NAG Library Function Document

nag_real_symm_general_eigenvalues (f02adc)

+− 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_real_symm_general_eigenvalues (f02adc) calculates all the eigenvalues of

A x = λ B x

, where

A

is a real symmetric matrix and

B

is a real symmetric positive definite matrix.

2 Specification

#include <nag.h>

#include <nagf02.h>

void	nag_real_symm_general_eigenvalues (Integer n, double a[], Integer tda, double b[], Integer tdb, double r[], NagError *fail)

3 Description

The problem is reduced to the standard symmetric eigenproblem using Cholesky's method to decompose

B

into triangular matrices,

B = {L L}^{T}

, where

L

is lower triangular. Then

A x = λ B x

implies

(L^{- 1} {A L}^{- T}) (L^{T} x) = λ (L^{T} x)

; hence the eigenvalues of

A x = λ B x

are those of

P y = λ y

where

P

is the symmetric matrix

L^{- 1} {A L}^{- T}

. Householder's method is used to tridiagonalise the matrix

P

and the eigenvalues are then found using the

Q L

algorithm.

4 References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrices $A$ and $B$ .
Constraint: $n \geq 1$ .
2: a[ $n \times tda$ ] – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the upper triangle of the $n$ by $n$ symmetric matrix $A$ . The elements of the array below the diagonal need not be set.

On exit: the lower triangle of the array is overwritten. The rest of the array is unchanged.
3: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
4: b[ $n \times tdb$ ] – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $B$ is stored in $b [(i - 1) \times tdb + j - 1]$ .

On entry: the upper triangle of the $n$ by $n$ symmetric positive definite matrix $B$ . The elements of the array below the diagonal need not be set.

On exit: the elements below the diagonal are overwritten. The rest of the array is unchanged.
5: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq n$ .
6: r[n] – doubleOutput: On exit: the eigenvalues in ascending order.
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
On entry, $tdb = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tdb \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
NE_NOT_POS_DEF: The matrix $B$ is not positive definite, possibly due to rounding errors.
NE_TOO_MANY_ITERATIONS: More than $⟨value⟩$ iterations are required to isolate all the eigenvalues.

7 Accuracy

In general this function is very accurate. However, if

B

is ill-conditioned with respect to inversion, the eigenvalues could be inaccurately determined. For a detailed error analysis see pages 310, 222 and 235 Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_real_symm_general_eigenvalues (f02adc) is approximately proportional to

n^{3}

10 Example

To calculate all the eigenvalues of the general symmetric eigenproblem

A x = λ B x