slag2(3)

NAME

SLAG2 - compute the eigenvalues of a 2 x 2 generalized

eigenvalue problem A - w B, with scaling as necessary to avoid

over-/underflow

SYNOPSIS

SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1,  SCALE2,
WR1, WR2, WI )
    INTEGER       LDA, LDB
    REAL          SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
    REAL          A( LDA, * ), B( LDB, * )

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 "s" 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 un

derflow, either.

ARGUMENTS

A (input) 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 (input) INTEGER

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

B (input) 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 di

agonals 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 (input) INTEGER

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

SAFMIN (input) REAL

The smallest positive number s.t. 1/SAFMIN does

not overflow. (This should always be SLAMCH('S') -- it is an ar

gument in order to avoid having to call SLAMCH frequently.)

SCALE1 (output) 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 threshhold if the

exact eigenvalue is sufficiently large.

SCALE2 (output) 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 eigen

values 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 threshhold if the exact

eigenvalue is sufficiently large.

WR1 (output) 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 (output) 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 (output) 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.

LAPACK version 3.0 15 June 2000

SLAG2(3)