##### laebz

###### 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.

Table 1.364. Dispatching of laebz

Value type of VectorD | LAPACK routine
--- | ---
`float` | SLAEBZ
`double` | DLAEBZ

###### Definition

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

###### Parameters or Requirements on Types

Parameters

MatrixA

  The definition of term 1

MatrixB

  The definition of term 2

MatrixC

  The definition of term 3.

  Definitions may contain paragraphs.

###### 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

- Originating Fortran source files slaebz.f and dlaebz.f at Netlib.