Internal methods

FiniteHorizonGramians._exp_and_gram_chol_init! — Method

_exp_and_gram_init(
    A::AbstractMatrix{T},
    B::AbstractMatrix{T},
    method::ExpAndGram{T,q},
    [cache=alloc_mem(A, B, method)],
    )

Computes the matrix exponential exp(A) and the controllability Grammian

\[∫_0^1 e^{A t} B B' e^{A'*t} dt,\]

using a Legendre expansion of the matrix exponential of order q.

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FiniteHorizonGramians._exp_and_gram_double! — Method

_exp_and_gram_double!(eA, U, s, cache)

Computes s iterations of the doubling recursions:

\[Φ_{k+1} = Φ_k^2,\]

and

\[U_{k+1}' U_{k+1} = Φ_k U_k' U_k Φ_k' + U_k' U_k\]

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FiniteHorizonGramians._dims_if_compatible — Method

_dims_if_compatible(A::AbstractMatrix, B::AbstractMatrix)

Throws DimensionMismatch if A is not square or if the number of rows of B does not equal the number of columns of A. Is equivalent to size(B) if no error is thrown.

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FiniteHorizonGramians.triu2cholesky_factor! — Method

triu2cholesky_factor!(A::AbstractMatrix{T})

If A is an upper triangular matrix, it computes the product QA in-place, where Q is a unitary transform such that QA is a valid Cholesky factor. If A is not an upper triangular matrix, returns garbage.

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FiniteHorizonGramians._symmetrize! — Method

_symmetrize!(A::AbstractMatrix{T}) where {T<:Number}

Discards the skew-Hermitian part of A in-place.

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