diff --git a/lapack-netlib/SRC/cgedmd.f90 b/lapack-netlib/SRC/cgedmd.f90 index 499489270d..1413130ec3 100644 --- a/lapack-netlib/SRC/cgedmd.f90 +++ b/lapack-netlib/SRC/cgedmd.f90 @@ -1,22 +1,526 @@ +!> \brief \b CGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices. +! +! =========== DOCUMENTATION =========== +! +! Definition: +! =========== +! +! SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & +! M, N, X, LDX, Y, LDY, NRNK, TOL, & +! K, EIGS, Z, LDZ, RES, B, LDB, & +! W, LDW, S, LDS, ZWORK, LZWORK, & +! RWORK, LRWORK, IWORK, LIWORK, INFO ) +!..... +! USE iso_fortran_env +! IMPLICIT NONE +! INTEGER, PARAMETER :: WP = real32 +! +!..... +! Scalar arguments +! CHARACTER, INTENT(IN) :: JOBS, JOBZ, JOBR, JOBF +! INTEGER, INTENT(IN) :: WHTSVD, M, N, LDX, LDY, & +! NRNK, LDZ, LDB, LDW, LDS, & +! LIWORK, LRWORK, LZWORK +! INTEGER, INTENT(OUT) :: K, INFO +! REAL(KIND=WP), INTENT(IN) :: TOL +! Array arguments +! COMPLEX(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*) +! COMPLEX(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), & +! W(LDW,*), S(LDS,*) +! COMPLEX(KIND=WP), INTENT(OUT) :: EIGS(*) +! COMPLEX(KIND=WP), INTENT(OUT) :: ZWORK(*) +! REAL(KIND=WP), INTENT(OUT) :: RES(*) +! REAL(KIND=WP), INTENT(OUT) :: RWORK(*) +! INTEGER, INTENT(OUT) :: IWORK(*) +! +!............................................................ +!> \par Purpose: +! ============= +!> \verbatim +!> CGEDMD 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, CGEDMD 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, CGEDMD 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) 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 :: CGESVD (the QR SVD algorithm) +!> 2 :: CGESDD (the Divide and Conquer algorithm; if enough +!> workspace available, this is the fastest option) +!> 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 +!> are the most accurate options) +!> 4 :: CGEJSV (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) COMPLEX(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) COMPLEX(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] EIGS +!> \verbatim +!> EIGS (output) COMPLEX(KIND=WP) N-by-1 array +!> The leading K (K<=N) entries of EIGS contain +!> the computed eigenvalues (Ritz values). +!> See the descriptions of K, and Z. +!> \endverbatim +!..... +!> \param[out] Z +!> \verbatim +!> Z (workspace/output) COMPLEX(KIND=WP) M-by-N array +!> If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) +!> is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. +!> If JOBZ == 'F', then the Z(:,i)'s are given implicitly as +!> the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) +!> is an eigenvector corresponding to EIGS(i). The columns +!> of W(1:k,1:K) are the computed eigenvectors of the +!> K-by-K Rayleigh quotient. +!> See the descriptions of EIGS, 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, +!> RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. +!> See the description of EIGS and Z. +!> \endverbatim +!..... +!> \param[out] B +!> \verbatim +!> B (output) COMPLEX(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) COMPLEX(KIND=WP) N-by-N array +!> On exit, W(1:K,1:K) contains the K computed +!> eigenvectors of the matrix Rayleigh quotient. +!> 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) COMPLEX(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 CGEEV. +!> 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] ZWORK +!> \verbatim +!> ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array +!> ZWORK is used as complex workspace in the complex SVD, as +!> specified by WHTSVD (1,2, 3 or 4) and for CGEEV for computing +!> the eigenvalues of a Rayleigh quotient. +!> If the call to CGEDMD is only workspace query, then +!> ZWORK(1) contains the minimal complex workspace length and +!> ZWORK(2) is the optimal complex workspace length. +!> Hence, the length of work is at least 2. +!> See the description of LZWORK. +!> \endverbatim +!..... +!> \param[in] LZWORK +!> \verbatim +!> LZWORK (input) INTEGER +!> The minimal length of the workspace vector ZWORK. +!> LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_CGEEV), +!> where LZWORK_CGEEV = MAX( 1, 2*N ) and the minimal +!> LZWORK_SVD is calculated as follows +!> If WHTSVD == 1 :: CGESVD :: +!> LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) +!> If WHTSVD == 2 :: CGESDD :: +!> LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) +!> If WHTSVD == 3 :: CGESVDQ :: +!> LZWORK_SVD = obtainable by a query +!> If WHTSVD == 4 :: CGEJSV :: +!> LZWORK_SVD = obtainable by a query +!> If on entry LZWORK = -1, then a workspace query is +!> assumed and the procedure only computes the minimal +!> and the optimal workspace lengths and returns them in +!> LZWORK(1) and LZWORK(2), respectively. +!> \endverbatim +!..... +!> \param[out] RWORK +!> \verbatim +!> RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array +!> On exit, RWORK(1:N) contains the singular values of +!> X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). +!> If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain +!> scaling factor RWORK(N+2)/RWORK(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 CGEDMD is only workspace query, then +!> RWORK(1) contains the minimal workspace length. +!> See the description of LRWORK. +!> \endverbatim +!..... +!> \param[in] LRWORK +!> \verbatim +!> LRWORK (input) INTEGER +!> The minimal length of the workspace vector RWORK. +!> LRWORK is calculated as follows: +!> LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_CGEEV), where +!> LRWORK_CGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace +!> for the SVD subroutine determined by the input parameter +!> WHTSVD. +!> If WHTSVD == 1 :: CGESVD :: +!> LRWORK_SVD = 5*MIN(M,N) +!> If WHTSVD == 2 :: CGESDD :: +!> LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), +!> 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) +!> If WHTSVD == 3 :: CGESVDQ :: +!> LRWORK_SVD = obtainable by a query +!> If WHTSVD == 4 :: CGEJSV :: +!> LRWORK_SVD = obtainable by a query +!> If on entry LRWORK = -1, then a workspace query is +!> assumed and the procedure only computes the minimal +!> real workspace length and returns it in RWORK(1). +!> \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 ZWORK, RWORK and +!> IWORK. See the descriptions of ZWORK, RWORK 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 CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & M, N, X, LDX, Y, LDY, NRNK, TOL, & K, EIGS, Z, LDZ, RES, B, LDB, & W, LDW, S, LDS, ZWORK, LZWORK, & RWORK, LRWORK, IWORK, LIWORK, INFO ) -! March 2023 +! +! -- 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 iso_fortran_env IMPLICIT NONE INTEGER, PARAMETER :: WP = real32 -!..... +! ! Scalar arguments +! ~~~~~~~~~~~~~~~~ CHARACTER, INTENT(IN) :: JOBS, JOBZ, JOBR, JOBF INTEGER, INTENT(IN) :: WHTSVD, M, N, LDX, LDY, & NRNK, LDZ, LDB, LDW, LDS, & LIWORK, LRWORK, LZWORK INTEGER, INTENT(OUT) :: K, INFO REAL(KIND=WP), INTENT(IN) :: TOL +! ! Array arguments +! ~~~~~~~~~~~~~~~ COMPLEX(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*) COMPLEX(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), & W(LDW,*), S(LDS,*) @@ -25,364 +529,14 @@ SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & REAL(KIND=WP), INTENT(OUT) :: RES(*) REAL(KIND=WP), INTENT(OUT) :: RWORK(*) INTEGER, INTENT(OUT) :: IWORK(*) -!............................................................ -! Purpose -! ======= -! CGEDMD 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, CGEDMD 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, CGEDMD 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. -! '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 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. -!..... -! 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. -!..... -! 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. -!..... -! WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } -! Allows for a selection of the SVD algorithm from the -! LAPACK library. -! 1 :: CGESVD (the QR SVD algorithm) -! 2 :: CGESDD (the Divide and Conquer algorithm; if enough -! workspace available, this is the fastest option) -! 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 -! are the most accurate options) -! 4 :: CGEJSV (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 row dimension of X, Y). -!..... -! N (input) INTEGER, 0 <= N <= M -! The number of data snapshot pairs -! (the number of columns of X and Y). -!..... -! X (input/output) COMPLEX(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. -!..... -! LDX (input) INTEGER, LDX >= M -! The leading dimension of the array X. -!..... -! Y (input/workspace/output) COMPLEX(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. -!..... -! LDY (input) INTEGER , LDY >= M -! 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 :: 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. -!..... -! 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 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. -!..... -! EIGS (output) COMPLEX(KIND=WP) N-by-1 array -! The leading K (K<=N) entries of EIGS contain -! the computed eigenvalues (Ritz values). -! See the descriptions of K, and Z. -!..... -! Z (workspace/output) COMPLEX(KIND=WP) M-by-N array -! If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) -! is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. -! If JOBZ == 'F', then the Z(:,i)'s are given implicitly as -! the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) -! is an eigenvector corresponding to EIGS(i). The columns -! of W(1:k,1:K) are the computed eigenvectors of the -! K-by-K Rayleigh quotient. -! See the descriptions of EIGS, X and W. -!..... -! LDZ (input) INTEGER , LDZ >= M -! The leading dimension of the array Z. -!..... -! RES (output) REAL(KIND=WP) N-by-1 array -! RES(1:K) contains the residuals for the K computed -! Ritz pairs, -! RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. -! See the description of EIGS and Z. -!..... -! B (output) COMPLEX(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. -!..... -! LDB (input) INTEGER, LDB >= M -! The leading dimension of the array B. -!..... -! W (workspace/output) COMPLEX(KIND=WP) N-by-N array -! On exit, W(1:K,1:K) contains the K computed -! eigenvectors of the matrix Rayleigh quotient. -! 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. -!..... -! LDW (input) INTEGER, LDW >= N -! The leading dimension of the array W. -!..... -! S (workspace/output) COMPLEX(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 CGEEV. -! See the description of K. -!..... -! LDS (input) INTEGER, LDS >= N -! The leading dimension of the array S. -!..... -! ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array -! ZWORK is used as complex workspace in the complex SVD, as -! specified by WHTSVD (1,2, 3 or 4) and for CGEEV for computing -! the eigenvalues of a Rayleigh quotient. -! If the call to CGEDMD is only workspace query, then -! ZWORK(1) contains the minimal complex workspace length and -! ZWORK(2) is the optimal complex workspace length. -! Hence, the length of work is at least 2. -! See the description of LZWORK. -!..... -! LZWORK (input) INTEGER -! The minimal length of the workspace vector ZWORK. -! LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_CGEEV), -! where LZWORK_CGEEV = MAX( 1, 2*N ) and the minimal -! LZWORK_SVD is calculated as follows -! If WHTSVD == 1 :: CGESVD :: -! LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) -! If WHTSVD == 2 :: CGESDD :: -! LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) -! If WHTSVD == 3 :: CGESVDQ :: -! LZWORK_SVD = obtainable by a query -! If WHTSVD == 4 :: CGEJSV :: -! LZWORK_SVD = obtainable by a query -! If on entry LZWORK = -1, then a workspace query is -! assumed and the procedure only computes the minimal -! and the optimal workspace lengths and returns them in -! LZWORK(1) and LZWORK(2), respectively. -!..... -! RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array -! On exit, RWORK(1:N) contains the singular values of -! X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). -! If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain -! scaling factor RWORK(N+2)/RWORK(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 CGEDMD is only workspace query, then -! RWORK(1) contains the minimal workspace length. -! See the description of LRWORK. -!..... -! LRWORK (input) INTEGER -! The minimal length of the workspace vector RWORK. -! LRWORK is calculated as follows: -! LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_CGEEV), where -! LRWORK_CGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace -! for the SVD subroutine determined by the input parameter -! WHTSVD. -! If WHTSVD == 1 :: CGESVD :: -! LRWORK_SVD = 5*MIN(M,N) -! If WHTSVD == 2 :: CGESDD :: -! LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), -! 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) -! If WHTSVD == 3 :: CGESVDQ :: -! LRWORK_SVD = obtainable by a query -! If WHTSVD == 4 :: CGEJSV :: -! LRWORK_SVD = obtainable by a query -! If on entry LRWORK = -1, then a workspace query is -! assumed and the procedure only computes the minimal -! real workspace length and returns it in RWORK(1). -!..... -! 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 -! 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 ZWORK, RWORK and -! IWORK. See the descriptions of ZWORK, RWORK 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 COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) COMPLEX(KIND=WP), PARAMETER :: ZZERO = ( 0.0_WP, 0.0_WP ) - +! ! Local scalars ! ~~~~~~~~~~~~~ REAL(KIND=WP) :: OFL, ROOTSC, SCALE, SMALL, & @@ -400,7 +554,7 @@ SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! Local arrays ! ~~~~~~~~~~~~ REAL(KIND=WP) :: RDUMMY(2) - +! ! External functions (BLAS and LAPACK) ! ~~~~~~~~~~~~~~~~~ REAL(KIND=WP) CLANGE, SLAMCH, SCNRM2 @@ -408,13 +562,13 @@ SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & INTEGER ICAMAX LOGICAL SISNAN, LSAME EXTERNAL SISNAN, LSAME - +! ! External subroutines (BLAS and LAPACK) ! ~~~~~~~~~~~~~~~~~~~~ EXTERNAL CAXPY, CGEMM, CSSCAL EXTERNAL CGEEV, CGEJSV, CGESDD, CGESVD, CGESVDQ, & CLACPY, CLASCL, CLASSQ, XERBLA - +! ! Intrinsic functions ! ~~~~~~~~~~~~~~~~~~~ INTRINSIC FLOAT, INT, MAX, SQRT @@ -607,7 +761,8 @@ SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & K = 0 DO i = 1, N !WORK(i) = SCNRM2( M, X(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL CLASSQ( M, X(1,i), 1, SCALE, SSUM ) IF ( SISNAN(SCALE) .OR. SISNAN(SSUM) ) THEN K = 0 @@ -680,7 +835,8 @@ SUBROUTINE CGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! carefully computed using CLASSQ. DO i = 1, N !RWORK(i) = SCNRM2( M, Y(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL CLASSQ( M, Y(1,i), 1, SCALE, SSUM ) IF ( SISNAN(SCALE) .OR. SISNAN(SSUM) ) THEN K = 0 diff --git a/lapack-netlib/SRC/dgedmd.f90 b/lapack-netlib/SRC/dgedmd.f90 index 20424808f9..15df48fe91 100644 --- a/lapack-netlib/SRC/dgedmd.f90 +++ b/lapack-netlib/SRC/dgedmd.f90 @@ -1,424 +1,574 @@ - 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 ) -! March 2023 +!> \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 iso_fortran_env - IMPLICIT NONE - INTEGER, PARAMETER :: WP = real64 +! USE iso_fortran_env +! 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 +! 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(*) -!............................................................ -! Purpose -! ======= -! 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]. -! -! 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. +! 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(*) ! -!...................................................................... -! 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 -!============================================================ !............................................................ +!> \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 ! ========= -! JOBS (input) 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. +! +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! M (input) INTEGER, M>= 0 -! The state space dimension (the row dimension of X, Y). +!> \param[in] M +!> \verbatim +!> M (input) INTEGER, M>= 0 +!> The state space dimension (the row dimension of X, Y). +!> \endverbatim !..... -! N (input) INTEGER, 0 <= N <= M -! The number of data snapshot pairs -! (the number of columns of X and Y). +!> \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 !..... -! 