expand_entries()

static QuadraticOperator.expand_entries(Hc)

Given \(\tilde{\H}\in\RR^{a \times r(r+1)/2}\), construct the matrix \(\Hhat\in\RR^{a\times r^2}\) such that \(\Hhat[\qhat\otimes\qhat] = \tilde{\H}[\qhat\,\hat{\otimes}\,\qhat]\) for all \(\qhat\in\RR^{r}\) where \(\hat{\otimes}\) is the compressed Kronecker product (see ckron()).

Parameters:

Hc(a, r(r+1)/2) ndarray

Matrix that acts on the compressed Kronecker product.

Returns:

H(a, r^2) ndarray

Matrix that acts on the full Kronecker product. This matrix is “symmetric” in the sense that H.reshape((a, r, r))[i] is symmetric for i = 0, …, r.

Examples

>>> from opinf.operators import QuadraticOperator
>>> r = 20
>>> Htilde = np.random.random((r, r * (r + 1) / 2))
>>> Htilde.shape
(20, 210)
>>> H = QuadraticOperator.expand_entries(Htilde)
>>> H.shape
(20, 400)
>>> q = np.random.random(r)
>>> Hq2 = H @ np.kron(q, q)
>>> np.allclose(Hq2, Htilde @ QuadraticOperator.ckron(q))
True
>>> np.all(QuadraticOperator.compress_entries(G) == Gtilde)
True