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Add functional C replacements for ?GEDMD?

tags/v0.3.24
Martin Kroeker GitHub 2 years ago
parent
174f4e65e3
commit
5cc1f7a0ba
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8 changed files with 7935 additions and 0 deletions
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  1. +1159
    -0
      lapack-netlib/SRC/cgedmd.c
  2. +778
    -0
      lapack-netlib/SRC/cgedmdq.c
  3. +1242
    -0
      lapack-netlib/SRC/dgedmd.c
  4. +789
    -0
      lapack-netlib/SRC/dgedmdq.c
  5. +1235
    -0
      lapack-netlib/SRC/sgedmd.c
  6. +785
    -0
      lapack-netlib/SRC/sgedmdq.c
  7. +1165
    -0
      lapack-netlib/SRC/zgedmd.c
  8. +782
    -0
      lapack-netlib/SRC/zgedmdq.c

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lapack-netlib/SRC/cgedmd.c
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lapack-netlib/SRC/cgedmdq.c View File

@@ -509,3 +509,781 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ



/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/



/* Table of constant values */

static integer c_n1 = -1;

/* Subroutine */ int cgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq,
char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n,
complex *f, integer *ldf, complex *x, integer *ldx, complex *y,
integer *ldy, integer *nrnk, real *tol, integer *k, complex *eigs,
complex *z__, integer *ldz, real *res, complex *b, integer *ldb,
complex *v, integer *ldv, complex *s, integer *lds, complex *zwork,
integer *lzwork, real *work, integer *lwork, integer *iwork, integer *
liwork, integer *info)
{
/* System generated locals */
integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1,
z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset,
i__1, i__2;

/* Local variables */
real zero;
integer info1;
extern logical lsame_(char *, char *);
char jobvl[1];
integer minmn;
logical wantq;
integer mlwqr, olwqr;
logical wntex;
complex zzero;
extern /* Subroutine */ int cgedmd_(char *, char *, char *, char *,
integer *, integer *, integer *, complex *, integer *, complex *,
integer *, integer *, real *, integer *, complex *, complex *,
integer *, real *, complex *, integer *, complex *, integer *,
complex *, integer *, complex *, integer *, real *, integer *,
integer *, integer *, integer *),
cgeqrf_(integer *, integer *, complex *, integer *, complex *,
complex *, integer *, integer *), clacpy_(char *, integer *,
integer *, complex *, integer *, complex *, integer *),
claset_(char *, integer *, integer *, complex *, complex *,
complex *, integer *), xerbla_(char *, integer *);
integer mlwdmd, olwdmd;
logical sccolx, sccoly;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
integer iminwr;
logical wntvec, wntvcf;
integer mlwgqr;
logical wntref;
integer mlwork, olwgqr, olwork, mlrwrk, mlwmqr, olwmqr;
logical lquery, wntres, wnttrf, wntvcq;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
real one;

/* March 2023 */
/* ..... */
/* USE iso_fortran_env */
/* INTEGER, PARAMETER :: WP = real32 */
/* ..... */
/* Scalar arguments */
/* Array arguments */
/* ..... */
/* Purpose */
/* ======= */
/* CGEDMDQ 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, CGEDMDQ 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, CGEDMDQ 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 */
/* ============================================================ */
/* ...................................................................... */
/* Arguments */
/* ========= */
/* 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 (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 discretised operator onto the span of */
/* the data snapshots. See the descriptions of F, V, Z. */
/* Q is from the inital QR facorization. */
/* 'N' :: The eigenvectors are not computed. */
/* ..... */
/* 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 (input) CHARACTER*1 */
/* Specifies whether to explicitly compute and return the */
/* unitary matrix from the QR factorization. */
/* 'Q' :: The matrix Q of the QR factorization of the data */
/* snapshot matrix is computed and stored in the */
/* array F. See the description of F. */
/* 'N' :: The matrix Q is not explicitly computed. */
/* ..... */
/* JOBT (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 (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 (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */
/* Allows for a selection of the SVD algorithm from the */
/* LAPACK library. */
/* 1 :: CGESVD (the QR SVD algorithm) */
/* 2 :: CGESDD (the Divide and Conquer algorithm; if enough */
/* workspace available, this is the fastest option) */
/* 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 */
/* are the most accurate options) */
/* 4 :: CGEJSV (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 (input) INTEGER, M >= 0 */
/* The state space dimension (the number of rows of F). */
/* ..... */
/* 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 (input/output) COMPLEX(KIND=WP) M-by-N array */
/* > On entry, */
/* the columns of F are the sequence of data snapshots */
/* from a single trajectory, taken at equidistant discrete */
/* times. It is assumed that the column norms of F are */
/* in the range of the normalized floating point numbers. */
/* < On exit, */
/* If JOBQ == 'Q', the array F contains the orthogonal */
/* matrix/factor of the QR factorization of the initial */
/* data snapshots matrix F. See the description of JOBQ. */
/* If JOBQ == 'N', the entries in F strictly below the main */
/* diagonal contain, column-wise, the information on the */
/* Householder vectors, as returned by CGEQRF. The */
/* remaining information to restore the orthogonal matrix */
/* of the initial QR factorization is stored in ZWORK(1:MIN(M,N)). */
/* See the description of ZWORK. */
/* ..... */
/* LDF (input) INTEGER, LDF >= M */
/* The leading dimension of the array F. */
/* ..... */
/* X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array */
/* X is used as workspace to hold representations of the */
/* leading N-1 snapshots in the orthonormal basis computed */
/* in the QR factorization of F. */
/* On exit, the leading K columns of X contain the leading */
/* K left singular vectors of the above described content */
/* of X. To lift them to the space of the left singular */
/* vectors U(:,1:K) of the input data, pre-multiply with the */
/* Q factor from the initial QR factorization. */
/* See the descriptions of F, K, V and Z. */
/* ..... */
/* LDX (input) INTEGER, LDX >= N */
/* The leading dimension of the array X. */
/* ..... */
/* Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array */
/* Y is used as workspace to hold representations of the */
/* trailing N-1 snapshots in the orthonormal basis computed */
/* in the QR factorization of F. */
/* On exit, */
/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */
/* triangular factor from the QR factorization of the data */
/* snapshot matrix F. */
/* ..... */
/* LDY (input) INTEGER , LDY >= N */
/* The leading dimension of the array Y. */
/* ..... */
/* 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 (input) REAL(KIND=WP), 0 <= TOL < 1 */
/* The tolerance for truncating small singular values. */
/* See the description of NRNK. */
/* ..... */
/* K (output) INTEGER, 0 <= K <= N */
/* The dimension of the SVD/POD basis for the leading N-1 */
/* data snapshots (columns of F) and the number of the */
/* computed Ritz pairs. The value of K is determined */
/* according to the rule set by the parameters NRNK and */
/* TOL. See the descriptions of NRNK and TOL. */
/* ..... */
/* EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array */
/* The leading K (K<=N-1) entries of EIGS contain */
/* the computed eigenvalues (Ritz values). */
/* See the descriptions of K, and Z. */
/* ..... */
/* Z (workspace/output) COMPLEX(KIND=WP) M-by-(N-1) array */
/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */
/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */
/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */
/* Z*V, where Z contains orthonormal matrix (the product of */
/* Q from the initial QR factorization and the SVD/POD_basis */
/* returned by CGEDMD in X) and the second factor (the */
/* eigenvectors of the Rayleigh quotient) is in the array V, */
/* as returned by CGEDMD. That is, X(:,1:K)*V(:,i) */
/* is an eigenvector corresponding to EIGS(i). The columns */
/* of V(1:K,1:K) are the computed eigenvectors of the */
/* K-by-K Rayleigh quotient. */
/* See the descriptions of EIGS, X and V. */
/* ..... */
/* LDZ (input) INTEGER , LDZ >= M */
/* The leading dimension of the array Z. */
/* ..... */
/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */
/* RES(1:K) contains the residuals for the K computed */
/* Ritz pairs, */
/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */
/* See the description of EIGS and Z. */
/* ..... */
/* B (output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array. */
/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */
/* be used for computing the refined vectors; see further */
/* details in the provided references. */
/* If JOBF == 'E', B(1:N,1;K) contains */
/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */
/* Exact DMD, up to scaling by the inverse eigenvalues. */
/* In both cases, the content of B can be lifted to the */
/* original dimension of the input data by pre-multiplying */
/* with the Q factor from the initial QR factorization. */
/* Here A denotes a compression of the underlying operator. */
/* See the descriptions of F and X. */
/* If JOBF =='N', then B is not referenced. */
/* ..... */
/* LDB (input) INTEGER, LDB >= MIN(M,N) */
/* The leading dimension of the array B. */
/* ..... */
/* V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
/* On exit, V(1:K,1:K) V contains the K eigenvectors of */
/* the Rayleigh quotient. The Ritz vectors */
/* (returned in Z) are the product of Q from the initial QR */
/* factorization (see the description of F) X (see the */
/* description of X) and V. */
/* ..... */
/* LDV (input) INTEGER, LDV >= N-1 */
/* The leading dimension of the array V. */
/* ..... */
/* S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
/* The array S(1:K,1:K) is used for the matrix Rayleigh */
/* quotient. This content is overwritten during */
/* the eigenvalue decomposition by CGEEV. */
/* See the description of K. */
/* ..... */
/* LDS (input) INTEGER, LDS >= N-1 */
/* The leading dimension of the array S. */
/* ..... */
/* ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array */
/* On exit, */
/* ZWORK(1:MIN(M,N)) contains the scalar factors of the */
/* elementary reflectors as returned by CGEQRF of the */
/* M-by-N input matrix F. */
/* If the call to CGEDMDQ is only workspace query, then */
/* ZWORK(1) contains the minimal complex workspace length and */
/* ZWORK(2) is the optimal complex workspace length. */
/* Hence, the length of work is at least 2. */
/* See the description of LZWORK. */
/* ..... */
/* LZWORK (input) INTEGER */
/* The minimal length of the workspace vector ZWORK. */
/* LZWORK is calculated as follows: */
/* Let MLWQR = N (minimal workspace for CGEQRF[M,N]) */
/* MLWDMD = minimal workspace for CGEDMD (see the */
/* description of LWORK in CGEDMD) */
/* MLWMQR = N (minimal workspace for */
/* ZUNMQR['L','N',M,N,N]) */
/* MLWGQR = N (minimal workspace for ZUNGQR[M,N,N]) */
/* MINMN = MIN(M,N) */
/* Then */
/* LZWORK = MAX(2, MIN(M,N)+MLWQR, MINMN+MLWDMD) */
/* is further updated as follows: */
/* if JOBZ == 'V' or JOBZ == 'F' THEN */
/* LZWORK = MAX( LZWORK, MINMN+MLWMQR ) */
/* if JOBQ == 'Q' THEN */
/* LZWORK = MAX( ZLWORK, MINMN+MLWGQR) */

