NAG Library Routine Document

F08JFF (DSTERF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

F08JFF (DSTERF) computes all the eigenvalues of a real symmetric tridiagonal matrix.

2 Specification

SUBROUTINE F08JFF (

N, D, E, INFO)

INTEGER	N, INFO
REAL (KIND=nag_wp)	D(), E()

The routine may be called by its LAPACK name dsterf.

3 Description

F08JFF (DSTERF) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the

Q R

algorithm.

The routine uses an explicit shift, and, like F08JEF (DSTEQR), switches between the

Q R

and

Q L

variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)).

4 References

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn17.pdf

Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

5 Parameters

1: $N$ – INTEGERInput: On entry: $n$ , the order of the matrix $T$ .

Constraint: $N \geq 0$ .
2: $D (*)$ – REAL (KIND=nag_wp) arrayInput/Output: Note: the dimension of the array D must be at least $\max (1, N)$ .

On entry: the diagonal elements of the tridiagonal matrix $T$ .

On exit: the $n$ eigenvalues in ascending order, unless $INFO > 0$ (in which case see Section 6).
3: $E (*)$ – REAL (KIND=nag_wp) arrayInput/Output: Note: the dimension of the array E must be at least $\max (1, N - 1)$ .

On entry: the off-diagonal elements of the tridiagonal matrix $T$ .

On exit: E is overwritten.
4: $INFO$ – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: The algorithm has failed to find all the eigenvalues after a total of $30 \times N$ iterations. If $INFO = i$ , then on exit $i$ elements of E have not converged to zero.

7 Accuracy

The computed eigenvalues are exact for a nearby matrix

(T + E)

, where

{‖E‖}_{2} = O (ε) {‖T‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq c (n) ε {‖T‖}_{2},

where

c (n)

is a modestly increasing function of

n

8 Parallelism and Performance

Not applicable.

9 Further Comments

The total number of floating-point operations is typically about

14 n^{2}

, but depends on how rapidly the algorithm converges. The operations are all performed in scalar mode.

There is no complex analogue of this routine.

10 Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix

T