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Cloned library LAPACK-3.11.0 with extra build files for internal package management.
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LAPACK-3.11.0/SRC/sgedmdq.f90

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SUBROUTINE SGEDMDQ( JOBS, JOBZ, JOBR, JOBQ, JOBT, JOBF, &
WHTSVD, M, N, F, LDF, X, LDX, Y, &
LDY, NRNK, TOL, K, REIG, IMEIG, &
Z, LDZ, RES, B, LDB, V, LDV, &
S, LDS, WORK, LWORK, IWORK, LIWORK, INFO )
! March 2023
!.....
USE iso_fortran_env
IMPLICIT NONE
INTEGER, PARAMETER :: WP = real32
!.....
! Scalar arguments
CHARACTER, INTENT(IN) :: JOBS, JOBZ, JOBR, JOBQ, &
JOBT, JOBF
INTEGER, INTENT(IN) :: WHTSVD, M, N, LDF, LDX, &
LDY, NRNK, LDZ, LDB, LDV, &
LDS, LWORK, LIWORK
INTEGER, INTENT(OUT) :: INFO, K
REAL(KIND=WP), INTENT(IN) :: TOL
! Array arguments
REAL(KIND=WP), INTENT(INOUT) :: F(LDF,*)
REAL(KIND=WP), INTENT(OUT) :: X(LDX,*), Y(LDY,*), &
Z(LDZ,*), B(LDB,*), &
V(LDV,*), S(LDS,*)
REAL(KIND=WP), INTENT(OUT) :: REIG(*), IMEIG(*), &
RES(*)
REAL(KIND=WP), INTENT(OUT) :: WORK(*)
INTEGER, INTENT(OUT) :: IWORK(*)
!.....
! Purpose
! =======
! SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for
! a pair of data snapshot matrices, using a QR factorization
! based compression of the data. For the input matrices
! X and Y such that Y = A*X with an unaccessible matrix
! A, SGEDMDQ 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, SGEDMDQ 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].
!
! References
! ==========
! [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.
!
! Developed and supported by:
! ===========================
! 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.
!============================================================
! Distribution Statement A:
! Approved for Public Release, Distribution Unlimited.
! Cleared by DARPA on September 29, 2022
!============================================================
!......................................................................
! Arguments
! =========
! JOBS (input) CHARACTER*1
! Determines whether the initial data snapshots are scaled
! by a diagonal matrix. The data snapshots are the columns
! of F. The leading N-1 columns of F are denoted X and the
! trailing N-1 columns are denoted Y.
! '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.
!.....
! 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 Z*V, where Z
! is orthonormal and V contains the eigenvectors
! of the corresponding Rayleigh quotient.
! See the descriptions of F, V, Z.
! 'Q' :: The eigenvectors (Koopman modes) will be returned
! in factored form as the product Q*Z, where Z
! contains the eigenvectors of the compression of the
! underlying discretized operator onto the span of
! the data snapshots. See the descriptions of F, V, Z.
! Q is from the initial QR factorization.
! 'N' :: The eigenvectors are not computed.
!.....
! 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.
!.....
! JOBQ (input) CHARACTER*1
! Specifies whether to explicitly compute and return the
! orthogonal matrix from the QR factorization.
! 'Q' :: The matrix Q of the QR factorization of the data
! snapshot matrix is computed and stored in the
! array F. See the description of F.
! 'N' :: The matrix Q is not explicitly computed.
!.....
! JOBT (input) CHARACTER*1
! Specifies whether to return the upper triangular factor
! from the QR factorization.
! 'R' :: The matrix R of the QR factorization of the data
! snapshot matrix F is returned in the array Y.
! See the description of Y and Further details.
! 'N' :: The matrix R is not returned.
!.....
! 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.
! To be useful on exit, this option needs JOBQ='Q'.
!.....
! WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 }
! Allows for a selection of the SVD algorithm from the
! LAPACK library.
! 1 :: SGESVD (the QR SVD algorithm)
! 2 :: SGESDD (the Divide and Conquer algorithm; if enough
! workspace available, this is the fastest option)
! 3 :: SGESVDQ (the preconditioned QR SVD ; this and 4
! are the most accurate options)
! 4 :: SGEJSV (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.
!.....
! M (input) INTEGER, M >= 0
! The state space dimension (the number of rows of F)
!.....