/* ..... */
/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */
/* On exit, */
/* WORK(1:N-1) contains the singular values of */
/* the input submatrix F(1:M,1:N-1). */
/* If the call to CGEDMDQ 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 (input) INTEGER */
/* The minimal length of the workspace vector WORK. */
/* LWORK is the same as in CGEDMD, because in CGEDMDQ */
/* only CGEDMD requires real workspace for snapshots */
/* of dimensions MIN(M,N)-by-(N-1). */
/* If on entry LWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace lengths for both WORK and */
/* IWORK. See the descriptions of WORK and 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 (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(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 (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. */
/* ............................................................. */
/* ............................................................. */
/* Parameters */
/* ~~~~~~~~~~ */
/* COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) */

/* Local scalars */
/* ~~~~~~~~~~~~~ */

/* External functions (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~ */

/* External subroutines (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* External subroutines */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* Intrinsic functions */
/* ~~~~~~~~~~~~~~~~~~~ */
/* .......................................................... */
/* Parameter adjustments */
f_dim1 = *ldf;
f_offset = 1 + f_dim1 * 1;
f -= f_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
--eigs;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--res;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
--zwork;
--work;
--iwork;

/* Function Body */
one = 1.f;
zero = 0.f;
zzero.r = 0.f, zzero.i = 0.f;

/* Test the input arguments */
wntres = lsame_(jobr, "R");
sccolx = lsame_(jobs, "S") || 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 = f2cmin(*m,*n);
*info = 0;
lquery = *lwork == -1 || *liwork == -1;

if (! (sccolx || sccoly || lsame_(jobs, "N"))) {
*info = -1;
} else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (wntres || lsame_(jobr, "N")) ||
wntres && lsame_(jobz, "N")) {
*info = -3;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -4;
} else if (! (wnttrf || lsame_(jobt, "N"))) {
*info = -5;
} else if (! (wntref || wntex || lsame_(jobf, "N")))
{
*info = -6;
} else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd ==
4)) {
*info = -7;
} else if (*m < 0) {
*info = -8;
} else if (*n < 0 || *n > *m + 1) {
*info = -9;
} else if (*ldf < *m) {
*info = -11;
} else if (*ldx < minmn) {
*info = -13;
} else if (*ldy < minmn) {
*info = -15;
} else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) {
*info = -16;
} else if (*tol < zero || *tol >= one) {
*info = -17;
} else if (*ldz < *m) {
*info = -21;
} else if ((wntref || wntex) && *ldb < minmn) {
*info = -24;
} else if (*ldv < *n - 1) {
*info = -26;
} else if (*lds < *n - 1) {
*info = -28;
}

if (wntvec || wntvcf || wntvcq) {
*(unsigned char *)jobvl = 'V';
} else {
*(unsigned char *)jobvl = 'N';
}
if (*info == 0) {
/* 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 || *n == 1) {
/* 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) {
iwork[1] = 1;
work[1] = 2.f;
work[2] = 2.f;
} else {
*k = 0;
}
*info = 1;
return 0;
}
mlrwrk = 2;
mlwork = 2;
olwork = 2;
iminwr = 1;
mlwqr = f2cmax(1,*n);
/* Minimal workspace length for CGEQRF. */
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
cgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[1], &c_n1, &
info1);
olwqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwqr;
olwork = f2cmax(i__1,i__2);
}
i__1 = *n - 1;
cgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset],
ldx, &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset]
, ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[
s_offset], lds, &zwork[1], lzwork, &work[1], &c_n1, &iwork[1],
liwork, &info1);
mlwdmd = (integer) zwork[1].r;
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwdmd;
mlwork = f2cmax(i__1,i__2);
/* Computing MAX */
i__1 = mlrwrk, i__2 = (integer) work[1];
mlrwrk = f2cmax(i__1,i__2);
iminwr = f2cmax(iminwr,iwork[1]);
if (lquery) {
olwdmd = (integer) zwork[2].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwdmd;
olwork = f2cmax(i__1,i__2);
}
if (wntvec || wntvcf) {
mlwmqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwmqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
cunmqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &zwork[1],
&z__[z_offset], ldz, &zwork[1], &c_n1, &info1);
olwmqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwmqr;
olwork = f2cmax(i__1,i__2);
}
}
if (wantq) {
mlwgqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwgqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
cungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &
zwork[1], &c_n1, &info1);
olwgqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwgqr;
olwork = f2cmax(i__1,i__2);
}
}
if (*liwork < iminwr && ! lquery) {
*info = -34;
}
if (*lwork < mlrwrk && ! lquery) {
*info = -32;
}
if (*lzwork < mlwork && ! lquery) {
*info = -30;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEDMDQ", &i__1);
return 0;
} else if (lquery) {
/* Return minimal and optimal workspace sizes */
iwork[1] = iminwr;
zwork[1].r = (real) mlwork, zwork[1].i = 0.f;
zwork[2].r = (real) olwork, zwork[2].i = 0.f;
work[1] = (real) mlrwrk;
work[2] = (real) mlrwrk;
return 0;
}
/* ..... */
/* 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. */

i__1 = *lzwork - minmn;
cgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[minmn + 1], &i__1, &
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. */
i__1 = *n - 1;
claset_("L", &minmn, &i__1, &zzero, &zzero, &x[x_offset], ldx);
i__1 = *n - 1;
clacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx);
i__1 = *n - 1;
clacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy);
if (*m >= 3) {
i__1 = minmn - 2;
i__2 = *n - 2;
claset_("L", &i__1, &i__2, &zzero, &zzero, &y[y_dim1 + 3], ldy);
}

