naginterfaces.library.lapacklin.dgttrf

naginterfaces.library.lapacklin.dgttrf(n, dl, d, du)

dgttrf computes the $LU$ factorization of a real $n\times n$ tridiagonal matrix $A$.

For full information please refer to the NAG Library document for f07cd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07cdf.html

Parameters

nint

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

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 $A$.

Returns

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

Is overwritten by the $(n-1)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.

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

Is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

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

Is overwritten by the $(n-1)$ elements of the first superdiagonal of $U$.

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

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

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

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Notes

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.

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