!> \brief \b SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
!
!  =========== DOCUMENTATION ===========
!
!  Definition:
!  ===========
!
!     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 )
!.....
!     USE                   iso_fortran_env 
!     IMPLICIT NONE
!     INTEGER, PARAMETER :: WP = real32     
!.....      
!     Scalar arguments       
!     CHARACTER, INTENT(IN)  :: JOBS, JOBZ, JOBR, JOBQ,    &
!                               JOBT, JOBF
!     INTEGER,   INTENT(IN)  :: WHTSVD, M, N,   LDF, LDX,  &
!                               LDY, NRNK, LDZ, LDB, LDV,  &
!                               LDS, LWORK,  LIWORK
!     INTEGER,   INTENT(OUT) :: INFO,   K      
!     REAL(KIND=WP), INTENT(IN)    ::   TOL     
!     Array arguments      
!     REAL(KIND=WP), INTENT(INOUT) :: F(LDF,*)
!     REAL(KIND=WP), INTENT(OUT)   :: X(LDX,*), Y(LDY,*),  &
!                                     Z(LDZ,*), B(LDB,*),  &
!                                     V(LDV,*), S(LDS,*)
!     REAL(KIND=WP), INTENT(OUT)   :: REIG(*),  IMEIG(*),  &
!                                     RES(*)
!     REAL(KIND=WP), INTENT(OUT)   :: WORK(*)  
!     INTEGER,       INTENT(OUT)   :: IWORK(*)
!
!............................................................
!>    \par Purpose:
!     =============
!>    \verbatim
!>    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].      
!>    \endverbatim
!............................................................
!>    \par References:
!     ================
!>    \verbatim
!>    [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.
!>    \endverbatim
!......................................................................
!>    \par Developed and supported by:
!     ================================
!>    \verbatim
!>    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.      
!>    \endverbatim
!......................................................................
!>    \par Distribution Statement A:
!     ==============================
!>    \verbatim
!>    Approved for Public Release, Distribution Unlimited.
!>    Cleared by DARPA on September 29, 2022
!>    \endverbatim
!......................................................................
!     Arguments
!     =========
!
!>    \param[in] JOBS
!>    \verbatim
!>    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.   
!>    \endverbatim
!.....
!>    \param[in] JOBZ
!>    \verbatim
!>    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.  
!>    \endverbatim
!.....
!>    \param[in] JOBR
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] JOBQ
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] JOBT
!>    \verbatim
!>    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.    
!>    \endverbatim
!.....
!>    \param[in] JOBF
!>    \verbatim
!>    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'.      
!>    \endverbatim
!.....
!>    \param[in] WHTSVD
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] M
!>    \verbatim
!>    M (input) INTEGER, M >= 0 
!>    The state space dimension (the number of rows of F)
!>    \endverbatim
!.....
!>    \param[in] N
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in,out] F
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDF
!>    \verbatim
!>    LDF (input) INTEGER, LDF >= M 
!>    The leading dimension of the array F.
!>    \endverbatim
!.....
!>    \param[in,out] X
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDX
!>    \verbatim
!>    LDX (input) INTEGER, LDX >= N  
!>    The leading dimension of the array X 
!>    \endverbatim
!.....
!>    \param[in,out] Y
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDY
!>    \verbatim
!>    LDY (input) INTEGER , LDY >= N
!>    The leading dimension of the array Y   
!>    \endverbatim
!.....
!>    \param[in] NRNK
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] TOL
!>    \verbatim
!>    TOL (input) REAL(KIND=WP), 0 <= TOL < 1
!>    The tolerance for truncating small singular values.
!>    See the description of NRNK.  
!>    \endverbatim
!.....
!>    \param[out] K
!>    \verbatim
!>    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. 
!>    \endverbatim
!.....
!>    \param[out] REIG
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[out] IMEIG
!>    \verbatim
!>    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.     
!>    \endverbatim
!.....