/* Compute the DMD of the projected snapshot pairs (X,Y) */
i__1 = *n - 1;
i__2 = *lzwork - minmn;
cgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx,
&y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset], ldz, &
res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[s_offset], lds, &
zwork[minmn + 1], &i__2, &work[1], lwork, &iwork[1], liwork, &
info1);
if (info1 == 2 || info1 == 3) {
/* Return with error code. See CGEDMD for details. */
*info = info1;
return 0;
} else {
*info = info1;
}

/* The Ritz vectors (Koopman modes) can be explicitly */
/* formed or returned in factored form. */
if (wntvec) {
/* Compute the eigenvectors explicitly. */
if (*m > minmn) {
i__1 = *m - minmn;
claset_("A", &i__1, k, &zzero, &zzero, &z__[minmn + 1 + z_dim1],
ldz);
}
i__1 = *lzwork - minmn;
cunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[
z_offset], ldz, &zwork[minmn + 1], &i__1, &info1);
} else if (wntvcf) {
/* 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 CGEDMD in X) and the */
/* second factor (the eigenvectors of the Rayleigh */
/* quotient) is in the array V, as returned by CGEDMD. */
clacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz);
if (*m > *n) {
i__1 = *m - *n;
claset_("A", &i__1, k, &zzero, &zzero, &z__[*n + 1 + z_dim1], ldz);
}
i__1 = *lzwork - minmn;
cunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[
z_offset], ldz, &zwork[minmn + 1], &i__1, &info1);
}

/* Some optional output variables: */

/* The upper triangular factor R in the initial QR */
/* factorization is optionally returned in the array Y. */
/* This is useful if this call to CGEDMDQ is to be */
/* followed by a streaming DMD that is implemented in a */
/* QR compressed form. */
if (wnttrf) {
/* Return the upper triangular R in Y */
claset_("A", &minmn, n, &zzero, &zzero, &y[y_offset], ldy);
clacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy);
}

/* The orthonormal/unitary factor Q 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) {
/* Q overwrites F */
i__1 = *lzwork - minmn;
cungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &zwork[minmn
+ 1], &i__1, &info1);
}

return 0;

} /* cgedmdq_ */


+ 1242
- 0
lapack-netlib/SRC/dgedmd.c
File diff suppressed because it is too large
View File


+ 789
- 0
lapack-netlib/SRC/dgedmdq.c View File

@@ -509,3 +509,792 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ



/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/



/* Table of constant values */

static integer c_n1 = -1;

/* Subroutine */ int dgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq,
char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n,
doublereal *f, integer *ldf, doublereal *x, integer *ldx, doublereal *
y, integer *ldy, integer *nrnk, doublereal *tol, integer *k,
doublereal *reig, doublereal *imeig, doublereal *z__, integer *ldz,
doublereal *res, doublereal *b, integer *ldb, doublereal *v, integer *
ldv, doublereal *s, integer *lds, doublereal *work, integer *lwork,
integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1,
z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset,
i__1, i__2;

/* Local variables */
doublereal zero;
integer info1;
extern logical lsame_(char *, char *);
char jobvl[1];
integer minmn;
logical wantq;
integer mlwqr, olwqr;
logical wntex;
extern /* Subroutine */ int dgedmd_(char *, char *, char *, char *,
integer *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *, integer *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, integer *, integer *, integer *, integer
*), dgeqrf_(integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *), dlacpy_(char *, integer *, integer *, doublereal *,
integer *, doublereal *, integer *), dlaset_(char *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
integer *), xerbla_(char *, integer *);
integer mlwdmd, olwdmd;
logical sccolx, sccoly;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *), dormqr_(char *, char *, integer *, integer *, integer
*, doublereal *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *, integer *);
integer iminwr;
logical wntvec, wntvcf;
integer mlwgqr;
logical wntref;
integer mlwork, olwgqr, olwork;
doublereal rdummy[2];
integer mlwmqr, olwmqr;
logical lquery, wntres, wnttrf, wntvcq;
doublereal one;

/* March 2023 */
/* ..... */
/* USE iso_fortran_env */
/* INTEGER, PARAMETER :: WP = real64 */
/* ..... */
/* Scalar arguments */
/* Array arguments */
/* ..... */
/* Purpose */
/* ======= */
/* DGEDMDQ 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, DGEDMDQ 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, DGEDMDQ 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 */
/* ============================================================ */
/* ...................................................................... */
/* Arguments */
/* ========= */
/* 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 (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 (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 (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 (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 (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 (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 (input) INTEGER, M >= 0 */
/* The state space dimension (the number of rows of F). */
/* ..... */
/* 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 (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 DGEQRF. 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 (input) INTEGER, LDF >= M */
/* The leading dimension of the array F. */
/* ..... */
/* 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 (input) INTEGER, LDX >= N */
/* The leading dimension of the array X. */
/* ..... */
/* 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 (input) INTEGER , LDY >= N */
/* The leading dimension of the array Y. */
/* ..... */
/* 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 (input) REAL(KIND=WP), 0 <= TOL < 1 */
/* The tolerance for truncating small singular values. */
/* See the description of NRNK. */
/* ..... */
/* 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 (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 (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 consequtive */
/* indices (i,i+1), with the positive imaginary part */
/* listed first. */
/* See the descriptions of K, REIG, 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 (input) INTEGER , LDZ >= M */
/* The leading dimension of the array Z. */
/* ..... */
/* 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 (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 (input) INTEGER, LDB >= MIN(M,N) */
/* The leading dimension of the array B. */
/* ..... */
/* 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 (input) INTEGER, LDV >= N-1 */
/* The leading dimension of the array V. */
/* ..... */
/* 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 DGEEV. */
/* See the description of K. */
/* ..... */
/* LDS (input) INTEGER, LDS >= N-1 */
/* The leading dimension of the array S. */
/* ..... */
/* 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 DGEQRF 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 DGEDMDQ 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 (input) INTEGER */
/* The minimal length of the workspace vector WORK. */
/* LWORK is calculated as follows: */
/* Let MLWQR = N (minimal workspace for DGEQRF[M,N]) */
/* MLWDMD = minimal workspace for DGEDMD (see the */
/* description of LWORK in DGEDMD) for */
/* snapshots of dimensions MIN(M,N)-by-(N-1) */
/* MLWMQR = N (minimal workspace for */
/* DORMQR['L','N',M,N,N]) */
/* MLWGQR = N (minimal workspace for DORGQR[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 (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 (input) INTEGER */
/* The minimal length of the workspace vector IWORK. */
/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */
/* Let M1=MIN(M,N), N1=N-1. Then */
/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1)) */
/* If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1) */
/* If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1) */
/* If on entry LIWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace lengths for both WORK and */
/* IWORK. See the descriptions of WORK and IWORK. */
/* ..... */
/* INFO (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. */
/* ............................................................. */
/* ............................................................. */
/* Parameters */
/* ~~~~~~~~~~ */

