SRC/sgedmdq.f90(3)

NAME
SRC/sgedmdq.f90

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine sgedmdq (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, real(kind=wp), dimension(ldf,*), intent(inout) f, integer, intent(in) ldf, real(kind=wp), dimension(ldx,*), intent(out) x, integer, intent(in) ldx, real(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, real(kind=wp), dimension(*), intent(out) reig, real(kind=wp), dimension(*), intent(out) imeig, real(kind=wp), dimension(ldz,*), intent(out) z, integer, intent(in) ldz, real(kind=wp), dimension(*), intent(out) res, real(kind=wp), dimension(ldb,*), intent(out) b, integer, intent(in) ldb, real(kind=wp), dimension(ldv,*), intent(out) v, integer, intent(in) ldv, real(kind=wp), dimension(lds,*), intent(out) s, integer, intent(in) lds, real(kind=wp), dimension(*), intent(out) work, integer, intent(in) lwork, integer, dimension(*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info)

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

Purpose:

        SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for
        a pair of data snapshot matrices, using a QR factorization
        based compression of the data. For the input matrices
        X and Y such that Y = A*X with an unaccessible matrix
        A, SGEDMDQ computes a certain number of Ritz pairs of A using
        the standard Rayleigh-Ritz extraction from a subspace of
        range(X) that is determined using the leading left singular
        vectors of X. Optionally, SGEDMDQ returns the residuals
        of the computed Ritz pairs, the information needed for
        a refinement of the Ritz vectors, or the eigenvectors of
        the Exact DMD.
        For further details see the references listed
        below. For more details of the implementation see [3].

References:

        [1] P. Schmid: Dynamic mode decomposition of numerical
            and experimental data,
            Journal of Fluid Mechanics 656, 5-28, 2010.
        [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal
            decompositions: analysis and enhancements,
            SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018.
        [3] Z. Drmac: A LAPACK implementation of the Dynamic
            Mode Decomposition I. Technical report. AIMDyn Inc.
            and LAPACK Working Note 298.
        [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L.
            Brunton, N. Kutz: On Dynamic Mode Decomposition:
            Theory and Applications, Journal of Computational
            Dynamics 1(2), 391 -421, 2014.

Developed and supported by:

        Developed and coded by Zlatko Drmac, Faculty of Science,
        University of Zagreb;  drmac@math.hr
        In cooperation with
        AIMdyn Inc., Santa Barbara, CA.
        and supported by
        - DARPA SBIR project 'Koopman Operator-Based Forecasting
        for Nonstationary Processes from Near-Term, Limited
        Observational Data' Contract No: W31P4Q-21-C-0007
        - DARPA PAI project 'Physics-Informed Machine Learning
        Methodologies' Contract No: HR0011-18-9-0033
        - DARPA MoDyL project 'A Data-Driven, Operator-Theoretic
        Framework for Space-Time Analysis of Process Dynamics'
        Contract No: HR0011-16-C-0116
        Any opinions, findings and conclusions or recommendations
        expressed in this material are those of the author and
        do not necessarily reflect the views of the DARPA SBIR
        Program Office.

Distribution Statement A:

        Approved for Public Release, Distribution Unlimited.
        Cleared by DARPA on September 29, 2022

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
        orthogonal matrix from the QR factorization.
        'Q' :: The matrix Q of the QR factorization of the data
               snapshot matrix is computed and stored in the
               array F. See the description of F.
        'N' :: The matrix Q is not explicitly computed.

    JOBT

        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 :: SGESVD (the QR SVD algorithm)
        2 :: SGESDD (the Divide and Conquer algorithm; if enough
             workspace available, this is the fastest option)
        3 :: SGESVDQ (the preconditioned QR SVD  ; this and 4
             are the most accurate options)
        4 :: SGEJSV (the preconditioned Jacobi SVD; this and 3
             are the most accurate options)
        For the four methods above, a significant difference in
        the accuracy of small singular values is possible if
        the snapshots vary in norm so that X is severely
        ill-conditioned. If small (smaller than EPS*||X||)
        singular values are of interest and JOBS=='N',  then
        the options (3, 4) give the most accurate results, where
        the option 4 is slightly better and with stronger
        theoretical background.
        If JOBS=='S', i.e. the columns of X will be normalized,
        then all methods give nearly equally accurate results.

    M

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

    LDF

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

    X

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

    LDX

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

    Y

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

    LDY

        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.

    REIG

        REIG (output) REAL(KIND=WP) (N-1)-by-1 array
        The leading K (K<=N) entries of REIG contain
        the real parts of the computed eigenvalues
        REIG(1:K) + sqrt(-1)*IMEIG(1:K).
        See the descriptions of K, IMEIG, Z.

