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: $d [\dim]$ – double Input/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: $e [\dim]$ – double Input/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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

f08jfc 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 function.

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}) .

f08jf: FL CL CPP AD

NAG CL Interfacef08jfc (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 CL Interface
f08jfc (dsterf)