/* Local scalars */
/* ~~~~~~~~~~~~~ */

/* Local array */
/* ~~~~~~~~~~~ */

/* External functions (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~ */

/* External subroutines (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* External subroutines */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* Intrinsic functions */
/* ~~~~~~~~~~~~~~~~~~~ */
/* .......................................................... */
/* Parameter adjustments */
f_dim1 = *ldf;
f_offset = 1 + f_dim1 * 1;
f -= f_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
--reig;
--imeig;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--res;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
--work;
--iwork;

/* Function Body */
zero = 0.f;
one = 1.f;

/* Test the input arguments */
wntres = lsame_(jobr, "R");
sccolx = lsame_(jobs, "S") || 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 = f2cmin(*m,*n);
*info = 0;
lquery = *lwork == -1 || *liwork == -1;

if (! (sccolx || sccoly || lsame_(jobs, "N"))) {
*info = -1;
} else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (wntres || lsame_(jobr, "N")) ||
wntres && lsame_(jobz, "N")) {
*info = -3;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -4;
} else if (! (wnttrf || lsame_(jobt, "N"))) {
*info = -5;
} else if (! (wntref || wntex || lsame_(jobf, "N")))
{
*info = -6;
} else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd ==
4)) {
*info = -7;
} else if (*m < 0) {
*info = -8;
} else if (*n < 0 || *n > *m + 1) {
*info = -9;
} else if (*ldf < *m) {
*info = -11;
} else if (*ldx < minmn) {
*info = -13;
} else if (*ldy < minmn) {
*info = -15;
} else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) {
*info = -16;
} else if (*tol < zero || *tol >= one) {
*info = -17;
} else if (*ldz < *m) {
*info = -22;
} else if ((wntref || wntex) && *ldb < minmn) {
*info = -25;
} else if (*ldv < *n - 1) {
*info = -27;
} else if (*lds < *n - 1) {
*info = -29;
}

if (wntvec || wntvcf || wntvcq) {
*(unsigned char *)jobvl = 'V';
} else {
*(unsigned char *)jobvl = 'N';
}
if (*info == 0) {
/* 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 || *n == 1) {
/* 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) {
iwork[1] = 1;
work[1] = 2.;
work[2] = 2.;
} else {
*k = 0;
}
*info = 1;
return 0;
}
mlwqr = f2cmax(1,*n);
/* Minimal workspace length for DGEQRF. */
mlwork = minmn + mlwqr;
if (lquery) {
dgeqrf_(m, n, &f[f_offset], ldf, &work[1], rdummy, &c_n1, &info1);
olwqr = (integer) rdummy[0];
olwork = f2cmin(*m,*n) + olwqr;
}
i__1 = *n - 1;
dgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset],
ldx, &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &
z__[z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset],
ldv, &s[s_offset], lds, &work[1], &c_n1, &iwork[1], liwork, &
info1);
mlwdmd = (integer) work[1];
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwdmd;
mlwork = f2cmax(i__1,i__2);
iminwr = iwork[1];
if (lquery) {
olwdmd = (integer) work[2];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwdmd;
olwork = f2cmax(i__1,i__2);
}
if (wntvec || wntvcf) {
mlwmqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + *n - 1 + mlwmqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
dormqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &work[1], &
z__[z_offset], ldz, &work[1], &c_n1, &info1);
olwmqr = (integer) work[1];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + *n - 1 + olwmqr;
olwork = f2cmax(i__1,i__2);
}
}
if (wantq) {
mlwgqr = *n;
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + *n - 1 + mlwgqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
dorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[
1], &c_n1, &info1);
olwgqr = (integer) work[1];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + *n - 1 + olwgqr;
olwork = f2cmax(i__1,i__2);
}
}
iminwr = f2cmax(1,iminwr);
mlwork = f2cmax(2,mlwork);
if (*lwork < mlwork && ! lquery) {
*info = -31;
}
if (*liwork < iminwr && ! lquery) {
*info = -33;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGEDMDQ", &i__1);
return 0;
} else if (lquery) {
/* Return minimal and optimal workspace sizes */
iwork[1] = iminwr;
work[1] = (doublereal) mlwork;
work[2] = (doublereal) olwork;
return 0;
}
/* ..... */
/* 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. */

i__1 = *lwork - minmn;
dgeqrf_(m, n, &f[f_offset], ldf, &work[1], &work[minmn + 1], &i__1, &
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. */
i__1 = *n - 1;
dlaset_("L", &minmn, &i__1, &zero, &zero, &x[x_offset], ldx);
i__1 = *n - 1;
dlacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx);
i__1 = *n - 1;
dlacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy);
if (*m >= 3) {
i__1 = minmn - 2;
i__2 = *n - 2;
dlaset_("L", &i__1, &i__2, &zero, &zero, &y[y_dim1 + 3], ldy);
}

/* Compute the DMD of the projected snapshot pairs (X,Y) */
i__1 = *n - 1;
i__2 = *lwork - minmn;
dgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx,
&y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &z__[
z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[
s_offset], lds, &work[minmn + 1], &i__2, &iwork[1], liwork, &
info1);
if (info1 == 2 || info1 == 3) {
/* Return with error code. See DGEDMD for details. */
*info = info1;
return 0;
} else {
*info = info1;
}

/* The Ritz vectors (Koopman modes) can be explicitly */
/* formed or returned in factored form. */
if (wntvec) {
/* Compute the eigenvectors explicitly. */
if (*m > minmn) {
i__1 = *m - minmn;
dlaset_("A", &i__1, k, &zero, &zero, &z__[minmn + 1 + z_dim1],
ldz);
}
i__1 = *lwork - (minmn + *n - 1);
dormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[
z_offset], ldz, &work[minmn + *n], &i__1, &info1);
} else if (wntvcf) {
/* 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 DGEDMD in X) and the */
/* second factor (the eigenvectors of the Rayleigh */
/* quotient) is in the array V, as returned by DGEDMD. */
dlacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz);
if (*m > *n) {
i__1 = *m - *n;
dlaset_("A", &i__1, k, &zero, &zero, &z__[*n + 1 + z_dim1], ldz);
}
i__1 = *lwork - (minmn + *n - 1);
dormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[
z_offset], ldz, &work[minmn + *n], &i__1, &info1);
}

/* Some optional output variables: */

/* The upper triangular factor R in the initial QR */
/* factorization is optionally returned in the array Y. */
/* This is useful if this call to DGEDMDQ is to be */
/* followed by a streaming DMD that is implemented in a */
/* QR compressed form. */
if (wnttrf) {
/* Return the upper triangular R in Y */
dlaset_("A", &minmn, n, &zero, &zero, &y[y_offset], ldy);
dlacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy);
}

/* The orthonormal/orthogonal factor Q 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) {
/* Q overwrites F */
i__1 = *lwork - (minmn + *n - 1);
dorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[minmn +
*n], &i__1, &info1);
}

return 0;

} /* dgedmdq_ */


+ 1235
- 0
lapack-netlib/SRC/sgedmd.c
File diff suppressed because it is too large
View File


+ 785
- 0
lapack-netlib/SRC/sgedmdq.c View File

@@ -509,3 +509,788 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ



/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/



/* Table of constant values */

static integer c_n1 = -1;

/* Subroutine */ int sgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq,
char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n, real
*f, integer *ldf, real *x, integer *ldx, real *y, integer *ldy,
integer *nrnk, real *tol, integer *k, real *reig, real *imeig, real *
z__, integer *ldz, real *res, real *b, integer *ldb, real *v, integer
*ldv, real *s, integer *lds, real *work, integer *lwork, integer *
iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1,
z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset,
i__1, i__2;

