naginterfaces.library.lapacklin.dptrfs

naginterfaces.library.lapacklin.dptrfs(nrhs, d, e, df, ef, b, x)

dptrfs computes error bounds and refines the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ symmetric positive definite tridiagonal matrix and $X$ and $B$ are $n\times r$ matrices, using the modified Cholesky factorization returned by dpttrf() and an initial solution returned by dpttrs(). Iterative refinement is used to reduce the backward error as much as possible.

For full information please refer to the NAG Library document for f07jh

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f07/f07jhf.html

Parameters

nrhsint

$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.

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

Must contain the $n$ diagonal elements of the matrix of $A$.

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

Must contain the $(n-1)$ subdiagonal elements of the matrix $A$.

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

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

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

Must contain the $(n-1)$ subdiagonal elements of the unit bidiagonal matrix $L$ from the $LD{L}^T$ factorization of $A$.

bfloat, array-like, shape $(n,nrhs)$

The $n\times r$ matrix of right-hand sides $B$.

xfloat, array-like, shape $(n,nrhs)$

The $n\times r$ initial solution matrix $X$.

Returns

xfloat, ndarray, shape $(n,nrhs)$

The $n\times r$ refined solution matrix $X$.

ferrfloat, ndarray, shape $\left(nrhs\right)$

Estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_j-x_j\Vert }_{\infty }/{\Vert {\hat{x}}_j\Vert }_{\infty }\le ferr[j-1]$, where ${\hat{x}}_j$ is the $j$th column of the computed solution returned in the array $x$ and $x_j$ is the corresponding column of the exact solution $X$. The estimate is almost always a slight overestimate of the true error.

berrfloat, ndarray, shape $\left(nrhs\right)$

Estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_j$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_j$ an exact solution).

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-2$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Notes

dptrfs should normally be preceded by calls to dpttrf() and dpttrs(). dpttrf() 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. dpttrs() then utilizes the factorization to compute a solution, $\hat{X}$, to the required equations. Letting $\hat{x}$ denote a column of $\hat{X}$, dptrfs computes a component-wise backward error, $\beta$, the smallest relative perturbation in each element of $A$ and $b$ such that $\hat{x}$ is the exact solution of a perturbed system

$$(A+E)\hat{x}=b+f\text{, with}\left|e_{ij}\right|\le \beta \left|a_{ij}\right|\text{, and}\left|f_j\right|\le \beta \left|b_j\right|\text{.}$$

The function also estimates a bound for the component-wise forward error in the computed solution defined by $max|x_i-\widehat{x_i}|/max\left|\widehat{x_i}\right|$, where $x$ is the corresponding column of the exact solution, $X$.

Note that the modified Cholesky factorization of $A$ can also be expressed as

$$A=U^{T}DU\text{,}$$

where $U$ is unit upper bidiagonal.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug