Library Functions Manual

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 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 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 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.

Definition at line 154 of file dlag2.f.

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.