*To*: mathgroup@smc.vnet.net*Subject*: [mg11032] Re: Is there a lowest Eigenvalues function around?*From*: Paul Abbott <paul@physics.uwa.edu.au>*Date*: Wed, 18 Feb 1998 20:31:33 -0500*Organization*: University of Western Australia*References*: <6c8q6s$ats@smc.vnet.net>

Christopher R. Carlen wrote: > I have to find the eigenvalues of very large matrices, ie 1024x1024 up > to over 10000x10000 . I run out of memory when trying to do more than > about 1600x1600 . I know there are algorithms to find the lowest or > highest eigenvalues of a matrix, but the Mathematica function > Eigenvalues[] finds all of them. > > Does anyone know if there is an implementation of a lowest-eigenvalues > function anywhere? I have looked aroung at www.wolfram.com but didn't > find anything. I set this as a problem for my students in their 1997 Computational Physics exam. The exam Notebook is available from http://www.pd.uwa.edu.au/Physics/Courses/Third_Year/Computational_Physics.html The largest or smallest eigenvalue of a matrix (and the corresponding eigenvector) can be computed using the power method [G H Golub and C F van Loan, Matrix Computations, Johns Hopkins Press, Baltimore, 1989]. Here is a simple implementation for finding the largest eigenvalue (and the corresponding eigenvector) using Nest. In[1]:= A = Table[Random[], {300}, {300}]; In[2]:= f[A_] := f[A] = Compile[{{q, _Real, 1}}, Module[{z = A.q}, z/Sqrt[z.z]]] Starting with In[3]:= q0 := Table[1., {Length[A]}] we Nest f a number of times to approximately obtain the eigenvector corresponding to the largest eigenvalue: In[4]:= (q = Nest[f[A], q0, 10];) //Timing//First Out[4]= 1.39 Second The largest eigenvalue is then In[5]:= q.A.q//Timing Out[5]= {0.47 Second,149.638} Compare this with In[6]:= Eigenvalues[A]//Sort//Last//Timing Out[6]= {14.55 Second,149.638} Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul@physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________