SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
$ WR2, WI )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* 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 "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.
*
* Arguments
* =========
*
* 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.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = ONE / TWO )
DOUBLE PRECISION FUZZY1
PARAMETER ( FUZZY1 = ONE+1.0D-5 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
$ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
$ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
$ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
$ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
$ WSCALE, WSIZE, WSMALL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
RTMIN = SQRT( SAFMIN )
RTMAX = ONE / RTMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A11 = ASCALE*A( 1, 1 )
A21 = ASCALE*A( 2, 1 )
A12 = ASCALE*A( 1, 2 )
A22 = ASCALE*A( 2, 2 )
*
* Perturb B if necessary to insure non-singularity
*
B11 = B( 1, 1 )
B12 = B( 1, 2 )
B22 = B( 2, 2 )
BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
IF( ABS( B11 ).LT.BMIN )
$ B11 = SIGN( BMIN, B11 )
IF( ABS( B22 ).LT.BMIN )
$ B22 = SIGN( BMIN, B22 )
*
* Scale B
*
BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
BSCALE = ONE / BSIZE
B11 = B11*BSCALE
B12 = B12*BSCALE
B22 = B22*BSCALE
*
* Compute larger eigenvalue by method described by C. van Loan
*
* ( AS is A shifted by -SHIFT*B )
*
BINV11 = ONE / B11
BINV22 = ONE / B22
S1 = A11*BINV11
S2 = A22*BINV22
IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
AS12 = A12 - S1*B12
AS22 = A22 - S1*B22
SS = A21*( BINV11*BINV22 )
ABI22 = AS22*BINV22 - SS*B12
PP = HALF*ABI22
SHIFT = S1
ELSE
AS12 = A12 - S2*B12
AS11 = A11 - S2*B11
SS = A21*( BINV11*BINV22 )
ABI22 = -SS*B12
PP = HALF*( AS11*BINV11+ABI22 )
SHIFT = S2
END IF
QQ = SS*AS12
IF( ABS( PP*RTMIN ).GE.ONE ) THEN
DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
R = SQRT( ABS( DISCR ) )*RTMAX
ELSE
IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
R = SQRT( ABS( DISCR ) )*RTMIN
ELSE
DISCR = PP**2 + QQ
R = SQRT( ABS( DISCR ) )
END IF
END IF
*
* Note: the test of R in the following IF is to cover the case when
* DISCR is small and negative and is flushed to zero during
* the calculation of R. On machines which have a consistent
* flush-to-zero threshhold and handle numbers above that
* threshhold correctly, it would not be necessary.
*
IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
SUM = PP + SIGN( R, PP )
DIFF = PP - SIGN( R, PP )
WBIG = SHIFT + SUM
*
* Compute smaller eigenvalue
*
WSMALL = SHIFT + DIFF
IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
WSMALL = WDET / WBIG
END IF
*
* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
* for WR1.
*
IF( PP.GT.ABI22 ) THEN
WR1 = MIN( WBIG, WSMALL )
WR2 = MAX( WBIG, WSMALL )
ELSE
WR1 = MAX( WBIG, WSMALL )
WR2 = MIN( WBIG, WSMALL )
END IF
WI = ZERO
ELSE
*
* Complex eigenvalues
*
WR1 = SHIFT + PP
WR2 = WR1
WI = R
END IF
*
* Further scaling to avoid underflow and overflow in computing
* SCALE1 and overflow in computing w*B.
*
* This scale factor (WSCALE) is bounded from above using C1 and C2,
* and from below using C3 and C4.
* C1 implements the condition s A must never overflow.
* C2 implements the condition w B must never overflow.
* C3, with C2,
* implement the condition that s A - w B must never overflow.
* C4 implements the condition s should not underflow.
* C5 implements the condition max(s,|w|) should be at least 2.
*
C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
C2 = SAFMIN*MAX( ONE, BNORM )
C3 = BSIZE*SAFMIN
IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
ELSE
C4 = ONE
END IF
IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
C5 = MIN( ONE, ASCALE*BSIZE )
ELSE
C5 = ONE
END IF
*
* Scale first eigenvalue
*
WABS = ABS( WR1 ) + ABS( WI )
WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
$ MIN( C4, HALF*MAX( WABS, C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR1 = WR1*WSCALE
IF( WI.NE.ZERO ) THEN
WI = WI*WSCALE
WR2 = WR1
SCALE2 = SCALE1
END IF
ELSE
SCALE1 = ASCALE*BSIZE
SCALE2 = SCALE1
END IF
*
* Scale second eigenvalue (if real)
*
IF( WI.EQ.ZERO ) THEN
WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
$ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR2 = WR2*WSCALE
ELSE
SCALE2 = ASCALE*BSIZE
END IF
END IF
*
* End of DLAG2
*
RETURN
END