NAG Library Function Document

nag_dsterf (f08jfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dsterf (Integer n, double d[], double e[], NagError *fail)

3 Description

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the

Q R

algorithm.

The function uses an explicit shift, and, like nag_dsteqr (f08jec), 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 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrix $T$ .
Constraint: $n \geq 0$ .
2: d[ $\dim$ ] – doubleInput/Output: Note: the dimension, dim, 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 $fail . code =$ NE_CONVERGENCE (in which case see Section 6).
3: e[ $\dim$ ] – doubleInput/Output: Note: the dimension, dim, 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: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: The algorithm has failed to find all the eigenvalues after a total of $30 \times n$ iterations; $⟨value⟩$ elements of e have not converged to zero.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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 function.

10 Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix

T