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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine sgedmdq | ( | character, intent(in) | jobs, |
| character, intent(in) | jobz, | ||
| character, intent(in) | jobr, | ||
| character, intent(in) | jobq, | ||
| character, intent(in) | jobt, | ||
| character, intent(in) | jobf, | ||
| integer, intent(in) | whtsvd, | ||
| integer, intent(in) | m, | ||
| integer, intent(in) | n, | ||
| real(kind=wp), dimension(ldf,*), intent(inout) | f, | ||
| integer, intent(in) | ldf, | ||
| real(kind=wp), dimension(ldx,*), intent(out) | x, | ||
| integer, intent(in) | ldx, | ||
| real(kind=wp), dimension(ldy,*), intent(out) | y, | ||
| integer, intent(in) | ldy, | ||
| integer, intent(in) | nrnk, | ||
| real(kind=wp), intent(in) | tol, | ||
| integer, intent(out) | k, | ||
| real(kind=wp), dimension(*), intent(out) | reig, | ||
| real(kind=wp), dimension(*), intent(out) | imeig, | ||
| real(kind=wp), dimension(ldz,*), intent(out) | z, | ||
| integer, intent(in) | ldz, | ||
| real(kind=wp), dimension(*), intent(out) | res, | ||
| real(kind=wp), dimension(ldb,*), intent(out) | b, | ||
| integer, intent(in) | ldb, | ||
| real(kind=wp), dimension(ldv,*), intent(out) | v, | ||
| integer, intent(in) | ldv, | ||
| real(kind=wp), dimension(lds,*), intent(out) | s, | ||
| integer, intent(in) | lds, | ||
| real(kind=wp), dimension(*), intent(out) | work, | ||
| integer, intent(in) | lwork, | ||
| integer, dimension(*), intent(out) | iwork, | ||
| integer, intent(in) | liwork, | ||
| integer, intent(out) | info ) |
SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
!> 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]. !>
!> [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 coded by Zlatko Drmac, Faculty of Science, !> University of Zagreb; drmac@math.hr !> In cooperation with !> AIMdyn Inc., Santa Barbara, CA. !> and supported by !> - DARPA SBIR project Contract No: W31P4Q-21-C-0007 !> - DARPA PAI project Contract No: HR0011-18-9-0033 !> - DARPA MoDyL project !> Contract No: HR0011-16-C-0116 !> Any opinions, findings and conclusions or recommendations !> expressed in this material are those of the author and !> do not necessarily reflect the views of the DARPA SBIR !> Program Office. !>
!> Approved for Public Release, Distribution Unlimited. !> Cleared by DARPA on September 29, 2022 !>
| [in] | JOBS | !> JOBS (input) CHARACTER*1 !> Determines whether the initial data snapshots are scaled !> by a diagonal matrix. The data snapshots are the columns !> of F. The leading N-1 columns of F are denoted X and the !> trailing N-1 columns are denoted Y. !> 'S' :: The data snapshots matrices X and Y are multiplied !> with a diagonal matrix D so that X*D has unit !> nonzero columns (in the Euclidean 2-norm) !> 'C' :: The snapshots are scaled as with the 'S' option. !> If it is found that an i-th column of X is zero !> vector and the corresponding i-th column of Y is !> non-zero, then the i-th column of Y is set to !> zero and a warning flag is raised. !> 'Y' :: The data snapshots matrices X and Y are multiplied !> by a diagonal matrix D so that Y*D has unit !> nonzero columns (in the Euclidean 2-norm) !> 'N' :: No data scaling. !> |
| [in] | JOBZ | !> JOBZ (input) CHARACTER*1 !> Determines whether the eigenvectors (Koopman modes) will !> be computed. !> 'V' :: The eigenvectors (Koopman modes) will be computed !> and returned in the matrix Z. !> See the description of Z. !> 'F' :: The eigenvectors (Koopman modes) will be returned !> in factored form as the product 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. !> |
| [in] | JOBR | !> JOBR (input) CHARACTER*1 !> Determines whether to compute the residuals. !> 'R' :: The residuals for the computed eigenpairs will !> be computed and stored in the array RES. !> See the description of RES. !> For this option to be legal, JOBZ must be 'V'. !> 'N' :: The residuals are not computed. !> |
| [in] | JOBQ | !> JOBQ (input) CHARACTER*1 !> Specifies whether to explicitly compute and return the !> orthogonal matrix from the QR factorization. !> 'Q' :: The matrix Q of the QR factorization of the data !> snapshot matrix is computed and stored in the !> array F. See the description of F. !> 'N' :: The matrix Q is not explicitly computed. !> |
| [in] | JOBT | !> JOBT (input) CHARACTER*1 !> Specifies whether to return the upper triangular factor !> from the QR factorization. !> 'R' :: The matrix R of the QR factorization of the data !> snapshot matrix F is returned in the array Y. !> See the description of Y and Further details. !> 'N' :: The matrix R is not returned. !> |
| [in] | JOBF | !> JOBF (input) CHARACTER*1 !> Specifies whether to store information needed for post- !> processing (e.g. computing refined Ritz vectors) !> 'R' :: The matrix needed for the refinement of the Ritz !> vectors is computed and stored in the array B. !> See the description of B. !> 'E' :: The unscaled eigenvectors of the Exact DMD are !> computed and returned in the array B. See the !> description of B. !> 'N' :: No eigenvector refinement data is computed. !> To be useful on exit, this option needs JOBQ='Q'. !> |
| [in] | WHTSVD | !> WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 }
!> Allows for a selection of the SVD algorithm from the
!> LAPACK library.
