table of contents
| gedmd(3) | LAPACK | gedmd(3) |
NAME¶
gedmd - DMD driver, Dynamic Mode Decomposition
SYNOPSIS¶
Functions¶
subroutine cgedmd (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)
CGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data
snapshot matrices. subroutine cgedmdq (jobs, jobz, jobr, jobq, jobt,
jobf, whtsvd, m, n, f, ldf, x, ldx, y, ldy, nrnk, tol, k, eigs, z, ldz, res,
b, ldb, v, ldv, s, lds, zwork, lzwork, work, lwork, iwork, liwork, info)
CGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of
data snapshot matrices. subroutine dgedmd (jobs, jobz, jobr, jobf,
whtsvd, m, n, x, ldx, y, ldy, nrnk, tol, k, reig, imeig, z, ldz, res, b,
ldb, w, ldw, s, lds, work, lwork, iwork, liwork, info)
DGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data
snapshot matrices. 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. subroutine sgedmd (jobs, jobz, jobr, jobf,
whtsvd, m, n, x, ldx, y, ldy, nrnk, tol, k, reig, imeig, z, ldz, res, b,
ldb, w, ldw, s, lds, work, lwork, iwork, liwork, info)
SGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data
snapshot matrices. subroutine sgedmdq (jobs, jobz, jobr, jobq, jobt,
jobf, whtsvd, m, n, f, ldf, x, ldx, y, ldy, nrnk, tol, k, reig, imeig, z,
ldz, res, b, ldb, v, ldv, s, lds, work, lwork, iwork, liwork, info)
SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of
data snapshot matrices. 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. subroutine zgedmdq (jobs, jobz, jobr, jobq, jobt,
jobf, whtsvd, m, n, f, ldf, x, ldx, y, ldy, nrnk, tol, k, eigs, z, ldz, res,
b, ldb, v, ldv, s, lds, zwork, lzwork, work, lwork, iwork, liwork, info)
ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of
data snapshot matrices.
Detailed Description¶
Function Documentation¶
subroutine cgedmd (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)¶
CGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
Purpose:
CGEDMD 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, CGEDMD 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, CGEDMD 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.
'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 :: 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
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 CGEEV.
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 CGEEV for computing
the eigenvalues of a Rayleigh quotient.
If the call to CGEDMD 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_CGEEV),
where LZWORK_CGEEV = MAX( 1, 2*N ) and the minimal
LZWORK_SVD is calculated as follows
If WHTSVD == 1 :: CGESVD ::
LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N))
If WHTSVD == 2 :: CGESDD ::
LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N)
If WHTSVD == 3 :: CGESVDQ ::
LZWORK_SVD = obtainable by a query
If WHTSVD == 4 :: CGEJSV ::
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 CGEDMD 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_CGEEV), where
LRWORK_CGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace
for the SVD subroutine determined by the input parameter
WHTSVD.
If WHTSVD == 1 :: CGESVD ::
LRWORK_SVD = 5*MIN(M,N)
If WHTSVD == 2 :: CGESDD ::
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 :: CGESVDQ ::
LRWORK_SVD = obtainable by a query
If WHTSVD == 4 :: CGEJSV ::
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
subroutine cgedmdq (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, complex(kind=wp), dimension(ldf,*), intent(inout) f, integer, intent(in) ldf, complex(kind=wp), dimension(ldx,*), intent(out) x, integer, intent(in) ldx, complex(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, 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(ldv,*), intent(out) v, integer, intent(in) ldv, 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) work, integer, intent(in) lwork, integer, dimension(*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info)¶
CGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
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.
Developed and supported by:
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 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
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
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
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 :: 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
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) 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
LDF (input) INTEGER, LDF >= M
The leading dimension of the array F.
X
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
LDX (input) INTEGER, LDX >= N
The leading dimension of the array X.
Y
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
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.
EIGS
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
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
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,
RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2.
See the description of EIGS and Z.
B
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
LDB (input) INTEGER, LDB >= MIN(M,N)
The leading dimension of the array B.
V
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
LDV (input) INTEGER, LDV >= N-1
The leading dimension of the array V.
