matrices
!> X and Y such that Y = A*X with an unaccessible matrix
!> A, DGEDMD computes a certain number of Ritz pairs of A using
!> the standard Rayleigh-Ritz extraction from a subspace of
!> range(X) that is determined using the leading left singular
!> vectors of X. Optionally, DGEDMD returns the residuals
!> of the computed Ritz pairs, the information needed for
!> a refinement of the Ritz vectors, or the eigenvectors of
!> the Exact DMD.
!> For further details see the references listed
!> below. For more details of the implementation see [3].
!> \endverbatim
!............................................................
!> \par References:
! ================
!> \verbatim
!> [1] P. Schmid: Dynamic mode decomposition of numerical
!> and experimental data,
!> Journal of Fluid Mechanics 656, 5-28, 2010.
!> [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal
!> decompositions: analysis and enhancements,
!> SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018.
!> [3] Z. Drmac: A LAPACK implementation of the Dynamic
!> Mode Decomposition I. Technical report. AIMDyn Inc.
!> and LAPACK Working Note 298.
!> [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L.
!> Brunton, N. Kutz: On Dynamic Mode Decomposition:
!> Theory and Applications, Journal of Computational
!> Dynamics 1(2), 391 -421, 2014.
!> \endverbatim
!......................................................................
!> \par Developed and supported by:
! ================================
!> \verbatim
!> Developed and coded by Zlatko Drmac, Faculty of Science,
!> University of Zagreb; drmac@math.hr
!> In cooperation with
!> AIMdyn Inc., Santa Barbara, CA.
!> and supported by
!> - DARPA SBIR project "Koopman Operator-Based Forecasting
!> for Nonstationary Processes from Near-Term, Limited
!> Observational Data" Contract No: W31P4Q-21-C-0007
!> - DARPA PAI project "Physics-Informed Machine Learning
!> Methodologies" Contract No: HR0011-18-9-0033
!> - DARPA MoDyL project "A Data-Driven, Operator-Theoretic
!> Framework for Space-Time Analysis of Process Dynamics"
!> Contract No: HR0011-16-C-0116
!> Any opinions, findings and conclusions or recommendations
!> expressed in this material are those of the author and
!> do not necessarily reflect the views of the DARPA SBIR
!> Program Office
!> \endverbatim
!......................................................................
!> \par Distribution Statement A:
! ==============================
!> \verbatim
!> Approved for Public Release, Distribution Unlimited.
!> Cleared by DARPA on September 29, 2022
!> \endverbatim
!......................................................................
! Arguments
! =========
!
!> \param[in] JOBS
!> \verbatim
!> JOBS (input) is CHARACTER*1
!> Determines whether the initial data snapshots are scaled
!> by a diagonal matrix.
!> 'S' :: The data snapshots matrices X and Y are multiplied
!> with a diagonal matrix D so that X*D has unit
!> nonzero columns (in the Euclidean 2-norm)
!> 'C' :: The snapshots are scaled as with the 'S' option.
!> If it is found that an i-th column of X is zero
!> vector and the corresponding i-th column of Y is
!> non-zero, then the i-th column of Y is set to
!> zero and a warning flag is raised.
!> 'Y' :: The data snapshots matrices X and Y are multiplied
!> by a diagonal matrix D so that Y*D has unit
!> nonzero columns (in the Euclidean 2-norm)
!> 'N' :: No data scaling.
!> \endverbatim
!.....
!> \param[in] JOBZ
!> \verbatim
!> JOBZ (input) CHARACTER*1
!> Determines whether the eigenvectors (Koopman modes) will
!> be computed.
!> 'V' :: The eigenvectors (Koopman modes) will be computed
!> and returned in the matrix Z.
!> See the description of Z.
!> 'F' :: The eigenvectors (Koopman modes) will be returned
!> in factored form as the product X(:,1:K)*W, where X
!> contains a POD basis (leading left singular vectors
!> of the data matrix X) and W contains the eigenvectors
!> of the corresponding Rayleigh quotient.
!> See the descriptions of K, X, W, Z.
!> 'N' :: The eigenvectors are not computed.
!> \endverbatim
!.....
!> \param[in] JOBR
!> \verbatim
!> JOBR (input) CHARACTER*1
!> Determines whether to compute the residuals.
!> 'R' :: The residuals for the computed eigenpairs will be
!> computed and stored in the array RES.
!> See the description of RES.
!> For this option to be legal, JOBZ must be 'V'.
!> 'N' :: The residuals are not computed.
!> \endverbatim
!.....
!> \param[in] JOBF
!> \verbatim
!> JOBF (input) CHARACTER*1
!> Specifies whether to store information needed for post-
!> processing (e.g. computing refined Ritz vectors)
!> 'R' :: The matrix needed for the refinement of the Ritz
!> vectors is computed and stored in the array B.
!> See the description of B.
!> 'E' :: The unscaled eigenvectors of the Exact DMD are
!> computed and returned in the array B. See the
!> description of B.
!> 'N' :: No eigenvector refinement data is computed.
!> \endverbatim
!.....
!> \param[in] WHTSVD
!> \verbatim
!> WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 }
!> Allows for a selection of the SVD algorithm from the
!> LAPACK library.
!> 1 :: DGESVD (the QR SVD algorithm)
!> 2 :: DGESDD (the Divide and Conquer algorithm; if enough
!> workspace available, this is the fastest option)
!> 3 :: DGESVDQ (the preconditioned QR SVD ; this and 4
!> are the most accurate options)
!> 4 :: DGEJSV (the preconditioned Jacobi SVD; this and 3
!> are the most accurate options)
!> For the four methods above, a significant difference in
!> the accuracy of small singular values is possible if
!> the snapshots vary in norm so that X is severely
!> ill-conditioned. If small (smaller than EPS*||X||)
!> singular values are of interest and JOBS=='N', then
!> the options (3, 4) give the most accurate results, where
!> the option 4 is slightly better and with stronger
!> theoretical background.
!> If JOBS=='S', i.e. the columns of X will be normalized,
!> then all methods give nearly equally accurate results.
!> \endverbatim
!.....
!> \param[in] M
!> \verbatim
!> M (input) INTEGER, M>= 0
!> The state space dimension (the row dimension of X, Y).
!> \endverbatim
!.....
!> \param[in] N
!> \verbatim
!> N (input) INTEGER, 0 <= N <= M
!> The number of data snapshot pairs
!> (the number of columns of X and Y).
!> \endverbatim
!.....
!> \param[in,out] X
!> \verbatim
!> X (input/output) REAL(KIND=WP) M-by-N array
!> > On entry, X contains the data snapshot matrix X. It is
!> assumed that the column norms of X are in the range of
!> the normalized floating point numbers.
!> < On exit, the leading K columns of X contain a POD basis,
!> i.e. the leading K left singular vectors of the input
!> data matrix X, U(:,1:K). All N columns of X contain all
!> left singular vectors of the input matrix X.
!> See the descriptions of K, Z and W.
!> \endverbatim
!.....
!> \param[in] LDX
!> \verbatim
!> LDX (input) INTEGER, LDX >= M
!> The leading dimension of the array X.
!> \endverbatim
!.....
!> \param[in,out] Y
!> \verbatim
!> Y (input/workspace/output) REAL(KIND=WP) M-by-N array
!> > On entry, Y contains the data snapshot matrix Y
!> < On exit,
!> If JOBR == 'R', the leading K columns of Y contain
!> the residual vectors for the computed Ritz pairs.
!> See the description of RES.
!> If JOBR == 'N', Y contains the original input data,
!> scaled according to the value of JOBS.
!> \endverbatim
!.....
!> \param[in] LDY
!> \verbatim
!> LDY (input) INTEGER , LDY >= M
!> The leading dimension of the array Y.
!> \endverbatim
!.....
!> \param[in] NRNK
!> \verbatim
!> NRNK (input) INTEGER
!> Determines the mode how to compute the numerical rank,
!> i.e. how to truncate small singular values of the input
!> matrix X. On input, if
!> NRNK = -1 :: i-th singular value sigma(i) is truncated
!> if sigma(i) <= TOL*sigma(1).
!> This option is recommended.
!> NRNK = -2 :: i-th singular value sigma(i) is truncated
!> if sigma(i) <= TOL*sigma(i-1)
!> This option is included for R&D purposes.
!> It requires highly accurate SVD, which
!> may not be feasible.
!>
!> The numerical rank can be enforced by using positive
!> value of NRNK as follows:
!> 0 < NRNK <= N :: at most NRNK largest singular values
!> will be used. If the number of the computed nonzero
!> singular values is less than NRNK, then only those
!> nonzero values will be used and the actually used
!> dimension is less than NRNK. The actual number of
!> the nonzero singular values is returned in the variable
!> K. See the descriptions of TOL and K.
!> \endverbatim
!.....
!> \param[in] TOL
!> \verbatim
!> TOL (input) REAL(KIND=WP), 0 <= TOL < 1
!> The tolerance for truncating small singular values.
!> See the description of NRNK.
!> \endverbatim
!.....
!> \param[out] K
!> \verbatim
!> K (output) INTEGER, 0 <= K <= N
!> The dimension of the POD basis for the data snapshot
!> matrix X and the number of the computed Ritz pairs.
!> The value of K is determined according to the rule set
!> by the parameters NRNK and TOL.
!> See the descriptions of NRNK and TOL.
!> \endverbatim
!.....
!> \param[out] REIG
!> \verbatim
!> REIG (output) REAL(KIND=WP) N-by-1 array
!> The leading K (K<=N) entries of REIG contain
!> the real parts of the computed eigenvalues
!> REIG(1:K) + sqrt(-1)*IMEIG(1:K).
!> See the descriptions of K, IMEIG, and Z.
!> \endverbatim
!.....
!> \param[out] IMEIG
!> \verbatim
!> IMEIG (output) REAL(KIND=WP) N-by-1 array
!> The leading K (K<=N) entries of IMEIG contain
!> the imaginary parts of the computed eigenvalues
!> REIG(1:K) + sqrt(-1)*IMEIG(1:K).
!> The eigenvalues are determined as follows:
!> If IMEIG(i) == 0, then the corresponding eigenvalue is
!> real, LAMBDA(i) = REIG(i).
!> If IMEIG(i)>0, then the corresponding complex
!> conjugate pair of eigenvalues reads
!> LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i)
!> LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i)
!> That is, complex conjugate pairs have consecutive
!> indices (i,i+1), with the positive imaginary part
!> listed first.
!> See the descriptions of K, REIG, and Z.
!> \endverbatim
!.....
!> \param[out] Z
!> \verbatim
!> Z (workspace/output) REAL(KIND=WP) M-by-N array
!> If JOBZ =='V' then
!> Z contains real Ritz vectors as follows:
!> If IMEIG(i)=0, then Z(:,i) is an eigenvector of
!> the i-th Ritz value; ||Z(:,i)||_2=1.
!> If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then
!> [Z(:,i) Z(:,i+1)] span an invariant subspace and
!> the Ritz values extracted from this subspace are
!> REIG(i) + sqrt(-1)*IMEIG(i) and
!> REIG(i) - sqrt(-1)*IMEIG(i).
!> The corresponding eigenvectors are
!> Z(:,i) + sqrt(-1)*Z(:,i+1) and
!> Z(:,i) - sqrt(-1)*Z(:,i+1), respectively.
!> || Z(:,i:i+1)||_F = 1.
!> If JOBZ == 'F', then the above descriptions hold for
!> the columns of X(:,1:K)*W(1:K,1:K), where the columns
!> of W(1:k,1:K) are the computed eigenvectors of the
!> K-by-K Rayleigh quotient. The columns of W(1:K,1:K)
!> are similarly structured: If IMEIG(i) == 0 then
!> X(:,1:K)*W(:,i) is an eigenvector, and if IMEIG(i)>0
!> then X(:,1:K)*W(:,i)+sqrt(-1)*X(:,1:K)*W(:,i+1) and
!> X(:,1:K)*W(:,i)-sqrt(-1)*X(:,1:K)*W(:,i+1)
!> are the eigenvectors of LAMBDA(i), LAMBDA(i+1).
!> See the descriptions of REIG, IMEIG, X and W.
!> \endverbatim
!.....
!> \param[in] LDZ
!> \verbatim
!> LDZ (input) INTEGER , LDZ >= M
!> The leading dimension of the array Z.
!> \endverbatim
!.....
!> \param[out] RES
!> \verbatim
!> RES (output) REAL(KIND=WP) N-by-1 array
!> RES(1:K) contains the residuals for the K computed
!> Ritz pairs.
!> If LAMBDA(i) is real, then
!> RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2.
!> If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair
!> then
!> RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F
!> where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ]
!> [-imag(LAMBDA(i)) real(LAMBDA(i)) ].
!> It holds that
!> RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2
!> RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2
!> where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1)
!> ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1)
!> See the description of REIG, IMEIG and Z.
!> \endverbatim
!.....
!> \param[out] B
!> \verbatim
!> B (output) REAL(KIND=WP) M-by-N array.
!> IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can
!> be used for computing the refined vectors; see further
!> details in the provided references.
!> If JOBF == 'E', B(1:M,1;K) contains
!> A*U(:,1:K)*W(1:K,1:K), which are the vectors from the
!> Exact DMD, up to scaling by the inverse eigenvalues.
!> If JOBF =='N', then B is not referenced.
!> See the descriptions of X, W, K.
!> \endverbatim
!.....
!> \param[in] LDB
!> \verbatim
!> LDB (input) INTEGER, LDB >= M
!> The leading dimension of the array B.
!> \endverbatim
!.....
!> \param[out] W
!> \verbatim
!> W (workspace/output) REAL(KIND=WP) N-by-N array
!> On exit, W(1:K,1:K) contains the K computed
!> eigenvectors of the matrix Rayleigh quotient (real and
!> imaginary parts for each complex conjugate pair of the
!> eigenvalues). The Ritz vectors (returned in Z) are the
!> product of X (containing a POD basis for the input
!> matrix X) and W. See the descriptions of K, S, X and Z.
!> W is also used as a workspace to temporarily store the
!> right singular vectors of X.
!> \endverbatim
!.....
!> \param[in] LDW
!> \verbatim
!> LDW (input) INTEGER, LDW >= N
!> The leading dimension of the array W.
!> \endverbatim
!.....
!> \param[out] S
!> \verbatim
!> S (workspace/output) REAL(KIND=WP) N-by-N array
!> The array S(1:K,1:K) is used for the matrix Rayleigh
!> quotient. This content is overwritten during
!> the eigenvalue decomposition by DGEEV.
!> See the description of K.
!> \endverbatim
!.....
!> \param[in] LDS
!> \verbatim
!> LDS (input) INTEGER, LDS >= N
!> The leading dimension of the array S.
!> \endverbatim
!.....
!> \param[out] WORK
!> \verbatim
!> WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
!> On exit, WORK(1:N) contains the singular values of
!> X (for JOBS=='N') or column scaled X (JOBS=='S', 'C').
!> If WHTSVD==4, then WORK(N+1) and WORK(N+2) contain
!> scaling factor WORK(N+2)/WORK(N+1) used to scale X
!> and Y to avoid overflow in the SVD of X.
!> This may be of interest if the scaling option is off
!> and as many as possible smallest eigenvalues are
!> desired to the highest feasible accuracy.
!> If the call to DGEDMD is only workspace query, then
!> WORK(1) contains the minimal workspace length and
!> WORK(2) is the optimal workspace length. Hence, the
!> leng of work is at least 2.
!> See the description of LWORK.
!> \endverbatim
!.....
!> \param[in] LWORK
!> \verbatim
!> LWORK (input) INTEGER
!> The minimal length of the workspace vector WORK.
!> LWORK is calculated as follows:
!> If WHTSVD == 1 ::
!> If JOBZ == 'V', then
!> LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)).
!> If JOBZ == 'N' then
!> LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)).
!> Here LWORK_SVD = MAX(1,3*N+M,5*N) is the minimal
!> workspace length of DGESVD.
!> If WHTSVD == 2 ::
!> If JOBZ == 'V', then
!> LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N))
!> If JOBZ == 'N', then
!> LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N))
!> Here LWORK_SVD = MAX(M, 5*N*N+4*N)+3*N*N is the
!> minimal workspace length of DGESDD.
!> If WHTSVD == 3 ::
!> If JOBZ == 'V', then
!> LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
!> If JOBZ == 'N', then
!> LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
!> Here LWORK_SVD = N+M+MAX(3*N+1,
!> MAX(1,3*N+M,5*N),MAX(1,N))
!> is the minimal workspace length of DGESVDQ.
!> If WHTSVD == 4 ::
!> If JOBZ == 'V', then
!> LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
!> If JOBZ == 'N', then
!> LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
!> Here LWORK_SVD = MAX(7,2*M+N,6*N+2*N*N) is the
!> minimal workspace length of DGEJSV.
!> The above expressions are not simplified in order to
!> make the usage of WORK more transparent, and for
!> easier checking. In any case, LWORK >= 2.
!> If on entry LWORK = -1, then a workspace query is
!> assumed and the procedure only computes the minimal
!> and the optimal workspace lengths for both WORK and
!> IWORK. See the descriptions of WORK and IWORK.
!> \endverbatim
!.....
!> \param[out] IWORK
!> \verbatim
!> IWORK (workspace/output) INTEGER LIWORK-by-1 array
!> Workspace that is required only if WHTSVD equals
!> 2 , 3 or 4. (See the description of WHTSVD).
!> If on entry LWORK =-1 or LIWORK=-1, then the
!> minimal length of IWORK is computed and returned in
!> IWORK(1). See the description of LIWORK.
!> \endverbatim
!.....
!> \param[in] LIWORK
!> \verbatim
!> LIWORK (input) INTEGER
!> The minimal length of the workspace vector IWORK.
!> If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1
!> If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N))
!> If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1)
!> If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N)
!> If on entry LIWORK = -1, then a workspace query is
!> assumed and the procedure only computes the minimal
!> and the optimal workspace lengths for both WORK and
!> IWORK. See the descriptions of WORK and IWORK.
