--- name: "zgedmd.f90" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/zgedmd.f90" --- # zgedmd.f90(3) ## NAME SRC/zgedmd.f90 ## SYNOPSIS ### Functions/Subroutines subroutine **zgedmd** (jobs, jobz, jobr, jobf, whtsvd, m, n, x, ldx, y, ldy, nrnk, tol, k, eigs, z, ldz, res, b, ldb, w, ldw, s, lds, zwork, lzwork, rwork, lrwork, iwork, liwork, info)\ **ZGEDMD** computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices. ## Function/Subroutine Documentation ### subroutine zgedmd (character, intent(in) jobs, character, intent(in) jobz, character, intent(in) jobr, character, intent(in) jobf, integer, intent(in) whtsvd, integer, intent(in) m, integer, intent(in) n, complex(kind=wp), dimension(ldx,\*), intent(inout) x, integer, intent(in) ldx, complex(kind=wp), dimension(ldy,\*), intent(inout) y, integer, intent(in) ldy, integer, intent(in) nrnk, real(kind=wp), intent(in) tol, integer, intent(out) k, complex(kind=wp), dimension(\*), intent(out) eigs, complex(kind=wp), dimension(ldz,\*), intent(out) z, integer, intent(in) ldz, real(kind=wp), dimension(\*), intent(out) res, complex(kind=wp), dimension(ldb,\*), intent(out) b, integer, intent(in) ldb, complex(kind=wp), dimension(ldw,\*), intent(out) w, integer, intent(in) ldw, complex(kind=wp), dimension(lds,\*), intent(out) s, integer, intent(in) lds, complex(kind=wp), dimension(\*), intent(out) zwork, integer, intent(in) lzwork, real(kind=wp), dimension(\*), intent(out) rwork, integer, intent(in) lrwork, integer, dimension(\*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info) **ZGEDMD** computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices. **Purpose:** ``` !> ZGEDMD computes the Dynamic Mode Decomposition (DMD) for !> a pair of data snapshot matrices. For the input matrices !> X and Y such that Y = A*X with an unaccessible matrix !> A, ZGEDMD 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, ZGEDMD returns the residuals !> of the computed Ritz pairs, the information needed for !> a refinement of the Ritz vectors, or the eigenvectors of !> the Exact DMD. !> For further details see the references listed !> below. For more details of the implementation see [3]. !> ``` **References:** ``` !> [1] P. Schmid: Dynamic mode decomposition of numerical !> and experimental data, !> Journal of Fluid Mechanics 656, 5-28, 2010. !> [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal !> decompositions: analysis and enhancements, !> SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. !> [3] Z. Drmac: A LAPACK implementation of the Dynamic !> Mode Decomposition I. Technical report. AIMDyn Inc. !> and LAPACK Working Note 298. !> [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. !> Brunton, N. Kutz: On Dynamic Mode Decomposition: !> Theory and Applications, Journal of Computational !> Dynamics 1(2), 391 -421, 2014. !> ``` **Developed and supported by:** ``` !> Developed and coded by Zlatko Drmac, Faculty of Science, !> University of Zagreb; drmac@math.hr !> In cooperation with !> AIMdyn Inc., Santa Barbara, CA. !> and supported by !> - DARPA SBIR project Contract No: W31P4Q-21-C-0007 !> - DARPA PAI project Contract No: HR0011-18-9-0033 !> - DARPA MoDyL project !> Contract No: HR0011-16-C-0116 !> Any opinions, findings and conclusions or recommendations !> expressed in this material are those of the author and !> do not necessarily reflect the views of the DARPA SBIR !> Program Office !> ``` **Distribution Statement A:** ``` !> Approved for Public Release, Distribution Unlimited. !> Cleared by DARPA on September 29, 2022 !> ``` **Parameters** *JOBS* ``` !> JOBS (input) CHARACTER*1 !> Determines whether the initial data snapshots are scaled !> by a diagonal matrix. !> 'S' :: The data snapshots matrices X and Y are multiplied !> with a diagonal matrix D so that X*D has unit !