S
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
LDS (input) INTEGER, LDS >= N-1
The leading dimension of the array S.
LZWORK
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
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
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
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
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(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
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
subroutine dgedmd (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, real(kind=wp), dimension(ldx,*), intent(inout) x, integer, intent(in) ldx, real(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, 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(ldw,*), intent(out) w, integer, intent(in) ldw, 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)¶
DGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
Purpose:
DGEDMD 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, DGEDMD 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, DGEDMD 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) is 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 :: 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 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) REAL(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) REAL(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.
REIG
REIG (output) REAL(KIND=WP) N-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, and Z.
IMEIG
IMEIG (output) REAL(KIND=WP) N-by-1 array
The leading K (K<=N) entries of IMEIG 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, and Z.
Z
Z (workspace/output) REAL(KIND=WP) M-by-N 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; ||Z(:,i)||_2=1.
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.
|| Z(:,i:i+1)||_F = 1.
If JOBZ == 'F', then the above descriptions hold for
the columns of X(:,1:K)*W(1:K,1:K), where the columns
of W(1:k,1:K) are the computed eigenvectors of the
K-by-K Rayleigh quotient. The columns of W(1:K,1:K)
are similarly structured: If IMEIG(i) == 0 then
X(:,1:K)*W(:,i) is an eigenvector, and if IMEIG(i)>0
then X(:,1:K)*W(:,i)+sqrt(-1)*X(:,1:K)*W(:,i+1) and
X(:,1:K)*W(:,i)-sqrt(-1)*X(:,1:K)*W(:,i+1)
are the eigenvectors of LAMBDA(i), LAMBDA(i+1).
See the descriptions of REIG, IMEIG, 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.
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 REIG, IMEIG and Z.
B
B (output) REAL(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) REAL(KIND=WP) N-by-N array
On exit, W(1:K,1:K) contains the K computed
eigenvectors of the matrix Rayleigh quotient (real and
imaginary parts for each complex conjugate pair of the
eigenvalues). 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) REAL(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 DGEEV.
See the description of K.
LDS
LDS (input) INTEGER, LDS >= N
The leading dimension of the array S.
WORK
WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
On exit, WORK(1:N) contains the singular values of
X (for JOBS=='N') or column scaled X (JOBS=='S', 'C').
If WHTSVD==4, then WORK(N+1) and WORK(N+2) contain
scaling factor WORK(N+2)/WORK(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 DGEDMD is only workspace query, then
WORK(1) contains the minimal workspace length and
WORK(2) is the optimal workspace length. Hence, the
leng 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:
If WHTSVD == 1 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)).
If JOBZ == 'N' then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)).
Here LWORK_SVD = MAX(1,3*N+M,5*N) is the minimal
workspace length of DGESVD.
If WHTSVD == 2 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N))
Here LWORK_SVD = MAX(M, 5*N*N+4*N)+3*N*N is the
minimal workspace length of DGESDD.
If WHTSVD == 3 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
Here LWORK_SVD = N+M+MAX(3*N+1,
MAX(1,3*N+M,5*N),MAX(1,N))
is the minimal workspace length of DGESVDQ.
If WHTSVD == 4 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
Here LWORK_SVD = MAX(7,2*M+N,6*N+2*N*N) is the
minimal workspace length of DGEJSV.
The above expressions are not simplified in order to
make the usage of WORK more transparent, and for
easier checking. In any case, LWORK >= 2.
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
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
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
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
subroutine sgedmd (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, real(kind=wp), dimension(ldx,*), intent(inout) x, integer, intent(in) ldx, real(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, 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(ldw,*), intent(out) w, integer, intent(in) ldw, 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)¶
SGEDMD computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
Purpose:
SGEDMD 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, SGEDMD 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, SGEDMD 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:
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 :: 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
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) REAL(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) REAL(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.
REIG
REIG (output) REAL(KIND=WP) N-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, and Z.
IMEIG
IMEIG (output) REAL(KIND=WP) N-by-1 array
The leading K (K<=N) entries of IMEIG 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, and Z.