    IMEIG

        IMEIG (output) REAL(KIND=WP) (N-1)-by-1 array
        The leading K (K<N) entries of REIG contain
        the imaginary parts of the computed eigenvalues
        REIG(1:K) + sqrt(-1)*IMEIG(1:K).
        The eigenvalues are determined as follows:
        If IMEIG(i) == 0, then the corresponding eigenvalue is
        real, LAMBDA(i) = REIG(i).
        If IMEIG(i)>0, then the corresponding complex
        conjugate pair of eigenvalues reads
        LAMBDA(i)   = REIG(i) + sqrt(-1)*IMAG(i)
        LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i)
        That is, complex conjugate pairs have consecutive
        indices (i,i+1), with the positive imaginary part
        listed first.
        See the descriptions of K, REIG, Z.

    Z

        Z (workspace/output) REAL(KIND=WP)  M-by-(N-1) array
        If JOBZ =='V' then
           Z contains real Ritz vectors as follows:
           If IMEIG(i)=0, then Z(:,i) is an eigenvector of
           the i-th Ritz value.
           If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then
           [Z(:,i) Z(:,i+1)] span an invariant subspace and
           the Ritz values extracted from this subspace are
           REIG(i) + sqrt(-1)*IMEIG(i) and
           REIG(i) - sqrt(-1)*IMEIG(i).
           The corresponding eigenvectors are
           Z(:,i) + sqrt(-1)*Z(:,i+1) and
           Z(:,i) - sqrt(-1)*Z(:,i+1), respectively.
        If JOBZ == 'F', then the above descriptions hold for
        the columns of Z*V, where the columns of V are the
        eigenvectors of the K-by-K Rayleigh quotient, and Z is
        orthonormal. The columns of V are similarly structured:
        If IMEIG(i) == 0 then Z*V(:,i) is an eigenvector, and if
        IMEIG(i) > 0 then Z*V(:,i)+sqrt(-1)*Z*V(:,i+1) and
                          Z*V(:,i)-sqrt(-1)*Z*V(:,i+1)
        are the eigenvectors of LAMBDA(i), LAMBDA(i+1).
        See the descriptions of REIG, IMEIG, X and V.

    LDZ

        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.
        If LAMBDA(i) is real, then
           RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2.
        If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair
        then
        RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F
        where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ]
                  [-imag(LAMBDA(i)) real(LAMBDA(i)) ].
        It holds that
        RES(i)   = || A*ZC(:,i)   - LAMBDA(i)  *ZC(:,i)   ||_2
        RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2
        where ZC(:,i)   =  Z(:,i) + sqrt(-1)*Z(:,i+1)
              ZC(:,i+1) =  Z(:,i) - sqrt(-1)*Z(:,i+1)
        See the description of Z.

    B

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

    LDB

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

    V

        V (workspace/output) REAL(KIND=WP) (N-1)-by-(N-1) array
        On exit, V(1:K,1:K) contains the K eigenvectors of
        the Rayleigh quotient. The eigenvectors of a complex
        conjugate pair of eigenvalues are returned in real form
        as explained in the description of Z. The Ritz vectors
        (returned in Z) are the product of X and V; see
        the descriptions of X and Z.

    LDV

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

    S

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

    LDS

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

    WORK

        WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
        On exit,
        WORK(1:MIN(M,N)) contains the scalar factors of the
        elementary reflectors as returned by SGEQRF of the
        M-by-N input matrix F.
        WORK(MIN(M,N)+1:MIN(M,N)+N-1) contains the singular values of
        the input submatrix F(1:M,1:N-1).
        If the call to SGEDMDQ is only workspace query, then
        WORK(1) contains the minimal workspace length and
        WORK(2) is the optimal workspace length. Hence, the
        length of work is at least 2.
        See the description of LWORK.

    LWORK

        LWORK (input) INTEGER
        The minimal length of the  workspace vector WORK.
        LWORK is calculated as follows:
        Let MLWQR  = N (minimal workspace for SGEQRF[M,N])
            MLWDMD = minimal workspace for SGEDMD (see the
                     description of LWORK in SGEDMD) for
                     snapshots of dimensions MIN(M,N)-by-(N-1)
            MLWMQR = N (minimal workspace for
                       SORMQR['L','N',M,N,N])
            MLWGQR = N (minimal workspace for SORGQR[M,N,N])
        Then
        LWORK = MAX(N+MLWQR, N+MLWDMD)
        is updated as follows:
           if   JOBZ == 'V' or JOBZ == 'F' THEN
                LWORK = MAX( LWORK,MIN(M,N)+N-1 +MLWMQR )
           if   JOBQ == 'Q' THEN
                LWORK = MAX( LWORK,MIN(M,N)+N-1+MLWGQR)
        If on entry LWORK = -1, then a workspace query is
        assumed and the procedure only computes the minimal
        and the optimal workspace lengths for both WORK and
        IWORK. See the descriptions of WORK and IWORK.

    IWORK

        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 worskpace query is
        assumed and the procedure only computes the minimal
        and the optimal workspace lengths for both WORK and
        IWORK. See the descriptions of WORK and IWORK.

    INFO

        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 571 of file sgedmdq.f90.

Author

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