SRC/zgedmdq.f90(3)

NAME
SRC/zgedmdq.f90

SYNOPSIS

  Functions/Subroutines
subroutine zgedmdq (jobs, jobz, jobr, jobq, jobt, jobf, whtsvd, m, n, f, ldf, x, ldx, y, ldy, nrnk, tol, k, eigs, z, ldz, res, b, ldb, v, ldv, s, lds, zwork, lzwork, work, lwork, iwork, liwork, info)
ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.

Function/Subroutine Documentation

  subroutine zgedmdq (character, intent(in) jobs, character, intent(in) jobz, character, intent(in) jobr, character, intent(in) jobq, character, intent(in) jobt, character, intent(in) jobf, integer, intent(in) whtsvd, integer, intent(in) m, integer, intent(in) n, complex(kind=wp), dimension(ldf,*), intent(inout) f, integer, intent(in) ldf, complex(kind=wp), dimension(ldx,*), intent(out) x, integer, intent(in) ldx, complex(kind=wp), dimension(ldy,*), intent(out) 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(ldv,*), intent(out) v, integer, intent(in) ldv, 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) work, integer, intent(in) lwork, integer, dimension(*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info)

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

Purpose:

        ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for
        a pair of data snapshot matrices, using a QR factorization
        based compression of the data. For the input matrices
        X and Y such that Y = A*X with an unaccessible matrix
        A, ZGEDMDQ 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, ZGEDMDQ 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.

Developed and supported by:

        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. The data snapshots are the columns
        of F. The leading N-1 columns of F are denoted X and the
        trailing N-1 columns are denoted Y.
        'S' :: The data snapshots matrices X and Y are multiplied
               with a diagonal matrix D so that X*D has unit
               nonzero columns (in the Euclidean 2-norm)
        'C' :: The snapshots are scaled as with the 'S' option.
               If it is found that an i-th column of X is zero
               vector and the corresponding i-th column of Y is
               non-zero, then the i-th column of Y is set to
               zero and a warning flag is raised.
        'Y' :: The data snapshots matrices X and Y are multiplied
               by a diagonal matrix D so that Y*D has unit
               nonzero columns (in the Euclidean 2-norm)
        'N' :: No data scaling.

    JOBZ

        JOBZ (input) CHARACTER*1
        Determines whether the eigenvectors (Koopman modes) will
        be computed.
        'V' :: The eigenvectors (Koopman modes) will be computed
               and returned in the matrix Z.
               See the description of Z.
        'F' :: The eigenvectors (Koopman modes) will be returned
               in factored form as the product Z*V, where Z
               is orthonormal and V contains the eigenvectors
               of the corresponding Rayleigh quotient.
               See the descriptions of F, V, Z.
        'Q' :: The eigenvectors (Koopman modes) will be returned
               in factored form as the product Q*Z, where Z
               contains the eigenvectors of the compression of the
               underlying discretized operator onto the span of
               the data snapshots. See the descriptions of F, V, Z.
               Q is from the initial QR factorization.
        'N' :: The eigenvectors are not computed.

    JOBR

        JOBR (input) CHARACTER*1
        Determines whether to compute the residuals.
        'R' :: The residuals for the computed eigenpairs will
               be computed and stored in the array RES.
               See the description of RES.
               For this option to be legal, JOBZ must be 'V'.
        'N' :: The residuals are not computed.

    JOBQ

        JOBQ (input) CHARACTER*1
        Specifies whether to explicitly compute and return the
        unitary matrix from the QR factorization.
        'Q' :: The matrix Q of the QR factorization of the data
               snapshot matrix is computed and stored in the
               array F. See the description of F.
        'N' :: The matrix Q is not explicitly computed.

    JOBT

        JOBT (input) CHARACTER*1
        Specifies whether to return the upper triangular factor
        from the QR factorization.
        'R' :: The matrix R of the QR factorization of the data
               snapshot matrix F is returned in the array Y.
               See the description of Y and Further details.
        'N' :: The matrix R is not returned.

    JOBF

        JOBF (input) CHARACTER*1
        Specifies whether to store information needed for post-
        processing (e.g. computing refined Ritz vectors)
        'R' :: The matrix needed for the refinement of the Ritz
               vectors is computed and stored in the array B.
               See the description of B.
        'E' :: The unscaled eigenvectors of the Exact DMD are
               computed and returned in the array B. See the
               description of B.
        'N' :: No eigenvector refinement data is computed.
        To be useful on exit, this option needs JOBQ='Q'.

    WHTSVD 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.
    M

        M (input) INTEGER, M >= 0
        The state space dimension (the number of rows of F).

    N

        N (input) INTEGER, 0 <= N <= M
        The number of data snapshots from a single trajectory,
        taken at equidistant discrete times. This is the
        number of columns of F.

    F

        F (input/output) COMPLEX(KIND=WP) M-by-N array
        > On entry,
        the columns of F are the sequence of data snapshots
        from a single trajectory, taken at equidistant discrete
        times. It is assumed that the column norms of F are
        in the range of the normalized floating point numbers.
        < On exit,
        If JOBQ == 'Q', the array F contains the orthogonal
        matrix/factor of the QR factorization of the initial
        data snapshots matrix F. See the description of JOBQ.
        If JOBQ == 'N', the entries in F strictly below the main
        diagonal contain, column-wise, the information on the
        Householder vectors, as returned by ZGEQRF. The
        remaining information to restore the orthogonal matrix
        of the initial QR factorization is stored in ZWORK(1:MIN(M,N)).
        See the description of ZWORK.