Z
Z (workspace/output) REAL(KIND=WP) M-by-N 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; ||Z(:,i)||_2=1.
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.
|| Z(:,i:i+1)||_F = 1.
If JOBZ == 'F', then the above descriptions hold for
the columns of X(:,1:K)*W(1:K,1:K), where the columns
of W(1:k,1:K) are the computed eigenvectors of the
K-by-K Rayleigh quotient. The columns of W(1:K,1:K)
are similarly structured: If IMEIG(i) == 0 then
X(:,1:K)*W(:,i) is an eigenvector, and if IMEIG(i)>0
then X(:,1:K)*W(:,i)+sqrt(-1)*X(:,1:K)*W(:,i+1) and
X(:,1:K)*W(:,i)-sqrt(-1)*X(:,1:K)*W(:,i+1)
are the eigenvectors of LAMBDA(i), LAMBDA(i+1).
See the descriptions of REIG, IMEIG, 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.
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 REIG, IMEIG and Z.
B
B (output) REAL(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) REAL(KIND=WP) N-by-N array
On exit, W(1:K,1:K) contains the K computed
eigenvectors of the matrix Rayleigh quotient (real and
imaginary parts for each complex conjugate pair of the
eigenvalues). 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
left singular vectors of X.
LDW
LDW (input) INTEGER, LDW >= N
The leading dimension of the array W.
S
S (workspace/output) REAL(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 SGEEV.
See the description of K.
LDS
LDS (input) INTEGER, LDS >= N
The leading dimension of the array S.
WORK
WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array
On exit, WORK(1:N) contains the singular values of
X (for JOBS=='N') or column scaled X (JOBS=='S', 'C').
If WHTSVD==4, then WORK(N+1) and WORK(N+2) contain
scaling factor WORK(N+2)/WORK(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 SGEDMD 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:
If WHTSVD == 1 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)).
If JOBZ == 'N' then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)).
Here LWORK_SVD = MAX(1,3*N+M,5*N) is the minimal
workspace length of SGESVD.
If WHTSVD == 2 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N))
Here LWORK_SVD = MAX(M, 5*N*N+4*N)+3*N*N is the
minimal workspace length of SGESDD.
If WHTSVD == 3 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
Here LWORK_SVD = N+M+MAX(3*N+1,
MAX(1,3*N+M,5*N),MAX(1,N))
is the minimal workspace length of SGESVDQ.
If WHTSVD == 4 ::
If JOBZ == 'V', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N))
If JOBZ == 'N', then
LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N))
Here LWORK_SVD = MAX(7,2*M+N,6*N+2*N*N) is the
minimal workspace length of SGEJSV.
The above expressions are not simplified in order to
make the usage of WORK more transparent, and for
easier checking. In any case, LWORK >= 2.
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
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
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
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.
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
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 :: 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
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 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
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 consecutive
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 SGEEV.
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 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
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
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 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
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
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 '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.
'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).
LDX
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 (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
subroutine zgedmdq (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, complex(kind=wp), dimension(ldf,*), intent(inout) f, integer, intent(in) ldf, complex(kind=wp), dimension(ldx,*), intent(out) x, integer, intent(in) ldx, complex(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, 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(ldv,*), intent(out) v, integer, intent(in) ldv, 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) work, integer, intent(in) lwork, integer, dimension(*), intent(out) iwork, integer, intent(in) liwork, integer, intent(out) info)¶
ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices.
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.
Developed and supported by:
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
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
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 ::
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 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) 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
LDF (input) INTEGER, LDF >= M
The leading dimension of the array F.
X
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
LDX (input) INTEGER, LDX >= N
The leading dimension of the array X.
Y
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
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.
EIGS
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
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
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,
RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2.
See the description of EIGS and Z.
B
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
LDB (input) INTEGER, LDB >= MIN(M,N)
The leading dimension of the array B.
V
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
LDV (input) INTEGER, LDV >= N-1
The leading dimension of the array V.
S
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
LDS (input) INTEGER, LDS >= N-1
The leading dimension of the array S.
LZWORK
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
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
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
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
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
Author¶
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