!> \endverbatim
!.....
!> \param[out] INFO
!> \verbatim
!> INFO (output) INTEGER
!> -i < 0 :: On entry, the i-th argument had an
!> illegal value
!> = 0 :: Successful return.
!> = 1 :: Void input. Quick exit (M=0 or N=0).
!> = 2 :: The SVD computation of X did not converge.
!> Suggestion: Check the input data and/or
!> repeat with different WHTSVD.
!> = 3 :: The computation of the eigenvalues did not
!> converge.
!> = 4 :: If data scaling was requested on input and
!> the procedure found inconsistency in the data
!> such that for some column index i,
!> X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set
!> to zero if JOBS=='C'. The computation proceeds
!> with original or modified data and warning
!> flag is set with INFO=4.
!> \endverbatim
!
! Authors:
! ========
!
!> \author Zlatko Drmac
!
!> \ingroup gedmd
!
!.............................................................
!.............................................................
SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, &
M, N, X, LDX, Y, LDY, NRNK, TOL, &
K, REIG, IMEIG, Z, LDZ, RES, &
B, LDB, W, LDW, S, LDS, &
WORK, LWORK, IWORK, LIWORK, INFO )
!
! -- LAPACK driver routine --
!
! -- LAPACK is a software package provided by University of --
! -- Tennessee, University of California Berkeley, University of --
! -- Colorado Denver and NAG Ltd.. --
!
!.....
USE, INTRINSIC :: iso_fortran_env, only: real64
IMPLICIT NONE
INTEGER, PARAMETER :: WP = real64
!
! Scalar arguments
! ~~~~~~~~~~~~~~~~
CHARACTER, INTENT(IN) :: JOBS, JOBZ, JOBR, JOBF
INTEGER, INTENT(IN) :: WHTSVD, M, N, LDX, LDY, &
NRNK, LDZ, LDB, LDW, LDS, &
LWORK, LIWORK
INTEGER, INTENT(OUT) :: K, INFO
REAL(KIND=WP), INTENT(IN) :: TOL
!
! Array arguments
! ~~~~~~~~~~~~~~~
REAL(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*)
REAL(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), &
W(LDW,*), S(LDS,*)
REAL(KIND=WP), INTENT(OUT) :: REIG(*), IMEIG(*), &
RES(*)
REAL(KIND=WP), INTENT(OUT) :: WORK(*)
INTEGER, INTENT(OUT) :: IWORK(*)
!
! Parameters
! ~~~~~~~~~~
REAL(KIND=WP), PARAMETER :: ONE = 1.0_WP
REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP
!
! Local scalars
! ~~~~~~~~~~~~~
REAL(KIND=WP) :: OFL, ROOTSC, SCALE, SMALL, &
SSUM, XSCL1, XSCL2
INTEGER :: i, j, IMINWR, INFO1, INFO2, &
LWRKEV, LWRSDD, LWRSVD, &
LWRSVQ, MLWORK, MWRKEV, MWRSDD, &
MWRSVD, MWRSVJ, MWRSVQ, NUMRNK, &
OLWORK
LOGICAL :: BADXY, LQUERY, SCCOLX, SCCOLY, &
WNTEX, WNTREF, WNTRES, WNTVEC
CHARACTER :: JOBZL, T_OR_N
CHARACTER :: JSVOPT
!
! Local arrays
! ~~~~~~~~~~~~
REAL(KIND=WP) :: AB(2,2), RDUMMY(2), RDUMMY2(2)
!
! External functions (BLAS and LAPACK)
! ~~~~~~~~~~~~~~~~~
REAL(KIND=WP) DLANGE, DLAMCH, DNRM2
EXTERNAL DLANGE, DLAMCH, DNRM2, IDAMAX
INTEGER IDAMAX
LOGICAL DISNAN, LSAME
EXTERNAL DISNAN, LSAME
!
! External subroutines (BLAS and LAPACK)
! ~~~~~~~~~~~~~~~~~~~~
EXTERNAL DAXPY, DGEMM, DSCAL
EXTERNAL DGEEV, DGEJSV, DGESDD, DGESVD, DGESVDQ, &
DLACPY, DLASCL, DLASSQ, XERBLA
!
! Intrinsic functions
! ~~~~~~~~~~~~~~~~~~~
INTRINSIC DBLE, INT, MAX, SQRT
!............................................................
!
! Test the input arguments
!
WNTRES = LSAME(JOBR,'R')
SCCOLX = LSAME(JOBS,'S') .OR. LSAME(JOBS,'C')
SCCOLY = LSAME(JOBS,'Y')
WNTVEC = LSAME(JOBZ,'V')
WNTREF = LSAME(JOBF,'R')
WNTEX = LSAME(JOBF,'E')
INFO = 0
LQUERY = ( ( LWORK == -1 ) .OR. ( LIWORK == -1 ) )
!
IF ( .NOT. (SCCOLX .OR. SCCOLY .OR. &
LSAME(JOBS,'N')) ) THEN
INFO = -1
ELSE IF ( .NOT. (WNTVEC .OR. LSAME(JOBZ,'N') &
.OR. LSAME(JOBZ,'F')) ) THEN
INFO = -2
ELSE IF ( .NOT. (WNTRES .OR. LSAME(JOBR,'N')) .OR. &
( WNTRES .AND. (.NOT.WNTVEC) ) ) THEN
INFO = -3
ELSE IF ( .NOT. (WNTREF .OR. WNTEX .OR. &
LSAME(JOBF,'N') ) ) THEN
INFO = -4
ELSE IF ( .NOT.((WHTSVD == 1) .OR. (WHTSVD == 2) .OR. &
(WHTSVD == 3) .OR. (WHTSVD == 4) )) THEN
INFO = -5
ELSE IF ( M < 0 ) THEN
INFO = -6
ELSE IF ( ( N < 0 ) .OR. ( N > M ) ) THEN
INFO = -7
ELSE IF ( LDX < M ) THEN
INFO = -9
ELSE IF ( LDY < M ) THEN
INFO = -11
ELSE IF ( .NOT. (( NRNK == -2).OR.(NRNK == -1).OR. &
((NRNK >= 1).AND.(NRNK <=N ))) ) THEN
INFO = -12
ELSE IF ( ( TOL < ZERO ) .OR. ( TOL >= ONE ) ) THEN
INFO = -13
ELSE IF ( LDZ < M ) THEN
INFO = -18
ELSE IF ( (WNTREF .OR. WNTEX ) .AND. ( LDB < M ) ) THEN
INFO = -21
ELSE IF ( LDW < N ) THEN
INFO = -23
ELSE IF ( LDS < N ) THEN
INFO = -25
END IF
!