!>    \param[out] Z
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDZ
!>    \verbatim
!>    LDZ (input) INTEGER , LDZ >= M
!>    The leading dimension of the array Z.
!>    \endverbatim
!.....
!>    \param[out] RES
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[out] B
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDB
!>    \verbatim
!>    LDB (input) INTEGER, LDB >= MIN(M,N)
!>    The leading dimension of the array B.
!>    \endverbatim
!.....
!>    \param[out] V
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDV
!>    \verbatim
!>    LDV (input) INTEGER, LDV >= N-1
!>    The leading dimension of the array V.
!>    \endverbatim
!.....
!>    \param[out] S
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LDS
!>    \verbatim
!>    LDS (input) INTEGER, LDS >= N-1        
!>    The leading dimension of the array S.
!>    \endverbatim
!.....
!>    \param[out] WORK
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LWORK
!>    \verbatim
!>    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.          
!>    \endverbatim
!.....
!>    \param[out] IWORK
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[in] LIWORK
!>    \verbatim
!>    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.
!>    \endverbatim
!.....
!>    \param[out] INFO
!>    \verbatim
!>    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.  
!>    \endverbatim
!
!  Authors:
!  ========
!
!> \author Zlatko Drmac
!
!> \ingroup gedmd
!
!.............................................................
!.............................................................
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 )
!
!  -- LAPACK driver routine                                           --
!
!  -- LAPACK is a software package provided by University of          --
!  -- Tennessee, University of California Berkeley, University of     --
!  -- Colorado Denver and NAG Ltd..                                   --
!
!.....
      USE                   iso_fortran_env 
      IMPLICIT NONE
      INTEGER, PARAMETER :: WP = real32     
!
!     Scalar arguments       
!     ~~~~~~~~~~~~~~~~
      CHARACTER, INTENT(IN)  :: JOBS, JOBZ, JOBR, JOBQ,    &
                                JOBT, JOBF
      INTEGER,   INTENT(IN)  :: WHTSVD, M, N,   LDF, LDX,  &
                                LDY, NRNK, LDZ, LDB, LDV,  &
                                LDS, LWORK,  LIWORK
      INTEGER,   INTENT(OUT) :: INFO,   K      
      REAL(KIND=WP), INTENT(IN)    ::   TOL     
!
!     Array arguments      
!     ~~~~~~~~~~~~~~~~
      REAL(KIND=WP), INTENT(INOUT) :: F(LDF,*)
      REAL(KIND=WP), INTENT(OUT)   :: X(LDX,*), Y(LDY,*),  &
                                      Z(LDZ,*), B(LDB,*),  &
                                      V(LDV,*), S(LDS,*)
      REAL(KIND=WP), INTENT(OUT)   :: REIG(*),  IMEIG(*),  &
                                      RES(*)
      REAL(KIND=WP), INTENT(OUT)   :: WORK(*)  
      INTEGER,       INTENT(OUT)   :: IWORK(*)
!
!     Parameters
!     ~~~~~~~~~~      
      REAL(KIND=WP), PARAMETER ::  ONE = 1.0_WP
      REAL(KIND=WP), PARAMETER :: ZERO = 0.0_WP
!      
!     Local scalars      
!     ~~~~~~~~~~~~~
      INTEGER           :: IMINWR, INFO1,  MLWDMD, MLWGQR, &
                           MLWMQR, MLWORK, MLWQR,  MINMN,  & 
                           OLWDMD, OLWGQR, OLWMQR, OLWORK, &
                           OLWQR
      LOGICAL           :: LQUERY, SCCOLX, SCCOLY, WANTQ,  &
                           WNTTRF, WNTRES, WNTVEC, WNTVCF, &
                           WNTVCQ, WNTREF, WNTEX
      CHARACTER(LEN=1)  :: JOBVL
!      
!     Local array      
!     ~~~~~~~~~~~      
      REAL(KIND=WP) :: RDUMMY(2)
!      
