Transfer Residual Errors to Backward Errors.

It can be shown that the computed eigenvalue and eigenvector(s) are the exact ones of a nearby matrix pair, i.e.,

$$\wtd\beta (A+E)\wtd x=\wtd\alpha (B+F)\wtd x,$$

and

$$\wtd\beta \wtd y^{\ast} (A+E)=\wtd\alpha\wtd y^{\ast} (B+F)$$

if $\wtd y$ is available, where error matrices $E$ and $F$ are small relative to the norms of $A$ and $B$.

1. Only $\wtd x$ is available but $\wtd y$ is not. Then the optimal error matrix $(E,F)$ (in both the 2-norm and the Frobenius norm) for which $(\wtd\alpha,\wtd\beta)$ and $\wtd x$ are an exact eigenvalue and its corresponding eigenvector of the pair $(A+E,B+F)$ satisfies
   

   $$\Vert(E,F)\Vert _2=\Vert(E,F)\Vert _{F}= \Vert r\Vert _2.$$ (241)

   
   

2. Both $\wtd x$ and $\wtd y$ are available. Then the optimal error matrices $(E_2,F_2)$ (in the 2-norm) and $(E_{F},F_{F})$
   
   (in the Frobenius norm) for which $(\wtd\alpha,\wtd\beta)$, $\wtd x$, and $\wtd y$ are an exact eigenvalue and its corresponding eigenvectors of the pair $\{A+E, B+F\}$ satisfy
   

   $$\Vert(E_2,F_2)\Vert _2= %%\frac{1}{\sqrt{\vert\wtd\alpha\ve... ...eta\vert^2}} \max\{\Vert r\Vert _2,\Vert s^{\ast}\Vert _2\}$$ (242)

   
   
   and
   

   $$\Vert(E_{F},F_{F})\Vert _{F} = %%\frac{1}{\sqrt{\vert\wtd\... ...eta\wtd y^{\ast} A\wtd x-\wtd\alpha\wtd y^{\ast} B\wtd x)^2}.$$ (243)

   
   

See [256,431,473].

We say the algorithm that delivers the approximate eigenpair $((\wtd\alpha,\wtd\beta),\wtd x)$ is $\tau$-backward stable for the pair with respect to the norm $\Vert\cdot\Vert$ if it is an exact eigenpair for $\{A+E, B+F\}$ with $\Vert(E,F)\Vert\le\tau$; analogously the algorithm that delivers the eigentriplet $((\wtd\alpha,\wtd\beta),\wtd x,\wtd y)$ is $\tau$-backward stable for the triplet with respect to the norm $\Vert\cdot\Vert$ if it is an exact eigentriplet for $\{A+E, B+F\}$ with $\Vert(E,F)\Vert\le\tau$. With these in mind, statements can be made about the backward stability of the algorithm which computes the eigenpair $((\wtd\alpha,\wtd\beta),\wtd x)$ or the eigentriplet $((\wtd\alpha,\wtd\beta),\wtd x,\wtd y)$. Conventionally, an algorithm is called backward stable if $\tau = O(\epsilon_M \Vert(A,B)\Vert)$.