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 https://www.netlib.org/lapack/lawnspdf/lawn17.pdf

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

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the order of the matrix $T$ .

Constraint: $n \geq 0$ .
2: $d (*)$ – Real (Kind=nag_wp) array Input/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) array Input/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: $info$ – Integer Output: 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; $〈value〉$ 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

f08jff is not threaded in any implementation.

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

, where

T = (\begin{array}{r} - 6.99 & - 0.44 & 0.00 & 0.00 \\ - 0.44 & 7.92 & - 2.63 & 0.00 \\ 0.00 & - 2.63 & 2.34 & - 1.18 \\ 0.00 & 0.00 & - 1.18 & 0.32 \end{array}) .

Interfaces: FL CL AD

NAG FL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08jf: FL CL AD

NAG FL Interfacef08jff (dsterf)

▸▿ 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

NAG FL Interface
f08jff (dsterf)