IF ( INFO == 0 ) THEN
! Compute the minimal and the optimal workspace
! requirements. Simulate running the code and
! determine minimal and optimal sizes of the
! workspace at any moment of the run.
IF ( N == 0 ) THEN
! Quick return. All output except K is void.
! INFO=1 signals the void input.
! In case of a workspace query, the default
! minimal workspace lengths are returned.
IF ( LQUERY ) THEN
IWORK(1) = 1
WORK(1) = 2
WORK(2) = 2
ELSE
K = 0
END IF
INFO = 1
RETURN
END IF
MLWORK = MAX(2,N)
OLWORK = MAX(2,N)
IMINWR = 1
SELECT CASE ( WHTSVD )
CASE (1)
! The following is specified as the minimal
! length of WORK in the definition of DGESVD:
! MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N))
MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N))
MLWORK = MAX(MLWORK,N + MWRSVD)
IF ( LQUERY ) THEN
CALL DGESVD( 'O', 'S', M, N, X, LDX, WORK, &
B, LDB, W, LDW, RDUMMY, -1, INFO1 )
LWRSVD = MAX( MWRSVD, INT( RDUMMY(1) ) )
OLWORK = MAX(OLWORK,N + LWRSVD)
END IF
CASE (2)
! The following is specified as the minimal
! length of WORK in the definition of DGESDD:
! MWRSDD = 3*MIN(M,N)*MIN(M,N) +
! MAX( MAX(M,N),5*MIN(M,N)*MIN(M,N)+4*MIN(M,N) )
! IMINWR = 8*MIN(M,N)
MWRSDD = 3*MIN(M,N)*MIN(M,N) + &
MAX( MAX(M,N),5*MIN(M,N)*MIN(M,N)+4*MIN(M,N) )
MLWORK = MAX(MLWORK,N + MWRSDD)
IMINWR = 8*MIN(M,N)
IF ( LQUERY ) THEN
CALL DGESDD( 'O', M, N, X, LDX, WORK, B, &
LDB, W, LDW, RDUMMY, -1, IWORK, INFO1 )
LWRSDD = MAX( MWRSDD, INT( RDUMMY(1) ) )
OLWORK = MAX(OLWORK,N + LWRSDD)
END IF
CASE (3)
!LWQP3 = 3*N+1
!LWORQ = MAX(N, 1)
!MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N))
!MWRSVQ = N + MAX( LWQP3, MWRSVD, LWORQ ) + MAX(M,2)
!MLWORK = N + MWRSVQ
!IMINWR = M+N-1
CALL DGESVDQ( 'H', 'P', 'N', 'R', 'R', M, N, &
X, LDX, WORK, Z, LDZ, W, LDW, &
NUMRNK, IWORK, LIWORK, RDUMMY, &
-1, RDUMMY2, -1, INFO1 )
IMINWR = IWORK(1)
MWRSVQ = INT(RDUMMY(2))
MLWORK = MAX(MLWORK,N+MWRSVQ+INT(RDUMMY2(1)))
IF ( LQUERY ) THEN
LWRSVQ = MAX( MWRSVQ, INT(RDUMMY(1)) )
OLWORK = MAX(OLWORK,N+LWRSVQ+INT(RDUMMY2(1)))
END IF
CASE (4)
JSVOPT = 'J'
!MWRSVJ = MAX( 7, 2*M+N, 6*N+2*N*N ) ! for JSVOPT='V'
MWRSVJ = MAX( 7, 2*M+N, 4*N+N*N, 2*N+N*N+6 )
MLWORK = MAX(MLWORK,N+MWRSVJ)
IMINWR = MAX( 3, M+3*N )
IF ( LQUERY ) THEN
OLWORK = MAX(OLWORK,N+MWRSVJ)
END IF
END SELECT
IF ( WNTVEC .OR. WNTEX .OR. LSAME(JOBZ,'F') ) THEN
JOBZL = 'V'
ELSE
JOBZL = 'N'
END IF
! Workspace calculation to the DGEEV call
IF ( LSAME(JOBZL,'V') ) THEN
MWRKEV = MAX( 1, 4*N )
ELSE
MWRKEV = MAX( 1, 3*N )
END IF
MLWORK = MAX(MLWORK,N+MWRKEV)
IF ( LQUERY ) THEN
CALL DGEEV( 'N', JOBZL, N, S, LDS, REIG, &
IMEIG, W, LDW, W, LDW, RDUMMY, -1, INFO1 )
LWRKEV = MAX( MWRKEV, INT(RDUMMY(1)) )
OLWORK = MAX( OLWORK, N+LWRKEV )
END IF
!
IF ( LIWORK < IMINWR .AND. (.NOT.LQUERY) ) INFO = -29
IF ( LWORK < MLWORK .AND. (.NOT.LQUERY) ) INFO = -27
END IF
!
IF( INFO /= 0 ) THEN
CALL XERBLA( 'DGEDMD', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
! Return minimal and optimal workspace sizes
IWORK(1) = IMINWR
WORK(1) = MLWORK
WORK(2) = OLWORK
RETURN
END IF
!............................................................
!
OFL = DLAMCH('O')
SMALL = DLAMCH('S')
BADXY = .FALSE.
!
! <1> Optional scaling of the snapshots (columns of X, Y)
! ==========================================================
IF ( SCCOLX ) THEN
! The columns of X will be normalized.
! To prevent overflows, the column norms of X are
! carefully computed using DLASSQ.