    LDF

        LDF (input) INTEGER, LDF >= M
        The leading dimension of the array F.

    X

        X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array
        X is used as workspace to hold representations of the
        leading N-1 snapshots in the orthonormal basis computed
        in the QR factorization of F.
        On exit, the leading K columns of X contain the leading
        K left singular vectors of the above described content
        of X. To lift them to the space of the left singular
        vectors U(:,1:K) of the input data, pre-multiply with the
        Q factor from the initial QR factorization.
        See the descriptions of F, K, V  and Z.

    LDX

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

    Y

        Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array
        Y is used as workspace to hold representations of the
        trailing N-1 snapshots in the orthonormal basis computed
        in the QR factorization of F.
        On exit,
        If JOBT == 'R', Y contains the MIN(M,N)-by-N upper
        triangular factor from the QR factorization of the data
        snapshot matrix F.

    LDY

        LDY (input) INTEGER , LDY >= N
        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-1 :: at most NRNK largest singular values
        will be used. If the number of the computed nonzero
        singular values is less than NRNK, then only those
        nonzero values will be used and the actually used
        dimension is less than NRNK. The actual number of
        the nonzero singular values is returned in the variable
        K. See the description of K.

    TOL

        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 SVD/POD basis for the leading N-1
        data snapshots (columns of F) and the number of the
        computed Ritz pairs. The value of K is determined
        according to the rule set by the parameters NRNK and
        TOL. See the descriptions of NRNK and TOL.

    EIGS

        EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array
        The leading K (K<=N-1) 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-1) 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
        Z*V, where Z contains orthonormal matrix (the product of
        Q from the initial QR factorization and the SVD/POD_basis
        returned by ZGEDMD in X) and the second factor (the
        eigenvectors of the Rayleigh quotient) is in the array V,
        as returned by ZGEDMD. That is,  X(:,1:K)*V(:,i)
        is an eigenvector corresponding to EIGS(i). The columns
        of V(1:K,1:K) are the computed eigenvectors of the
        K-by-K Rayleigh quotient.
        See the descriptions of EIGS, X and V.

    LDZ

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

    RES

        RES (output) REAL(KIND=WP) (N-1)-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)  MIN(M,N)-by-(N-1) array.
        IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can
        be used for computing the refined vectors; see further
        details in the provided references.
        If JOBF == 'E', B(1:N,1;K) contains
        A*U(:,1:K)*W(1:K,1:K), which are the vectors from the
        Exact DMD, up to scaling by the inverse eigenvalues.
        In both cases, the content of B can be lifted to the
        original dimension of the input data by pre-multiplying
        with the Q factor from the initial QR factorization.
        Here A denotes a compression of the underlying operator.
        See the descriptions of F and X.
        If JOBF =='N', then B is not referenced.

    LDB

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

    V

        V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array
        On exit, V(1:K,1:K) V contains the K eigenvectors of
        the Rayleigh quotient. The Ritz vectors
        (returned in Z) are the product of Q from the initial QR
        factorization (see the description of F) X (see the
        description of X) and V.

    LDV

        LDV (input) INTEGER, LDV >= N-1
        The leading dimension of the array V.

    S

        S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array
        The array S(1:K,1:K) is used for the matrix Rayleigh
        quotient. This content is overwritten during
        the eigenvalue decomposition by ZGEEV.
        See the description of K.

    LDS

        LDS (input) INTEGER, LDS >= N-1
        The leading dimension of the array S.

    LZWORK

        ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array
        On exit,
        ZWORK(1:MIN(M,N)) contains the scalar factors of the
        elementary reflectors as returned by ZGEQRF of the
        M-by-N input matrix F.
        If the call to ZGEDMDQ 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 follows:
        Let MLWQR  = N (minimal workspace for ZGEQRF[M,N])
            MLWDMD = minimal workspace for ZGEDMD (see the
                     description of LWORK in ZGEDMD)
            MLWMQR = N (minimal workspace for
                       ZUNMQR['L','N',M,N,N])
            MLWGQR = N (minimal workspace for ZUNGQR[M,N,N])
            MINMN  = MIN(M,N)
        Then
        LZWORK = MAX(2, MIN(M,N)+MLWQR, MINMN+MLWDMD)
        is further updated as follows:
           if   JOBZ == 'V' or JOBZ == 'F' THEN
                LZWORK = MAX(LZWORK, MINMN+MLWMQR)
           if   JOBQ == 'Q' THEN
                LZWORK = MAX(ZLWORK, MINMN+MLWGQR)

    WORK

        WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
        On exit,
        WORK(1:N-1) contains the singular values of
        the input submatrix F(1:M,1:N-1).
        If the call to ZGEDMDQ is only workspace query, then
        WORK(1) contains the minimal workspace length and
        WORK(2) is the optimal workspace length. hence, the
        length of work is at least 2.
        See the description of LWORK.

    LWORK

        LWORK (input) INTEGER
        The minimal length of the  workspace vector WORK.
        LWORK is the same as in ZGEDMD, because in ZGEDMDQ
        only ZGEDMD requires real workspace for snapshots
        of dimensions MIN(M,N)-by-(N-1).
        If on entry LWORK = -1, then a workspace query is
        assumed and the procedure only computes the minimal
        and the optimal workspace length for WORK.

    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
        Let M1=MIN(M,N), N1=N-1. Then
        If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1))
        If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1)
        If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1)
        If on entry LIWORK = -1, then a 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.

    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 550 of file zgedmdq.f90.

Author

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