From help-octave-request at bevo dot che dot wisc dot edu Sat Jan  4 07:05:17 2003
Subject: Re: solving Ax=b
From: A S Hodel <a dot s dot hodel at eng dot auburn dot edu>
To: Sven Khatri <skhatri at cds dot caltech dot edu>
Cc: help-octave at bevo dot che dot wisc dot edu
Date: Sat, 4 Jan 2003 07:04:21 -0600

For your problem I suspect that "easy" is a relative term (like NASA's 
goal of "low-cost" access to space).

There are numerous iterative methods for large, sparse A x = b 
problems.  Most of these will require work on your part.  See Golub and 
Van Loan, "Matrix Computations," Johns-Hopkins University press, for an 
introduction to some of them.

As a starting point, you may wish to look at the GMRES method by Youcef 
Saad (I think this is published in SIAM J. Sci. Stat. Comput. 1986).  
The octave routine krylov may be a useful starting point, but you'll 
have to modify it to use your f(A,x) instead of A*X.  If you can look at 
other splittings A = M + N besides M= I, N = (A-I), then some of Gene 
Wachspress's ADI (Alternating Direction Implicit) methods of the 60's 
may be of use.  I haven't kept up with the latest and greatest very 
large sparse solver methods, so I'd suggest a visit to the last few 
years of SIAM J. Matrix Analysis and SIAM J. Sci Stat Comput. to see 
what you can find.

On Saturday, January 4, 2003, at 03:00 AM, Sven Khatri wrote:

> Hi All,
>
> I'm looking to solve a problem that is equivalent to solving for x in
> the system of linear equations:
>   Ax = b
> where A \in R^{n \times n}, x \in R^n, b \in R^n. But the catch is
> that n is too large to explicitly construct A (even in Matlab's sparse
> implementation) but I can construct a function f so that f(x) = Ax. Is
> there an easy way (within octave) to solve for x? I can solve the 
> problem
> by introducing \tilde{A} = I - A, so that the solution to
>   x = \tilde{A}x + b
> is a solution to the above problem and \tilde{A} is a contraction and
> solving the problem iteratively but this computation is VERY slow.
>
> thanx...
> Sven
>
> PS I hope you all are comfortable with the above latex notation
>
>
>
> -------------------------------------------------------------
> Octave is freely available under the terms of the GNU GPL.
>
> Octave's home on the web:  http://www.octave.org
> How to fund new projects:  http://www.octave.org/funding.html
> Subscription information:  http://www.octave.org/archive.html
> -------------------------------------------------------------
>



-------------------------------------------------------------
Octave is freely available under the terms of the GNU GPL.

Octave's home on the web:  http://www.octave.org
How to fund new projects:  http://www.octave.org/funding.html
Subscription information:  http://www.octave.org/archive.html
-------------------------------------------------------------