?hptrd

Syntax

FORTRAN 77:

call chptrd(uplo, n, ap, d, e, tau, info)

call zhptrd(uplo, n, ap, d, e, tau, info)

Fortran 95:

call hptrd(ap, tau [,uplo] [,info])

C:

lapack_int LAPACKE_chptrd( int matrix_order, char uplo, lapack_int n, lapack_complex_float* ap, float* d, float* e, lapack_complex_float* tau );

lapack_int LAPACKE_zhptrd( int matrix_order, char uplo, lapack_int n, lapack_complex_double* ap, double* d, double* e, lapack_complex_double* tau );

Include Files

The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.

Description

The routine reduces a packed complex Hermitian matrix A to symmetric tridiagonal form T by a unitary similarity transformation: A = Q*T*Q^H. The unitary matrix Q is not formed explicitly but is represented as a product of n-1 elementary reflectors. Routines are provided for working with Q in this representation (see Application Notes below).

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', ap stores the packed upper triangle of A.

If uplo = 'L', ap stores the packed lower triangle of A.

n

INTEGER. The order of the matrix A (n ≥ 0).

ap

COMPLEX for chptrd

DOUBLE COMPLEX for zhptrd.

Array, DIMENSION at least max(1, n(n+1)/2). Contains either upper or lower triangle of A (as specified by uplo) in the packed form described in "Matrix Arguments" in Appendix "Routine and Function Arguments".

Output Parameters

ap

Overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q, as specified by uplo.

d, e

REAL for chptrd

DOUBLE PRECISION for zhptrd.

Arrays:

d(*) contains the diagonal elements of the matrix T.

The dimension of d must be at least max(1, n).

e(*) contains the off-diagonal elements of T.

The dimension of e must be at least max(1, n-1).

tau

COMPLEX for chptrd

DOUBLE COMPLEX for zhptrd.

Arrays, DIMENSION at least max(1, n-1). Contains further details of the orthogonal matrix Q.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine hptrd interface are the following:

ap

Holds the array A of size (n*(n+1)/2).

tau

Holds the vector with the number of elements n - 1.

uplo

Must be 'U' or 'L'. The default value is 'U'.

Note that diagonal (d) and off-diagonal (e) elements of the matrix T are omitted because they are kept in the matrix A on exit.

Application Notes

The computed matrix T is exactly similar to a matrix A + E, where ||E||₂ = c(n)*ε*||A||₂, c(n) is a modestly increasing function of n, and ε is the machine precision.

The approximate number of floating-point operations is (16/3)n³.

After calling this routine, you can call the following:

?upgtr

to form the computed matrix Q explicitly

?upmtr

to multiply a complex matrix by Q.

The real counterpart of this routine is ?sptrd.