naginterfaces.library.matop.real_​gen_​tridiag_​lu

naginterfaces.library.matop.real_gen_tridiag_lu(a, lamda, b, c, tol)

real_gen_tridiag_lu computes an $LU$ factorization of a real tridiagonal matrix, using Gaussian elimination with partial pivoting.

For full information please refer to the NAG Library document for f01le

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01lef.html

Parameters

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

The diagonal elements of $T$.

lamdafloat

The scalar $\lambda$. real_gen_tridiag_lu factorizes $T-\lambda I$.

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

The superdiagonal elements of $T$, stored in $b\left[1\right]$ to $b[n-1]$; $b\left[0\right]$ is not used.

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

The subdiagonal elements of $T$, stored in $c\left[1\right]$ to $c[n-1]$; $c\left[0\right]$ is not used.

tolfloat

A relative tolerance used to indicate whether or not the matrix ($T-\lambda I$) is nearly singular. $tol$ should normally be chosen as approximately the largest relative error in the elements of $T$. For example, if the elements of $T$ are correct to about $4$ significant figures, then $tol$ should be set to about $5\times {10}^{-4}$. See Further Comments for further details on how $tol$ is used. If $tol$ is supplied as less than $ϵ$, where $ϵ$ is the machine precision, then the value $ϵ$ is used in place of $tol$.

Returns

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

The diagonal elements of the upper triangular matrix $U$.

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

The elements of the first superdiagonal of $U$, stored in $b\left[1\right]$ to $b[n-1]$.

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

The subdiagonal elements of $L$, stored in $c\left[1\right]$ to $c[n-1]$.

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

The elements of the second superdiagonal of $U$, stored in $d\left[2\right]$ to $d[n-1]$; $d\left[0\right]$ and $d\left[1\right]$ are not used.

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

Details of the permutation matrix $P$. If an interchange occurred at the $k$th step of the elimination, then $ipiv[k-1]=1$, otherwise $ipiv[k-1]=0$. If a diagonal element of $U$ is small, indicating that $(T-\lambda I)$ is nearly singular, then the element $ipiv[n-1]$ is returned as positive. Otherwise $ipiv[n-1]$ is returned as $0$. See Further Comments for further details. If the application is such that it is important that $(T-\lambda I)$ is not nearly singular, then it is strongly recommended that $ipiv[n-1]$ is inspected on return.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The matrix $T-\lambda I$, where $T$ is a real $n\times n$ tridiagonal matrix, is factorized as

$$T-\lambda I=PLU\text{,}$$

where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix with at most one nonzero subdiagonal element per column, and $U$ is an upper triangular matrix with at most two nonzero superdiagonal elements per column.

The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling.

An indication of whether or not the matrix $T-\lambda I$ is nearly singular is returned in the $n$th element of the array $ipiv$. If it is important that $T-\lambda I$ is nonsingular, as is usually the case when solving a system of tridiagonal equations, then it is strongly recommended that $ipiv[n-1]$ is inspected on return from real_gen_tridiag_lu. (See the argument $ipiv$ and Further Comments for further details.)

The argument $\lambda$ is included in the function so that real_gen_tridiag_lu may be used, in conjunction with linsys.real_tridiag_fac_solve, to obtain eigenvectors of $T$ by inverse iteration.

References

Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag