LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dgedmd()

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
[in]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.
!>
[in]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.
!>
[in]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.
!>
[in]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.
!>
[in]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.
!>
[in]M
!>    M (input) INTEGER, M>= 0
!>    The state space dimension (the row dimension of X, Y).
!>
[in]N
!>    N (input) INTEGER, 0 <= N <= M
!>    The number of data snapshot pairs
!>    (the number of columns of X and Y).
!>
[in,out]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.
!>
[in]LDX
!>    LDX (input) INTEGER, LDX >= M
!>    The leading dimension of the array X.
!>
[in,out]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.
!>
[in]LDY
!>    LDY (input) INTEGER , LDY >= M
!>    The leading dimension of the array Y.
!>
[in]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.
!>
[in]TOL
!>    TOL (input) REAL(KIND=WP), 0 <= TOL < 1
!>    The tolerance for truncating small singular values.
!>    See the description of NRNK.
!>
[out]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.
!>
[out]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.
!>
[out]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.
!>
[out]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.
!>
[in]LDZ
!>    LDZ (input) INTEGER , LDZ >= M
!>    The leading dimension of the array Z.
!>
[out]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.
!>
[out]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.
!>
[in]LDB
!>    LDB (input) INTEGER, LDB >= M
!>    The leading dimension of the array B.
!>
[out]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.
!>
[in]LDW
!>    LDW (input) INTEGER, LDW >= N
!>    The leading dimension of the array W.
!>
[out]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.
!>
[in]LDS
!>    LDS (input) INTEGER, LDS >= N
!>    The leading dimension of the array S.
!>
[out]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.
!>
[in]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.
!>
[out]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.
!>
[in]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.
!>
[out]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