NAME

       lasq2 - lasq2: dqds step

SYNOPSIS

   Functions
       subroutine dlasq2 (n, z, info)
           DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal
           matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and
           sstegr.
       subroutine slasq2 (n, z, info)
           SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal
           matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and
           sstegr.

Detailed Description

Function Documentation

   subroutine dlasq2 (integer n, double precision, dimension( * ) z, integer info)
       DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix
       associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

       Purpose:

            DLASQ2 computes all the eigenvalues of the symmetric positive
            definite tridiagonal matrix associated with the qd array Z to high
            relative accuracy are computed to high relative accuracy, in the
            absence of denormalization, underflow and overflow.

            To see the relation of Z to the tridiagonal matrix, let L be a
            unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
            let U be an upper bidiagonal matrix with 1's above and diagonal
            Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
            symmetric tridiagonal to which it is similar.

            Note : DLASQ2 defines a logical variable, IEEE, which is true
            on machines which follow ieee-754 floating-point standard in their
            handling of infinities and NaNs, and false otherwise. This variable
            is passed to DLASQ3.

       Parameters
           N

                     N is INTEGER
                   The number of rows and columns in the matrix. N >= 0.

           Z

                     Z is DOUBLE PRECISION array, dimension ( 4*N )
                   On entry Z holds the qd array. On exit, entries 1 to N hold
                   the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
                   trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
                   N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
                   holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
                   shifts that failed.

           INFO

                     INFO is INTEGER
                   = 0: successful exit
                   < 0: if the i-th argument is a scalar and had an illegal
                        value, then INFO = -i, if the i-th argument is an
                        array and the j-entry had an illegal value, then
                        INFO = -(i*100+j)
                   > 0: the algorithm failed
                         = 1, a split was marked by a positive value in E
                         = 2, current block of Z not diagonalized after 100*N
                              iterations (in inner while loop).  On exit Z holds
                              a qd array with the same eigenvalues as the given Z.
                         = 3, termination criterion of outer while loop not met
                              (program created more than N unreduced blocks)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Local Variables: I0:N0 defines a current unreduced segment of Z.
             The shifts are accumulated in SIGMA. Iteration count is in ITER.
             Ping-pong is controlled by PP (alternates between 0 and 1).

   subroutine slasq2 (integer n, real, dimension( * ) z, integer info)
       SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix
       associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

       Purpose:

            SLASQ2 computes all the eigenvalues of the symmetric positive
            definite tridiagonal matrix associated with the qd array Z to high
            relative accuracy are computed to high relative accuracy, in the
            absence of denormalization, underflow and overflow.

            To see the relation of Z to the tridiagonal matrix, let L be a
            unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
            let U be an upper bidiagonal matrix with 1's above and diagonal
            Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
            symmetric tridiagonal to which it is similar.

            Note : SLASQ2 defines a logical variable, IEEE, which is true
            on machines which follow ieee-754 floating-point standard in their
            handling of infinities and NaNs, and false otherwise. This variable
            is passed to SLASQ3.

       Parameters
           N

                     N is INTEGER
                   The number of rows and columns in the matrix. N >= 0.

           Z

                     Z is REAL array, dimension ( 4*N )
                   On entry Z holds the qd array. On exit, entries 1 to N hold
                   the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
                   trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
                   N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
                   holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
                   shifts that failed.

           INFO

                     INFO is INTEGER
                   = 0: successful exit
                   < 0: if the i-th argument is a scalar and had an illegal
                        value, then INFO = -i, if the i-th argument is an
                        array and the j-entry had an illegal value, then
                        INFO = -(i*100+j)
                   > 0: the algorithm failed
                         = 1, a split was marked by a positive value in E
                         = 2, current block of Z not diagonalized after 100*N
                              iterations (in inner while loop).  On exit Z holds
                              a qd array with the same eigenvalues as the given Z.
                         = 3, termination criterion of outer while loop not met
                              (program created more than N unreduced blocks)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Local Variables: I0:N0 defines a current unreduced segment of Z.
             The shifts are accumulated in SIGMA. Iteration count is in ITER.
             Ping-pong is controlled by PP (alternates between 0 and 1).

Author

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