F01LEF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F01LEF computes an LU factorization of a real tridiagonal matrix, using Gaussian elimination with partial pivoting.

2  Specification

REAL (KIND=nag_wp)  A(N), LAMBDA, B(N), C(N), TOL, D(N)

3  Description

The matrix T-λI, where T is a real n by n tridiagonal matrix, is factorized as
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-λI is nearly singular is returned in the nth element of the array IPIV. If it is important that T-λI is nonsingular, as is usually the case when solving a system of tridiagonal equations, then it is strongly recommended that IPIVn is inspected on return from F01LEF. (See the parameter IPIV and Section 8 for further details.)
The parameter λ is included in the routine so that F01LEF may be used, in conjunction with F04LEF, to obtain eigenvectors of T by inverse iteration.

4  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

5  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix T.
Constraint: N1.
2:     A(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the diagonal elements of T.
On exit: the diagonal elements of the upper triangular matrix U.
3:     LAMBDA – REAL (KIND=nag_wp)Input
On entry: the scalar λ. F01LEF factorizes T-λI.
4:     B(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the superdiagonal elements of T, stored in B2 to Bn; B1 is not used.
On exit: the elements of the first superdiagonal of U, stored in B2 to Bn.
5:     C(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the subdiagonal elements of T, stored in C2 to Cn; C1 is not used.
On exit: the subdiagonal elements of L, stored in C2 to Cn.
6:     TOL – REAL (KIND=nag_wp)Input
On entry: a relative tolerance used to indicate whether or not the matrix (T-λI) 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 5×10-4. See Section 8 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.
7:     D(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the elements of the second superdiagonal of U, stored in D3 to Dn; D1 and D2 are not used.
8:     IPIV(N) – INTEGER arrayOutput
On exit: details of the permutation matrix P. If an interchange occurred at the kth step of the elimination, then IPIVk=1, otherwise IPIVk=0. If a diagonal element of U is small, indicating that T-λI is nearly singular, then the element IPIVn is returned as positive. Otherwise IPIVn is returned as 0. See Section 8 for further details. If the application is such that it is important that T-λI is not nearly singular, then it is strongly recommended that IPIVn is inspected on return.
9:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry,N<1.

7  Accuracy

The computed factorization will satisfy the equation
E1 9×maxij lij,lij2 ε T-λ I1
where ε is the machine precision.

8  Further Comments

The time taken by F01LEF is approximately proportional to n.
The factorization of a tridiagonal matrix proceeds in n-1 steps, each step eliminating one subdiagonal element of the tridiagonal matrix. In order to avoid small pivot elements and to prevent growth in the size of the elements of L, rows k and (k+1) will, if necessary, be interchanged at the kth step prior to the elimination.
The element IPIVn returns the smallest integer, j, for which
where T-λIj1 denotes the sum of the absolute values of the jth row of the matrix (T-λI). If no such j exists, then IPIVn is returned as zero. If such a j exists, then ujj is small and hence (T-λI) is singular or nearly singular.
This routine may be followed by F04LEF to solve systems of tridiagonal equations. If you wish to solve single systems of tridiagonal equations you should be aware of F07CAF (DGTSV), which solves tridiagonal systems with a single call. F07CAF (DGTSV) requires less storage and will generally be faster than the combination of F01LEF and F04LEF, but no test for near singularity is included in F07CAF (DGTSV) and so it should only be used when the equations are known to be nonsingular.

9  Example

This example factorizes the tridiagonal matrix T where
T= 3.0 2.1 0 0 0 3.4 2.3 -1.0 0 0 0 3.6 -5.0 1.9 0 0 0 7.0 -0.9 8.0 0 0 0 -6.0 7.1
and then to solve the equations Tx=y, where
y= 2.7 -0.5 2.6 0.6 2.7
by a call to F04LEF. The example program sets TOL=5×10-5 and, of course, sets LAMBDA=0.

9.1  Program Text

Program Text (f01lefe.f90)

9.2  Program Data

Program Data (f01lefe.d)

9.3  Program Results

Program Results (f01lefe.r)

F01LEF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012