lansvd.txt

LANSVD  Compute a few singular values and singular vectors.
   LANSVD computes singular triplets (u,v,sigma) such that
   A*u = sigma*v and  A'*v = sigma*u. Only a few singular values 
   and singular vectors are computed  using the Lanczos 
   bidiagonalization algorithm with partial reorthogonalization (BPRO). 

   S = LANSVD(A) 
   S = LANSVD('Afun','Atransfun',M,N)  

   The first input argument is either a  matrix or a
   string containing the name of an M-file which applies a linear
   operator to the columns of a given matrix.  In the latter case,
   the second input must be the name of an M-file which applies the
   transpose of the same operator to the columns of a given matrix,  
   and the third and fourth arguments must be M and N, the dimensions 
   of the problem.

   [U,S,V] = LANSVD(A,K,'L',...) computes the K largest singular values.

   [U,S,V] = LANSVD(A,K,'S',...) computes the K smallest singular values.

   The full calling sequence is

   [U,S,V] = LANSVD(A,K,SIGMA,OPTIONS) 
   [U,S,V] = LANSVD('Afun','Atransfun',M,N,K,SIGMA,OPTIONS)

   where K is the number of singular values desired and 
   SIGMA is 'L' or 'S'.

   The OPTIONS structure specifies certain parameters in the algorithm.
    Field name      Parameter                              Default
   
    OPTIONS.tol     Convergence tolerance                  16*eps
    OPTIONS.lanmax  Dimension of the Lanczos basis.
    OPTIONS.p0      Starting vector for the Lanczos        rand(n,1)-0.5
                    iteration.
    OPTIONS.delta   Level of orthogonality among the       sqrt(eps/K)
                    Lanczos vectors.
    OPTIONS.eta     Level of orthogonality after           10*eps^(3/4)
                    reorthogonalization. 
    OPTIONS.cgs     reorthogonalization method used        0
                    '0' : iterated modified Gram-Schmidt 
                    '1' : iterated classical Gram-Schmidt
    OPTIONS.elr     If equal to 1 then extended local      1
                    reorthogonalization is enforced. 

   See also LANBPRO, SVDS, SVD

 References: 
 R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.

 B. N. Parlett, ``The Symmetric Eigenvalue Problem'', 
 Prentice-Hall, Englewood Cliffs, NJ, 1980.

 H. D. Simon, ``The Lanczos algorithm with partial reorthogonalization'',
 Math. Comp. 42 (1984), no. 165, 115--142.

 Rasmus Munk Larsen, DAIMI, 1998