SRC/cgedmd.f90(3)

NAME
SRC/cgedmd.f90

SYNOPSIS

  Functions/Subroutines
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)
CGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.

Function/Subroutine Documentation

  subroutine cgedmd (character, intent(in) jobs, character, intent(in) jobz, character, intent(in) jobr, character, intent(in) jobf, integer, intent(in) whtsvd, integer, intent(in) m, integer, intent(in) n, complex(kind=wp), dimension(ldx,*), intent(inout) x, integer, intent(in) ldx, complex(kind=wp), dimension(ldy,*), intent(inout) y, integer, intent(in) ldy, integer, intent(in) nrnk, real(kind=wp), intent(in) tol, integer, intent(out) k, complex(kind=wp), dimension(*), intent(out) eigs, complex(kind=wp), dimension(ldz,*), intent(out) z, integer, intent(in) ldz, real(kind=wp), dimension(*), intent(out) res, complex(kind=wp), dimension(ldb,*), intent(out) b, integer, intent(in) ldb, complex(kind=wp), dimension(ldw,*), intent(out) w, integer, intent(in) ldw, complex(kind=wp), dimension(lds,*), intent(out) s, integer, intent(in) lds, complex(kind=wp), dimension(*), intent(out) zwork, integer, intent(in) lzwork, real(kind=wp), dimension(*), intent(out) rwork, integer, intent(in) lrwork, integer, dimension(*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info)

CGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.

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

Parameters
    JOBS

        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

        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

        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

        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

        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

        M (input) INTEGER, M>= 0
        The state space dimension (the row dimension of X, Y).

    N

        N (input) INTEGER, 0 <= N <= M
        The number of data snapshot pairs
        (the number of columns of X and Y).

    X

        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

        LDX (input) INTEGER, LDX >= M
        The leading dimension of the array X.

    Y

        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

        LDY (input) INTEGER , LDY >= M
        The leading dimension of the array Y.

    NRNK

        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

        TOL (input) REAL(KIND=WP), 0 <= TOL < 1
        The tolerance for truncating small singular values.
        See the description of NRNK.

    K

        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

        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

        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

        LDZ (input) INTEGER , LDZ >= M
        The leading dimension of the array Z.

    RES

        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

        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

        LDB (input) INTEGER, LDB >= M
        The leading dimension of the array B.

    W

        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

        LDW (input) INTEGER, LDW >= N
        The leading dimension of the array W.

    S

        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

        LDS (input) INTEGER, LDS >= N
        The leading dimension of the array S.

    ZWORK

        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

        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

        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

        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

        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

        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

        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.

Author
    Zlatko Drmac

Definition at line 496 of file cgedmd.f90.

Author

Generated automatically by Doxygen for LAPACK from the source code.