> nonzero columns (in the Euclidean 2-norm) !> 'C' :: The snapshots are scaled as with the 'S' option. !> If it is found that an i-th column of X is zero !> vector and the corresponding i-th column of Y is !> non-zero, then the i-th column of Y is set to !> zero and a warning flag is raised. !> 'Y' :: The data snapshots matrices X and Y are multiplied !> by a diagonal matrix D so that Y*D has unit !> nonzero columns (in the Euclidean 2-norm) !> 'N' :: No data scaling. !> ``` *JOBZ* ``` !> JOBZ (input) CHARACTER*1 !> Determines whether the eigenvectors (Koopman modes) will !> be computed. !> 'V' :: The eigenvectors (Koopman modes) will be computed !> and returned in the matrix Z. !> See the description of Z. !> 'F' :: The eigenvectors (Koopman modes) will be returned !> in factored form as the product X(:,1:K)*W, where X !> contains a POD basis (leading left singular vectors !> of the data matrix X) and W contains the eigenvectors !> of the corresponding Rayleigh quotient. !> See the descriptions of K, X, W, Z. !> 'N' :: The eigenvectors are not computed. !> ``` *JOBR* ``` !> JOBR (input) CHARACTER*1 !> Determines whether to compute the residuals. !> 'R' :: The residuals for the computed eigenpairs will be !> computed and stored in the array RES. !> See the description of RES. !> For this option to be legal, JOBZ must be 'V'. !> 'N' :: The residuals are not computed. !> ``` *JOBF* ``` !> JOBF (input) CHARACTER*1 !> Specifies whether to store information needed for post- !> processing (e.g. computing refined Ritz vectors) !> 'R' :: The matrix needed for the refinement of the Ritz !> vectors is computed and stored in the array B. !> See the description of B. !> 'E' :: The unscaled eigenvectors of the Exact DMD are !> computed and returned in the array B. See the !> description of B. !> 'N' :: No eigenvector refinement data is computed. !> ``` *WHTSVD* ``` !> WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } !> Allows for a selection of the SVD algorithm from the !> LAPACK library. !> 1 :: 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* ``` !> M (input) INTEGER, M>= 0 !> The state space dimension (the row dimension of X, Y). !> ``` *N* ``` !> N (input) INTEGER, 0 <= N <= M !> The number of data snapshot pairs !> (the number of columns of X and Y). !> ``` *X* ``` !> X (input/output) COMPLEX(KIND=WP) M-by-N array !> > On entry, X contains the data snapshot matrix X. It is !> assumed that the column norms of X are in the range of !> the normalized floating point numbers. !> < On exit, the leading K columns of X contain a POD basis, !> i.e. the leading K left singular vectors of the input !> data matrix X, U(:,1:K). All N columns of X contain all !> left singular vectors of the input matrix X. !> See the descriptions of K, Z and W. !> ``` *LDX* ``` !> LDX (input) INTEGER, LDX >= M !> The leading dimension of the array X. !> ``` *Y* ``` !> Y (input/workspace/output) COMPLEX(KIND=WP) M-by-N array !> > On entry, Y contains the data snapshot matrix Y !> < On exit, !> If JOBR == 'R', the leading K columns of Y contain !> the residual vectors for the computed Ritz pairs. !> See the description of RES. !> If JOBR == 'N', Y contains the original input data, !> scaled according to the value of JOBS. !> ``` *LDY* ``` !> LDY (input) INTEGER , LDY >= M !> The leading dimension of the array Y. !> ``` *NRNK* ``` !> NRNK (input) INTEGER !> Determines the mode how to compute the numerical rank, !> i.e. how to truncate small singular values of the input !> matrix X. On input, if !> NRNK = -1 :: i-th singular value sigma(i) is truncated !> if sigma(i) <= TOL*sigma(1) !> This option is recommended. !> NRNK = -2 :: i-th singular value sigma(i) is truncated !> if sigma(i) <= TOL*sigma(i-1) !> This option is included for R&D purposes. !