NAG Library Function Document

nag_real_eigensystem (f02agc)

is first balanced and then reduced to upper Hessenberg form using real stabilised elementary similarity transformations. The eigenvalues and eigenvectors of the Hessenberg matrix are calculated using the

Q R

algorithm. The eigenvectors of the Hessenberg matrix are back-transformed to give the eigenvectors of the original matrix

A

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 matrix $A$ .
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 $n$ by $n$ matrix $A$ .

On exit: a is overwritten.
3: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
4: r[n] – ComplexOutput: On exit: the eigenvalues.
5: v[ $n \times tdv$ ] – ComplexOutput: Note: the $(i, j)$ th element of the matrix $V$ is stored in $v [(i - 1) \times tdv + j - 1]$ .

On exit: the eigenvectors, stored by columns. The $i$ th column corresponds to the $i$ th eigenvalue. The eigenvectors are normalized so that the sum of the squares of the moduli of the elements is equal to 1 and the element of largest modulus is real. This ensures that real eigenvalues have real eigenvectors.
6: tdv – IntegerInput: On entry: the stride separating matrix column elements in the array v.

Constraint: $tdv \geq n$ .
7: iter[n] – IntegerOutput: On exit: $iter [i - 1]$ contains the number of iterations used to find the $i$ th eigenvalue. If $iter [i - 1]$ is negative, the $i$ th eigenvalue is the second of a pair found simultaneously.
Note: the eigenvalues are found in reverse order, starting with the $n$ th.
8: 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, $tdv = ⟨value⟩$ 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_TOO_MANY_ITERATIONS: More than $⟨value⟩$ iterations are required to isolate all the eigenvalues.

7 Accuracy

The accuracy of the results depends on the original matrix and the multiplicity of the roots. For a detailed error analysis see pages 352 and 390 Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_real_eigensystem (f02agc) is approximately proportional to

n^{3}

10 Example

To calculate all the eigenvalues and eigenvectors of the real matrix

(\begin{matrix} 1.5 & 0.1 & 4.5 & - 1.5 \\ - 22.5 & 3.5 & 12.5 & - 2.5 \\ - 2.5 & 0.3 & 4.5 & - 2.5 \\ - 2.5 & 0.1 & 4.5 & 2.5 \end{matrix}) .

NAG Library Function Documentnag_real_eigensystem (f02agc)

+− Contents

1 Purpose

2 Specification

3 Description