Minimizing the norm related with iteration method

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

• It would help if you gave some idea of what problem you are solving? Clearly you can choose $X_{k+1} = X_k$, and this will minimize the norm, but I am sure there are other conditions involved which you haven't stated. May 25, 2012 at 5:22
• @copper.hat i have edited sir May 25, 2012 at 6:07
• What is the question? How to choose $\beta$? May 25, 2012 at 6:12
• Sorry, what is the question? Are you asking how to analyze the sequence $X_k$? May 25, 2012 at 6:15
• @copper.hat sorry to trouble you. is there any alternate way to choose beta or initial approximation so that my condition of convergence remains the same. May 25, 2012 at 6:20

As far as I can see, your main concern is due to the large number of steps that your method needs to converge. Hence, you want to use a better initial approximation $X_0$ to arrive at the convergence phase faster.
It should be noted that the main point lies in the iterative methods that you are using, i.e. $X_{k+1}=X_k+\beta X_k(I-AX_k)$, where $\beta$ is a free parameter. This is only the dampled Newton's method on the matrix equation $AX=I$. On the other hand, when you are dealing with matrix inversion, there is no need to use the dampled Newton method, since although it has a free parameter, its convergecne order is mostly LINEARLY. Hence, the best way is to use the quadratical method by choosing $\beta=1$, which results in the well-known Schulz method $$X_{k+1}=X_k(2I-AX_k).$$ Before proceeding, I want to state that low order methods (even the above second-order method) require too much steps to arrive at the convergence phase. Hence I finally would like to suggest you to use the cubically method of Chebyshev given below, or any of the other high order methods (of orders 7,8,9,10) to have a better feedback: $$X_{k+1}=X_k(3I-AX_k(3I-AX_k)).$$
$$X_0=\frac{2}{\sigma_1^2+\sigma_r^2}A^*,$$
wherein $\sigma_1$ and $\sigma_r$ stand for the largest and the smallest singular values of the input matrix $A$.