K = 0
DO i = 1, N
!WORK(i) = DNRM2( M, X(1,i), 1 )
SSUM = ONE
SCALE = ZERO
CALL DLASSQ( M, X(1,i), 1, SCALE, SSUM )
IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN
K = 0
INFO = -8
CALL XERBLA('DGEDMD',-INFO)
END IF
IF ( (SCALE /= ZERO) .AND. (SSUM /= ZERO) ) THEN
ROOTSC = SQRT(SSUM)
IF ( SCALE .GE. (OFL / ROOTSC) ) THEN
! Norm of X(:,i) overflows. First, X(:,i)
! is scaled by
! ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2.
! Next, the norm of X(:,i) is stored without
! overflow as WORK(i) = - SCALE * (ROOTSC/M),
! the minus sign indicating the 1/M factor.
! Scaling is performed without overflow, and
! underflow may occur in the smallest entries
! of X(:,i). The relative backward and forward
! errors are small in the ell_2 norm.
CALL DLASCL( 'G', 0, 0, SCALE, ONE/ROOTSC, &
M, 1, X(1,i), M, INFO2 )
WORK(i) = - SCALE * ( ROOTSC / DBLE(M) )
ELSE
! X(:,i) will be scaled to unit 2-norm
WORK(i) = SCALE * ROOTSC
CALL DLASCL( 'G',0, 0, WORK(i), ONE, M, 1, &
X(1,i), M, INFO2 ) ! LAPACK CALL
! X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC
END IF
ELSE
WORK(i) = ZERO
K = K + 1
END IF
END DO
IF ( K == N ) THEN
! All columns of X are zero. Return error code -8.
! (the 8th input variable had an illegal value)
K = 0
INFO = -8
CALL XERBLA('DGEDMD',-INFO)
RETURN
END IF
DO i = 1, N
! Now, apply the same scaling to the columns of Y.
IF ( WORK(i) > ZERO ) THEN
CALL DSCAL( M, ONE/WORK(i), Y(1,i), 1 ) ! BLAS CALL
! Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC
ELSE IF ( WORK(i) < ZERO ) THEN
CALL DLASCL( 'G', 0, 0, -WORK(i), &
ONE/DBLE(M), M, 1, Y(1,i), M, INFO2 ) ! LAPACK CALL
ELSE IF ( Y(IDAMAX(M, Y(1,i),1),i ) &
/= ZERO ) THEN
! X(:,i) is zero vector. For consistency,
! Y(:,i) should also be zero. If Y(:,i) is not
! zero, then the data might be inconsistent or
! corrupted. If JOBS == 'C', Y(:,i) is set to
! zero and a warning flag is raised.
! The computation continues but the
! situation will be reported in the output.
BADXY = .TRUE.
IF ( LSAME(JOBS,'C')) &
CALL DSCAL( M, ZERO, Y(1,i), 1 ) ! BLAS CALL
END IF
END DO
END IF
!
IF ( SCCOLY ) THEN
! The columns of Y will be normalized.
! To prevent overflows, the column norms of Y are
! carefully computed using DLASSQ.
DO i = 1, N
!WORK(i) = DNRM2( M, Y(1,i), 1 )
SSUM = ONE
SCALE = ZERO
CALL DLASSQ( M, Y(1,i), 1, SCALE, SSUM )
IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN
K = 0
INFO = -10
CALL XERBLA('DGEDMD',-INFO)
END IF
IF ( SCALE /= ZERO .AND. (SSUM /= ZERO) ) THEN
ROOTSC = SQRT(SSUM)
IF ( SCALE .GE. (OFL / ROOTSC) ) THEN
! Norm of Y(:,i) overflows. First, Y(:,i)
! is scaled by
! ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2.
! Next, the norm of Y(:,i) is stored without
! overflow as WORK(i) = - SCALE * (ROOTSC/M),
! the minus sign indicating the 1/M factor.
! Scaling is performed without overflow, and
! underflow may occur in the smallest entries
! of Y(:,i). The relative backward and forward
! errors are small in the ell_2 norm.
CALL DLASCL( 'G', 0, 0, SCALE, ONE/ROOTSC, &
M, 1, Y(1,i), M, INFO2 )
WORK(i) = - SCALE * ( ROOTSC / DBLE(M) )
ELSE
! X(:,i) will be scaled to unit 2-norm
WORK(i) = SCALE * ROOTSC
CALL DLASCL( 'G',0, 0, WORK(i), ONE, M, 1, &
Y(1,i), M, INFO2 ) ! LAPACK CALL
! Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC
END IF
ELSE
WORK(i) = ZERO
END IF
END DO
DO i = 1, N
! Now, apply the same scaling to the columns of X.
IF ( WORK(i) > ZERO ) THEN
CALL DSCAL( M, ONE/WORK(i), X(1,i), 1 ) ! BLAS CALL
! X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC
ELSE IF ( WORK(i) < ZERO ) THEN
CALL DLASCL( 'G', 0, 0, -WORK(i), &
ONE/DBLE(M), M, 1, X(1,i), M, INFO2 ) ! LAPACK CALL
ELSE IF ( X(IDAMAX(M, X(1,i),1),i ) &
/= ZERO ) THEN
! Y(:,i) is zero vector. If X(:,i) is not
! zero, then a warning flag is raised.
! The computation continues but the
! situation will be reported in the output.
BADXY = .TRUE.
END IF
END DO
END IF
!
! <2> SVD of the data snapshot matrix X.
! =====================================
! The left singular vectors are stored in the array X.
! The right singular vectors are in the array W.
! The array W will later on contain the eigenvectors
! of a Rayleigh quotient.