!     External functions (BLAS and LAPACK)
!     ~~~~~~~~~~~~~~~~~
      LOGICAL       LSAME
      EXTERNAL      LSAME 
!
!     External subroutines (BLAS and LAPACK)
!     ~~~~~~~~~~~~~~~~~~~~
      EXTERNAL      SGEMM 
      EXTERNAL      SGEDMD, SGEQRF, SLACPY, SLASET, SORGQR, & 
                    SORMQR, XERBLA
!      
!     Intrinsic functions
!     ~~~~~~~~~~~~~~~~~~~
      INTRINSIC      MAX, MIN, INT         
!..........................................................  
!
!     Test the input arguments    
      WNTRES = LSAME(JOBR,'R')
      SCCOLX = LSAME(JOBS,'S') .OR. LSAME( JOBS, 'C' )
      SCCOLY = LSAME(JOBS,'Y')
      WNTVEC = LSAME(JOBZ,'V') 
      WNTVCF = LSAME(JOBZ,'F')
      WNTVCQ = LSAME(JOBZ,'Q')
      WNTREF = LSAME(JOBF,'R') 
      WNTEX  = LSAME(JOBF,'E')
      WANTQ  = LSAME(JOBQ,'Q')
      WNTTRF = LSAME(JOBT,'R')     
      MINMN  = MIN(M,N)
      INFO = 0 
      LQUERY = ( ( LWORK == -1 ) .OR. ( LIWORK == -1 ) )
!       
      IF ( .NOT. (SCCOLX .OR. SCCOLY .OR. LSAME(JOBS,'N')) )  THEN 
          INFO = -1
      ELSE IF ( .NOT. (WNTVEC .OR. WNTVCF .OR. WNTVCQ       &
                              .OR. LSAME(JOBZ,'N')) ) THEN
          INFO = -2
      ELSE IF ( .NOT. (WNTRES .OR. LSAME(JOBR,'N')) .OR.    & 
          ( WNTRES .AND. LSAME(JOBZ,'N') ) ) THEN
          INFO = -3
      ELSE IF ( .NOT. (WANTQ .OR. LSAME(JOBQ,'N')) ) THEN
          INFO = -4                 
      ELSE IF ( .NOT. ( WNTTRF .OR. LSAME(JOBT,'N') ) )  THEN
          INFO = -5
      ELSE IF ( .NOT. (WNTREF .OR. WNTEX .OR.             & 
                LSAME(JOBF,'N') ) )                    THEN
          INFO = -6    
      ELSE IF ( .NOT. ((WHTSVD == 1).OR.(WHTSVD == 2).OR.   &
                       (WHTSVD == 3).OR.(WHTSVD == 4)) ) THEN
          INFO = -7
      ELSE IF ( M < 0 ) THEN
          INFO = -8
      ELSE IF ( ( N < 0 ) .OR. ( N > M+1 ) ) THEN
          INFO = -9
      ELSE IF ( LDF < M ) THEN
          INFO = -11
      ELSE IF ( LDX < MINMN ) THEN
          INFO = -13
      ELSE IF ( LDY < MINMN ) THEN
          INFO = -15
      ELSE IF ( .NOT. (( NRNK == -2).OR.(NRNK == -1).OR.    & 
                       ((NRNK >= 1).AND.(NRNK <=N ))) )  THEN
          INFO = -16
      ELSE IF ( ( TOL < ZERO ) .OR. ( TOL >= ONE ) ) THEN
          INFO = -17
      ELSE IF ( LDZ < M ) THEN
          INFO = -22
      ELSE IF ( (WNTREF.OR.WNTEX ).AND.( LDB < MINMN ) ) THEN
          INFO = -25
      ELSE IF ( LDV < N-1 ) THEN
          INFO = -27
      ELSE IF ( LDS < N-1 ) THEN
          INFO = -29
      END IF
!      
