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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

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 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.

9 Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix

T