! N (input) INTEGER, 0 <= N <= M
! The number of data snapshots from a single trajectory,
! taken at equidistant discrete times. This is the
! number of columns of F.
!.....
! F (input/output) REAL(KIND=WP) M-by-N array
! > On entry,
! the columns of F are the sequence of data snapshots
! from a single trajectory, taken at equidistant discrete
! times. It is assumed that the column norms of F are
! in the range of the normalized floating point numbers.
! < On exit,
! If JOBQ == 'Q', the array F contains the orthogonal
! matrix/factor of the QR factorization of the initial
! data snapshots matrix F. See the description of JOBQ.
! If JOBQ == 'N', the entries in F strictly below the main
! diagonal contain, column-wise, the information on the
! Householder vectors, as returned by SGEQRF. The
! remaining information to restore the orthogonal matrix
! of the initial QR factorization is stored in WORK(1:N).
! See the description of WORK.
!.....
! LDF (input) INTEGER, LDF >= M
! The leading dimension of the array F.
!.....
! X (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array
! X is used as workspace to hold representations of the
! leading N-1 snapshots in the orthonormal basis computed
! in the QR factorization of F.
! On exit, the leading K columns of X contain the leading
! K left singular vectors of the above described content
! of X. To lift them to the space of the left singular
! vectors U(:,1:K)of the input data, pre-multiply with the
! Q factor from the initial QR factorization.
! See the descriptions of F, K, V and Z.
!.....
! LDX (input) INTEGER, LDX >= N
! The leading dimension of the array X
!.....
! Y (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array
! Y is used as workspace to hold representations of the
! trailing N-1 snapshots in the orthonormal basis computed
! in the QR factorization of F.
! On exit,
! If JOBT == 'R', Y contains the MIN(M,N)-by-N upper
! triangular factor from the QR factorization of the data
! snapshot matrix F.
!.....
! LDY (input) INTEGER , LDY >= N
! The leading dimension of the array Y
!.....
! 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-1 :: 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 description of K.
!.....
! TOL (input) REAL(KIND=WP), 0 <= TOL < 1
! The tolerance for truncating small singular values.
! See the description of NRNK.
!.....
! K (output) INTEGER, 0 <= K <= N
! The dimension of the SVD/POD basis for the leading N-1
! data snapshots (columns of F) 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.
!.....
! REIG (output) REAL(KIND=WP) (N-1)-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, Z.
!.....
! IMEIG (output) REAL(KIND=WP) (N-1)-by-1 array
! The leading K (K<N) entries of REIG 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, Z.
!.....
! Z (workspace/output) REAL(KIND=WP) M-by-(N-1) 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.
! 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.
! If JOBZ == 'F', then the above descriptions hold for
! the columns of Z*V, where the columns of V are the
! eigenvectors of the K-by-K Rayleigh quotient, and Z is
! orthonormal. The columns of V are similarly structured:
! If IMEIG(i) == 0 then Z*V(:,i) is an eigenvector, and if
! IMEIG(i) > 0 then Z*V(:,i)+sqrt(-1)*Z*V(:,i+1) and
! Z*V(:,i)-sqrt(-1)*Z*V(:,i+1)
! are the eigenvectors of LAMBDA(i), LAMBDA(i+1).
! See the descriptions of REIG, IMEIG, X and V.
!.....
! LDZ (input) INTEGER , LDZ >= M
! The leading dimension of the array Z.
!.....
! RES (output) REAL(KIND=WP) (N-1)-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 Z.
!.....
! B (output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array.
! IF JOBF =='R', B(1:N,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:N,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.
! In both cases, the content of B can be lifted to the
! original dimension of the input data by pre-multiplying
! with the Q factor from the initial QR factorization.
! Here A denotes a compression of the underlying operator.
! See the descriptions of F and X.
! If JOBF =='N', then B is not referenced.
!.....
! LDB (input) INTEGER, LDB >= MIN(M,N)
! The leading dimension of the array B.
!.....
! V (workspace/output) REAL(KIND=WP) (N-1)-by-(N-1) array
! On exit, V(1:K,1:K) contains the K eigenvectors of
! the Rayleigh quotient. The eigenvectors of a complex
! conjugate pair of eigenvalues are returned in real form
! as explained in the description of Z. The Ritz vectors
! (returned in Z) are the product of X and V; see
! the descriptions of X and Z.
!.....
