naginterfaces.library.lapacklin.dgtsvx

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

dgtsvx uses the $LU$ factorization to compute the solution to a real system of linear equations

$$AX=B\text{or}{A}^{T}X=B\text{,}$$

where $A$ is a tridiagonal matrix of order $n$ and $X$ and $B$ are $n\times r$ matrices. Error bounds on the solution and a condition estimate are also provided.

For full information please refer to the NAG Library document for f07cb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07cbf.html

Parameters

factstr, length 1

Specifies whether or not the factorized form of the matrix $A$ has been supplied.

$fact=\text{'F'}$

$dlf$, $df$, $duf$, $du2$ and $ipiv$ contain the factorized form of the matrix $A$. $dlf$, $df$, $duf$, $du2$ and $ipiv$ will not be modified.

$fact=\text{'N'}$

The matrix $A$ will be copied to $dlf$, $df$ and $duf$ and factorized.

transstr, length 1

Specifies the form of the system of equations.

$trans=\text{'N'}$

$AX=B$ (No transpose).

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

$A^{T}X=B$ (Transpose).

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 $(n-1)$

The $(n-1)$ subdiagonal elements of $A$.

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

The $n$ diagonal elements of $A$.

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

The $(n-1)$ superdiagonal elements of $A$.

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

If $fact=\text{'F'}$, $dlf$ contains the $(n-1)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.

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

If $fact=\text{'F'}$, $df$ contains the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

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

If $fact=\text{'F'}$, $duf$ contains the $(n-1)$ elements of the first superdiagonal of $U$.

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

If $fact=\text{'F'}$, $du2$ contains the ($n-2$) elements of the second superdiagonal of $U$.

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

If $fact=\text{'F'}$, $ipiv$ contains the pivot indices from the $LU$ factorization of $A$.

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

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

Returns

dlffloat, ndarray, shape $(n-1)$

If $fact=\text{'N'}$, $dlf$ contains the $(n-1)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.

dffloat, ndarray, shape $\left(n\right)$

If $fact=\text{'N'}$, $df$ contains the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

duffloat, ndarray, shape $(n-1)$

If $fact=\text{'N'}$, $duf$ contains the $(n-1)$ elements of the first superdiagonal of $U$.

du2float, ndarray, shape $(n-2)$

If $fact=\text{'N'}$, $du2$ contains the ($n-2$) elements of the second superdiagonal of $U$.

ipivint, ndarray, shape $\left(n\right)$

If $fact=\text{'N'}$, $ipiv$ contains the pivot indices from the $LU$ factorization of $A$; row $i$ of the matrix was interchanged with row $ipiv[i-1]$. $ipiv[i-1]$ will always be either $i$ or $i+1$; $ipiv[i-1]=i$ indicates a row interchange was not required.

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

If the function exits successfully or $errno$ = $n$ + 1, the $n\times r$ solution matrix $X$.

rcondfloat

The estimate of the reciprocal condition number of the matrix $A$. If $rcond=0.0$, the matrix may be exactly singular. This condition is indicated by $errno$ in 1 … $n$. Otherwise, if $rcond$ is less than the machine precision, the matrix is singular to working precision. This condition is indicated by $errno$ = $n$ + 1.

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

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_j-x_j\Vert }_{\infty }/{\Vert x_j\Vert }_{\infty }\le ferr[j-1]$ 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 as reliable as the estimate for $rcond$, and is almost always a slight overestimate of the true error.

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

If the function exits successfully or $errno$ = $n$ + 1, an 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 $fact$.

Constraint: $fact=\text{'F'}$ or $\text{'N'}$.

(errno $-2$)

On entry, error in parameter $trans$.

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

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i<n)$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has not been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond=0.0$ is returned.

(errno $(i>0)\text{and}\left(n\right)$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond=0.0$ is returned.

(errno $n+1$)

$U$ is nonsingular, but $rcond$ is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of $rcond$ would suggest.

Notes

dgtsvx performs the following steps:

1. If $fact=\text{'N'}$, the $LU$ decomposition is used to factor the matrix $A$ as $A=LU$, where $L$ is a product of permutation and unit lower bidiagonal matrices and $U$ is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.

2. If some $u_{ii}=0$, so that $U$ is exactly singular, then the function returns with $\text{errno}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, $errno$ = $n$ + 1 is returned as a warning, but the function still goes on to solve for $X$ and compute error bounds as described below.

3. The system of equations is solved for $X$ using the factored form of $A$.

4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

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