> It requires highly accurate SVD, which !> may not be feasible. !> The numerical rank can be enforced by using positive !> value of NRNK as follows: !> 0 < NRNK <= N :: at most NRNK largest singular values !> will be used. If the number of the computed nonzero !> singular values is less than NRNK, then only those !> nonzero values will be used and the actually used !> dimension is less than NRNK. The actual number of !> the nonzero singular values is returned in the variable !> K. See the descriptions of TOL and K. !> ``` *TOL* ``` !> TOL (input) REAL(KIND=WP), 0 <= TOL < 1 !> The tolerance for truncating small singular values. !> See the description of NRNK. !> ``` *K* ``` !> K (output) INTEGER, 0 <= K <= N !> The dimension of the POD basis for the data snapshot !> matrix X and the number of the computed Ritz pairs. !> The value of K is determined according to the rule set !> by the parameters NRNK and TOL. !> See the descriptions of NRNK and TOL. !> ``` *EIGS* ``` !> EIGS (output) COMPLEX(KIND=WP) N-by-1 array !> The leading K (K<=N) entries of EIGS contain !> the computed eigenvalues (Ritz values). !> See the descriptions of K, and Z. !> ``` *Z* ``` !> Z (workspace/output) COMPLEX(KIND=WP) M-by-N 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 !> the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) !> is an eigenvector corresponding to EIGS(i). The columns !> of W(1:k,1:K) are the computed eigenvectors of the !> K-by-K Rayleigh quotient. !> See the descriptions of EIGS, X and W. !> ``` *LDZ* ``` !> LDZ (input) INTEGER , LDZ >= M !> The leading dimension of the array Z. !> ``` *RES* ``` !> RES (output) REAL(KIND=WP) N-by-1 array !> RES(1:K) contains the residuals for the K computed !> Ritz pairs, !> RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. !> See the description of EIGS and Z. !> ``` *B* ``` !> B (output) COMPLEX(KIND=WP) M-by-N array. !> IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can !> be used for computing the refined vectors; see further !> details in the provided references. !> If JOBF == 'E', B(1:M,1:K) contains !> A*U(:,1:K)*W(1:K,1:K), which are the vectors from the !> Exact DMD, up to scaling by the inverse eigenvalues. !> If JOBF =='N', then B is not referenced. !> See the descriptions of X, W, K. !> ``` *LDB* ``` !> LDB (input) INTEGER, LDB >= M !> The leading dimension of the array B. !> ``` *W* ``` !> W (workspace/output) COMPLEX(KIND=WP) N-by-N array !> On exit, W(1:K,1:K) contains the K computed !> eigenvectors of the matrix Rayleigh quotient. !> The Ritz vectors (returned in Z) are the !> product of X (containing a POD basis for the input !> matrix X) and W. See the descriptions of K, S, X and Z. !> W is also used as a workspace to temporarily store the !> right singular vectors of X. !> ``` *LDW* ``` !> LDW (input) INTEGER, LDW >= N !> The leading dimension of the array W. !> ``` *S* ``` !> S (workspace/output) COMPLEX(KIND=WP) N-by-N array !> The array S(1:K,1:K) is used for the matrix Rayleigh !> quotient. This content is overwritten during !> the eigenvalue decomposition by ZGEEV. !> See the description of K. !> ``` *LDS* ``` !> LDS (input) INTEGER, LDS >= N !> The leading dimension of the array S. !> ``` *ZWORK* ``` !> ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array !> ZWORK is used as complex workspace in the complex SVD, as !> specified by WHTSVD (1,2, 3 or 4) and for ZGEEV for computing !> the eigenvalues of a Rayleigh quotient. !> If the call to ZGEDMD 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* ``` !> LZWORK (input) INTEGER !> The minimal length of the workspace vector ZWORK. !> LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_ZGEEV), !> where LZWORK_ZGEEV = MAX( 1, 2*N ) and the minimal !> LZWORK_SVD is calculated as follows !