! LDV (input) INTEGER, LDV >= N-1
! The leading dimension of the array V.
!.....
! S (output) REAL(KIND=WP) (N-1)-by-(N-1) array
! The array S(1:K,1:K) is used for the matrix Rayleigh
! quotient. This content is overwritten during
! the eigenvalue decomposition by SGEEV.
! See the description of K.
!.....
! LDS (input) INTEGER, LDS >= N-1
! The leading dimension of the array S.
!.....
! WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
! On exit,
! WORK(1:MIN(M,N)) contains the scalar factors of the
! elementary reflectors as returned by SGEQRF of the
! M-by-N input matrix F.
! WORK(MIN(M,N)+1:MIN(M,N)+N-1) contains the singular values of
! the input submatrix F(1:M,1:N-1).
! If the call to SGEDMDQ is only workspace query, then
! WORK(1) contains the minimal workspace length and
! WORK(2) is the optimal workspace length. Hence, the
! length of work is at least 2.
! See the description of LWORK.
!.....
! LWORK (input) INTEGER
! The minimal length of the workspace vector WORK.
! LWORK is calculated as follows:
! Let MLWQR = N (minimal workspace for SGEQRF[M,N])
! MLWDMD = minimal workspace for SGEDMD (see the
! description of LWORK in SGEDMD) for
! snapshots of dimensions MIN(M,N)-by-(N-1)
! MLWMQR = N (minimal workspace for
! SORMQR['L','N',M,N,N])
! MLWGQR = N (minimal workspace for SORGQR[M,N,N])
! Then
! LWORK = MAX(N+MLWQR, N+MLWDMD)
! is updated as follows:
! if JOBZ == 'V' or JOBZ == 'F' THEN
! LWORK = MAX( LWORK,MIN(M,N)+N-1 +MLWMQR )
! if JOBQ == 'Q' THEN
! LWORK = MAX( LWORK,MIN(M,N)+N-1+MLWGQR)
! 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.
!.....
! 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.
!.....
! LIWORK (input) INTEGER
! The minimal length of the workspace vector IWORK.
! If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1
! Let M1=MIN(M,N), N1=N-1. Then
! If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1))
! If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1)
! If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1)
! If on entry LIWORK = -1, then a worskpace 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.
!.....
! 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.
!.............................................................
!.............................................................
! Parameters
! ~~~~~~~~~~
REAL(KIND=WP), PARAMETER :: ONE = 1.0_WP
REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP
!
! Local scalars
! ~~~~~~~~~~~~~
INTEGER :: IMINWR, INFO1, MLWDMD, MLWGQR, &
MLWMQR, MLWORK, MLWQR, MINMN, &
OLWDMD, OLWGQR, OLWMQR, OLWORK, &
OLWQR
LOGICAL :: LQUERY, SCCOLX, SCCOLY, WANTQ, &
WNTTRF, WNTRES, WNTVEC, WNTVCF, &
WNTVCQ, WNTREF, WNTEX
CHARACTER(LEN=1) :: JOBVL
!
! Local array
! ~~~~~~~~~~~
REAL(KIND=WP) :: RDUMMY(2)
!
! External functions (BLAS and LAPACK)
! ~~~~~~~~~~~~~~~~~
LOGICAL LSAME
EXTERNAL LSAME
!
! External subroutines (BLAS and LAPACK)
! ~~~~~~~~~~~~~~~~~~~~
EXTERNAL SGEMM
EXTERNAL SGEQRF, SLACPY, SLASET, SORGQR, &
SORMQR, XERBLA
! External subroutines
! ~~~~~~~~~~~~~~~~~~~~
EXTERNAL SGEDMD
! Intrinsic functions
! ~~~~~~~~~~~~~~~~~~~
INTRINSIC MAX, MIN, INT
!..........................................................
!
