lag2(3)

NAME
lag2 - lag2: 2x2 eig

SYNOPSIS

  Functions

subroutine dlag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi)
DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
subroutine slag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Detailed Description

Function Documentation

  subroutine dlag2 (double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision safmin, double precision scale1, double precision scale2, double precision wr1, double precision wr2, double precision wi)

DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Purpose:

    !>
    !> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
    !> problem  A - w B, with scaling as necessary to avoid over-/underflow.
    !>
    !> The scaling factor  results in a modified eigenvalue equation
    !>
    !>     s A - w B
    !>
    !> where  s  is a non-negative scaling factor chosen so that  w,  w B,
    !> and  s A  do not overflow and, if possible, do not underflow, either.
    !>

Parameters
    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, 2)
    !>          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
    !>          is less than 1/SAFMIN.  Entries less than
    !>          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 2.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, 2)
    !>          On entry, the 2 x 2 upper triangular matrix B.  It is
    !>          assumed that the one-norm of B is less than 1/SAFMIN.  The
    !>          diagonals should be at least sqrt(SAFMIN) times the largest
    !>          element of B (in absolute value); if a diagonal is smaller
    !>          than that, then  +/- sqrt(SAFMIN) will be used instead of
    !>          that diagonal.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 2.
    !>

    SAFMIN

    !>          SAFMIN is DOUBLE PRECISION
    !>          The smallest positive number s.t. 1/SAFMIN does not
    !>          overflow.  (This should always be DLAMCH('S') -- it is an
    !>          argument in order to avoid having to call DLAMCH frequently.)
    !>

    SCALE1

    !>          SCALE1 is DOUBLE PRECISION
    !>          A scaling factor used to avoid over-/underflow in the
    !>          eigenvalue equation which defines the first eigenvalue.  If
    !>          the eigenvalues are complex, then the eigenvalues are
    !>          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
    !>          exponent range of the machine), SCALE1=SCALE2, and SCALE1
    !>          will always be positive.  If the eigenvalues are real, then
    !>          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
    !>          overflow or underflow, and in fact, SCALE1 may be zero or
    !>          less than the underflow threshold if the exact eigenvalue
    !>          is sufficiently large.
    !>

    SCALE2

    !>          SCALE2 is DOUBLE PRECISION
    !>          A scaling factor used to avoid over-/underflow in the
    !>          eigenvalue equation which defines the second eigenvalue.  If
    !>          the eigenvalues are complex, then SCALE2=SCALE1.  If the
    !>          eigenvalues are real, then the second (real) eigenvalue is
    !>          WR2 / SCALE2 , but this may overflow or underflow, and in
    !>          fact, SCALE2 may be zero or less than the underflow
    !>          threshold if the exact eigenvalue is sufficiently large.
    !>

    WR1

    !>          WR1 is DOUBLE PRECISION
    !>          If the eigenvalue is real, then WR1 is SCALE1 times the
    !>          eigenvalue closest to the (2,2) element of A B**(-1).  If the
    !>          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
    !>          part of the eigenvalues.
    !>

    WR2

    !>          WR2 is DOUBLE PRECISION
    !>          If the eigenvalue is real, then WR2 is SCALE2 times the
    !>          other eigenvalue.  If the eigenvalue is complex, then
    !>          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
    !>

    WI

    !>          WI is DOUBLE PRECISION
    !>          If the eigenvalue is real, then WI is zero.  If the
    !>          eigenvalue is complex, then WI is SCALE1 times the imaginary
    !>          part of the eigenvalues.  WI will always be non-negative.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine slag2 (real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real safmin, real scale1, real scale2, real wr1, real wr2, real wi)

SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Purpose:

    !>
    !> SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
    !> problem  A - w B, with scaling as necessary to avoid over-/underflow.
    !>
    !> The scaling factor  results in a modified eigenvalue equation
    !>
    !>     s A - w B
    !>
    !> where  s  is a non-negative scaling factor chosen so that  w,  w B,
    !> and  s A  do not overflow and, if possible, do not underflow, either.
    !>

Parameters
    A

    !>          A is REAL array, dimension (LDA, 2)
    !>          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
    !>          is less than 1/SAFMIN.  Entries less than
    !>          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 2.
    !>

    B

    !>          B is REAL array, dimension (LDB, 2)
    !>          On entry, the 2 x 2 upper triangular matrix B.  It is
    !>          assumed that the one-norm of B is less than 1/SAFMIN.  The
    !>          diagonals should be at least sqrt(SAFMIN) times the largest
    !>          element of B (in absolute value); if a diagonal is smaller
    !>          than that, then  +/- sqrt(SAFMIN) will be used instead of
    !>          that diagonal.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 2.
    !>

    SAFMIN

    !>          SAFMIN is REAL
    !>          The smallest positive number s.t. 1/SAFMIN does not
    !>          overflow.  (This should always be SLAMCH('S') -- it is an
    !>          argument in order to avoid having to call SLAMCH frequently.)
    !>

    SCALE1

    !>          SCALE1 is REAL
    !>          A scaling factor used to avoid over-/underflow in the
    !>          eigenvalue equation which defines the first eigenvalue.  If
    !>          the eigenvalues are complex, then the eigenvalues are
    !>          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
    !>          exponent range of the machine), SCALE1=SCALE2, and SCALE1
    !>          will always be positive.  If the eigenvalues are real, then
    !>          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
    !>          overflow or underflow, and in fact, SCALE1 may be zero or
    !>          less than the underflow threshold if the exact eigenvalue
    !>          is sufficiently large.
    !>

    SCALE2

    !>          SCALE2 is REAL
    !>          A scaling factor used to avoid over-/underflow in the
    !>          eigenvalue equation which defines the second eigenvalue.  If
    !>          the eigenvalues are complex, then SCALE2=SCALE1.  If the
    !>          eigenvalues are real, then the second (real) eigenvalue is
    !>          WR2 / SCALE2 , but this may overflow or underflow, and in
    !>          fact, SCALE2 may be zero or less than the underflow
    !>          threshold if the exact eigenvalue is sufficiently large.
    !>

    WR1

    !>          WR1 is REAL
    !>          If the eigenvalue is real, then WR1 is SCALE1 times the
    !>          eigenvalue closest to the (2,2) element of A B**(-1).  If the
    !>          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
    !>          part of the eigenvalues.
    !>

    WR2

    !>          WR2 is REAL
    !>          If the eigenvalue is real, then WR2 is SCALE2 times the
    !>          other eigenvalue.  If the eigenvalue is complex, then
    !>          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
    !>

    WI

    !>          WI is REAL
    !>          If the eigenvalue is real, then WI is zero.  If the
    !>          eigenvalue is complex, then WI is SCALE1 times the imaginary
    !>          part of the eigenvalues.  WI will always be non-negative.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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