f02 Chapter Contents
f02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_real_symm_general_eigenvalues (f02adc)

## 1  Purpose

nag_real_symm_general_eigenvalues (f02adc) calculates all the eigenvalues of $Ax=\lambda Bx$, where $A$ is a real symmetric matrix and $B$ is a real symmetric positive definite matrix.

## 2  Specification

 #include #include
 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, $B={LL}^{\mathrm{T}}$, where $L$ is lower triangular. Then $Ax=\lambda Bx$ implies $\left({L}^{-1}{AL}^{-T}\right)\left({L}^{\mathrm{T}}x\right)=\lambda \left({L}^{\mathrm{T}}x\right)$; hence the eigenvalues of $Ax=\lambda Bx$ are those of $Py=\lambda y$ where $P$ is the symmetric matrix ${L}^{-1}{AL}^{-T}$. Householder's method is used to tridiagonalise the matrix $P$ and the eigenvalues are then found using the $QL$ algorithm.

## 4  References

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

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{a}\left[{\mathbf{n}}×{\mathbf{tda}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{tda}}+j-1\right]$.
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:    $\mathbf{tda}$IntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: ${\mathbf{tda}}\ge {\mathbf{n}}$.
4:    $\mathbf{b}\left[{\mathbf{n}}×{\mathbf{tdb}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{tdb}}+j-1\right]$.
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:    $\mathbf{tdb}$IntegerInput
On entry: the stride separating matrix column elements in the array b.
Constraint: ${\mathbf{tdb}}\ge {\mathbf{n}}$.
6:    $\mathbf{r}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the eigenvalues in ascending order.
7:    $\mathbf{fail}$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, ${\mathbf{tda}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{tdb}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdb}}\ge {\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_NOT_POS_DEF
The matrix $B$ is not positive definite, possibly due to rounding errors.
NE_TOO_MANY_ITERATIONS
More than $〈\mathit{\text{value}}〉$ 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.

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 $Ax=\lambda Bx$ where $A$ is the symmetric matrix
 $0.5 1.5 6.6 4.8 1.5 6.5 16.2 8.6 6.6 16.2 37.6 9.8 4.8 8.6 9.8 -17.1$
and $B$ is the symmetric positive definite matrix
 $1 3 4 1 3 13 16 11 4 16 24 18 1 11 18 27 .$