! Test the input arguments
WNTRES = LSAME(JOBR,'R')
SCCOLX = LSAME(JOBS,'S') .OR. LSAME( JOBS, 'C' )
SCCOLY = LSAME(JOBS,'Y')
WNTVEC = LSAME(JOBZ,'V')
WNTVCF = LSAME(JOBZ,'F')
WNTVCQ = LSAME(JOBZ,'Q')
WNTREF = LSAME(JOBF,'R')
WNTEX = LSAME(JOBF,'E')
WANTQ = LSAME(JOBQ,'Q')
WNTTRF = LSAME(JOBT,'R')
MINMN = MIN(M,N)
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. WNTVCF .OR. WNTVCQ &
.OR. LSAME(JOBZ,'N')) ) THEN
INFO = -2
ELSE IF ( .NOT. (WNTRES .OR. LSAME(JOBR,'N')) .OR. &
( WNTRES .AND. LSAME(JOBZ,'N') ) ) THEN
INFO = -3
ELSE IF ( .NOT. (WANTQ .OR. LSAME(JOBQ,'N')) ) THEN
INFO = -4
ELSE IF ( .NOT. ( WNTTRF .OR. LSAME(JOBT,'N') ) ) THEN
INFO = -5
ELSE IF ( .NOT. (WNTREF .OR. WNTEX .OR. &
LSAME(JOBF,'N') ) ) THEN
INFO = -6
ELSE IF ( .NOT. ((WHTSVD == 1).OR.(WHTSVD == 2).OR. &
(WHTSVD == 3).OR.(WHTSVD == 4)) ) THEN
INFO = -7
ELSE IF ( M < 0 ) THEN
INFO = -8
ELSE IF ( ( N < 0 ) .OR. ( N > M+1 ) ) THEN
INFO = -9
ELSE IF ( LDF < M ) THEN
INFO = -11
ELSE IF ( LDX < MINMN ) THEN
INFO = -13
ELSE IF ( LDY < MINMN ) THEN
INFO = -15
ELSE IF ( .NOT. (( NRNK == -2).OR.(NRNK == -1).OR. &
((NRNK >= 1).AND.(NRNK <=N ))) ) THEN
INFO = -16
ELSE IF ( ( TOL < ZERO ) .OR. ( TOL >= ONE ) ) THEN
INFO = -17
ELSE IF ( LDZ < M ) THEN
INFO = -22
ELSE IF ( (WNTREF.OR.WNTEX ).AND.( LDB < MINMN ) ) THEN
INFO = -25
ELSE IF ( LDV < N-1 ) THEN
INFO = -27
ELSE IF ( LDS < N-1 ) THEN
INFO = -29
END IF
!
IF ( WNTVEC .OR. WNTVCF ) THEN
JOBVL = 'V'
ELSE
JOBVL = 'N'
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 ) .OR. ( N == 1 ) ) THEN
! All output except K is void. INFO=1 signals
! the void input. In case of a workspace query,
! the 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
MLWQR = MAX(1,N) ! Minimal workspace length for SGEQRF.
MLWORK = MIN(M,N) + MLWQR
IF ( LQUERY ) THEN
CALL SGEQRF( M, N, F, LDF, WORK, RDUMMY, -1, &
INFO1 )
OLWQR = INT(RDUMMY(1))
OLWORK = MIN(M,N) + OLWQR
END IF
CALL SGEDMD( JOBS, JOBVL, JOBR, JOBF, WHTSVD, MINMN,&
N-1, X, LDX, Y, LDY, NRNK, TOL, K, &
REIG, IMEIG, Z, LDZ, RES, B, LDB, &
V, LDV, S, LDS, WORK, -1, IWORK, &
LIWORK, INFO1 )
MLWDMD = INT(WORK(1))
MLWORK = MAX(MLWORK, MINMN + MLWDMD)
IMINWR = IWORK(1)
IF ( LQUERY ) THEN
OLWDMD = INT(WORK(2))
OLWORK = MAX(OLWORK, MINMN+OLWDMD)
END IF
IF ( WNTVEC .OR. WNTVCF ) THEN
MLWMQR = MAX(1,N)
MLWORK = MAX(MLWORK,MINMN+N-1+MLWMQR)
IF ( LQUERY ) THEN
CALL SORMQR( 'L','N', M, N, MINMN, F, LDF, &
WORK, Z, LDZ, WORK, -1, INFO1 )
OLWMQR = INT(WORK(1))
OLWORK = MAX(OLWORK,MINMN+N-1+OLWMQR)
END IF
END IF
IF ( WANTQ ) THEN
MLWGQR = N
MLWORK = MAX(MLWORK,MINMN+N-1+MLWGQR)
IF ( LQUERY ) THEN
CALL SORGQR( M, MINMN, MINMN, F, LDF, WORK, &
WORK, -1, INFO1 )
OLWGQR = INT(WORK(1))
OLWORK = MAX(OLWORK,MINMN+N-1+OLWGQR)
END IF
END IF
IMINWR = MAX( 1, IMINWR )
MLWORK = MAX( 2, MLWORK )
IF ( LWORK < MLWORK .AND. (.NOT.LQUERY) ) INFO = -31
IF ( LIWORK < IMINWR .AND. (.NOT.LQUERY) ) INFO = -33
END IF
IF( INFO /= 0 ) THEN
CALL XERBLA( 'SGEDMDQ', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
! Return minimal and optimal workspace sizes
IWORK(1) = IMINWR
WORK(1) = MLWORK
WORK(2) = OLWORK
RETURN
END IF
!.....