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. +!> \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 !..... -! LDX (input) INTEGER, LDX >= M -! The leading dimension of the array X. +!> \param[in] LDX +!> \verbatim +!> LDX (input) INTEGER, LDX >= M +!> The leading dimension of the array X. +!> \endverbatim !..... -! 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. +!> \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 !..... -! LDY (input) INTEGER , LDY >= M -! The leading dimension of the array Y. +!> \param[in] LDY +!> \verbatim +!> LDY (input) INTEGER , LDY >= M +!> The leading dimension of the array Y. +!> \endverbatim !..... -! 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. +!> \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 !..... -! TOL (input) REAL(KIND=WP), 0 <= TOL < 1 -! The tolerance for truncating small singular values. -! See the description of NRNK. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! LDZ (input) INTEGER , LDZ >= M -! The leading dimension of the array Z. +!> \param[in] LDZ +!> \verbatim +!> LDZ (input) INTEGER , LDZ >= M +!> The leading dimension of the array Z. +!> \endverbatim !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! LDB (input) INTEGER, LDB >= M -! The leading dimension of the array B. +!> \param[in] LDB +!> \verbatim +!> LDB (input) INTEGER, LDB >= M +!> The leading dimension of the array B. +!> \endverbatim !..... -! 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. +!> \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 !..... -! LDW (input) INTEGER, LDW >= N -! The leading dimension of the array W. +!> \param[in] LDW +!> \verbatim +!> LDW (input) INTEGER, LDW >= N +!> The leading dimension of the array W. +!> \endverbatim !..... -! 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. +!> \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 !..... -! LDS (input) INTEGER, LDS >= N -! The leading dimension of the array S. +!> \param[in] LDS +!> \verbatim +!> LDS (input) INTEGER, LDS >= N +!> The leading dimension of the array S. +!> \endverbatim !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 iso_fortran_env + 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, & @@ -432,10 +582,11 @@ SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & 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 @@ -443,13 +594,13 @@ SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & 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 @@ -632,7 +783,8 @@ SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & K = 0 DO i = 1, N !WORK(i) = DNRM2( M, X(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL DLASSQ( M, X(1,i), 1, SCALE, SSUM ) IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN K = 0 @@ -705,7 +857,8 @@ SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! carefully computed using DLASSQ. DO i = 1, N !WORK(i) = DNRM2( M, Y(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL DLASSQ( M, Y(1,i), 1, SCALE, SSUM ) IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN K = 0 @@ -1051,4 +1204,3 @@ SUBROUTINE DGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & RETURN ! ...... END SUBROUTINE DGEDMD - diff --git a/lapack-netlib/SRC/sgedmd.f90 b/lapack-netlib/SRC/sgedmd.f90 index 49cb11527c..4860e88983 100644 --- a/lapack-netlib/SRC/sgedmd.f90 +++ b/lapack-netlib/SRC/sgedmd.f90 @@ -1,423 +1,573 @@ - SUBROUTINE SGEDMD( 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 ) -! March 2023 +!> \brief \b SGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices. +! +! =========== DOCUMENTATION =========== +! +! Definition: +! =========== +! +! SUBROUTINE SGEDMD( 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 iso_fortran_env - IMPLICIT NONE - INTEGER, PARAMETER :: WP = real32 +! USE iso_fortran_env +! IMPLICIT NONE +! INTEGER, PARAMETER :: WP = real32 !..... ! 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 +! 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(*) -!............................................................ -! Purpose -! ======= -! SGEDMD 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, SGEDMD 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, SGEDMD 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. +! 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 +!> SGEDMD 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, SGEDMD 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, SGEDMD 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 !...................................................................... -! 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 -!============================================================ +!> \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 +!> Distribution Statement A: +!> Approved for Public Release, Distribution Unlimited. +!> Cleared by DARPA on September 29, 2022 +!> \endverbatim +!============================================================ ! Arguments ! ========= -! JOBS (input) 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. +! +!> \param[in] JOBS +!> \verbatim +!> JOBS (input) 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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 :: 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. +!> \endverbatim !..... -! M (input) INTEGER, M>= 0 -! The state space dimension (the row dimension of X, Y). +!> \param[in] M +!> \verbatim +!> M (input) INTEGER, M>= 0 +!> The state space dimension (the row dimension of X, Y). +!