SRC/dgedmd.f90(3)

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💅 SRC/dgedmd.f90(3) — Library Functions Manual 💅

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🌸 NAME 🌸

SRC/dgedmd.f90 — that's the name, no cap 💖.

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

  • subroutine dgedmd (...) — DGEDMD 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. 💫

🫶 Function/Subroutine Documentation 🫶

💜 subroutine dgedmd 💜

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 🌟

Okay so picture this: you've got the input matrices X and Y such that Y = A*X, and plot twist — A is an unaccessible matrix 😭. 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 Contract No: W31P4Q-21-C-0007
  • DARPA PAI project Contract No: HR0011-18-9-0033
  • DARPA MoDyL project, Contract No: HR0011-16-C-0116

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the DARPA SBIR Program Office. 🫶

🚀 Distribution Statement A: 🚀

Approved for Public Release, Distribution Unlimited.

Cleared by DARPA on September 29, 2022. 💖

🎀 Parameters 🎀

JOBS

JOBS (input) is CHARACTER*1. It 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. It 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. It 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. It 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 }. It 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. It 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

Okay bestie, TOL is the input tolerance for truncating small singular values. It’s a REAL(KIND=WP) and it lives in 0 <= TOL < 1. Need the full lore? See the description of NRNK. 💅

✨ K (output) INTEGER, 0 <= K <= N

K is the dimension of the POD basis for the data snapshot matrix X and the number of the computed Ritz pairs. Its value is determined according to the rule set by the parameters NRNK and TOL. See the descriptions of NRNK and TOL.

💖 REIG (output) REAL(KIND=WP) N-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 (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 (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 (input) INTEGER, LDZ >= M

The leading dimension of the array Z. Slay.

🔥 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 (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 (input) INTEGER, LDB >= M

The leading dimension of the array B.

🧚 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 (input) INTEGER, LDW >= N

The leading dimension of the array W.

🌸 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 (input) INTEGER, LDS >= N

The leading dimension of the array S.

💖 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 (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 (workspace/output) INTEGER LIWORK-by-1 array

Workspace that is required only if WHTSVD equals 2, 3 or 4. (See the description of WHTSVD.)

If on entry LWORK =-1 or LIWORK=-1, then the minimal length of IWORK is computed and returned in IWORK(1). See the description of LIWORK.

💀 LIWORK (input) INTEGER

The minimal length of the workspace vector IWORK.

If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1

If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N))

If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1)

If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N)

If on entry LIWORK = -1, then a workspace query is assumed and the procedure only computes the minimal and the optimal workspace lengths for both WORK and IWORK. See the descriptions of WORK and IWORK.

🚨 INFO (output) INTEGER

Here’s the tea:

  • -i < 0 :: On entry, the i-th argument had an illegal value.
  • = 0 :: Successful return.
  • = 1 :: Void input. Quick exit (M=0 or N=0).
  • = 2 :: The SVD computation of X did not converge. Suggestion: Check the input data and/or repeat with different WHTSVD.
  • = 3 :: The computation of the eigenvalues did not converge.
  • = 4 :: If data scaling was requested on input and the procedure found inconsistency in the data such that for some column index i, X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set to zero if JOBS=='C'. The computation proceeds with original or modified data and warning flag is set with INFO=4.

👑 Author

Zlatko Drmac

Definition at line 530 of file dgedmd.f90.

👑 Author (Doxygen edition)

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