quimb.linalg.autoblock

Numba accelerated functions for finding charge sectors and subselecting submatrices.

Module Contents

Functions

get_nz(A)
compute_blocks(ix, jx, d)	Find the charge sectors (blocks in matrix terms) given element
subselect(A, p)	Select only the intersection of rows and columns of A matching the
subselect_set(A, B, p)	Set only the intersection of rows and colums of A matching the
_eigh_autoblocked(A[, sort])
_eigvalsh_autoblocked(A[, sort])
eigensystem_autoblocked(A[, sort, return_vecs, isherm])	Perform Hermitian eigen-decomposition, automatically identifying and

quimb.linalg.autoblock.get_nz(A)

quimb.linalg.autoblock.compute_blocks(ix, jx, d)

Find the charge sectors (blocks in matrix terms) given element coordinates ix and jx and total size d.

Parameters:

• ix (array of int) – The row coordinates of non-zero elements.

• jx (array of int) – The column coordinates of non-zero elements.

• d (int) – The total size of the operator.

Returns:

sectors – The list of charge sectors. Each element is itself a sorted list of the basis numbers that make up that sector. The permutation that would block diagonalize the operator is then np.concatenate(sectors).

Return type:

list[list[int]]

Examples

>>> H = ham_hubbard_hardcore(4, sparse=True)
>>> ix, jx = H.nonzero()
>>> d = H.shape[0]
>>> sectors = compute_blocks(ix, jx, d)
>>> sectors
[[0], [1, 2, 4, 8], [3, 5, 6, 9, 10, 12], [7, 11, 13, 14], [15]]

quimb.linalg.autoblock.subselect(A, p)

Select only the intersection of rows and columns of A matching the basis indices p. Faster than double numpy slicing.

Parameters:

• A (2D-array) – Dense matrix to select from.

• p (sequence of int) – The basis indices to select.

Returns:

B – The matrix, of size (len(p), len(p)).

Return type:

2D-array

Examples

>>> A = np.arange(25).reshape(5, 5)
>>> A
array([[ 0,  1,  2,  3,  4],
       [ 5,  6,  7,  8,  9],
       [10, 11, 12, 13, 14],
       [15, 16, 17, 18, 19],
       [20, 21, 22, 23, 24]])

>>> subselect(A, [1, 3])
array([[ 6,  8],
       [16, 18]])

quimb.linalg.autoblock.subselect_set(A, B, p)

Set only the intersection of rows and colums of A matching the basis indices p to B.

Parameters:

• A (array with shape (d, d)) – The matrix to set elements in.

• B (array with shape (dp, dp)) – The matrix to set elements from.

• p (sequence of size dp) – The basis indices.

Examples

>>> A = np.zeros((5, 5))
>>> B = np.random.randn(3, 3)
>>> p = [0, 2, 4]
>>> subselect_set(A, B, p)
array([[-0.31888218,  0.        ,  0.39293245,  0.        ,  0.21822712],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.66674486,  0.        ,  1.03388035,  0.        ,  1.7319345 ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
       [-0.94542733,  0.        , -0.37211882,  0.        ,  0.51951555]])

quimb.linalg.autoblock._eigh_autoblocked(A, sort=True)

quimb.linalg.autoblock._eigvalsh_autoblocked(A, sort=True)

quimb.linalg.autoblock.eigensystem_autoblocked(A, sort=True, return_vecs=True, isherm=True)

Perform Hermitian eigen-decomposition, automatically identifying and exploiting symmetries appearing in the current basis as block diagonals formed via permutation of rows and columns. The whole process is accelerated using numba.

Parameters:

• A (array_like) – The operator to eigen-decompose.

• sort (bool, optional) – Whether to sort into ascending order, default True.

• isherm (bool, optional) – Whether A is hermitian, default True.

• return_vecs (bool, optional) – Whether to return the eigenvectors, default True.

Returns:

• evals (1D-array) – The eigenvalues.

• evecs (qarray) – If return_vecs=True, the eigenvectors.