/* Local variables */
real zero;
integer info1;
extern logical lsame_(char *, char *);
char jobvl[1];
integer minmn;
logical wantq;
integer mlwqr, olwqr;
logical wntex;
extern /* Subroutine */ int sgedmd_(char *, char *, char *, char *,
integer *, integer *, integer *, real *, integer *, real *,
integer *, integer *, real *, integer *, real *, real *, real *,
integer *, real *, real *, integer *, real *, integer *, real *,
integer *, real *, integer *, integer *, integer *, integer *), xerbla_(char *, integer *);
integer mlwdmd, olwdmd;
extern /* Subroutine */ int sgeqrf_(integer *, integer *, real *, integer
*, real *, real *, integer *, integer *);
logical sccolx, sccoly;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), slaset_(char *, integer *,
integer *, real *, real *, real *, integer *);
integer iminwr;
logical wntvec, wntvcf;
integer mlwgqr;
logical wntref;
integer mlwork, olwgqr, olwork;
real rdummy[2];
integer mlwmqr, olwmqr;
logical lquery, wntres, wnttrf, wntvcq;
extern /* Subroutine */ int sorgqr_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *, integer *), sormqr_(char
*, char *, integer *, integer *, integer *, real *, integer *,
real *, real *, integer *, real *, integer *, integer *);
real one;

/* March 2023 */
/* ..... */
/* USE iso_fortran_env */
/* INTEGER, PARAMETER :: WP = real32 */
/* ..... */
/* Scalar arguments */
/* Array arguments */
/* ..... */
/* 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 */
/* ============================================================ */
/* ...................................................................... */
/* Arguments */
/* ========= */
/* 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 (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 (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 (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 (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 (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 (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 (input) INTEGER, M >= 0 */
/* The state space dimension (the number of rows of F) */
/* ..... */
/* 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 (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 (input) INTEGER, LDF >= M */
/* The leading dimension of the array F. */
/* ..... */
/* 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 (input) INTEGER, LDX >= N */
/* The leading dimension of the array X */
/* ..... */
/* 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 (input) INTEGER , LDY >= N */
/* The leading dimension of the array Y */
/* ..... */
/* 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 (input) REAL(KIND=WP), 0 <= TOL < 1 */
/* The tolerance for truncating small singular values. */
/* See the description of NRNK. */
/* ..... */
/* 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 (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 (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 (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 (input) INTEGER , LDZ >= M */
/* The leading dimension of the array Z. */
/* ..... */
/* 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 (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 (input) INTEGER, LDB >= MIN(M,N) */
/* The leading dimension of the array B. */
/* ..... */
/* 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 (input) INTEGER, LDV >= N-1 */
/* The leading dimension of the array V. */
/* ..... */
/* 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 (input) INTEGER, LDS >= N-1 */
/* The leading dimension of the array S. */
/* ..... */
/* 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 (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 (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 (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 (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. */
/* ............................................................. */
/* ............................................................. */
/* Parameters */
/* ~~~~~~~~~~ */

/* Local scalars */
/* ~~~~~~~~~~~~~ */

/* Local array */
/* ~~~~~~~~~~~ */

/* External functions (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~ */

/* External subroutines (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* External subroutines */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* Intrinsic functions */
/* ~~~~~~~~~~~~~~~~~~~ */
/* Parameter adjustments */
f_dim1 = *ldf;
f_offset = 1 + f_dim1 * 1;
f -= f_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
--reig;
--imeig;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--res;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
--work;
--iwork;

/* Function Body */
one = 1.f;
zero = 0.f;
/* .......................................................... */

/* Test the input arguments */
wntres = lsame_(jobr, "R");
sccolx = lsame_(jobs, "S") || 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 = f2cmin(*m,*n);
*info = 0;
lquery = *lwork == -1 || *liwork == -1;

if (! (sccolx || sccoly || lsame_(jobs, "N"))) {
*info = -1;
} else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (wntres || lsame_(jobr, "N")) ||
wntres && lsame_(jobz, "N")) {
*info = -3;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -4;
} else if (! (wnttrf || lsame_(jobt, "N"))) {
*info = -5;
} else if (! (wntref || wntex || lsame_(jobf, "N")))
{
*info = -6;
} else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd ==
4)) {
*info = -7;
} else if (*m < 0) {
*info = -8;
} else if (*n < 0 || *n > *m + 1) {
*info = -9;
} else if (*ldf < *m) {
*info = -11;
} else if (*ldx < minmn) {
*info = -13;
} else if (*ldy < minmn) {
*info = -15;
} else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) {
*info = -16;
} else if (*tol < zero || *tol >= one) {
*info = -17;
} else if (*ldz < *m) {
*info = -22;
} else if ((wntref || wntex) && *ldb < minmn) {
*info = -25;
} else if (*ldv < *n - 1) {
*info = -27;
} else if (*lds < *n - 1) {
*info = -29;
}

if (wntvec || wntvcf) {
*(unsigned char *)jobvl = 'V';
} else {
*(unsigned char *)jobvl = 'N';
}
if (*info == 0) {
/* 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 || *n == 1) {
/* 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) {
iwork[1] = 1;
work[1] = 2.f;
work[2] = 2.f;
} else {
*k = 0;
}
*info = 1;
return 0;
}
mlwqr = f2cmax(1,*n);
/* Minimal workspace length for SGEQRF. */
mlwork = f2cmin(*m,*n) + mlwqr;
if (lquery) {
sgeqrf_(m, n, &f[f_offset], ldf, &work[1], rdummy, &c_n1, &info1);
olwqr = (integer) rdummy[0];
olwork = f2cmin(*m,*n) + olwqr;
}
i__1 = *n - 1;
sgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset],
ldx, &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &
z__[z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset],
ldv, &s[s_offset], lds, &work[1], &c_n1, &iwork[1], liwork, &
info1);
mlwdmd = (integer) work[1];
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwdmd;
mlwork = f2cmax(i__1,i__2);
iminwr = iwork[1];
if (lquery) {
olwdmd = (integer) work[2];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwdmd;
olwork = f2cmax(i__1,i__2);
}
if (wntvec || wntvcf) {
mlwmqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + *n - 1 + mlwmqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
sormqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &work[1], &
z__[z_offset], ldz, &work[1], &c_n1, &info1);
olwmqr = (integer) work[1];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + *n - 1 + olwmqr;
olwork = f2cmax(i__1,i__2);
}
}
if (wantq) {
mlwgqr = *n;
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + *n - 1 + mlwgqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
sorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[
1], &c_n1, &info1);
olwgqr = (integer) work[1];
/* Computing MAX */
i__1 = olwork, i__2 = minmn + *n - 1 + olwgqr;
olwork = f2cmax(i__1,i__2);
}
}
iminwr = f2cmax(1,iminwr);
mlwork = f2cmax(2,mlwork);
if (*lwork < mlwork && ! lquery) {
*info = -31;
}
if (*liwork < iminwr && ! lquery) {
*info = -33;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGEDMDQ", &i__1);
return 0;
} else if (lquery) {
/* Return minimal and optimal workspace sizes */
iwork[1] = iminwr;
work[1] = (real) mlwork;
work[2] = (real) olwork;
return 0;
}
/* ..... */
/* 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. */

i__1 = *lwork - minmn;
sgeqrf_(m, n, &f[f_offset], ldf, &work[1], &work[minmn + 1], &i__1, &
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. */
i__1 = *n - 1;
slaset_("L", &minmn, &i__1, &zero, &zero, &x[x_offset], ldx);
i__1 = *n - 1;
slacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx);
i__1 = *n - 1;
slacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy);
if (*m >= 3) {
i__1 = minmn - 2;
i__2 = *n - 2;
slaset_("L", &i__1, &i__2, &zero, &zero, &y[y_dim1 + 3], ldy);
}