> \endverbatim !..... -! N (input) INTEGER, 0 <= N <= M -! The number of data snapshot pairs -! (the number of columns of X and Y). +!> \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 !..... -! 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. +!> \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 !..... -! LDX (input) INTEGER, LDX >= M -! The leading dimension of the array X. +!> \param[in] LDX +!> \verbatim +!> LDX (input) INTEGER, LDX >= M +!> The leading dimension of the array X. +!> \endverbatim !..... -! 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. +!> \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 !..... -! LDY (input) INTEGER , LDY >= M -! The leading dimension of the array Y. +!> \param[in] LDY +!> \verbatim +!> LDY (input) INTEGER , LDY >= M +!> The leading dimension of the array Y. +!> \endverbatim !..... -! 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. +!> \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 !..... -! TOL (input) REAL(KIND=WP), 0 <= TOL < 1 -! The tolerance for truncating small singular values. -! See the description of NRNK. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! LDZ (input) INTEGER , LDZ >= M -! The leading dimension of the array Z. +!> \param[in] LDZ +!> \verbatim +!> LDZ (input) INTEGER , LDZ >= M +!> The leading dimension of the array Z. +!> \endverbatim !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! LDB (input) INTEGER, LDB >= M -! The leading dimension of the array B. +!> \param[in] LDB +!> \verbatim +!> LDB (input) INTEGER, LDB >= M +!> The leading dimension of the array B. +!> \endverbatim !..... -! 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 -! left singular vectors of X. +!> \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 +!> left singular vectors of X. +!> \endverbatim !..... -! LDW (input) INTEGER, LDW >= N -! The leading dimension of the array W. +!> \param[in] LDW +!> \verbatim +!> LDW (input) INTEGER, LDW >= N +!> The leading dimension of the array W. +!> \endverbatim !..... -! 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 SGEEV. -! See the description of K. +!> \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 SGEEV. +!> See the description of K. +!> \endverbatim !..... -! LDS (input) INTEGER, LDS >= N -! The leading dimension of the array S. +!> \param[in] LDS +!> \verbatim +!> LDS (input) INTEGER, LDS >= N +!> The leading dimension of the array S. +!> \endverbatim !..... -! 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 SGEDMD 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. +!> \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 SGEDMD 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. +!> \endverbatim !..... -! 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 SGESVD. -! 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 SGESDD. -! 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 SGESVDQ. -! 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 SGEJSV. -! 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. +!> \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 SGESVD. +!> 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 SGESDD. +!> 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 SGESVDQ. +!> 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 SGEJSV. +!> 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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 SGEDMD( 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 iso_fortran_env + IMPLICIT NONE + INTEGER, PARAMETER :: WP = real32 +! +! 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, & @@ -431,11 +581,11 @@ SUBROUTINE SGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & 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) SLANGE, SLAMCH, SNRM2 @@ -443,13 +593,13 @@ SUBROUTINE SGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & INTEGER ISAMAX LOGICAL SISNAN, LSAME EXTERNAL SISNAN, LSAME - +! ! External subroutines (BLAS and LAPACK) ! ~~~~~~~~~~~~~~~~~~~~ EXTERNAL SAXPY, SGEMM, SSCAL EXTERNAL SGEEV, SGEJSV, SGESDD, SGESVD, SGESVDQ, & SLACPY, SLASCL, SLASSQ, XERBLA - +! ! Intrinsic functions ! ~~~~~~~~~~~~~~~~~~~ INTRINSIC INT, FLOAT, MAX, SQRT @@ -632,7 +782,8 @@ SUBROUTINE SGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & K = 0 DO i = 1, N !WORK(i) = DNRM2( M, X(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL SLASSQ( M, X(1,i), 1, SCALE, SSUM ) IF ( SISNAN(SCALE) .OR. SISNAN(SSUM) ) THEN K = 0 @@ -705,7 +856,8 @@ SUBROUTINE SGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! carefully computed using SLASSQ. DO i = 1, N !WORK(i) = DNRM2( M, Y(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL SLASSQ( M, Y(1,i), 1, SCALE, SSUM ) IF ( SISNAN(SCALE) .OR. SISNAN(SSUM) ) THEN K = 0 diff --git a/lapack-netlib/SRC/zgedmd.f90 b/lapack-netlib/SRC/zgedmd.f90 index 090641ad84..5045cb166c 100644 --- a/lapack-netlib/SRC/zgedmd.f90 +++ b/lapack-netlib/SRC/zgedmd.f90 @@ -1,389 +1,539 @@ - SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & - M, N, X, LDX, Y, LDY, NRNK, TOL, & - K, EIGS, Z, LDZ, RES, B, LDB, & - W, LDW, S, LDS, ZWORK, LZWORK, & - RWORK, LRWORK, IWORK, LIWORK, INFO ) -! March 2023 -!..... - USE iso_fortran_env - 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, & - LIWORK, LRWORK, LZWORK - INTEGER, INTENT(OUT) :: K, INFO - REAL(KIND=WP), INTENT(IN) :: TOL -! Array arguments - COMPLEX(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*) - COMPLEX(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), & - W(LDW,*), S(LDS,*) - COMPLEX(KIND=WP), INTENT(OUT) :: EIGS(*) - COMPLEX(KIND=WP), INTENT(OUT) :: ZWORK(*) - REAL(KIND=WP), INTENT(OUT) :: RES(*) - REAL(KIND=WP), INTENT(OUT) :: RWORK(*) - INTEGER, INTENT(OUT) :: IWORK(*) -!............................................................ -! Purpose -! ======= -! ZGEDMD 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, ZGEDMD 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, ZGEDMD 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. +!> \brief \b ZGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices. +! +! =========== DOCUMENTATION =========== +! +! Definition: +! =========== ! +! SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & +! M, N, X, LDX, Y, LDY, NRNK, TOL, & +! K, EIGS, Z, LDZ, RES, B, LDB, & +! W, LDW, S, LDS, ZWORK, LZWORK, & +! RWORK, LRWORK, IWORK, LIWORK, INFO ) +!...... +! USE iso_fortran_env +! 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, & +! LIWORK, LRWORK, LZWORK +! INTEGER, INTENT(OUT) :: K, INFO +! REAL(KIND=WP), INTENT(IN) :: TOL +! Array arguments +! COMPLEX(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*) +! COMPLEX(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), & +! W(LDW,*), S(LDS,*) +! COMPLEX(KIND=WP), INTENT(OUT) :: EIGS(*) +! COMPLEX(KIND=WP), INTENT(OUT) :: ZWORK(*) +! REAL(KIND=WP), INTENT(OUT) :: RES(*) +! REAL(KIND=WP), INTENT(OUT) :: RWORK(*) +! INTEGER, INTENT(OUT) :: IWORK(*) +! +!............................................................ +!> \par Purpose: +! ============= +!> \verbatim +!> ZGEDMD 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, ZGEDMD 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, ZGEDMD 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 !...................................................................... -! 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 -!============================================================ +!> \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 ! ========= -! JOBS (input) 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. +! +!> \param[in] JOBS +!> \verbatim +!> JOBS (input) 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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! 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. +!> \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 !..... -! WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } -! Allows for a selection of the SVD algorithm from the -! LAPACK library. -! 1 :: ZGESVD (the QR SVD algorithm) -! 2 :: ZGESDD (the Divide and Conquer algorithm; if enough -! workspace available, this is the fastest option) -! 3 :: ZGESVDQ (the preconditioned QR SVD ; this and 4 -! are the most accurate options) -! 4 :: ZGEJSV (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. +!> \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 :: ZGESVD (the QR SVD algorithm) +!> 2 :: ZGESDD (the Divide and Conquer algorithm; if enough +!> workspace available, this is the fastest option) +!> 3 :: ZGESVDQ (the preconditioned QR SVD ; this and 4 +!> are the most accurate options) +!> 4 :: ZGEJSV (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 !..... -! M (input) INTEGER, M>= 0 -! The state space dimension (the row dimension of X, Y). +!> \param[in] M +!> \verbatim +!> M (input) INTEGER, M>= 0 +!> The state space dimension (the row dimension of X, Y). +!> \endverbatim !..... -! N (input) INTEGER, 0 <= N <= M -! The number of data snapshot pairs -! (the number of columns of X and Y). +!> \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 !..... -! X (input/output) COMPLEX(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. +!> \param[in] LDX +!> \verbatim +!> X (input/output) COMPLEX(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. !..... -! LDX (input) INTEGER, LDX >= M -! The leading dimension of the array X. +!> LDX (input) INTEGER, LDX >= M +!> The leading dimension of the array X. +!> \endverbatim !..... -! Y (input/workspace/output) COMPLEX(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. +!> \param[in,out] Y +!> \verbatim +!> Y (input/workspace/output) COMPLEX(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 !..... -! LDY (input) INTEGER , LDY >= M -! The leading dimension of the array Y. +!> \param[in] LDY +!> \verbatim +!> LDY (input) INTEGER , LDY >= M +!> The leading dimension of the array Y. +!> \endverbatim !..... -! 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. +!> \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 !..... -! TOL (input) REAL(KIND=WP), 0 <= TOL < 1 -! The tolerance for truncating small singular values. -! See the description of NRNK. +!> \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 !..... -! 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. +!> \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 !..... -! EIGS (output) COMPLEX(KIND=WP) N-by-1 array -! The leading K (K<=N) entries of EIGS contain -! the computed eigenvalues (Ritz values). -! See the descriptions of K, and Z. +!> \param[out] EIGS +!> \verbatim +!> EIGS (output) COMPLEX(KIND=WP) N-by-1 array +!> The leading K (K<=N) entries of EIGS contain +!> the computed eigenvalues (Ritz values). +!> See the descriptions of K, and Z. +!> \endverbatim !..... -! Z (workspace/output) COMPLEX(KIND=WP) M-by-N array -! If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) -! is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. -! If JOBZ == 'F', then the Z(:,i)'s are given implicitly as -! the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) -! is an eigenvector corresponding to EIGS(i). The columns -! of W(1:k,1:K) are the computed eigenvectors of the -! K-by-K Rayleigh quotient. -! See the descriptions of EIGS, X and W. +!