DLAG2 - compute the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow

SUBROUTINE DLAG2(

A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI )

INTEGER LDA, LDB

DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2

DOUBLE PRECISION A( LDA, * ), B( LDB, * )

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 "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 underflow, either.

A (input) 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 (input) INTEGER

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

B (input) 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 (input) INTEGER

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

SAFMIN (input) 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 (output) 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 threshhold if the exact eigenvalue is sufficiently large.

SCALE2 (output) 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 threshhold if the exact eigenvalue is sufficiently large.

WR1 (output) 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 (output) 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 (output) 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.