/* Compute the DMD of the projected snapshot pairs (X,Y) */
i__1 = *n - 1;
i__2 = *lwork - minmn;
sgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx,
&y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &z__[
z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[
s_offset], lds, &work[minmn + 1], &i__2, &iwork[1], liwork, &
info1);
if (info1 == 2 || info1 == 3) {
/* Return with error code. */
*info = info1;
return 0;
} else {
*info = info1;
}

/* The Ritz vectors (Koopman modes) can be explicitly */
/* formed or returned in factored form. */
if (wntvec) {
/* Compute the eigenvectors explicitly. */
if (*m > minmn) {
i__1 = *m - minmn;
slaset_("A", &i__1, k, &zero, &zero, &z__[minmn + 1 + z_dim1],
ldz);
}
i__1 = *lwork - (minmn + *n - 1);
sormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[
z_offset], ldz, &work[minmn + *n], &i__1, &info1);
} else if (wntvcf) {
/* 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. */
slacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz);
if (*m > *n) {
i__1 = *m - *n;
slaset_("A", &i__1, k, &zero, &zero, &z__[*n + 1 + z_dim1], ldz);
}
i__1 = *lwork - (minmn + *n - 1);
sormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[
z_offset], ldz, &work[minmn + *n], &i__1, &info1);
}

/* 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) {
/* Return the upper triangular R in Y */
slaset_("A", &minmn, n, &zero, &zero, &y[y_offset], ldy);
slacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy);
}

/* 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) {
/* Q overwrites F */
i__1 = *lwork - (minmn + *n - 1);
sorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[minmn +
*n], &i__1, &info1);
}

return 0;

} /* sgedmdq_ */


+ 1165
- 0
lapack-netlib/SRC/zgedmd.c
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+ 782
- 0
lapack-netlib/SRC/zgedmdq.c View File

@@ -509,3 +509,785 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ



/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/



/* Table of constant values */

static integer c_n1 = -1;

/* Subroutine */ int zgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq,
char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n,
doublecomplex *f, integer *ldf, doublecomplex *x, integer *ldx,
doublecomplex *y, integer *ldy, integer *nrnk, doublereal *tol,
integer *k, doublecomplex *eigs, doublecomplex *z__, integer *ldz,
doublereal *res, doublecomplex *b, integer *ldb, doublecomplex *v,
integer *ldv, doublecomplex *s, integer *lds, doublecomplex *zwork,
integer *lzwork, doublereal *work, integer *lwork, integer *iwork,
integer *liwork, integer *info)
{
/* System generated locals */
integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1,
z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset,
i__1, i__2;

/* Local variables */
doublereal zero;
integer info1;
extern logical lsame_(char *, char *);
char jobvl[1];
integer minmn;
logical wantq;
integer mlwqr, olwqr;
logical wntex;
doublecomplex zzero;
extern /* Subroutine */ int zgedmd_(char *, char *, char *, char *,
integer *, integer *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, integer *, doublereal *, integer *,
doublecomplex *, doublecomplex *, integer *, doublereal *,
doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *,
doublereal *, integer *, integer *, integer *, integer *), xerbla_(char *, integer *);
integer mlwdmd, olwdmd;
logical sccolx, sccoly;
extern /* Subroutine */ int zgeqrf_(integer *, integer *, doublecomplex *,
integer *, doublecomplex *, doublecomplex *, integer *, integer *
), zlacpy_(char *, integer *, integer *, doublecomplex *, integer
*, doublecomplex *, integer *), zlaset_(char *, integer *,
integer *, doublecomplex *, doublecomplex *, doublecomplex *,
integer *);
integer iminwr;
logical wntvec, wntvcf;
integer mlwgqr;
logical wntref;
integer mlwork, olwgqr, olwork, mlrwrk, mlwmqr, olwmqr;
logical lquery, wntres, wnttrf, wntvcq;
extern /* Subroutine */ int zungqr_(integer *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *, integer *), zunmqr_(char *, char *, integer *, integer
*, integer *, doublecomplex *, integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *, integer *);
doublereal one;

/* March 2023 */
/* ..... */
/* USE iso_fortran_env */
/* INTEGER, PARAMETER :: WP = real64 */
/* ..... */
/* Scalar arguments */
/* Array arguments */
/* ..... */
/* Purpose */
/* ======= */
/* ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */
/* a pair of data snapshot matrices, using a QR factorization */
/* based compression of the data. For the input matrices */
/* X and Y such that Y = A*X with an unaccessible matrix */
/* A, ZGEDMDQ computes a certain number of Ritz pairs of A using */
/* the standard Rayleigh-Ritz extraction from a subspace of */
/* range(X) that is determined using the leading left singular */
/* vectors of X. Optionally, ZGEDMDQ returns the residuals */
/* of the computed Ritz pairs, the information needed for */
/* a refinement of the Ritz vectors, or the eigenvectors of */
/* the Exact DMD. */
/* For further details see the references listed */
/* below. For more details of the implementation see [3]. */