! Initial QR factorization that is used to represent the
! snapshots as elements of lower dimensional subspace.
! For large scale computation with M >>N , at this place
! one can use an out of core QRF.
!
CALL SGEQRF( M, N, F, LDF, WORK, &
WORK(MINMN+1), LWORK-MINMN, INFO1 )
!
! Define X and Y as the snapshots representations in the
! orthogonal basis computed in the QR factorization.
! X corresponds to the leading N-1 and Y to the trailing
! N-1 snapshots.
CALL SLASET( 'L', MINMN, N-1, ZERO, ZERO, X, LDX )
CALL SLACPY( 'U', MINMN, N-1, F, LDF, X, LDX )
CALL SLACPY( 'A', MINMN, N-1, F(1,2), LDF, Y, LDY )
IF ( M >= 3 ) THEN
CALL SLASET( 'L', MINMN-2, N-2, ZERO, ZERO, &
Y(3,1), LDY )
END IF
!
! Compute the DMD of the projected snapshot pairs (X,Y)
CALL SGEDMD( JOBS, JOBVL, JOBR, JOBF, WHTSVD, MINMN, &
N-1, X, LDX, Y, LDY, NRNK, TOL, K, &
REIG, IMEIG, Z, LDZ, RES, B, LDB, V, &
LDV, S, LDS, WORK(MINMN+1), LWORK-MINMN, IWORK, &
LIWORK, INFO1 )
IF ( INFO1 == 2 .OR. INFO1 == 3 ) THEN
! Return with error code.
INFO = INFO1
RETURN
ELSE
INFO = INFO1
END IF
!
! The Ritz vectors (Koopman modes) can be explicitly
! formed or returned in factored form.
IF ( WNTVEC ) THEN
! Compute the eigenvectors explicitly.
IF ( M > MINMN ) CALL SLASET( 'A', M-MINMN, K, ZERO, &
ZERO, Z(MINMN+1,1), LDZ )
CALL SORMQR( 'L','N', M, K, MINMN, F, LDF, WORK, Z, &
LDZ, WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )
ELSE IF ( WNTVCF ) THEN
! Return the Ritz vectors (eigenvectors) in factored
! form Z*V, where Z contains orthonormal matrix (the
! product of Q from the initial QR factorization and
! the SVD/POD_basis returned by SGEDMD in X) and the
! second factor (the eigenvectors of the Rayleigh
! quotient) is in the array V, as returned by SGEDMD.
CALL SLACPY( 'A', N, K, X, LDX, Z, LDZ )
IF ( M > N ) CALL SLASET( 'A', M-N, K, ZERO, ZERO, &
Z(N+1,1), LDZ )
CALL SORMQR( 'L','N', M, K, MINMN, F, LDF, WORK, Z, &
LDZ, WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )
END IF
!
! Some optional output variables:
!
! The upper triangular factor in the initial QR
! factorization is optionally returned in the array Y.
! This is useful if this call to SGEDMDQ is to be
! followed by a streaming DMD that is implemented in a
! QR compressed form.
IF ( WNTTRF ) THEN ! Return the upper triangular R in Y
CALL SLASET( 'A', MINMN, N, ZERO, ZERO, Y, LDY )
CALL SLACPY( 'U', MINMN, N, F, LDF, Y, LDY )
END IF
!
! The orthonormal/orthogonal factor in the initial QR
! factorization is optionally returned in the array F.
! Same as with the triangular factor above, this is
! useful in a streaming DMD.
IF ( WANTQ ) THEN ! Q overwrites F
CALL SORGQR( M, MINMN, MINMN, F, LDF, WORK, &
WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )
END IF
!
RETURN
!
END SUBROUTINE SGEDMDQ