lasq2(3)

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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