*> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAS2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX ) * * .. Scalar Arguments .. * DOUBLE PRECISION F, G, H, SSMAX, SSMIN * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAS2 computes the singular values of the 2-by-2 matrix *> [ F G ] *> [ 0 H ]. *> On return, SSMIN is the smaller singular value and SSMAX is the *> larger singular value. *> \endverbatim * * Arguments: * ========== * *> \param[in] F *> \verbatim *> F is DOUBLE PRECISION *> The (1,1) element of the 2-by-2 matrix. *> \endverbatim *> *> \param[in] G *> \verbatim *> G is DOUBLE PRECISION *> The (1,2) element of the 2-by-2 matrix. *> \endverbatim *> *> \param[in] H *> \verbatim *> H is DOUBLE PRECISION *> The (2,2) element of the 2-by-2 matrix. *> \endverbatim *> *> \param[out] SSMIN *> \verbatim *> SSMIN is DOUBLE PRECISION *> The smaller singular value. *> \endverbatim *> *> \param[out] SSMAX *> \verbatim *> SSMAX is DOUBLE PRECISION *> The larger singular value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup OTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> Barring over/underflow, all output quantities are correct to within *> a few units in the last place (ulps), even in the absence of a guard *> digit in addition/subtraction. *> *> In IEEE arithmetic, the code works correctly if one matrix element is *> infinite. *> *> Overflow will not occur unless the largest singular value itself *> overflows, or is within a few ulps of overflow. (On machines with *> partial overflow, like the Cray, overflow may occur if the largest *> singular value is within a factor of 2 of overflow.) *> *> Underflow is harmless if underflow is gradual. Otherwise, results *> may correspond to a matrix modified by perturbations of size near *> the underflow threshold. *> \endverbatim *> * ===================================================================== SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX ) * * -- 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..-- * December 2016 * * .. Scalar Arguments .. DOUBLE PRECISION F, G, H, SSMAX, SSMIN * .. * * ==================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) * .. * .. Local Scalars .. DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * FA = ABS( F ) GA = ABS( G ) HA = ABS( H ) FHMN = MIN( FA, HA ) FHMX = MAX( FA, HA ) IF( FHMN.EQ.ZERO ) THEN SSMIN = ZERO IF( FHMX.EQ.ZERO ) THEN SSMAX = GA ELSE SSMAX = MAX( FHMX, GA )*SQRT( ONE+ \$ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 ) END IF ELSE IF( GA.LT.FHMX ) THEN AS = ONE + FHMN / FHMX AT = ( FHMX-FHMN ) / FHMX AU = ( GA / FHMX )**2 C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) ) SSMIN = FHMN*C SSMAX = FHMX / C ELSE AU = FHMX / GA IF( AU.EQ.ZERO ) THEN * * Avoid possible harmful underflow if exponent range * asymmetric (true SSMIN may not underflow even if * AU underflows) * SSMIN = ( FHMN*FHMX ) / GA SSMAX = GA ELSE AS = ONE + FHMN / FHMX AT = ( FHMX-FHMN ) / FHMX C = ONE / ( SQRT( ONE+( AS*AU )**2 )+ \$ SQRT( ONE+( AT*AU )**2 ) ) SSMIN = ( FHMN*C )*AU SSMIN = SSMIN + SSMIN SSMAX = GA / ( C+C ) END IF END IF END IF RETURN * * End of DLAS2 * END