NAG Library Function Document

nag_real_symm_eigensystem (f02abc)

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

On entry: the lower triangle of the $n$ by $n$ symmetric matrix $A$ . The elements of the array above the diagonal need not be set. See also Section 9
3: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
4: r[n] – doubleOutput: On exit: the eigenvalues in ascending order.
5: 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 are normalized so that the sum of squares of the elements is equal to 1.
6: tdv – IntegerInput: On entry: the stride separating matrix column elements in the array v.

Constraint: $tdv \geq n$ .
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, $tdv = ⟨value⟩$ 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 eigenvectors are always accurately orthogonal but the accuracy of the individual eigenvectors is dependent on their inherent sensitivity to changes in the original matrix. For a detailed error analysis see pages 222 and 235 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_real_symm_eigensystem (f02abc) 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.

10 Example

To calculate all the eigenvalues and eigenvectors of the real symmetric matrix

(\begin{matrix} 0.5 & 0.0 & 2.3 & - 2.6 \\ 0.0 & 0.5 & - 1.4 & - 0.7 \\ 2.3 & - 1.4 & 0.5 & 0.0 \\ - 2.6 & - 0.7 & 0.0 & 0.5 \end{matrix}) .

NAG Library Function Documentnag_real_symm_eigensystem (f02abc)

+− 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 Library Function Document

nag_real_symm_eigensystem (f02abc)