SRC/dgedmd.f90(3)

NAME
SRC/dgedmd.f90

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine dgedmd (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, real(kind=wp), dimension(ldx,*), intent(inout) x, integer, intent(in) ldx, real(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, 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(ldw,*), intent(out) w, integer, intent(in) ldw, 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)

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

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.
    !>

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  Contract No: W31P4Q-21-C-0007
    !>    - DARPA PAI project  Contract No: HR0011-18-9-0033
    !>    - DARPA MoDyL project
    !>    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) 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.
    !>

    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 :: 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.
    !>

    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) 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.
    !>

    LDX

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

    Y

    !>    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.
    !>

    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.
    !>

    REIG

    !>    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.
    !>

    IMEIG

    !>    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.
    !>

    Z

    !>    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.
    !>

    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.
    !>    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.
    !>

    B

    !>    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.
    !>

    LDB

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

    W

    !>    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.
    !>

    LDW

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

    S

    !>    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.
    !>

    LDS

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

    WORK

    !>    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.
    !>

    LWORK

    !>    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.
    !>

    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 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 530 of file dgedmd.f90.

Author

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