naginterfaces.library.lapacklin.dptcon

naginterfaces.library.lapacklin.dptcon(n, d, e, anorm)

dptcon computes the reciprocal condition number of a real $n\times n$ symmetric positive definite tridiagonal matrix $A$, using the $LD{L}^T$ factorization returned by dpttrf().

For full information please refer to the NAG Library document for f07jg

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07jgf.html

Parameters

nint

$n$, the order of the matrix $A$.

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

Must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^T$ factorization of $A$.

efloat, array-like, shape $(n-1)$

Must contain the $(n-1)$ subdiagonal elements of the unit lower bidiagonal matrix $L$. ($e$ can also be regarded as the superdiagonal of the unit upper bidiagonal matrix $U$ from the $U^{T}DU$ factorization of $A$.)

anormfloat

The $1$-norm of the original matrix $A$, which may be computed by calling blas.dlanst with its argument $\text{norm}=\text{'1'}$. $anorm$ must be computed either before calling dpttrf() or else from a copy of the original matrix $A$.

Returns

rcondfloat

The reciprocal condition number, $1/{\kappa }_1\left(A\right)=1/\left({\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\right)$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $anorm$.

Constraint: $anorm\ge 0.0$.

Notes

dptcon should be preceded by a call to dpttrf(), which computes a modified Cholesky factorization of the matrix $A$ as

$$A=LD{L}^T\text{,}$$

where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. dptcon then utilizes the factorization to compute ${\Vert A^{-1}\Vert }_1$ by a direct method, from which the reciprocal of the condition number of $A$, $1/\kappa \left(A\right)$ is computed as

$$1/{\kappa }_1\left(A\right)=1/\left({\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\right)\text{.}$$

$1/\kappa \left(A\right)$ is returned, rather than $\kappa \left(A\right)$, since when $A$ is singular $\kappa \left(A\right)$ is infinite.

References

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia