naginterfaces.library.lapacklin.dgtrfs

naginterfaces.library.lapacklin.dgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, x)

dgtrfs computes error bounds and refines the solution to a real system of linear equations $AX=B$ or $A^{T}X=B$, where $A$ is an $n\times n$ tridiagonal matrix and $X$ and $B$ are $n\times r$ matrices, using the $LU$ factorization returned by dgttrf() and an initial solution returned by dgttrs(). Iterative refinement is used to reduce the backward error as much as possible.

For full information please refer to the NAG Library document for f07ch

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07chf.html

Parameters

transstr, length 1

Specifies the equations to be solved as follows:

$trans=\text{'N'}$

Solve $AX=B$ for $X$.

$trans=\text{'T'}$ or $\text{'C'}$

Solve $A^{T}X=B$ for $X$.

nint

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

nrhsint

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

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

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

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

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

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

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

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

Must contain the $(n-1)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.

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

Must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

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

Must contain the $(n-1)$ elements of the first superdiagonal of $U$.

du2float, array-like, shape $(n-2)$

Must contain the $(n-2)$ elements of the second superdiagonal of $U$.

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

Must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row $ipiv[i-1]$, and $ipiv[i-1]$ must always be either $i$ or $(i+1)$, $ipiv[i-1]=i$ indicating that a row interchange was not performed.

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 $trans$.

Constraint: $trans=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

(errno $-2$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Notes

dgtrfs should normally be preceded by calls to dgttrf() and dgttrs(). dgttrf() uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as

$$A=PLU\text{,}$$

where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. dgttrs() then utilizes the factorization to compute a solution, $\hat{X}$, to the required equations. Letting $\hat{x}$ denote a column of $\hat{X}$, dgtrfs 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, $X$.

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