      IF ( WNTVEC .OR. WNTVCF ) THEN
          JOBVL = 'V'
      ELSE
          JOBVL = 'N'
      END IF     
      IF ( INFO == 0 ) THEN  
          ! Compute the minimal and the optimal workspace
          ! requirements. Simulate running the code and 
          ! determine minimal and optimal sizes of the 
          ! workspace at any moment of the run.         
         IF ( ( N == 0 ) .OR. ( N == 1 ) ) THEN
             ! All output except K is void. INFO=1 signals
             ! the void input. In case of a workspace query,
             ! the minimal workspace lengths are returned.
            IF ( LQUERY ) THEN  
               IWORK(1) = 1
                WORK(1) = 2
                WORK(2) = 2
            ELSE                
               K = 0
            END IF             
            INFO = 1  
            RETURN
         END IF     
         MLWQR  = MAX(1,N)  ! Minimal workspace length for SGEQRF.
         MLWORK = MIN(M,N) + MLWQR 
         IF ( LQUERY ) THEN 
             CALL SGEQRF( M, N, F, LDF, WORK, RDUMMY, -1, &
                          INFO1 )
             OLWQR = INT(RDUMMY(1))
             OLWORK = MIN(M,N) + OLWQR           
         END IF
         CALL SGEDMD( JOBS, JOBVL, JOBR, JOBF, WHTSVD, MINMN,& 
                      N-1, X, LDX, Y, LDY, NRNK, TOL, K,     & 
                      REIG, IMEIG, Z, LDZ, RES,  B, LDB,     & 
                      V, LDV, S, LDS, WORK, -1, IWORK,       &
                      LIWORK, INFO1 )
         MLWDMD = INT(WORK(1))
         MLWORK = MAX(MLWORK, MINMN + MLWDMD)
         IMINWR = IWORK(1)
         IF ( LQUERY ) THEN 
             OLWDMD = INT(WORK(2))
             OLWORK = MAX(OLWORK, MINMN+OLWDMD)
         END IF
         IF ( WNTVEC .OR. WNTVCF ) THEN
            MLWMQR = MAX(1,N) 
            MLWORK = MAX(MLWORK,MINMN+N-1+MLWMQR)
            IF ( LQUERY ) THEN
               CALL SORMQR( 'L','N', M, N, MINMN, F, LDF,  & 
                            WORK, Z, LDZ, WORK, -1, INFO1 )
               OLWMQR = INT(WORK(1))
               OLWORK = MAX(OLWORK,MINMN+N-1+OLWMQR)
            END IF
         END IF  
         IF ( WANTQ ) THEN
            MLWGQR = N
            MLWORK = MAX(MLWORK,MINMN+N-1+MLWGQR)
            IF ( LQUERY ) THEN 
                CALL SORGQR( M, MINMN, MINMN, F, LDF, WORK, &
                             WORK, -1, INFO1 )        
                OLWGQR = INT(WORK(1))
                OLWORK = MAX(OLWORK,MINMN+N-1+OLWGQR)
            END IF            
         END IF   
         IMINWR = MAX( 1, IMINWR )
         MLWORK = MAX( 2, MLWORK )      
         IF (  LWORK < MLWORK .AND. (.NOT.LQUERY) ) INFO = -31
         IF ( LIWORK < IMINWR .AND. (.NOT.LQUERY) ) INFO = -33
      END IF  
      IF( INFO /= 0 ) THEN
         CALL XERBLA( 'SGEDMDQ', -INFO )
         RETURN
      ELSE IF ( LQUERY ) THEN
!     Return minimal and optimal workspace sizes
          IWORK(1) = IMINWR
          WORK(1)  = MLWORK
          WORK(2)  = OLWORK
          RETURN
      END IF   
!.....	  
!     Initial QR factorization that is used to represent the
!     snapshots as elements of lower dimensional subspace.
!     For large scale computation with M >>N , at this place 
!     one can use an out of core QRF.
!   
      CALL SGEQRF( M, N, F, LDF, WORK,         & 
                   WORK(MINMN+1), LWORK-MINMN, INFO1 )
!      
!     Define X and Y as the snapshots representations in the
!     orthogonal basis computed in the QR factorization.
!     X corresponds to the leading N-1 and Y to the trailing
!     N-1 snapshots.
      CALL SLASET( 'L', MINMN, N-1, ZERO,  ZERO, X, LDX )
      CALL SLACPY( 'U', MINMN, N-1, F,      LDF, X, LDX )
      CALL SLACPY( 'A', MINMN, N-1, F(1,2), LDF, Y, LDY )
      IF ( M >= 3 ) THEN
          CALL SLASET( 'L', MINMN-2, N-2, ZERO,  ZERO, &
                       Y(3,1), LDY )  
      END IF
!
!     Compute the DMD of the projected snapshot pairs (X,Y)   
      CALL SGEDMD( JOBS, JOBVL, JOBR, JOBF, WHTSVD, MINMN,  &
                   N-1, X, LDX, Y, LDY, NRNK,   TOL, K,     &
                   REIG, IMEIG, Z, LDZ, RES, B, LDB, V,     &
                   LDV, S, LDS, WORK(MINMN+1), LWORK-MINMN, IWORK,  & 
                   LIWORK, INFO1 )
      IF ( INFO1 == 2 .OR. INFO1 == 3 ) THEN
          ! Return with error code.
          INFO = INFO1
          RETURN
      ELSE
          INFO = INFO1
      END IF    
!      
!     The Ritz vectors (Koopman modes) can be explicitly 
!     formed or returned in factored form.
      IF ( WNTVEC ) THEN
        ! Compute the eigenvectors explicitly.  
        IF ( M > MINMN ) CALL SLASET( 'A', M-MINMN, K, ZERO, &
                                     ZERO, Z(MINMN+1,1), LDZ )
        CALL SORMQR( 'L','N', M, K, MINMN, F, LDF, WORK, Z,  &
                     LDZ, WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )
      ELSE IF ( WNTVCF ) THEN   
        !   Return the Ritz vectors (eigenvectors) in factored
        !   form Z*V, where Z contains orthonormal matrix (the
        !   product of Q from the initial QR factorization and 
        !   the SVD/POD_basis returned by SGEDMD in X) and the 
        !   second factor (the eigenvectors of the Rayleigh 
        !   quotient) is in the array V, as returned by SGEDMD.
        CALL SLACPY( 'A', N, K, X, LDX, Z, LDZ )
        IF ( M > N ) CALL SLASET( 'A', M-N, K, ZERO, ZERO,   & 
                                  Z(N+1,1), LDZ )
        CALL SORMQR( 'L','N', M, K, MINMN, F, LDF, WORK, Z,  &
             LDZ, WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )
      END IF
!     
!     Some optional output variables:
!
!     The upper triangular factor in the initial QR 
!     factorization is optionally returned in the array Y.
!     This is useful if this call to SGEDMDQ is to be 
!     followed by a streaming DMD that is implemented in a 
!     QR compressed form.
      IF ( WNTTRF ) THEN ! Return the upper triangular R in Y 
         CALL SLASET( 'A', MINMN, N, ZERO,  ZERO, Y, LDY )
         CALL SLACPY( 'U', MINMN, N, F, LDF,      Y, LDY )
      END IF    
!
!     The orthonormal/orthogonal factor in the initial QR 
!     factorization is optionally returned in the array F. 
!     Same as with the triangular factor above, this is 
!     useful in a streaming DMD.
      IF ( WANTQ ) THEN  ! Q overwrites F 
         CALL SORGQR( M, MINMN, MINMN, F, LDF, WORK, &
              WORK(MINMN+N), LWORK-(MINMN+N-1), INFO1 )  
      END IF
!      
      RETURN
!      
      END SUBROUTINE SGEDMDQ
    