NUMRNK = N
SELECT CASE ( WHTSVD )
CASE (1)
CALL DGESVD( 'O', 'S', M, N, X, LDX, WORK, B, &
LDB, W, LDW, WORK(N+1), LWORK-N, INFO1 ) ! LAPACK CALL
T_OR_N = 'T'
CASE (2)
CALL DGESDD( 'O', M, N, X, LDX, WORK, B, LDB, W, &
LDW, WORK(N+1), LWORK-N, IWORK, INFO1 ) ! LAPACK CALL
T_OR_N = 'T'
CASE (3)
CALL DGESVDQ( 'H', 'P', 'N', 'R', 'R', M, N, &
X, LDX, WORK, Z, LDZ, W, LDW, &
NUMRNK, IWORK, LIWORK, WORK(N+MAX(2,M)+1),&
LWORK-N-MAX(2,M), WORK(N+1), MAX(2,M), INFO1) ! LAPACK CALL
CALL DLACPY( 'A', M, NUMRNK, Z, LDZ, X, LDX ) ! LAPACK CALL
T_OR_N = 'T'
CASE (4)
CALL DGEJSV( 'F', 'U', JSVOPT, 'N', 'N', 'P', M, &
N, X, LDX, WORK, Z, LDZ, W, LDW, &
WORK(N+1), LWORK-N, IWORK, INFO1 ) ! LAPACK CALL
CALL DLACPY( 'A', M, N, Z, LDZ, X, LDX ) ! LAPACK CALL
T_OR_N = 'N'
XSCL1 = WORK(N+1)
XSCL2 = WORK(N+2)
IF ( XSCL1 /= XSCL2 ) THEN
! This is an exceptional situation. If the
! data matrices are not scaled and the
! largest singular value of X overflows.
! In that case DGEJSV can return the SVD
! in scaled form. The scaling factor can be used
! to rescale the data (X and Y).
CALL DLASCL( 'G', 0, 0, XSCL1, XSCL2, M, N, Y, LDY, INFO2 )
END IF
END SELECT
!
IF ( INFO1 > 0 ) THEN
! The SVD selected subroutine did not converge.
! Return with an error code.
INFO = 2
RETURN
END IF
!
IF ( WORK(1) == ZERO ) THEN
! The largest computed singular value of (scaled)
! X is zero. Return error code -8
! (the 8th input variable had an illegal value).
K = 0
INFO = -8
CALL XERBLA('DGEDMD',-INFO)
RETURN
END IF
!
!<3> Determine the numerical rank of the data
! snapshots matrix X. This depends on the
! parameters NRNK and TOL.
SELECT CASE ( NRNK )
CASE ( -1 )
K = 1
DO i = 2, NUMRNK
IF ( ( WORK(i) <= WORK(1)*TOL ) .OR. &
( WORK(i) <= SMALL ) ) EXIT
K = K + 1
END DO
CASE ( -2 )
K = 1
DO i = 1, NUMRNK-1
IF ( ( WORK(i+1) <= WORK(i)*TOL ) .OR. &
( WORK(i) <= SMALL ) ) EXIT
K = K + 1
END DO
CASE DEFAULT
K = 1
DO i = 2, NRNK
IF ( WORK(i) <= SMALL ) EXIT
K = K + 1
END DO
END SELECT
! Now, U = X(1:M,1:K) is the SVD/POD basis for the
! snapshot data in the input matrix X.
!<4> Compute the Rayleigh quotient S = U^T * A * U.
! Depending on the requested outputs, the computation
! is organized to compute additional auxiliary
! matrices (for the residuals and refinements).
!
! In all formulas below, we need V_k*Sigma_k^(-1)
! where either V_k is in W(1:N,1:K), or V_k^T is in
! W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)).
IF ( LSAME(T_OR_N, 'N') ) THEN
DO i = 1, K
CALL DSCAL( N, ONE/WORK(i), W(1,i), 1 ) ! BLAS CALL
! W(1:N,i) = (ONE/WORK(i)) * W(1:N,i) ! INTRINSIC
END DO
ELSE
! This non-unit stride access is due to the fact
! that DGESVD, DGESVDQ and DGESDD return the
! transposed matrix of the right singular vectors.
!DO i = 1, K
! CALL DSCAL( N, ONE/WORK(i), W(i,1), LDW ) ! BLAS CALL
! ! W(i,1:N) = (ONE/WORK(i)) * W(i,1:N) ! INTRINSIC
!END DO
DO i = 1, K
WORK(N+i) = ONE/WORK(i)
END DO
DO j = 1, N
DO i = 1, K
W(i,j) = (WORK(N+i))*W(i,j)
END DO
END DO
END IF
!
IF ( WNTREF ) THEN
!
! Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K)))
! for computing the refined Ritz vectors
! (optionally, outside DGEDMD).
CALL DGEMM( 'N', T_OR_N, M, K, N, ONE, Y, LDY, W, &
LDW, ZERO, Z, LDZ ) ! BLAS CALL
! Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRINSIC, for T_OR_N=='T'
! Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! INTRINSIC, for T_OR_N=='N'
!
! At this point Z contains
! A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and
! this is needed for computing the residuals.
! This matrix is returned in the array B and
! it can be used to compute refined Ritz vectors.
CALL DLACPY( 'A', M, K, Z, LDZ, B, LDB ) ! BLAS CALL
! B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC
CALL DGEMM( 'T', 'N', K, K, M, ONE, X, LDX, Z, &
LDZ, ZERO, S, LDS ) ! BLAS CALL
! S(1:K,1:K) = MATMUL(TANSPOSE(X(1:M,1:K)),Z(1:M,1:K)) ! INTRINSIC
! At this point S = U^T * A * U is the Rayleigh quotient.
ELSE
! A * U(:,1:K) is not explicitly needed and the
! computation is organized differently. The Rayleigh
! quotient is computed more efficiently.
CALL DGEMM( 'T', 'N', K, N, M, ONE, X, LDX, Y, LDY, &
ZERO, Z, LDZ ) ! BLAS CALL
! Z(1:K,1:N) = MATMUL( TRANSPOSE(X(1:M,1:K)), Y(1:M,1:N) ) ! INTRINSIC
! In the two DGEMM calls here, can use K for LDZ.
CALL DGEMM( 'N', T_OR_N, K, K, N, ONE, Z, LDZ, W, &
LDW, ZERO, S, LDS ) ! BLAS CALL
! S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRINSIC, for T_OR_N=='T'
! S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! INTRINSIC, for T_OR_N=='N'
! At this point S = U^T * A * U is the Rayleigh quotient.
! If the residuals are requested, save scaled V_k into Z.