> \param[out] Z +!> \verbatim +!> Z (workspace/output) COMPLEX(KIND=WP) M-by-N array +!> If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) +!> is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. +!> If JOBZ == 'F', then the Z(:,i)'s are given implicitly as +!> the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) +!> is an eigenvector corresponding to EIGS(i). The columns +!> of W(1:k,1:K) are the computed eigenvectors of the +!> K-by-K Rayleigh quotient. +!> See the descriptions of EIGS, X and W. +!> \endverbatim !..... -! LDZ (input) INTEGER , LDZ >= M -! The leading dimension of the array Z. +!> \param[in] LDZ +!> \verbatim +!> LDZ (input) INTEGER , LDZ >= M +!> The leading dimension of the array Z. +!> \endverbatim !..... -! RES (output) REAL(KIND=WP) N-by-1 array -! RES(1:K) contains the residuals for the K computed -! Ritz pairs, -! RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. -! See the description of EIGS and Z. +!> \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, +!> RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. +!> See the description of EIGS and Z. +!> \endverbatim !..... -! B (output) COMPLEX(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. +!> \param[out] B +!> \verbatim +!> B (output) COMPLEX(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 !..... -! LDB (input) INTEGER, LDB >= M -! The leading dimension of the array B. +!> \param[in] LDB +!> \verbatim +!> LDB (input) INTEGER, LDB >= M +!> The leading dimension of the array B. +!> \endverbatim !..... -! W (workspace/output) COMPLEX(KIND=WP) N-by-N array -! On exit, W(1:K,1:K) contains the K computed -! eigenvectors of the matrix Rayleigh quotient. -! 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. +!> \param[out] W +!> \verbatim +!> W (workspace/output) COMPLEX(KIND=WP) N-by-N array +!> On exit, W(1:K,1:K) contains the K computed +!> eigenvectors of the matrix Rayleigh quotient. +!> 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 !..... -! LDW (input) INTEGER, LDW >= N -! The leading dimension of the array W. +!> \param[in] LDW +!> \verbatim +!> LDW (input) INTEGER, LDW >= N +!> The leading dimension of the array W. +!> \endverbatim !..... -! S (workspace/output) COMPLEX(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 ZGEEV. -! See the description of K. +!> \param[out] S +!> \verbatim +!> S (workspace/output) COMPLEX(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 ZGEEV. +!> See the description of K. +!> \endverbatim !..... -! LDS (input) INTEGER, LDS >= N -! The leading dimension of the array S. +!> \param[in] LDS +!> \verbatim +!> LDS (input) INTEGER, LDS >= N +!> The leading dimension of the array S. +!> \endverbatim !..... -! ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array -! ZWORK is used as complex workspace in the complex SVD, as -! specified by WHTSVD (1,2, 3 or 4) and for ZGEEV for computing -! the eigenvalues of a Rayleigh quotient. -! If the call to ZGEDMD is only workspace query, then -! ZWORK(1) contains the minimal complex workspace length and -! ZWORK(2) is the optimal complex workspace length. -! Hence, the length of work is at least 2. -! See the description of LZWORK. +!> \param[out] ZWORK +!> \verbatim +!> ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array +!> ZWORK is used as complex workspace in the complex SVD, as +!> specified by WHTSVD (1,2, 3 or 4) and for ZGEEV for computing +!> the eigenvalues of a Rayleigh quotient. +!> If the call to ZGEDMD is only workspace query, then +!> ZWORK(1) contains the minimal complex workspace length and +!> ZWORK(2) is the optimal complex workspace length. +!> Hence, the length of work is at least 2. +!> See the description of LZWORK. +!> \endverbatim !..... -! LZWORK (input) INTEGER -! The minimal length of the workspace vector ZWORK. -! LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_ZGEEV), -! where LZWORK_ZGEEV = MAX( 1, 2*N ) and the minimal -! LZWORK_SVD is calculated as follows -! If WHTSVD == 1 :: ZGESVD :: -! LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) -! If WHTSVD == 2 :: ZGESDD :: -! LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) -! If WHTSVD == 3 :: ZGESVDQ :: -! LZWORK_SVD = obtainable by a query -! If WHTSVD == 4 :: ZGEJSV :: -! LZWORK_SVD = obtainable by a query -! If on entry LZWORK = -1, then a workspace query is -! assumed and the procedure only computes the minimal -! and the optimal workspace lengths and returns them in -! LZWORK(1) and LZWORK(2), respectively. +!> \param[in] LZWORK +!> \verbatim +!> LZWORK (input) INTEGER +!> The minimal length of the workspace vector ZWORK. +!> LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_ZGEEV), +!> where LZWORK_ZGEEV = MAX( 1, 2*N ) and the minimal +!> LZWORK_SVD is calculated as follows +!> If WHTSVD == 1 :: ZGESVD :: +!> LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) +!> If WHTSVD == 2 :: ZGESDD :: +!> LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) +!> If WHTSVD == 3 :: ZGESVDQ :: +!> LZWORK_SVD = obtainable by a query +!> If WHTSVD == 4 :: ZGEJSV :: +!> LZWORK_SVD = obtainable by a query +!> If on entry LZWORK = -1, then a workspace query is +!> assumed and the procedure only computes the minimal +!> and the optimal workspace lengths and returns them in +!> LZWORK(1) and LZWORK(2), respectively. +!