# dgedmdq.f90 - Man Page

SRC/dgedmdq.f90

## Synopsis

### Functions/Subroutines

subroutine **dgedmdq** (jobs, jobz, jobr, jobq, jobt, jobf, whtsvd, m, n, f, ldf, x, ldx, y, ldy, nrnk, tol, k, reig, imeig, z, ldz, res, b, ldb, v, ldv, s, lds, work, lwork, iwork, liwork, info)**DGEDMDQ** computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.

## Function/Subroutine Documentation

### subroutine dgedmdq (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)

**DGEDMDQ** computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.

**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

**Parameters***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.

*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.

*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.

*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.

*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.

*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'.

*WHTSVD*WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } Allows for a selection of the SVD algorithm from the LAPACK library. 1 :: DGESVD (the QR SVD algorithm) 2 :: DGESDD (the Divide and Conquer algorithm; if enough workspace available, this is the fastest option) 3 :: DGESVDQ (the preconditioned QR SVD ; this and 4 are the most accurate options) 4 :: DGEJSV (the preconditioned Jacobi SVD; this and 3 are the most accurate options) For the four methods above, a significant difference in the accuracy of small singular values is possible if the snapshots vary in norm so that X is severely ill-conditioned. If small (smaller than EPS*||X||) singular values are of interest and JOBS=='N', then the options (3, 4) give the most accurate results, where the option 4 is slightly better and with stronger theoretical background. If JOBS=='S', i.e. the columns of X will be normalized, then all methods give nearly equally accurate results.

*M*M (input) INTEGER, M >= 0 The state space dimension (the number of rows of F).

*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.

*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*LDF (input) INTEGER, LDF >= M The leading dimension of the array F.

*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.

*LDX*LDX (input) INTEGER, LDX >= N The leading dimension of the array X.

*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.

*LDY*LDY (input) INTEGER , LDY >= N 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-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*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 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*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*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*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*LDZ (input) INTEGER , LDZ >= M The leading dimension of the array Z.

*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.

*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.

*LDB*LDB (input) INTEGER, LDB >= MIN(M,N) The leading dimension of the array B.

*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.

*LDV*LDV (input) INTEGER, LDV >= N-1 The leading dimension of the array V.

*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 DGEEV. See the description of K.

*LDS*LDS (input) INTEGER, LDS >= N-1 The leading dimension of the array S.

*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 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*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*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 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*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 **571** of file **dgedmdq.f90**.

## Author

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## Referenced By

The man page dgedmdq(3) is an alias of dgedmdq.f90(3).

Tue Nov 28 2023 12:08:41 Version 3.12.0 LAPACK