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[section pteqr]

[heading Prototype]
There is one prototype of `pteqr` available, please see below.
``
pteqr( const char compz, VectorD& d, VectorE& e, MatrixZ& z );
``


[heading Description]

`pteqr` (short for $FRIENDLY_NAME) provides a C++
interface to LAPACK routines SPTEQR, DPTEQR, CPTEQR, and ZPTEQR. 
`pteqr` computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor.

This routine computes the eigenvalues of the positive definite
tridiagonal matrix to high relative accuracy. This means that if the
eigenvalues range over many orders of magnitude in size, then the
small eigenvalues and corresponding eigenvectors will be computed
more accurately than, for example, with the standard QR method.

The eigenvectors of a full or band positive definite Hermitian matrix
can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to
reduce this matrix to tridiagonal form. (The reduction to
tridiagonal form, however, may preclude the possibility of obtaining
high relative accuracy in the small eigenvalues of the original
matrix, if these eigenvalues range over many orders of magnitude.)

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 Dispatching of pteqr
[  [ Value type of VectorD ] [LAPACK routine] ]
[  [`float`][SPTEQR] ]
[  [`double`][DPTEQR] ]
[  [`complex<float>`][CPTEQR] ]
[  [`complex<double>`][ZPTEQR] ]

]


[heading Definition]
Defined in header [headerref boost/numeric/bindings/lapack/computational/pteqr.hpp].


[heading Parameters or Requirements on Types]

[variablelist Parameters
    [[MatrixA] [The definition of term 1]]
    [[MatrixB] [The definition of term 2]]
    [[MatrixC] [
    The definition of term 3.

    Definitions may contain paragraphs.
    ]]
]


[heading Complexity]


[heading Example]
``
#include <boost/numeric/bindings/lapack/computational/pteqr.hpp>
using namespace boost::numeric::bindings;

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

``

this will output

``
[5] 0 1 2 3 4 5
``



[heading Notes]


[heading See Also]

* Originating Fortran source files [@http://www.netlib.org/lapack/single/spteqr.f spteqr.f], [@http://www.netlib.org/lapack/double/dpteqr.f dpteqr.f], [@http://www.netlib.org/lapack/complex/cpteqr.f cpteqr.f], and [@http://www.netlib.org/lapack/complex16/zpteqr.f zpteqr.f] at Netlib.

[endsect]
