NAME

       larfgp - larfgp: generate Householder reflector, beta ≥ 0

SYNOPSIS

   Functions
       subroutine clarfgp (n, alpha, x, incx, tau)
           CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
       subroutine dlarfgp (n, alpha, x, incx, tau)
           DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
       subroutine slarfgp (n, alpha, x, incx, tau)
           SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
       subroutine zlarfgp (n, alpha, x, incx, tau)
           ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

Detailed Description

Function Documentation

   subroutine clarfgp (integer n, complex alpha, complex, dimension( * ) x, integer incx, complex
       tau)
       CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

       Purpose:

            CLARFGP generates a complex elementary reflector H of order n, such
            that

                  H**H * ( alpha ) = ( beta ),   H**H * H = I.
                         (   x   )   (   0  )

            where alpha and beta are scalars, beta is real and non-negative, and
            x is an (n-1)-element complex vector.  H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**H ) ,
                                ( v )

            where tau is a complex scalar and v is a complex (n-1)-element
            vector. Note that H is not hermitian.

            If the elements of x are all zero and alpha is real, then tau = 0
            and H is taken to be the unit matrix.

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is COMPLEX
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is COMPLEX array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is COMPLEX
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dlarfgp (integer n, double precision alpha, double precision, dimension( * ) x,
       integer incx, double precision tau)
       DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

       Purpose:

            DLARFGP generates a real elementary reflector H of order n, such
            that

                  H * ( alpha ) = ( beta ),   H**T * H = I.
                      (   x   )   (   0  )

            where alpha and beta are scalars, beta is non-negative, and x is
            an (n-1)-element real vector.  H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**T ) ,
                                ( v )

            where tau is a real scalar and v is a real (n-1)-element
            vector.

            If the elements of x are all zero, then tau = 0 and H is taken to be
            the unit matrix.

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is DOUBLE PRECISION
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is DOUBLE PRECISION array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is DOUBLE PRECISION
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine slarfgp (integer n, real alpha, real, dimension( * ) x, integer incx, real tau)
       SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

       Purpose:

            SLARFGP generates a real elementary reflector H of order n, such
            that

                  H * ( alpha ) = ( beta ),   H**T * H = I.
                      (   x   )   (   0  )

            where alpha and beta are scalars, beta is non-negative, and x is
            an (n-1)-element real vector.  H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**T ) ,
                                ( v )

            where tau is a real scalar and v is a real (n-1)-element
            vector.

            If the elements of x are all zero, then tau = 0 and H is taken to be
            the unit matrix.

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is REAL
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is REAL array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is REAL
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zlarfgp (integer n, complex*16 alpha, complex*16, dimension( * ) x, integer incx,
       complex*16 tau)
       ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

       Purpose:

            ZLARFGP generates a complex elementary reflector H of order n, such
            that

                  H**H * ( alpha ) = ( beta ),   H**H * H = I.
                         (   x   )   (   0  )

            where alpha and beta are scalars, beta is real and non-negative, and
            x is an (n-1)-element complex vector.  H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**H ) ,
                                ( v )

            where tau is a complex scalar and v is a complex (n-1)-element
            vector. Note that H is not hermitian.

            If the elements of x are all zero and alpha is real, then tau = 0
            and H is taken to be the unit matrix.

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is COMPLEX*16
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is COMPLEX*16 array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is COMPLEX*16
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

Author

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