*> \brief \b SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAG2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, * WR2, WI ) * * .. Scalar Arguments .. * INTEGER LDA, LDB * REAL SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 * .. * .. Array Arguments .. * REAL A( LDA, * ), B( LDB, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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 underflow, either. *> \endverbatim * * Arguments: * ========== * *> \param[in] A *> \verbatim *> 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= 2. *> \endverbatim *> *> \param[in] B *> \verbatim *> 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. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= 2. *> \endverbatim *> *> \param[in] SAFMIN *> \verbatim *> 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.) *> \endverbatim *> *> \param[out] SCALE1 *> \verbatim *> 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. *> \endverbatim *> *> \param[out] SCALE2 *> \verbatim *> 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. *> \endverbatim *> *> \param[out] WR1 *> \verbatim *> 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. *> \endverbatim *> *> \param[out] WR2 *> \verbatim *> 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. *> \endverbatim *> *> \param[out] WI *> \verbatim *> 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup realOTHERauxiliary * * ===================================================================== SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, \$ WR2, WI ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. INTEGER LDA, LDB REAL SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 * .. * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) REAL HALF PARAMETER ( HALF = ONE / TWO ) REAL FUZZY1 PARAMETER ( FUZZY1 = ONE+1.0E-5 ) * .. * .. Local Scalars .. REAL 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 threshold and handle numbers above that * threshold 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 SLAG2 * RETURN END