- DynamicModeDecomposition
- DynamicModeDecompositionQR
DynamicModeDecompositionQR
Compute the Dynamic Mode Decomposition (DMD) for a pair of data snapshot matrices, using a QR factorization based compression of the data. Lapack function GEDMDQ. For the input matrices X and Y such that Y = A*X with an unaccessible matrix A, GEDMDQ 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, GEDMDQ 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.
The input M-by-N matrix F. 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.
Computing for type matrix<double>
bool matrix::DynamicModeDecompositionQR(
|
Computing for type matrix<float>
bool matrixf::DynamicModeDecompositionQR(
|
Computing for type matrix<complex>
bool matrixc::DynamicModeDecompositionQR(
|
Computing for type matrix<complexf>
bool matrixcf::DynamicModeDecompositionQR(
|
Parameters
jobs
[in] Value from the ENUM_DMD_SCALE enumeration which 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.
jobz
[in] Value from the ENUM_DMDQ_EIGV enumeration which determines whether the eigenvectors (Koopman modes) will be computed.
jobr
[in] Value from the ENUM_DMD_RESIDUALS enumeration which determines whether to compute the residuals.
jobq
[in] Value from the ENUM_DMDQ_Q enumeration which specifies whether to explicitly compute and return the unitary matrix from the QR factorization of the data snapshot matrix.
jobt
[in] Value from the ENUM_DMDQ_R enumeration which specifies whether to return the upper triangular factor from the QR factorization of the data snapshot matrix.
jobf
[in] Value from the ENUM_DMD_REFINE enumeration which specifies whether to store information needed for post-processing (e.g. computing refined Ritz vectors).
whtsvd
[in] Value from the ENUM_SVD_ALG enumeration which allows for a selection of the SVD algorithm from the LAPACK library.
nrnk
[in] Value determines the mode how to compute the numerical rank, i.e. how to truncate small singular values of the input matrix X. 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.
tol
[in] The tolerance for truncating small singular values. 0 <= TOL < 1
eigen_values
[out] Vector of eigenvalues of size K. The leading K (K<N) entries of EIGS contain the computed eigenvalues (Ritz values).
Q
[out] Orthogonal matrix/factor of the QR factorization of the initial data snapshots matrix F.
R key
[out] N-by-N upper triangular factor from the QR factorization of the data snapshot matrix F.
left_vectors
[out] Matrix of left singular vectors. The leading K columns of X contain the leading K left singular vectors of the hold representations of the leading N-1 snapshots in the orthonormal basis computed in the QR factorization of F. 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.
Z
[out] M-by-K matrix
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 GEDMD in X) and the second factor (the eigenvectors of the Rayleigh quotient) is in the matrix V, as returned by GEDMD. 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.
residuals
[out] Residuals for the K computed Ritz pairs.
B
[out] N-by-K matrix (K<N).
If JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can be used for computing the refined vectors.
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.
If JOBF =='N', then B is not referenced.
V
[out] K-by-K matrix (K<N). Contains the the K eigenvectors of the Rayleigh quotient. The Ritz vectors (returned in Z) are the product of Q from the initial QR factorization.
S
[out] K-by-K matrix (K<N). The array S(1:K,1:K) is used for the matrix Rayleigh quotient. This content is overwritten during the eigenvalue decomposition by GEEV.
Return Value
The function returns 'true' on success or 'false' if an error occurs.
Note
The number of matrices rows (M) must not be less than the number of columns (N).
ENUM_DMD_SCALE
An enumeration that determines whether the initial data snapshots are scaled by a diagonal matrix.
ID |
Description |
|---|---|
DMDSCALE_S |
'S': The data snapshots matrices X and Y are multiplied with a diagonal matrix D so that X*D has unit nonzero columns |
DMDSCALE_C |
'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 |
DMDSCALE_Y |
'Y': The data snapshots matrices X and Y are multiplied with a diagonal matrix D so that Y*D has unit nonzero columns |
DMDSCALE_N |
'N': No data scaling |
ENUM_DMDQ_EIGV
An enumeration which determines whether the eigenvectors (Koopman modes) will be computed.
ID |
Description |
|---|---|
DMDQEIGV_V |
'V': The eigenvectors (Koopman modes) will be computed |
DMDQEIGV_F |
'F': The eigenvectors (Koopman modes) will be returned in factored form as the product Z*V |
DMDQEIGV_Q |
'Q': The eigenvectors (Koopman modes) will be returned in factored form as the product Q*Z |
DMDQEIGV_N |
'N': The eigenvectors are not computed |
ENUM_DMD_RESIDUALS
An enumeration which determines whether to compute the residuals.
ID |
Description |
|---|---|
DMDRESIDUALS_R |
'R': The residuals for the computed eigenpairs will be computed |
DMDRESIDUALS_N |
'N': The residuals are not computed |
An enumeration which specifies whether to explicitly compute and return the unitary matrix from the QR factorization.
ID |
Description |
|---|---|
DMDQQ_Q |
'Q': The matrix Q of the QR factorization of the data snapshot matrix is computed |
DMDQQ_N |
'N': The matrix Q is not explicitly computed |
An enumeration whichspecifies whether to return the upper triangular factor from the QR factorization.
ID |
Description |
|---|---|
DMDQR_R |
'R': The matrix R of the QR factorization of the data snapshot matrix is computed |
DMDQR_N |
'N': The matrix R is not explicitly computed |
ENUM_DMD_REFINE
An enumeration which specifies whether to store information needed for post-processing.
ID |
Description |
|---|---|
DMDREFINE_R |
'R': The matrix needed for the refinement of the Ritz vectors is computed and stored in the array B |
DMDREFINE_E |
'E': The unscaled eigenvectors of the Exact DMD are computed and returned in the array B |
DMDREFINE_N |
'N': No eigenvector refinement data is computed |
ENUM_SVD_ALG
An enumeration selecting the SVD algorythm.