NAG Library Function Document

nag_real_symm_general_eigensystem (f02aec)

void	nag_real_symm_general_eigensystem (Integer n, double a[], Integer tda, double b[], Integer tdb, double r[], double v[], Integer tdv, 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 found using the

Q L

algorithm. An eigenvector

z

of the derived problem is related to an eigenvector

x

of the original problem by

z = L^{T} x

. The eigenvectors

z

are determined using the

Q L

algorithm and are normalized so that

z^{T} z = 1

; the eigenvectors of the original problem are then determined by solving

L^{T} x = z

, and are normalized so that

x^{T} B x = 1

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. See also Section 9
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: v[ $n \times tdv$ ] – doubleOutput: Note: the $(i, j)$ th element of the matrix $V$ is stored in $v [(i - 1) \times tdv + j - 1]$ .

On exit: the normalized eigenvectors, stored by columns; the $i$ th column corresponds to the $i$ th eigenvalue. The eigenvectors $x$ are normalized so that $x^{T} B x = 1$ . See also Section 9
8: tdv – IntegerInput: On entry: the stride separating matrix column elements in the array v.

Constraint: $tdv \geq n$ .
9: 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 $tdb \geq n$ .
On entry, $tdv = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tdv \geq n$ .
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 eigenvectors could be inaccurately determined. For a detailed error analysis see pages 310, 222 and 235 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_real_symm_general_eigensystem (f02aec) is approximately proportional to

n^{3}

The function may be called with the same actual array supplied for arguments a and v, in which case the eigenvectors will overwrite the original matrix

A

10 Example

To calculate all the eigenvalues and eigenvectors of the general symmetric eigenproblem

A x = λ B x

where

A

is the symmetric matrix

(\begin{matrix} 0.5 & 1.5 & 6.6 & 4.8 \\ 1.5 & 6.5 & 16.2 & 8.6 \\ 6.6 & 16.2 & 37.6 & 9.8 \\ 4.8 & 8.6 & 9.8 & - 17.1 \end{matrix})

and

B

is the symmetric positive definite matrix

(\begin{matrix} 1 & 3 & 4 & 1 \\ 3 & 13 & 16 & 11 \\ 4 & 16 & 24 & 18 \\ 1 & 11 & 18 & 27 \end{matrix}) .

NAG Library Function Documentnag_real_symm_general_eigensystem (f02aec)

+− Contents

1 Purpose

2 Specification