LANBPRO Lanczos bidiagonalization with partial reorthogonalization.

LANBPRO computes the Lanczos bidiagonalization of a rectangular matrix
using partial reorthogonalization.

[U_k,B_k,V_k,R,ierr,work] = LANBPRO(A,K,R0,OPTIONS,U_old,B_old,V_old) 
[U_k,B_k,V_k,R,ierr,work] = LANBPRO('Afun','Atransfun',M,N,K,R0, ...
                                    OPTIONS,U_old,B_old,V_old) 

Computes K steps of the Lanczos bidiagonalization algorithm with
partial reorthogonalization (BPRO) on the M-by-N matrix A with M-by-1
starting vector R0, producing a lower bidiagonal K-by-K matrix B_k, an
N-by-K matrix V_k, an M-by-K matrix U_k and a M-by-1 vector R such
that
     A*V_k = U_k*B_k + R
Partial reorthogonalization is used to keep the columns of V_K and U_k
semiorthogonal:
      MAX(MAX(ABS(EYE(K) - V_K'*V_K))) <= OPTIONS.delta 
and 
      MAX(MAX(ABS(EYE(K) - U_K'*U_K))) <= OPTIONS.delta.

B

B_k = LANBPRO(...) returns the bidiagonal matrix only.

The first input argument is either a real 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 linear operator to the columns of a given matrix,  
and the third and fourth arguments must be M and N, the dimensions 
of then problem.

The OPTIONS structure is used to control the reorthogonalization:
  OPTIONS.delta:  Desired level of orthogonality 
                  (default = sqrt(eps/K)).
  OPTIONS.eta  :  Level of orthogonality after reorthogonalization 
                  (default = eps^(3/4)/sqrt(K)).
  OPTIONS.cgs  :  Flag for switching between different reorthogonalization
                  algorithms:
                   0 = iterated modified Gram-Schmidt  (default)
                   1 = iterated classical Gram-Schmidt 
  OPTIONS.elr  :  If OPTIONS.elr = 1 (default) then extended local
                  reorthogonalization is enforced.
  OPTIONS.onesided
               :  If OPTIONS.onesided = 0 (default) then both the left
                  (U) and right (V) Lanczos vectors are kept 
                  semiorthogonal. 
                  OPTIONS.onesided = 1 then only the columns of U are
                  are reorthogonalized.
                  OPTIONS.onesided = -1 then only the columns of V are
                  are reorthogonalized.
  OPTIONS.waitbar
               :  The progress of the algorithm is display graphically.

If both R0, U_old, B_old, and V_old are provided, they must
contain a partial Lanczos bidiagonalization of A on the form

     A V_old = U_old B_old + R0 .  

In this case the factorization is extended to dimension K x K by
continuing the Lanczos bidiagonalization algorithm with R0 as a 
starting vector.

The output array work contains information about the work used in
reorthogonalizing the u- and v-vectors.
   work = [ RU  PU ]
          [ RV  PV ] 
where
   RU = Number of reorthogonalizations of U.
   PU = Number of inner products used in reorthogonalizing U.
   RV = Number of reorthogonalizations of V.
   PV = Number of inner products used in reorthogonalizing V.

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

G. H. Golub & C. F. Van Loan, "Matrix Computations",
3. Ed., Johns Hopkins, 1996.  Section 9.3.4.

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, Stanford University, 2000