> \endverbatim !..... -! RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array -! On exit, RWORK(1:N) contains the singular values of -! X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). -! If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain -! scaling factor RWORK(N+2)/RWORK(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 ZGEDMD is only workspace query, then -! RWORK(1) contains the minimal workspace length. -! See the description of LRWORK. +!> \param[out] RWORK +!> \verbatim +!> RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array +!> On exit, RWORK(1:N) contains the singular values of +!> X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). +!> If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain +!> scaling factor RWORK(N+2)/RWORK(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 ZGEDMD is only workspace query, then +!> RWORK(1) contains the minimal workspace length. +!> See the description of LRWORK. +!> \endverbatim !..... -! LRWORK (input) INTEGER -! The minimal length of the workspace vector RWORK. -! LRWORK is calculated as follows: -! LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_ZGEEV), where -! LRWORK_ZGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace -! for the SVD subroutine determined by the input parameter -! WHTSVD. -! If WHTSVD == 1 :: ZGESVD :: -! LRWORK_SVD = 5*MIN(M,N) -! If WHTSVD == 2 :: ZGESDD :: -! LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), -! 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) -! If WHTSVD == 3 :: ZGESVDQ :: -! LRWORK_SVD = obtainable by a query -! If WHTSVD == 4 :: ZGEJSV :: -! LRWORK_SVD = obtainable by a query -! If on entry LRWORK = -1, then a workspace query is -! assumed and the procedure only computes the minimal -! real workspace length and returns it in RWORK(1). +!> \param[in] LRWORK +!> \verbatim +!> LRWORK (input) INTEGER +!> The minimal length of the workspace vector RWORK. +!> LRWORK is calculated as follows: +!> LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_ZGEEV), where +!> LRWORK_ZGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace +!> for the SVD subroutine determined by the input parameter +!> WHTSVD. +!> If WHTSVD == 1 :: ZGESVD :: +!> LRWORK_SVD = 5*MIN(M,N) +!> If WHTSVD == 2 :: ZGESDD :: +!> LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), +!> 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) +!> If WHTSVD == 3 :: ZGESVDQ :: +!> LRWORK_SVD = obtainable by a query +!> If WHTSVD == 4 :: ZGEJSV :: +!> LRWORK_SVD = obtainable by a query +!> If on entry LRWORK = -1, then a workspace query is +!> assumed and the procedure only computes the minimal +!> real workspace length and returns it in RWORK(1). +!> \endverbatim !..... -! 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. +!> \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 !..... -! 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 ZWORK, RWORK and -! IWORK. See the descriptions of ZWORK, RWORK and IWORK. +!> \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 ZWORK, RWORK and +!> IWORK. See the descriptions of ZWORK, RWORK and IWORK. +!> \endverbatim !..... -! 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. +!> \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 ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & + M, N, X, LDX, Y, LDY, NRNK, TOL, & + K, EIGS, Z, LDZ, RES, B, LDB, & + W, LDW, S, LDS, ZWORK, LZWORK, & + RWORK, LRWORK, 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 iso_fortran_env + 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, & + LIWORK, LRWORK, LZWORK + INTEGER, INTENT(OUT) :: K, INFO + REAL(KIND=WP), INTENT(IN) :: TOL +! +! Array arguments +! ~~~~~~~~~~~~~~~ + COMPLEX(KIND=WP), INTENT(INOUT) :: X(LDX,*), Y(LDY,*) + COMPLEX(KIND=WP), INTENT(OUT) :: Z(LDZ,*), B(LDB,*), & + W(LDW,*), S(LDS,*) + COMPLEX(KIND=WP), INTENT(OUT) :: EIGS(*) + COMPLEX(KIND=WP), INTENT(OUT) :: ZWORK(*) + REAL(KIND=WP), INTENT(OUT) :: RES(*) + REAL(KIND=WP), INTENT(OUT) :: RWORK(*) + INTEGER, INTENT(OUT) :: IWORK(*) +! ! Parameters ! ~~~~~~~~~~ REAL(KIND=WP), PARAMETER :: ONE = 1.0_WP REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) COMPLEX(KIND=WP), PARAMETER :: ZZERO = ( 0.0_WP, 0.0_WP ) - +! ! Local scalars ! ~~~~~~~~~~~~~ REAL(KIND=WP) :: OFL, ROOTSC, SCALE, SMALL, & @@ -401,7 +551,7 @@ SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! Local arrays ! ~~~~~~~~~~~~ REAL(KIND=WP) :: RDUMMY(2) - +! ! External functions (BLAS and LAPACK) ! ~~~~~~~~~~~~~~~~~ REAL(KIND=WP) ZLANGE, DLAMCH, DZNRM2 @@ -409,13 +559,13 @@ SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & INTEGER IZAMAX LOGICAL DISNAN, LSAME EXTERNAL DISNAN, LSAME - +! ! External subroutines (BLAS and LAPACK) ! ~~~~~~~~~~~~~~~~~~~~ EXTERNAL ZAXPY, ZGEMM, ZDSCAL EXTERNAL ZGEEV, ZGEJSV, ZGESDD, ZGESVD, ZGESVDQ, & ZLACPY, ZLASCL, ZLASSQ, XERBLA - +! ! Intrinsic functions ! ~~~~~~~~~~~~~~~~~~~ INTRINSIC DBLE, INT, MAX, SQRT @@ -608,7 +758,8 @@ SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & K = 0 DO i = 1, N !WORK(i) = DZNRM2( M, X(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL ZLASSQ( M, X(1,i), 1, SCALE, SSUM ) IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN K = 0 @@ -681,7 +832,8 @@ SUBROUTINE ZGEDMD( JOBS, JOBZ, JOBR, JOBF, WHTSVD, & ! carefully computed using ZLASSQ. DO i = 1, N !RWORK(i) = DZNRM2( M, Y(1,i), 1 ) - SCALE = ZERO + SSUM = ONE + SCALE = ZERO CALL ZLASSQ( M, Y(1,i), 1, SCALE, SSUM ) IF ( DISNAN(SCALE) .OR. DISNAN(SSUM) ) THEN K = 0