naginterfaces.library.blas.deig2

naginterfaces.library.blas.deig2(a, b, c)

deig2 returns an eigenvalue of a $2\times 2$ real symmetric matrix.

For full information please refer to the NAG Library document for f06bp

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06bpf.html

Parameters

afloat

The value $a$, the $(1,1)$ element of the input matrix.

bfloat

The value $b$, the $(1,2)$ or $(2,1)$ element of the input matrix.

cfloat

The value $c$, the $(2,2)$ element of the input matrix.

Returns

evfloat

The eigenvalue.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

deig2 returns an eigenvalue of the $2\times 2$ real symmetric matrix

$$\begin{array}{c}\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)\text{,}\text{.}\end{array}$$

The result is intended for use as a shift in symmetric eigenvalue functions.

The eigenvalue is computed as

$$c-\frac{b}{f+sign\left(f\right)\times \sqrt{1+f^2}}\text{,}$$

where $f=\frac{a-c}{2b}$.

This is the eigenvalue nearer to $c$ if $a\ne c$, and is equal to $c-b$ if $a=c$.