!> 1 :: SGESVD (the QR SVD algorithm)
!> 2 :: SGESDD (the Divide and Conquer algorithm; if enough
!> workspace available, this is the fastest option)
!> 3 :: SGESVDQ (the preconditioned QR SVD ; this and 4
!> are the most accurate options)
!> 4 :: SGEJSV (the preconditioned Jacobi SVD; this and 3
!> are the most accurate options)
!> For the four methods above, a significant difference in
!> the accuracy of small singular values is possible if
!> the snapshots vary in norm so that X is severely
!> ill-conditioned. If small (smaller than EPS*||X||)
!> singular values are of interest and JOBS=='N', then
!> the options (3, 4) give the most accurate results, where
!> the option 4 is slightly better and with stronger
!> theoretical background.
!> If JOBS=='S', i.e. the columns of X will be normalized,
!> then all methods give nearly equally accurate results.
!> |
| [in] | M | !> M (input) INTEGER, M >= 0 !> The state space dimension (the number of rows of F) !> |
| [in] | N | !> N (input) INTEGER, 0 <= N <= M !> The number of data snapshots from a single trajectory, !> taken at equidistant discrete times. This is the !> number of columns of F. !> |
| [in,out] | 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. !> |
| [in] | LDF | !> LDF (input) INTEGER, LDF >= M !> The leading dimension of the array F. !> |
| [in,out] | X | !> X (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array !> X is used as workspace to hold representations of the !> leading N-1 snapshots in the orthonormal basis computed !> in the QR factorization of F. !> On exit, the leading K columns of X contain the leading !> K left singular vectors of the above described content !> of X. To lift them to the space of the left singular !> vectors U(:,1:K)of the input data, pre-multiply with the !> Q factor from the initial QR factorization. !> See the descriptions of F, K, V and Z. !> |
| [in] | LDX | !> LDX (input) INTEGER, LDX >= N !> The leading dimension of the array X !> |
| [in,out] | Y | !> Y (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array !> Y is used as workspace to hold representations of the !> trailing N-1 snapshots in the orthonormal basis computed !> in the QR factorization of F. !> On exit, !> If JOBT == 'R', Y contains the MIN(M,N)-by-N upper !> triangular factor from the QR factorization of the data !> snapshot matrix F. !> |
| [in] | LDY | !> LDY (input) INTEGER , LDY >= N !> The leading dimension of the array Y !> |
| [in] | NRNK | !> NRNK (input) INTEGER !> Determines the mode how to compute the numerical rank, !> i.e. how to truncate small singular values of the input !> matrix X. On input, if !> NRNK = -1 :: i-th singular value sigma(i) is truncated !> if sigma(i) <= TOL*sigma(1) !> This option is recommended. !> NRNK = -2 :: i-th singular value sigma(i) is truncated !> if sigma(i) <= TOL*sigma(i-1) !> This option is included for R&D purposes. !> It requires highly accurate SVD, which !> may not be feasible. !> The numerical rank can be enforced by using positive !> value of NRNK as follows: !> 0 < NRNK <= N-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. !> |
| [in] | TOL | !> TOL (input) REAL(KIND=WP), 0 <= TOL < 1 !> The tolerance for truncating small singular values. !> See the description of NRNK. !> |
| [out] | K | !> K (output) INTEGER, 0 <= K <= N !> The dimension of the 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. !> |
| [out] | REIG | !> REIG (output) REAL(KIND=WP) (N-1)-by-1 array !> The leading K (K<=N) entries of REIG contain !> the real parts of the computed eigenvalues !> REIG(1:K) + sqrt(-1)*IMEIG(1:K). !> See the descriptions of K, IMEIG, Z. !> |
| [out] | IMEIG | !> IMEIG (output) REAL(KIND=WP) (N-1)-by-1 array !> The leading K (K<N) entries of REIG contain !> the imaginary parts of the computed eigenvalues !> REIG(1:K) + sqrt(-1)*IMEIG(1:K). !> The eigenvalues are determined as follows: !> If IMEIG(i) == 0, then the corresponding eigenvalue is !> real, LAMBDA(i) = REIG(i). !> If IMEIG(i)>0, then the corresponding complex !> conjugate pair of eigenvalues reads !> LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i) !> LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i) !> That is, complex conjugate pairs have consecutive !> indices (i,i+1), with the positive imaginary part !> listed first. !> See the descriptions of K, REIG, Z. !> |
| [out] | 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. !> |