> If WHTSVD == 1 :: ZGESVD :: !> LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) !> If WHTSVD == 2 :: ZGESDD :: !> LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) !> If WHTSVD == 3 :: ZGESVDQ :: !> LZWORK_SVD = obtainable by a query !> If WHTSVD == 4 :: ZGEJSV :: !> LZWORK_SVD = obtainable by a query !> If on entry LZWORK = -1, then a workspace query is !> assumed and the procedure only computes the minimal !> and the optimal workspace lengths and returns them in !> LZWORK(1) and LZWORK(2), respectively. !> ``` *RWORK* ``` !> RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array !> On exit, RWORK(1:N) contains the singular values of !> X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). !> If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain !> scaling factor RWORK(N+2)/RWORK(N+1) used to scale X !> and Y to avoid overflow in the SVD of X. !> This may be of interest if the scaling option is off !> and as many as possible smallest eigenvalues are !> desired to the highest feasible accuracy. !> If the call to ZGEDMD is only workspace query, then !> RWORK(1) contains the minimal workspace length. !> See the description of LRWORK. !> ``` *LRWORK* ``` !> LRWORK (input) INTEGER !> The minimal length of the workspace vector RWORK. !> LRWORK is calculated as follows: !> LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_ZGEEV), where !> LRWORK_ZGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace !> for the SVD subroutine determined by the input parameter !> WHTSVD. !> If WHTSVD == 1 :: ZGESVD :: !> LRWORK_SVD = 5*MIN(M,N) !> If WHTSVD == 2 :: ZGESDD :: !> LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), !> 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) !> If WHTSVD == 3 :: ZGESVDQ :: !> LRWORK_SVD = obtainable by a query !> If WHTSVD == 4 :: ZGEJSV :: !> LRWORK_SVD = obtainable by a query !> If on entry LRWORK = -1, then a workspace query is !> assumed and the procedure only computes the minimal !> real workspace length and returns it in RWORK(1). !> ``` *IWORK* ``` !> IWORK (workspace/output) INTEGER LIWORK-by-1 array !> Workspace that is required only if WHTSVD equals !> 2 , 3 or 4. (See the description of WHTSVD). !> If on entry LWORK =-1 or LIWORK=-1, then the !> minimal length of IWORK is computed and returned in !> IWORK(1). See the description of LIWORK. !> ``` *LIWORK* ``` !> LIWORK (input) INTEGER !> The minimal length of the workspace vector IWORK. !> If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 !> If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) !> If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) !> If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) !> If on entry LIWORK = -1, then a workspace query is !> assumed and the procedure only computes the minimal !> and the optimal workspace lengths for ZWORK, RWORK and !> IWORK. See the descriptions of ZWORK, RWORK and IWORK. !> ``` *INFO* ``` !> INFO (output) INTEGER !> -i < 0 :: On entry, the i-th argument had an !> illegal value !> = 0 :: Successful return. !> = 1 :: Void input. Quick exit (M=0 or N=0). !> = 2 :: The SVD computation of X did not converge. !> Suggestion: Check the input data and/or !> repeat with different WHTSVD. !> = 3 :: The computation of the eigenvalues did not !> converge. !> = 4 :: If data scaling was requested on input and !> the procedure found inconsistency in the data !> such that for some column index i, !> X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set !> to zero if JOBS=='C'. The computation proceeds !> with original or modified data and warning !> flag is set with INFO=4. !> ``` **Author** Zlatko Drmac Definition at line **496** of file **zgedmd.f90**. ## Author Generated automatically by Doxygen for LAPACK from the source code.