helpers

Functions

block_view(A, b)
Q_from_VanLoan(→ tinygp.helpers.JAXArray)	Compute the process noise covariance via the Van Loan method.
Phibar_from_VanLoan(→ tinygp.helpers.JAXArray)	Compute the integrated transition matrix via the Van Loan method.
VanLoan(→ dict[str, tinygp.helpers.JAXArray])	Compute all submatrices of the Van Loan matrix exponential.

Module Contents

smolgp.helpers.block_view(A, b)

smolgp.helpers.Q_from_VanLoan(F: tinygp.helpers.JAXArray, L: tinygp.helpers.JAXArray, Qc: tinygp.helpers.JAXArray, dt: tinygp.helpers.JAXArray) → tinygp.helpers.JAXArray

Compute the process noise covariance via the Van Loan method.

Evaluates

\[Q_k = \int_0^{\Delta t} e^{F(\Delta t - s)}\, L\, Q_c\, L^T\, e^{F^T(\Delta t - s)}\, ds\]

See Van Loan (1978), “Computing Integrals Involving the Matrix Exponential” (PDF).

Parameters:

• F – Feedback (design) matrix \(F\) from design_matrix().

• L – Noise effect matrix \(L\) from noise_effect_matrix().

• Qc – Spectral density \(Q_c\) from noise().

• dt – Time step \(\Delta t = X_2 - X_1\).

Returns:

Process noise covariance matrix \(Q_k\) over time step \(\Delta t\).

smolgp.helpers.Phibar_from_VanLoan(F: tinygp.helpers.JAXArray, dt: tinygp.helpers.JAXArray) → tinygp.helpers.JAXArray

Compute the integrated transition matrix via the Van Loan method.

Evaluates

\[\bar{\Phi} = \int_0^{\Delta t} e^{F s}\, ds\]

See Van Loan (1978), “Computing Integrals Involving the Matrix Exponential” (PDF).

Parameters:

• F – Feedback (design) matrix \(F\) from design_matrix().

• dt – Time step \(\Delta t = X_2 - X_1\).

Returns:

Integrated transition matrix \(\bar{\Phi}\) over time step \(\Delta t\).

smolgp.helpers.VanLoan(F: tinygp.helpers.JAXArray, L: tinygp.helpers.JAXArray, Qc: tinygp.helpers.JAXArray, dt: tinygp.helpers.JAXArray) → dict[str, tinygp.helpers.JAXArray]

Compute all submatrices of the Van Loan matrix exponential.

Assembles the block matrix \(C\) and returns its matrix exponential, partitioned into the submatrices F1-F4, G1-G3, H1-H2, K1 (see Van Loan 1978 for notation), from which various integrals such as Q_from_VanLoan() and Phibar_from_VanLoan() can be derived.

See Van Loan (1978), “Computing Integrals Involving the Matrix Exponential” (PDF).

Parameters:

• F – Feedback (design) matrix \(F\).

• L – Noise effect matrix \(L\).

• Qc – Spectral density \(Q_c\).

• dt – Time step \(\Delta t = X_2 - X_1\).

Returns:

Dictionary of named submatrices of the Van Loan exponential.