LANPRO Lanczos tridiagonalization with partial reorthogonalization
LANPRO computes the Lanczos tridiagonalization of a real symmetric
matrix using the symmetric Lanczos algorithm with partial
reorthogonalization.
[Q_K,T_K,R,ANORM,IERR,WORK] = LANPRO(A,K,R0,OPTIONS,Q_old,T_old)
[Q_K,T_K,R,ANORM,IERR,WORK] = LANPRO('Afun',N,K,R0,OPTIONS,Q_old,T_old)
Computes K steps of the Lanczos algorithm with starting vector R0,
and returns the K x K tridiagonal T_K, the N x K matrix Q_K
with semiorthonormal columns and the residual vector R such that
A*Q_K = Q_K*T_K + R .
Partial reorthogonalization is used to keep the columns of Q_K
semiorthogonal:
MAX(MAX(ABS(EYE(k) - Q_K'*Q_K))) <= OPTIONS.delta.
The first input argument is either a real symmetric matrix, a struct with
components A.L and A.U 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 argument must be N, the order of the problem.
If A is a struct with components A.L and A.U, such that
L*U = (A - sigma*I), a shift-and-invert Lanczos iteration is performed
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.Y : The lanczos vectors are reorthogonalized against
the columns of the matrix OPTIONS.Y.
If both R0, Q_old and T_old are provided, they must contain
a partial Lanczos tridiagonalization of A on the form
A Q_old = Q_old T_old + R0 .
In this case the factorization is extended to dimension K x K by
continuing the Lanczos algorithm with R0 as starting vector.
On exit ANORM contains an approximation to ||A||_2.
IERR = 0 : K steps were performed succesfully.
IERR > 0 : K steps were performed succesfully, but the algorithm
switched to full reorthogonalization after IERR steps.
IERR < 0 : Iteration was terminated after -IERR steps because an
invariant subspace was found, and 3 deflation attempts
were unsuccessful.
On exit WORK(1) contains the number of reorthogonalizations performed, and
WORK(2) contains the number of inner products performed in the
reorthogonalizations.
See also LANEIG, REORTH, COMPUTE_INT
References:
R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.
G. H. Golub & C. F. Van Loan, "Matrix Computations",
3. Ed., Johns Hopkins, 1996. Chapter 9.
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