Prototype

There is one prototype of laebz available, please see below.

laebz( const int_t ijob, const int_t nitmax,
        const int_t n, const int_t minp,
        const int_t nbmin, const Scalar >, const Scalar >,
        const Scalar >, const VectorD& d, const VectorE& e,
        const VectorE2& e2, VectorNVAL& nval, MatrixAB& ab, VectorC& c,
        int_t& mout, MatrixNAB& nab );

Description

laebz (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines SLAEBZ and DLAEBZ. laebz contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops:

IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.

IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval.

Note that the intervals are in all cases half-open intervals, i.e., of the form (a,b] , which includes b but not a .

To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow*(1/2) * underflow*(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either.

See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966

Note: the arguments are, in general, not checked for unreasonable values.

The selection of the LAPACK routine is done during compile-time, and is determined by the type of values contained in type VectorD. The type of values is obtained through the value_type meta-function typename value_type<VectorD>::type. The dispatching table below illustrates to which specific routine the code path will be generated.

Definition

Defined in header boost/numeric/bindings/lapack/auxiliary/laebz.hpp.

Parameters or Requirements on Types

Complexity

Example

#include <boost/numeric/bindings/lapack/auxiliary/laebz.hpp>
using namespace boost::numeric::bindings;

lapack::laebz( x, y, z );

this will output

[5] 0 1 2 3 4 5

Notes

See Also