modified_cholesky

modified_cholesky(mat, *, error_threshold=1e-08, max_vecs=None)

Modified Cholesky decomposition.

The modified Cholesky decomposition of a square matrix \(M\) has the form

\[M = \sum_{i=1}^N v_i v_i^\dagger\]

where each \(v_i\) is a vector. \(M\) must be positive definite. No checking is performed to verify whether \(M\) is positive definite. The number of terms \(N\) in the decomposition depends on the allowed error threshold. A larger error threshold may yield a smaller number of terms. Furthermore, the max_vecs parameter specifies an optional upper bound on \(N\). The max_vecs parameter is always respected, so if it is too small, then the error of the decomposition may exceed the specified error threshold.

Warning

No checking is performed to verify whether the input matrix is positive definite. If the input matrix is not positive definite, then the decomposition returned will be invalid.

References

• Ab initio computations of molecular systems by the auxiliary-field quantum Monte Carlo method

• Simplifications in the generation and transformation of two-electron integrals in molecular calculations

Parameters:

• mat (np.ndarray) – The matrix to decompose.

• error_threshold (float) – Threshold for allowed error in the decomposition. The error is defined as the maximum absolute difference between an element of the original tensor and the corresponding element of the reconstructed tensor.

• max_vecs (int | None) – The maximum number of vectors to include in the decomposition.

Returns:

The Cholesky vectors v_i assembled into a 2-dimensional Numpy array whose columns are the vectors.

Return type:

np.ndarray