NAME

       lapll - lapll: linear dependence of 2 vectors

SYNOPSIS

   Functions
       subroutine clapll (n, x, incx, y, incy, ssmin)
           CLAPLL measures the linear dependence of two vectors.
       subroutine dlapll (n, x, incx, y, incy, ssmin)
           DLAPLL measures the linear dependence of two vectors.
       subroutine slapll (n, x, incx, y, incy, ssmin)
           SLAPLL measures the linear dependence of two vectors.
       subroutine zlapll (n, x, incx, y, incy, ssmin)
           ZLAPLL measures the linear dependence of two vectors.

Detailed Description

Function Documentation

   subroutine clapll (integer n, complex, dimension( * ) x, integer incx, complex, dimension( * )
       y, integer incy, real ssmin)
       CLAPLL measures the linear dependence of two vectors.

       Purpose:

            Given two column vectors X and Y, let

                                 A = ( X Y ).

            The subroutine first computes the QR factorization of A = Q*R,
            and then computes the SVD of the 2-by-2 upper triangular matrix R.
            The smaller singular value of R is returned in SSMIN, which is used
            as the measurement of the linear dependency of the vectors X and Y.

       Parameters
           N

                     N is INTEGER
                     The length of the vectors X and Y.

           X

                     X is COMPLEX array, dimension (1+(N-1)*INCX)
                     On entry, X contains the N-vector X.
                     On exit, X is overwritten.

           INCX

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

           Y

                     Y is COMPLEX array, dimension (1+(N-1)*INCY)
                     On entry, Y contains the N-vector Y.
                     On exit, Y is overwritten.

           INCY

                     INCY is INTEGER
                     The increment between successive elements of Y. INCY > 0.

           SSMIN

                     SSMIN is REAL
                     The smallest singular value of the N-by-2 matrix A = ( X Y ).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dlapll (integer n, double precision, dimension( * ) x, integer incx, double
       precision, dimension( * ) y, integer incy, double precision ssmin)
       DLAPLL measures the linear dependence of two vectors.

       Purpose:

            Given two column vectors X and Y, let

                                 A = ( X Y ).

            The subroutine first computes the QR factorization of A = Q*R,
            and then computes the SVD of the 2-by-2 upper triangular matrix R.
            The smaller singular value of R is returned in SSMIN, which is used
            as the measurement of the linear dependency of the vectors X and Y.

       Parameters
           N

                     N is INTEGER
                     The length of the vectors X and Y.

           X

                     X is DOUBLE PRECISION array,
                                    dimension (1+(N-1)*INCX)
                     On entry, X contains the N-vector X.
                     On exit, X is overwritten.

           INCX

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

           Y

                     Y is DOUBLE PRECISION array,
                                    dimension (1+(N-1)*INCY)
                     On entry, Y contains the N-vector Y.
                     On exit, Y is overwritten.

           INCY

                     INCY is INTEGER
                     The increment between successive elements of Y. INCY > 0.

           SSMIN

                     SSMIN is DOUBLE PRECISION
                     The smallest singular value of the N-by-2 matrix A = ( X Y ).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine slapll (integer n, real, dimension( * ) x, integer incx, real, dimension( * ) y,
       integer incy, real ssmin)
       SLAPLL measures the linear dependence of two vectors.

       Purpose:

            Given two column vectors X and Y, let

                                 A = ( X Y ).

            The subroutine first computes the QR factorization of A = Q*R,
            and then computes the SVD of the 2-by-2 upper triangular matrix R.
            The smaller singular value of R is returned in SSMIN, which is used
            as the measurement of the linear dependency of the vectors X and Y.

       Parameters
           N

                     N is INTEGER
                     The length of the vectors X and Y.

           X

                     X is REAL array,
                                    dimension (1+(N-1)*INCX)
                     On entry, X contains the N-vector X.
                     On exit, X is overwritten.

           INCX

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

           Y

                     Y is REAL array,
                                    dimension (1+(N-1)*INCY)
                     On entry, Y contains the N-vector Y.
                     On exit, Y is overwritten.

           INCY

                     INCY is INTEGER
                     The increment between successive elements of Y. INCY > 0.

           SSMIN

                     SSMIN is REAL
                     The smallest singular value of the N-by-2 matrix A = ( X Y ).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zlapll (integer n, complex*16, dimension( * ) x, integer incx, complex*16,
       dimension( * ) y, integer incy, double precision ssmin)
       ZLAPLL measures the linear dependence of two vectors.

       Purpose:

            Given two column vectors X and Y, let

                                 A = ( X Y ).

            The subroutine first computes the QR factorization of A = Q*R,
            and then computes the SVD of the 2-by-2 upper triangular matrix R.
            The smaller singular value of R is returned in SSMIN, which is used
            as the measurement of the linear dependency of the vectors X and Y.

       Parameters
           N

                     N is INTEGER
                     The length of the vectors X and Y.

           X

                     X is COMPLEX*16 array, dimension (1+(N-1)*INCX)
                     On entry, X contains the N-vector X.
                     On exit, X is overwritten.

           INCX

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

           Y

                     Y is COMPLEX*16 array, dimension (1+(N-1)*INCY)
                     On entry, Y contains the N-vector Y.
                     On exit, Y is overwritten.

           INCY

                     INCY is INTEGER
                     The increment between successive elements of Y. INCY > 0.

           SSMIN

                     SSMIN is DOUBLE PRECISION
                     The smallest singular value of the N-by-2 matrix A = ( X Y ).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

Author

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