naginterfaces.library.matop.real_​mod_​chol_​perturbed_​a

naginterfaces.library.matop.real_mod_chol_perturbed_a(uplo, a, offdiag, ipiv)

real_mod_chol_perturbed_a computes the positive definite perturbed matrix $A+E$ from the factors of a modified Cholesky factorization of a real symmetric matrix, $A$.

For full information please refer to the NAG Library document for f01me

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01mef.html

Parameters

uplostr, length 1

Indicates whether the upper or lower triangular part of $A$ was stored and how it was factorized.

$uplo=\text{'U'}$

The upper triangular part of $A$ was stored and we compute $A+E$ such that $P^T(A+E)P=UD{U}^T$.

$uplo=\text{'L'}$

The lower triangular part of $A$ was stored and we compute $A+E$ such that $P^T(A+E)P=LD{L}^T$.

afloat, array-like, shape $(n,n)$

The modified Cholesky factor of $A$, as returned by real_modified_cholesky().

offdiagfloat, array-like, shape $\left(n\right)$

The array $\text{offdiag}$ as returned by real_modified_cholesky().

ipivint, array-like, shape $\left(n\right)$

The array $\text{ipiv}$ as returned by real_modified_cholesky().

Returns

afloat, ndarray, shape $(n,n)$

If $uplo=\text{'U'}$, the upper triangular part of $A+E$ is returned and the elements of the array below the diagonal are not referenced.

If $uplo=\text{'L'}$, the lower triangular part of $A+E$ is returned and the elements of the array above the diagonal are not referenced.

Raises

NagValueError

(errno $1$)

On entry, $uplo=⟨value⟩$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

Notes

real_mod_chol_perturbed_a computes the positive definite perturbed matrix $A+E$ from the factors provided by a previous call to real_modified_cholesky(). For a symmetric, possibly indefinite matrix $A$, real_modified_cholesky() finds the Cheng–Higham modified Cholesky factorization

$$P^T(A+E)P=LD{L}^T\text{,}$$

when $uplo=\text{'L'}$. Here $L$ is a unit lower triangular matrix, $P$ is a permutation matrix, $D$ is a symmetric block diagonal matrix (with blocks of order $1$ or $2$). The matrix $E$ is not explicitly formed.

If $uplo=\text{'U'}$, we compute $A+E$ from the factorization $P^T(A+E)P=UD{U}^T$, where $U$ is a unit upper triangular matrix.

References

Cheng, S H and Higham, N J, 1998, A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J. Matrix Anal. Appl. (19(4)), 1097–1110