I am working on a iteration method to compute the generalized inverse of a matrix $A$ of rank $r$
Iteration method is
$X_{k+1} = X_{k} + \beta X_{k} (I - A X_{k}) $
where notations are as follows
$A$ is given $n\times n$ matrix
$\beta$ is any non zero scalar such that $ 0< \beta\leq 1 $
and my initial approximation is $X_{0} = \beta A ^t$ and my condition for convergent is as follows
$X_{k}$ are sequence of approximations to compute generalized inverse say $X$of a matrix $A$. Though my method is convergent but number of iterations it is taking is higher than usual one. My condition of convergence if $\max_{1\leq i\leq r} | 1 - \lambda_i (\beta AA^t) |<1$. $\lambda$ denotes eigenvalue of matrix.
Can anybody suggest me any other value of $X_{0}$ and $\beta $? Any kind of help or hint will be helpful to me. Please pardon me if my question is inappropriate for this community.
I have edited my question Thanks