DynamicModeDecomposition

Compute the Dynamic Mode Decomposition(по-русски "разложение по динамическим модам") (DMD) for a pair of data snapshot matrices. Lapack function GEDMD. For the input matrices X and Y such that Y = A*X with an unaccessible matrix A, GEDMD 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, GEDMD 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. Both matrices must be the same size.

Computing for type matrix<double>

bool  matrix::DynamicModeDecomposition(
  matrix&             B,                      // second snapshot matrix in the pair
  ENUM_DMD_SCALE       jobs,                    // determines whether the initial data snapshots are scaled by a diagonal matrix
  ENUM_DMD_EIGV       jobz,                    // determines whether the eigenvectors (Koopman modes) will be computed
  ENUM_DMD_RESIDUALS   jobr,                    // determines whether to compute the residuals
  ENUM_DMD_REFINE     jobf,                    // specifies whether to store information needed for post-processing
  ENUM_SVD_ALG         whtsvd,                  // allows for a selection of the SVD algorithm from the LAPACK library
  long                 nrnk,                    // determines the mode how to compute the numerical rank
  double             tol,                    // the tolerance for truncating small singular values
  vectorc&             eigen_values,            // vector of eigenvalues (Ritz values)
  matrix&               left_vectors,            // matrix of left singular vectors
  matrix&               Z                        // matrix of Ritz vectors
  vector&             residuals,               // residuals for the Ritz pairs
  matrix&             res_vectors,             // residual vectors for the Ritz pairs
  matrix&             B,                      // computed vectors
  matrix&             W,                      // eigenvectors of the matrix Rayleigh quotient
  matrix&             S                        // computed vectors
  );

Computing for type matrix<float>

bool  matrixf::DynamicModeDecomposition(
  matrixf&             B,                      // second snapshot matrix in the pair
  ENUM_DMD_SCALE       jobs,                    // determines whether the initial data snapshots are scaled by a diagonal matrix
  ENUM_DMD_EIGV       jobz,                    // determines whether the eigenvectors (Koopman modes) will be computed
  ENUM_DMD_RESIDUALS   jobr,                    // determines whether to compute the residuals
  ENUM_DMD_REFINE     jobf,                    // specifies whether to store information needed for post-processing
  ENUM_SVD_ALG         whtsvd,                  // allows for a selection of the SVD algorithm from the LAPACK library
  long                 nrnk,                    // determines the mode how to compute the numerical rank
  float               tol,                    // the tolerance for truncating small singular values
  vectorcf&           eigen_values,            // vector of eigenvalues (Ritz values)
  matrixf&             left_vectors,            // matrix of left singular vectors
  matrixf&             Z                        // matrix of Ritz vectors
  vectorf&             residuals,               // residuals for the Ritz pairs
  matrixf&             res_vectors,             // residual vectors for the Ritz pairs
  matrixf&           B,                      // computed vectors
  matrixf&           W,                      // eigenvectors of the matrix Rayleigh quotient
  matrixf&           S                        // computed vectors
  );

Computing for type matrix<complex>

bool  matrixc::DynamicModeDecomposition(
  matrixc&           B,                      // second snapshot matrix in the pair
  ENUM_DMD_SCALE       jobs,                    // determines whether the initial data snapshots are scaled by a diagonal matrix
  ENUM_DMD_EIGV       jobz,                    // determines whether the eigenvectors (Koopman modes) will be computed
  ENUM_DMD_RESIDUALS   jobr,                    // determines whether to compute the residuals
  ENUM_DMD_REFINE     jobf,                    // specifies whether to store information needed for post-processing
  ENUM_SVD_ALG         whtsvd,                  // allows for a selection of the SVD algorithm from the LAPACK library
  long                 nrnk,                    // determines the mode how to compute the numerical rank
  double             tol,                    // the tolerance for truncating small singular values
  vectorc&             eigen_values,            // vector of eigenvalues (Ritz values)
  matrixc&             left_vectors,            // matrix of left singular vectors
  matrixc&             Z                        // matrix of Ritz vectors
  vector&             residuals,               // residuals for the Ritz pairs
  matrixc&             res_vectors,             // residual vectors for the Ritz pairs
  matrixc&             B,                      // computed vectors
  matrixc&             W,                      // eigenvectors of the matrix Rayleigh quotient
  matrixc&             S                        // computed vectors
  );

Computing for type matrix<complexf>

bool  matrixcf::DynamicModeDecomposition(
  matrixcf&           B,                      // second snapshot matrix in the pair
  ENUM_DMD_SCALE       jobs,                    // determines whether the initial data snapshots are scaled by a diagonal matrix
  ENUM_DMD_EIGV       jobz,                    // determines whether the eigenvectors (Koopman modes) will be computed
  ENUM_DMD_RESIDUALS   jobr,                    // determines whether to compute the residuals
  ENUM_DMD_REFINE     jobf,                    // specifies whether to store information needed for post-processing
  ENUM_SVD_ALG         whtsvd,                  // allows for a selection of the SVD algorithm from the LAPACK library
  long                 nrnk,                    // determines the mode how to compute the numerical rank
  float               tol,                    // the tolerance for truncating small singular values
  vectorcf&           eigen_values,            // vector of eigenvalues (Ritz values)
  matrixcf&             left_vectors,            // matrix of left singular vectors
  matrixcf&             Z                        // matrix of Ritz vectors
  vectorf&             residuals,               // residuals for the Ritz pairs
  matrixcf&           res_vectors,             // residual vectors for the Ritz pairs
  matrixcf&           B,                      // computed vectors
  matrixcf&           W,                      // eigenvectors of the matrix Rayleigh quotient
  matrixcf&           S                        // computed vectors
  );

Parameters

Y

[in]  The second snapshot matrix in the pair.

jobs

[in]  Value from the ENUM_DMD_SCALE enumeration which determines whether the initial data snapshots are scaled by a diagonal matrix.

jobz

[in]   Value from the ENUM_DMD_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.

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 :: 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).

left_vectors

[out] Matrix of left singular vectors. The leading K columns contain a POD basis (POD - Proper Orthogonal Decomposition, правильная ортогональная декомпозиция), i.e. the leading K left singular vectors of the input data matrix X  All N columns of contain all left singular vectors of the input matrix X.

Z

[out] Matrix of Ritz vectors. 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.

residuals

[out]  Residuals for the K computed Ritz pairs.

res_vectors

[out]  Residual vectors for the K computed Ritz pairs.

B

[out]  M-by-K matrix (K<=N).

        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.

W

[out]  K-by-K matrix (K<=N). 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.

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_DMD_EIGV

An enumeration which determines whether the eigenvectors (Koopman modes) will be computed.

ID

Description

DMDEIGV_V

'V': The eigenvectors (Koopman modes) will be computed

DMDEIGV_F

'F': The eigenvectors (Koopman modes) will be returned in factored form

DMDEIGV_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

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.

ID

Description

SVDALG_1

1: GESVD (the QR SVD algorithm)

SVDALG_2

2: GESDD (the Divide and Conquer algorithm)

SVDALG_3

3: GESVDQ (the preconditioned QR SVD; this and 4 are the most accurate options)

SVDALG_4

4: GEJSV (the preconditioned Jacobi SVD; this and 3 are the most accurate options)

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