! Recall that V_k or V_k^T is stored in W.
IF ( WNTRES .OR. WNTEX ) THEN
IF ( LSAME(T_OR_N, 'N') ) THEN
CALL DLACPY( 'A', N, K, W, LDW, Z, LDZ )
ELSE
CALL DLACPY( 'A', K, N, W, LDW, Z, LDZ )
END IF
END IF
END IF
!
!<5> Compute the Ritz values and (if requested) the
! right eigenvectors of the Rayleigh quotient.
!
CALL DGEEV( 'N', JOBZL, K, S, LDS, REIG, IMEIG, W, &
LDW, W, LDW, WORK(N+1), LWORK-N, INFO1 ) ! LAPACK CALL
!
! W(1:K,1:K) contains the eigenvectors of the Rayleigh
! quotient. Even in the case of complex spectrum, all
! computation is done in real arithmetic. REIG and
! IMEIG are the real and the imaginary parts of the
! eigenvalues, so that the spectrum is given as
! REIG(:) + sqrt(-1)*IMEIG(:). Complex conjugate pairs
! are listed at consecutive positions. For such a
! complex conjugate pair of the eigenvalues, the
! corresponding eigenvectors are also a complex
! conjugate pair with the real and imaginary parts
! stored column-wise in W at the corresponding
! consecutive column indices. See the description of Z.
! Also, see the description of DGEEV.
IF ( INFO1 > 0 ) THEN
! DGEEV failed to compute the eigenvalues and
! eigenvectors of the Rayleigh quotient.
INFO = 3
RETURN
END IF
!
! <6> Compute the eigenvectors (if requested) and,
! the residuals (if requested).
!
IF ( WNTVEC .OR. WNTEX ) THEN
IF ( WNTRES ) THEN
IF ( WNTREF ) THEN
! Here, if the refinement is requested, we have
! A*U(:,1:K) already computed and stored in Z.
! For the residuals, need Y = A * U(:,1;K) * W.
CALL DGEMM( 'N', 'N', M, K, K, ONE, Z, LDZ, W, &
LDW, ZERO, Y, LDY ) ! BLAS CALL
! Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC
! This frees Z; Y contains A * U(:,1:K) * W.
ELSE
! Compute S = V_k * Sigma_k^(-1) * W, where
! V_k * Sigma_k^(-1) is stored in Z
CALL DGEMM( T_OR_N, 'N', N, K, K, ONE, Z, LDZ, &
W, LDW, ZERO, S, LDS)
! Then, compute Z = Y * S =
! = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) =
! = A * U(:,1:K) * W(1:K,1:K)
CALL DGEMM( 'N', 'N', M, K, N, ONE, Y, LDY, S, &
LDS, ZERO, Z, LDZ)
! Save a copy of Z into Y and free Z for holding
! the Ritz vectors.
CALL DLACPY( 'A', M, K, Z, LDZ, Y, LDY )
IF ( WNTEX ) CALL DLACPY( 'A', M, K, Z, LDZ, B, LDB )
END IF
ELSE IF ( WNTEX ) THEN
! Compute S = V_k * Sigma_k^(-1) * W, where
! V_k * Sigma_k^(-1) is stored in Z
CALL DGEMM( T_OR_N, 'N', N, K, K, ONE, Z, LDZ, &
W, LDW, ZERO, S, LDS )
! Then, compute Z = Y * S =
! = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) =
! = A * U(:,1:K) * W(1:K,1:K)
CALL DGEMM( 'N', 'N', M, K, N, ONE, Y, LDY, S, &
LDS, ZERO, B, LDB )
! The above call replaces the following two calls
! that were used in the developing-testing phase.
! CALL DGEMM( 'N', 'N', M, K, N, ONE, Y, LDY, S, &
! LDS, ZERO, Z, LDZ)
! Save a copy of Z into B and free Z for holding
! the Ritz vectors.
! CALL DLACPY( 'A', M, K, Z, LDZ, B, LDB )
END IF
!
! Compute the real form of the Ritz vectors
IF ( WNTVEC ) CALL DGEMM( 'N', 'N', M, K, K, ONE, X, LDX, W, LDW, &
ZERO, Z, LDZ ) ! BLAS CALL
! Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRINSIC
!
IF ( WNTRES ) THEN
i = 1
DO WHILE ( i <= K )
IF ( IMEIG(i) == ZERO ) THEN
! have a real eigenvalue with real eigenvector
CALL DAXPY( M, -REIG(i), Z(1,i), 1, Y(1,i), 1 ) ! BLAS CALL
! Y(1:M,i) = Y(1:M,i) - REIG(i) * Z(1:M,i) ! INTRINSIC
RES(i) = DNRM2( M, Y(1,i), 1) ! BLAS CALL
i = i + 1
ELSE
! Have a complex conjugate pair
! REIG(i) +- sqrt(-1)*IMEIG(i).
! Since all computation is done in real
! arithmetic, the formula for the residual
! is recast for real representation of the
! complex conjugate eigenpair. See the
! description of RES.
AB(1,1) = REIG(i)
AB(2,1) = -IMEIG(i)
AB(1,2) = IMEIG(i)
AB(2,2) = REIG(i)
CALL DGEMM( 'N', 'N', M, 2, 2, -ONE, Z(1,i), &
LDZ, AB, 2, ONE, Y(1,i), LDY ) ! BLAS CALL
! Y(1:M,i:i+1) = Y(1:M,i:i+1) - Z(1:M,i:i+1) * AB ! INTRINSIC
RES(i) = DLANGE( 'F', M, 2, Y(1,i), LDY, &
WORK(N+1) ) ! LAPACK CALL
RES(i+1) = RES(i)
i = i + 2
END IF
END DO
END IF
END IF
!
IF ( WHTSVD == 4 ) THEN
WORK(N+1) = XSCL1
WORK(N+2) = XSCL2
END IF
!
! Successful exit.
IF ( .NOT. BADXY ) THEN
INFO = 0
ELSE
! A warning on possible data inconsistency.
! This should be a rare event.
INFO = 4
END IF
!............................................................
RETURN
! ......
END SUBROUTINE DGEDMD
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