nitrogen.linalg¶

Linear algebra routines, including iterative eigensolvers. This module uses the SciPy LinearOperator class to represent matrix-vector product routines.

class nitrogen.linalg.RealOrthogonalProjector(*args, **kwargs)¶

Bases: scipy.sparse.linalg.interface.LinearOperator

A projection operator to orthogonalize against a given set of vectors

\[\hat{P} = 1 - U U^T\]

Parameters: U ((N,n) ndarray) – n column vectors to orthogonalize against. The columns of U are assumed to be real and orthonormal.

nitrogen.linalg.aslinearoperator(A)¶

Return A as a LinearOperator.

‘A’ may be any of the following types:

ndarray
matrix
sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
LinearOperator
An object with .shape and .matvec attributes

See the LinearOperator documentation for additional information.

Notes

If ‘A’ has no .dtype attribute, the data type is determined by calling LinearOperator.matvec() - set the .dtype attribute to prevent this call upon the linear operator creation.

Examples

>>> from scipy.sparse.linalg import aslinearoperator
>>> M = np.array([[1,2,3],[4,5,6]], dtype=np.int32)
>>> aslinearoperator(M)
<2x3 MatrixLinearOperator with dtype=int32>

nitrogen.linalg.bounds(H, k=10, m=3)¶

Estimate lower and upper bounds to the spectral range of the Hermitian operator H.

Parameters

H (LinearOperator) – A Hermitian operator.
k (int, optional) – The number of Lanczos steps to take before estimating the bounds. The default is 10.
m (int, optional) – The safe-guard level for the bound estimate. This must be less than or equal to k. The default is 3.

Returns

low (float) – An estimate of the lower bound of the smallest eigenvalue.
high (float) – An estimate of the upper bound of the largest eigenvalue.

Notes

These spectral bound estimates are based the work of Y. Zhou and R.-C. Li [ZL11]. Their bounds (2.6), (2.7), and (2.8) correspond to m = 1, k, and 3, respectively.

In general, a larger value of m will provide a safer, but looser, bound on the spectral range.

References

ZL11: Y. Zhou and R.-C. Li, “Bound the spectrum of large Hermitian matrices.” Linear Algebra and its Applications, 435, 480-493 (2011). https://doi.org/10.1016/j.laa.2010.06.034

nitrogen.linalg.chebauto(H, K, v0=None)¶

Calculate the auto-correlation function of a wavepacket via the Chebyshev propagator of of a Hermitian operator H.

Parameters

H (LinearOperator) – A Hermitian operator.
K (int) – The number of propagation steps.
v0 (ndarray, optional) – The initial vector. If None, a uniform vector is used. The default is None.

Returns

C (ndarray) – A (2*K+1,) array containing the auto-correlation function of v0 in the Chebyshev order domain.
scale ((float, float)) – The mean energy and half-width, (ebar, de) used to scale the Hamiltonian. The unscaled energy is related to the Chebyshev angle parameter as E = de * cos(theta) + ebar.

nitrogen.linalg.packed¶

Packed-storage matrices and routines. The lower triangle is stored in row-major order. (For symmetric matrices, this is equivalent to upper triangle column-major order. For Hermitian matrices, this is the conjugate of the upper triangle in column-major order.)

nitrogen.linalg.packed.IJ2k(I, J)¶: Assuming a symmetric array, calculate the 1D packed storage index for the (I,J) = (J,I) element

nitrogen.linalg.packed.chol_sp(A, out=None)¶

Cholesky decomposition of a symmetric matrix in packed format. If real symmetric, then A should be positive definite. If complex symmetric (not Hermitian), then H should have non-zero pivots.

Parameters

A ((np, ...) ndarray) – A is stored in 1D packed format (see nitrogen.linalg.packed)
out ((np, ...) ndarray) – Output buffer. If None, this will be created. If out = H, then in-place decomposition is performed

Returns

out – The lower triangle Cholesky decomposition L in packed storage. H = L @ L.T

Return type

ndarray

nitrogen.linalg.packed.inv_sp(A, out=None)¶

Inverse of a real symmetric, positive definite matrix in packed format. (Complex symmetric matrices may not be defined.)

Parameters

A ((np, ...) ndarray) – The packed storage array.
out ((np, ...) ndarray) – Output buffer. If None, this will be created. If out = A, then in-place inversion is performed

Returns

out – The result.

Return type

ndarray

nitrogen.linalg.packed.inv_tp(L, out=None)¶

Invert a triangular matrix in lower row-packed (or upper column-packed) storage.

Parameters

L ((np, ...) ndarray) – L is stored in 1D packed format (see nitrogen.linalg.packed)
out ((np, ...) ndarray) – Output buffer. If None, this will be created. If out = L, then in-place inversion is performed

Returns

out – The inverse of the triangular matrix in packed storage.

Return type

ndarray

nitrogen.linalg.packed.k2IJ(k)¶

Calculate the full 2D index (I,`J`) for a given packed 1D index k. The returned index is always in the lower triangle.

Parameters: k (int) – The packed 1D index.
Returns: I,J – The unpacked 2D index.
Return type: np.uint64

nitrogen.linalg.packed.llt_tp(L, out=None)¶

L @ L.T of a lower triangular matrix.

Parameters

L ((np, ...) ndarray) – Lower triangular matrix in packed storage.
out ((np, ...) ndarray) – Output buffer. If None, this will be created. If out = L, then in-place multiplication is performed

Returns

out – The symmetric result in packed storage.

Return type

ndarray

nitrogen.linalg.packed.ltl_tp(L, out=None)¶

L.T @ L with a lower triangular matrix.

Parameters

L ((np, ...) ndarray) – Lower triangular matrix in packed storage.
out ((np, ...) ndarray) – Output buffer. If None, this will be created. If out = L, then in-place multiplication is performed

Returns

out – The symmetric result in packed storage.

Return type

ndarray

nitrogen.linalg.packed.n2N(n)¶

Calculate the square matrix rank N for a packed storage size n

Parameters: n (int) – The packed length.
Returns: N – The matrix rank.
Return type: np.uint64

nitrogen.linalg.packed.symfull(P)¶

Return the full array for a symmetric array in packed storage

Parameters: P (ndarray) – The lower triangle of a symmetric array in packed storage.
Returns: The full symmetric matrix
Return type: ndarray

nitrogen.linalg.packed.trAB_sp(A, B, out=None)¶

The trace of A @ B, in symmetric packed stroage.

Parameters

A ((np, ...) ndarray) – Symmetric matrix in packed storage.
B ((np, ...) ndarray) – Symmetric matrix in packed storage.
out ((...) ndarray, optional) – Output buffer. If None, this will be created. If scalar, this is ignored.

Returns

out – The result.

Return type

ndarray or scalar

nitrogen.linalg.packed.trilfull(L)¶

Return the full array for a lower triangle array. in packed storage

Parameters: L (ndarray) – The lower triangle in packed storage
Returns: The full lower triangle array.
Return type: ndarray

nitrogen.linalg¶

nitrogen.linalg.packed¶

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