naginterfaces.library.linsys.real_​tridiag_​fac_​solve

naginterfaces.library.linsys.real_tridiag_fac_solve(job, a, b, c, d, ipiv, y, tol)

real_tridiag_fac_solve solves a system of tridiagonal equations following the factorization by matop.real_gen_tridiag_lu. This function is intended for applications such as inverse iteration as well as straightforward linear equation applications.

For full information please refer to the NAG Library document for f04le

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f04/f04lef.html

Parameters

jobint

Must specify the equations to be solved.

$job=1$

The equations $(T-\lambda I)x=y$ are to be solved, but diagonal elements of $U$ are not to be perturbed.

$job=-1$

The equations $(T-\lambda I)x=y$ are to be solved and, if overflow would otherwise occur, diagonal elements of $U$ are to be perturbed. See argument $tol$.

$job=2$

The equations ${(T-\lambda I)}^{T}x=y$ are to be solved, but diagonal elements of $U$ are not to be perturbed.

$job=-2$

The equations ${(T-\lambda I)}^{T}x=y$ are to be solved and, if overflow would otherwise occur, diagonal elements of $U$ are to be perturbed. See argument $tol$.

$job=3$

The equations $Ux=y$ are to be solved, but diagonal elements of $U$ are not to be perturbed.

$job=-3$

The equations $Ux=y$ are to be solved and, if overflow would otherwise occur, diagonal elements of $U$ are to be perturbed. See argument $tol$.

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

The diagonal elements of $U$ as returned by real_tridiag_fac_solve.

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

The elements of the first superdiagonal of $U$ as returned by real_tridiag_fac_solve.

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

The subdiagonal elements of $L$ as returned by real_tridiag_fac_solve.

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

The elements of the second superdiagonal of $U$ as returned by real_tridiag_fac_solve.

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

Details of the matrix $P$ as returned by matop.real_gen_tridiag_lu.

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

The right-hand side vector $y$.

tolfloat

The minimum perturbation to be made to very small diagonal elements of $U$. $tol$ is only referenced when $job$ is negative. $tol$ should normally be chosen as about $ϵ\parallel U\parallel$, where $ϵ$ is the machine precision, but if $tol$ is supplied as non-positive, it is reset to $ϵmax\left(\left|u_{ij}\right|\right)$.

Returns

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

$y$ is overwritten by the solution vector $x$.

tolfloat

If on entry $tol$ is non-positive, it is reset as just described. Otherwise $tol$ is unchanged.

Raises

NagValueError

(errno $i>1$)

Overflow would occur computing the element $I$ in the solution. $I=⟨value⟩$.

(errno $1$)

On entry, $job=⟨value⟩$.

Constraint: $job\ne 0$ and $job\ge -3$ and $job\le 3$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

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

Following the factorization of the $n\times n$ tridiagonal matrix $(T-\lambda I)$ as

$$T-\lambda I=PLU$$

by matop.real_gen_tridiag_lu, real_tridiag_fac_solve may be used to solve any of the equations

$$(T-\lambda I)x=y\text{,}{(T-\lambda I)}^{T}x=y\text{,}Ux=y$$

for $x$, the choice of equation being controlled by the argument $job$. In each case there is an option to perturb zero or very small diagonal elements of $U$, this option being intended for use in applications such as 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