Different Inverse Approach As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[m,n]$ and $b$ is $[1,n]$ matrix. These all inverse methods inverse multiplying proccess.
I want to use another inverse method that can inverse explorasial or logarithm proccess.
For example, How can I solve linear equation when $log (A, x)=b.$ I am searching log-inverse method.Is there any method like this in the literature, or can somebody help me to advance ?
Thanks for answers.
 A: Although I'm still a bit unsure as to what the question actually is, I think the following may help clarify some things.
First, I should mention there are some great books that treat matrix functions:


*

*Functions of Matrices by Nicholas Higham

*Matrix Analysis by Roger Horn and Charles Johnson

*Theory of Matrices by Peter Lancaster

*Matrix Computations by Gene Golub and Charles Van Loan


Function of Matrices is the most recent and most comprehensive; the other three contain proofs that Higham cites, but matrix functions are not their main focus.
Secondly, you have to consider how a scalar function like $\log(z)$ can be generalized to take a matrix argument.  For polynomials like $x^2 + 1$ and square matrices, this is very natural: $A^2 + I$.  But for things like $\log(z)$, just as you would take branch cuts for complex arguments $z$, you would have to do something similar for a matrix $A$, and this is where things get messy.  Also, matrix products don't make sense for rectangular $A$.  So, you would need to expand the definition of a matrix function somehow to include these idiosyncrasies.
For rectangular $A$, the easiest way to extend the definition is by adding rows of zeros to $A$ to make it square.  Then matrix function definitions apply more naturally.  (For the Jordan definition, you would not need to do this, since it can handle rectangular matrices; however, it appears that Matlab's algorithms are based off the polynomial definitions, which require square arguments.)
As for whether there's a special method for inverses of $\log(A)$, you would have to explore the scalar function first.  The scalar exponential has lots of great properties, many of which transfer nicely to the matrix exponential; in particular, its inverse relation $(e^z)^{-1} = e^{-z}$.  But $\log(z)$ is not so special.  Perhaps you could make use of a series expansion or Taylor polynomial, but it is known that these aren't very accurate for computing matrix functions, unless the condition number of $A$ is small.  It appears Matlab actually uses Padé approximants, which are great, but difficult to come up with.  A Cauchy integral representation of $\frac{1}{\log(z)}$ may also be useful, but then you have to choose the right curve as well as the right numerical quadrature.
You may also want to consider Krylov subspace techniques, if your matrix $A$ is large and sparse.  This is an area of active research.  For example, see http://www-ai.math.uni-wuppertal.de/SciComp/en/Publications/70_2014.php, which investigates restarted Arnoldi's method.