| [in] | LDZ | !> LDZ (input) INTEGER , LDZ >= M !> The leading dimension of the array Z. !> |
| [out] | RES | !> RES (output) REAL(KIND=WP) (N-1)-by-1 array !> RES(1:K) contains the residuals for the K computed !> Ritz pairs. !> If LAMBDA(i) is real, then !> RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2. !> If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair !> then !> RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F !> where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ] !> [-imag(LAMBDA(i)) real(LAMBDA(i)) ]. !> It holds that !> RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2 !> RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2 !> where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1) !> ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1) !> See the description of Z. !> |
| [out] | B | !> B (output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array. !> IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can !> be used for computing the refined vectors; see further !> details in the provided references. !> If JOBF == 'E', B(1:N,1;K) contains !> A*U(:,1:K)*W(1:K,1:K), which are the vectors from the !> Exact DMD, up to scaling by the inverse eigenvalues. !> In both cases, the content of B can be lifted to the !> original dimension of the input data by pre-multiplying !> with the Q factor from the initial QR factorization. !> Here A denotes a compression of the underlying operator. !> See the descriptions of F and X. !> If JOBF =='N', then B is not referenced. !> |
| [in] | LDB | !> LDB (input) INTEGER, LDB >= MIN(M,N) !> The leading dimension of the array B. !> |
| [out] | V | !> V (workspace/output) REAL(KIND=WP) (N-1)-by-(N-1) array !> On exit, V(1:K,1:K) contains the K eigenvectors of !> the Rayleigh quotient. The eigenvectors of a complex !> conjugate pair of eigenvalues are returned in real form !> as explained in the description of Z. The Ritz vectors !> (returned in Z) are the product of X and V; see !> the descriptions of X and Z. !> |
| [in] | LDV | !> LDV (input) INTEGER, LDV >= N-1 !> The leading dimension of the array V. !> |
| [out] | S | !> S (output) REAL(KIND=WP) (N-1)-by-(N-1) array !> The array S(1:K,1:K) is used for the matrix Rayleigh !> quotient. This content is overwritten during !> the eigenvalue decomposition by SGEEV. !> See the description of K. !> |
| [in] | LDS | !> LDS (input) INTEGER, LDS >= N-1 !> The leading dimension of the array S. !> |
| [out] | WORK | !> WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array !> On exit, !> WORK(1:MIN(M,N)) contains the scalar factors of the !> elementary reflectors as returned by SGEQRF of the !> M-by-N input matrix F. !> WORK(MIN(M,N)+1:MIN(M,N)+N-1) contains the singular values of !> the input submatrix F(1:M,1:N-1). !> If the call to SGEDMDQ is only workspace query, then !> WORK(1) contains the minimal workspace length and !> WORK(2) is the optimal workspace length. Hence, the !> length of work is at least 2. !> See the description of LWORK. !> |
| [in] | 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. !> |
| [out] | IWORK | !> IWORK (workspace/output) INTEGER LIWORK-by-1 array !> Workspace that is required only if WHTSVD equals !> 2 , 3 or 4. (See the description of WHTSVD). !> If on entry LWORK =-1 or LIWORK=-1, then the !> minimal length of IWORK is computed and returned in !> IWORK(1). See the description of LIWORK. !> |
| [in] | LIWORK | !> LIWORK (input) INTEGER !> The minimal length of the workspace vector IWORK. !> If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 !> 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. !> |
| [out] | INFO | !> INFO (output) INTEGER !> -i < 0 :: On entry, the i-th argument had an !> illegal value !> = 0 :: Successful return. !> = 1 :: Void input. Quick exit (M=0 or N=0). !> = 2 :: The SVD computation of X did not converge. !> Suggestion: Check the input data and/or !> repeat with different WHTSVD. !> = 3 :: The computation of the eigenvalues did not !> converge. !> = 4 :: If data scaling was requested on input and !> the procedure found inconsistency in the data !> such that for some column index i, !> X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set !> to zero if JOBS=='C'. The computation proceeds !> with original or modified data and warning !> flag is set with INFO=4. !> |
Definition at line 571 of file sgedmdq.f90.
1.11.0