naginterfaces.library.matop.real_​symm_​posdef_​inv

naginterfaces.library.matop.real_symm_posdef_inv(a)

real_symm_posdef_inv calculates the accurate inverse of a real symmetric positive definite matrix, using a Cholesky factorization and iterative refinement.

For full information please refer to the NAG Library document for f01ab

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01abf.html

Parameters

afloat, array-like, shape $(n+1,n)$

The upper triangle of the $n\times n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.

Returns

afloat, ndarray, shape $(n+1,n)$

The lower triangle of the inverse matrix $X$ is stored in the elements of the array below the diagonal, in rows $2$ to $n+1$; $x_{ij}$ is stored in $a[i,j-1]$ for $i\ge j$. The upper triangle of the original matrix is unchanged.

bfloat, ndarray, shape $(n,n)$

The lower triangle of the inverse matrix $X$, with $x_{ij}$ stored in $b[i-1,j-1]$, for $i\ge j$.

Raises

NagValueError

(errno $1$)

The matrix $A$ is not positive definite, possibly due to rounding errors.

(errno $2$)

The refinement process failed to converge. The matrix $A$ is ill-conditioned.

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

To compute the inverse $X$ of a real symmetric positive definite matrix $A$, real_symm_posdef_inv first computes a Cholesky factorization of $A$ as $A=L{L}^T$, where $L$ is lower triangular. An approximation to $X$ is found by computing $L^{-1}$ and then the product $L^{-T}{L}^{-1}$. The residual matrix $R=I-AX$ is calculated using additional precision, and a correction $D$ to $X$ is found by solving $L{L}^{T}D=R$. $X$ is replaced by $X+D$, and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.

References

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag