naginterfaces.library.lapacklin.zgtsv

naginterfaces.library.lapacklin.zgtsv(dl, d, du, b)

zgtsv computes the solution to a complex system of linear equations

$$AX=B\text{,}$$

where $A$ is an $n\times n$ tridiagonal matrix and $X$ and $B$ are $n\times r$ matrices.

For full information please refer to the NAG Library document for f07cn

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07cnf.html

Parameters

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

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

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

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

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

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

bcomplex, array-like, shape $(n,\text{nrhs})$

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

Returns

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

If no constraints are violated, $dl$ is overwritten by the ($n-2$) elements of the second superdiagonal of the upper triangular matrix $U$ from the $LU$ factorization of $A$, in $dl\left[0\right],dl\left[1\right],\dots ,dl[n-3]$.

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

If no constraints are violated, $d$ is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

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

If no constraints are violated, $du$ is overwritten by the $(n-1)$ elements of the first superdiagonal of $U$.

bcomplex, ndarray, shape $(n,\text{nrhs})$

If no exception or warning is raised, the $n\times r$ solution matrix $X$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $n\ge 0$.

(errno $-2$)

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

Constraint: $\text{nrhs}\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless $n=⟨value⟩$.

Notes

zgtsv uses Gaussian elimination with partial pivoting and row interchanges to solve the equations $AX=B$. The matrix $A$ is factorized as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element per column, and $U$ is an upper triangular band matrix, with two superdiagonals.

Note that the equations $A^{T}X=B$ may be solved by interchanging the order of the arguments $du$ and $dl$.

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