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

/* Developed and supported by: */
/* =========================== */
/* Developed and coded by Zlatko Drmac, Faculty of Science, */
/* University of Zagreb; drmac@math.hr */
/* In cooperation with */
/* AIMdyn Inc., Santa Barbara, CA. */
/* and supported by */
/* - DARPA SBIR project "Koopman Operator-Based Forecasting */
/* for Nonstationary Processes from Near-Term, Limited */
/* Observational Data" Contract No: W31P4Q-21-C-0007 */
/* - DARPA PAI project "Physics-Informed Machine Learning */
/* Methodologies" Contract No: HR0011-18-9-0033 */
/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */
/* Framework for Space-Time Analysis of Process Dynamics" */
/* Contract No: HR0011-16-C-0116 */
/* Any opinions, findings and conclusions or recommendations */
/* expressed in this material are those of the author and */
/* do not necessarily reflect the views of the DARPA SBIR */
/* Program Office. */
/* ============================================================ */
/* Distribution Statement A: */
/* Approved for Public Release, Distribution Unlimited. */
/* Cleared by DARPA on September 29, 2022 */
/* ============================================================ */
/* ...................................................................... */
/* Arguments */
/* ========= */
/* 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 (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 (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 (input) CHARACTER*1 */
/* Specifies whether to explicitly compute and return the */
/* unitary matrix from the QR factorization. */
/* 'Q' :: The matrix Q of the QR factorization of the data */
/* snapshot matrix is computed and stored in the */
/* array F. See the description of F. */
/* 'N' :: The matrix Q is not explicitly computed. */
/* ..... */
/* JOBT (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 (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 (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */
/* Allows for a selection of the SVD algorithm from the */
/* LAPACK library. */
/* 1 :: ZGESVD (the QR SVD algorithm) */
/* 2 :: ZGESDD (the Divide and Conquer algorithm; if enough */
/* workspace available, this is the fastest option) */
/* 3 :: ZGESVDQ (the preconditioned QR SVD ; this and 4 */
/* are the most accurate options) */
/* 4 :: ZGEJSV (the preconditioned Jacobi SVD; this and 3 */
/* are the most accurate options) */
/* For the four methods above, a significant difference in */
/* the accuracy of small singular values is possible if */
/* the snapshots vary in norm so that X is severely */
/* ill-conditioned. If small (smaller than EPS*||X||) */
/* singular values are of interest and JOBS=='N', then */
/* the options (3, 4) give the most accurate results, where */
/* the option 4 is slightly better and with stronger */
/* theoretical background. */
/* If JOBS=='S', i.e. the columns of X will be normalized, */
/* then all methods give nearly equally accurate results. */
/* ..... */
/* M (input) INTEGER, M >= 0 */
/* The state space dimension (the number of rows of F). */
/* ..... */
/* 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 (input/output) COMPLEX(KIND=WP) M-by-N array */
/* > On entry, */
/* the columns of F are the sequence of data snapshots */
/* from a single trajectory, taken at equidistant discrete */
/* times. It is assumed that the column norms of F are */
/* in the range of the normalized floating point numbers. */
/* < On exit, */
/* If JOBQ == 'Q', the array F contains the orthogonal */
/* matrix/factor of the QR factorization of the initial */
/* data snapshots matrix F. See the description of JOBQ. */
/* If JOBQ == 'N', the entries in F strictly below the main */
/* diagonal contain, column-wise, the information on the */
/* Householder vectors, as returned by ZGEQRF. The */
/* remaining information to restore the orthogonal matrix */
/* of the initial QR factorization is stored in ZWORK(1:MIN(M,N)). */
/* See the description of ZWORK. */
/* ..... */
/* LDF (input) INTEGER, LDF >= M */
/* The leading dimension of the array F. */
/* ..... */
/* X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array */
/* X is used as workspace to hold representations of the */
/* leading N-1 snapshots in the orthonormal basis computed */
/* in the QR factorization of F. */
/* On exit, the leading K columns of X contain the leading */
/* K left singular vectors of the above described content */
/* of X. To lift them to the space of the left singular */
/* vectors U(:,1:K) of the input data, pre-multiply with the */
/* Q factor from the initial QR factorization. */
/* See the descriptions of F, K, V and Z. */
/* ..... */
/* LDX (input) INTEGER, LDX >= N */
/* The leading dimension of the array X. */
/* ..... */
/* Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array */
/* Y is used as workspace to hold representations of the */
/* trailing N-1 snapshots in the orthonormal basis computed */
/* in the QR factorization of F. */
/* On exit, */
/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */
/* triangular factor from the QR factorization of the data */
/* snapshot matrix F. */
/* ..... */
/* LDY (input) INTEGER , LDY >= N */
/* The leading dimension of the array Y. */
/* ..... */
/* 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 (input) REAL(KIND=WP), 0 <= TOL < 1 */
/* The tolerance for truncating small singular values. */
/* See the description of NRNK. */
/* ..... */
/* K (output) INTEGER, 0 <= K <= N */
/* The dimension of the SVD/POD basis for the leading N-1 */
/* data snapshots (columns of F) and the number of the */
/* computed Ritz pairs. The value of K is determined */
/* according to the rule set by the parameters NRNK and */
/* TOL. See the descriptions of NRNK and TOL. */
/* ..... */
/* EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array */
/* The leading K (K<=N-1) entries of EIGS contain */
/* the computed eigenvalues (Ritz values). */
/* See the descriptions of K, and Z. */
/* ..... */
/* Z (workspace/output) COMPLEX(KIND=WP) M-by-(N-1) array */
/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */
/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */
/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */
/* Z*V, where Z contains orthonormal matrix (the product of */
/* Q from the initial QR factorization and the SVD/POD_basis */
/* returned by ZGEDMD in X) and the second factor (the */
/* eigenvectors of the Rayleigh quotient) is in the array V, */
/* as returned by ZGEDMD. That is, X(:,1:K)*V(:,i) */
/* is an eigenvector corresponding to EIGS(i). The columns */
/* of V(1:K,1:K) are the computed eigenvectors of the */
/* K-by-K Rayleigh quotient. */
/* See the descriptions of EIGS, X and V. */
/* ..... */
/* LDZ (input) INTEGER , LDZ >= M */
/* The leading dimension of the array Z. */
/* ..... */
/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */
/* RES(1:K) contains the residuals for the K computed */
/* Ritz pairs, */
/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */
/* See the description of EIGS and Z. */
/* ..... */
/* B (output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array. */
/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */
/* be used for computing the refined vectors; see further */
/* details in the provided references. */
/* If JOBF == 'E', B(1:N,1;K) contains */
/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */
/* Exact DMD, up to scaling by the inverse eigenvalues. */
/* In both cases, the content of B can be lifted to the */
/* original dimension of the input data by pre-multiplying */
/* with the Q factor from the initial QR factorization. */
/* Here A denotes a compression of the underlying operator. */
/* See the descriptions of F and X. */
/* If JOBF =='N', then B is not referenced. */
/* ..... */
/* LDB (input) INTEGER, LDB >= MIN(M,N) */
/* The leading dimension of the array B. */
/* ..... */
/* V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
/* On exit, V(1:K,1:K) V contains the K eigenvectors of */
/* the Rayleigh quotient. The Ritz vectors */
/* (returned in Z) are the product of Q from the initial QR */
/* factorization (see the description of F) X (see the */
/* description of X) and V. */
/* ..... */
/* LDV (input) INTEGER, LDV >= N-1 */
/* The leading dimension of the array V. */
/* ..... */
/* S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
/* The array S(1:K,1:K) is used for the matrix Rayleigh */
/* quotient. This content is overwritten during */
/* the eigenvalue decomposition by ZGEEV. */
/* See the description of K. */
/* ..... */
/* LDS (input) INTEGER, LDS >= N-1 */
/* The leading dimension of the array S. */
/* ..... */
/* ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array */
/* On exit, */
/* ZWORK(1:MIN(M,N)) contains the scalar factors of the */
/* elementary reflectors as returned by ZGEQRF of the */
/* M-by-N input matrix F. */
/* If the call to ZGEDMDQ is only workspace query, then */
/* ZWORK(1) contains the minimal complex workspace length and */
/* ZWORK(2) is the optimal complex workspace length. */
/* Hence, the length of work is at least 2. */
/* See the description of LZWORK. */
/* ..... */
/* LZWORK (input) INTEGER */
/* The minimal length of the workspace vector ZWORK. */
/* LZWORK is calculated as follows: */
/* Let MLWQR = N (minimal workspace for ZGEQRF[M,N]) */
/* MLWDMD = minimal workspace for ZGEDMD (see the */
/* description of LWORK in ZGEDMD) */
/* MLWMQR = N (minimal workspace for */
/* ZUNMQR['L','N',M,N,N]) */
/* MLWGQR = N (minimal workspace for ZUNGQR[M,N,N]) */
/* MINMN = MIN(M,N) */
/* Then */
/* LZWORK = MAX(2, MIN(M,N)+MLWQR, MINMN+MLWDMD) */
/* is further updated as follows: */
/* if JOBZ == 'V' or JOBZ == 'F' THEN */
/* LZWORK = MAX(LZWORK, MINMN+MLWMQR) */
/* if JOBQ == 'Q' THEN */
/* LZWORK = MAX(ZLWORK, MINMN+MLWGQR) */

/* ..... */
/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */
/* On exit, */
/* WORK(1:N-1) contains the singular values of */
/* the input submatrix F(1:M,1:N-1). */
/* If the call to ZGEDMDQ is only workspace query, then */
/* WORK(1) contains the minimal workspace length and */
/* WORK(2) is the optimal workspace length. hence, the */
/* length of work is at least 2. */
/* See the description of LWORK. */
/* ..... */
/* LWORK (input) INTEGER */
/* The minimal length of the workspace vector WORK. */
/* LWORK is the same as in ZGEDMD, because in ZGEDMDQ */
/* only ZGEDMD requires real workspace for snapshots */
/* of dimensions MIN(M,N)-by-(N-1). */
/* If on entry LWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace length for WORK. */
/* ..... */
/* IWORK (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 (input) INTEGER */
/* The minimal length of the workspace vector IWORK. */
/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */
/* Let M1=MIN(M,N), N1=N-1. Then */
/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1)) */
/* If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1) */
/* If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1) */
/* If on entry LIWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace lengths for both WORK and */
/* IWORK. See the descriptions of WORK and IWORK. */
/* ..... */
/* INFO (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. */
/* ............................................................. */
/* ............................................................. */
/* Parameters */
/* ~~~~~~~~~~ */
/* COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) */

/* Local scalars */
/* ~~~~~~~~~~~~~ */

/* External functions (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~ */

/* External subroutines (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* External subroutines */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* Intrinsic functions */
/* ~~~~~~~~~~~~~~~~~~~ */
/* .......................................................... */
/* Parameter adjustments */
f_dim1 = *ldf;
f_offset = 1 + f_dim1 * 1;
f -= f_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
--eigs;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--res;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
--zwork;
--work;
--iwork;

/* Function Body */
one = 1.f;
zero = 0.f;
zzero.r = 0.f, zzero.i = 0.f;

/* Test the input arguments */
wntres = lsame_(jobr, "R");
sccolx = lsame_(jobs, "S") || 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 = f2cmin(*m,*n);
*info = 0;
lquery = *lzwork == -1 || *lwork == -1 || *liwork == -1;

if (! (sccolx || sccoly || lsame_(jobs, "N"))) {
*info = -1;
} else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) {
*info = -2;
} else if (! (wntres || lsame_(jobr, "N")) ||
wntres && lsame_(jobz, "N")) {
*info = -3;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -4;
} else if (! (wnttrf || lsame_(jobt, "N"))) {
*info = -5;
} else if (! (wntref || wntex || lsame_(jobf, "N")))
{
*info = -6;
} else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd ==
4)) {
*info = -7;
} else if (*m < 0) {
*info = -8;
} else if (*n < 0 || *n > *m + 1) {
*info = -9;
} else if (*ldf < *m) {
*info = -11;
} else if (*ldx < minmn) {
*info = -13;
} else if (*ldy < minmn) {
*info = -15;
} else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) {
*info = -16;
} else if (*tol < zero || *tol >= one) {
*info = -17;
} else if (*ldz < *m) {
*info = -21;
} else if ((wntref || wntex) && *ldb < minmn) {
*info = -24;
} else if (*ldv < *n - 1) {
*info = -26;
} else if (*lds < *n - 1) {
*info = -28;
}

if (wntvec || wntvcf || wntvcq) {
*(unsigned char *)jobvl = 'V';
} else {
*(unsigned char *)jobvl = 'N';
}
if (*info == 0) {
/* 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 || *n == 1) {
/* 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) {
iwork[1] = 1;
zwork[1].r = 2., zwork[1].i = 0.;
zwork[2].r = 2., zwork[2].i = 0.;
work[1] = 2.;
work[2] = 2.;
} else {
*k = 0;
}
*info = 1;
return 0;
}
mlrwrk = 2;
mlwork = 2;
olwork = 2;
iminwr = 1;
mlwqr = f2cmax(1,*n);
/* Minimal workspace length for ZGEQRF. */
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
zgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[1], &c_n1, &
info1);
olwqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwqr;
olwork = f2cmax(i__1,i__2);
}
i__1 = *n - 1;
zgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset],
ldx, &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset]
, ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[
s_offset], lds, &zwork[1], &c_n1, &work[1], &c_n1, &iwork[1],
&c_n1, &info1);
mlwdmd = (integer) zwork[1].r;
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwdmd;
mlwork = f2cmax(i__1,i__2);
/* Computing MAX */
i__1 = mlrwrk, i__2 = (integer) work[1];
mlrwrk = f2cmax(i__1,i__2);
iminwr = f2cmax(iminwr,iwork[1]);
if (lquery) {
olwdmd = (integer) zwork[2].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwdmd;
olwork = f2cmax(i__1,i__2);
}
if (wntvec || wntvcf) {
mlwmqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwmqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
zunmqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &zwork[1],
&z__[z_offset], ldz, &zwork[1], &c_n1, &info1);
olwmqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwmqr;
olwork = f2cmax(i__1,i__2);
}
}
if (wantq) {
mlwgqr = f2cmax(1,*n);
/* Computing MAX */
i__1 = mlwork, i__2 = minmn + mlwgqr;
mlwork = f2cmax(i__1,i__2);
if (lquery) {
zungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &
zwork[1], &c_n1, &info1);
olwgqr = (integer) zwork[1].r;
/* Computing MAX */
i__1 = olwork, i__2 = minmn + olwgqr;
olwork = f2cmax(i__1,i__2);
}
}
if (*liwork < iminwr && ! lquery) {
*info = -34;
}
if (*lwork < mlrwrk && ! lquery) {
*info = -32;
}
if (*lzwork < mlwork && ! lquery) {
*info = -30;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGEDMDQ", &i__1);
return 0;
} else if (lquery) {
/* Return minimal and optimal workspace sizes */
iwork[1] = iminwr;
zwork[1].r = (doublereal) mlwork, zwork[1].i = 0.;
zwork[2].r = (doublereal) olwork, zwork[2].i = 0.;
work[1] = (doublereal) mlrwrk;
work[2] = (doublereal) mlrwrk;
return 0;
}
/* ..... */
/* 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. */

i__1 = *lzwork - minmn;
zgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[minmn + 1], &i__1, &
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. */
i__1 = *n - 1;
zlaset_("L", &minmn, &i__1, &zzero, &zzero, &x[x_offset], ldx);
i__1 = *n - 1;
zlacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx);
i__1 = *n - 1;
zlacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy);
if (*m >= 3) {
i__1 = minmn - 2;
i__2 = *n - 2;
zlaset_("L", &i__1, &i__2, &zzero, &zzero, &y[y_dim1 + 3], ldy);
}

/* Compute the DMD of the projected snapshot pairs (X,Y) */
i__1 = *n - 1;
i__2 = *lzwork - minmn;
zgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx,
&y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset], ldz, &
res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[s_offset], lds, &
zwork[minmn + 1], &i__2, &work[1], lwork, &iwork[1], liwork, &
info1);
if (info1 == 2 || info1 == 3) {
/* Return with error code. See ZGEDMD for details. */
*info = info1;
return 0;
} else {
*info = info1;
}

/* The Ritz vectors (Koopman modes) can be explicitly */
/* formed or returned in factored form. */
if (wntvec) {
/* Compute the eigenvectors explicitly. */
if (*m > minmn) {
i__1 = *m - minmn;
zlaset_("A", &i__1, k, &zzero, &zzero, &z__[minmn + 1 + z_dim1],
ldz);
}
i__1 = *lzwork - minmn;
zunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[
z_offset], ldz, &zwork[minmn + 1], &i__1, &info1);
} else if (wntvcf) {
/* 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 ZGEDMD in X) and the */
/* second factor (the eigenvectors of the Rayleigh */
/* quotient) is in the array V, as returned by ZGEDMD. */
zlacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz);
if (*m > *n) {
i__1 = *m - *n;
zlaset_("A", &i__1, k, &zzero, &zzero, &z__[*n + 1 + z_dim1], ldz);
}
i__1 = *lzwork - minmn;
zunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[
z_offset], ldz, &zwork[minmn + 1], &i__1, &info1);
}

/* Some optional output variables: */

/* The upper triangular factor R in the initial QR */
/* factorization is optionally returned in the array Y. */
/* This is useful if this call to ZGEDMDQ is to be */
/* followed by a streaming DMD that is implemented in a */
/* QR compressed form. */
if (wnttrf) {
/* Return the upper triangular R in Y */
zlaset_("A", &minmn, n, &zzero, &zzero, &y[y_offset], ldy);
zlacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy);
}

/* The orthonormal/unitary factor Q 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) {
/* Q overwrites F */
i__1 = *lzwork - minmn;
zungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &zwork[minmn
+ 1], &i__1, &info1);
}

return 0;

} /* zgedmdq_ */


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