LAPACK 3.3.0

slas2.f

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00001       SUBROUTINE SLAS2( F, G, H, SSMIN, SSMAX )
00002 *
00003 *  -- LAPACK auxiliary routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       REAL               F, G, H, SSMAX, SSMIN
00010 *     ..
00011 *
00012 *  Purpose
00013 *  =======
00014 *
00015 *  SLAS2  computes the singular values of the 2-by-2 matrix
00016 *     [  F   G  ]
00017 *     [  0   H  ].
00018 *  On return, SSMIN is the smaller singular value and SSMAX is the
00019 *  larger singular value.
00020 *
00021 *  Arguments
00022 *  =========
00023 *
00024 *  F       (input) REAL
00025 *          The (1,1) element of the 2-by-2 matrix.
00026 *
00027 *  G       (input) REAL
00028 *          The (1,2) element of the 2-by-2 matrix.
00029 *
00030 *  H       (input) REAL
00031 *          The (2,2) element of the 2-by-2 matrix.
00032 *
00033 *  SSMIN   (output) REAL
00034 *          The smaller singular value.
00035 *
00036 *  SSMAX   (output) REAL
00037 *          The larger singular value.
00038 *
00039 *  Further Details
00040 *  ===============
00041 *
00042 *  Barring over/underflow, all output quantities are correct to within
00043 *  a few units in the last place (ulps), even in the absence of a guard
00044 *  digit in addition/subtraction.
00045 *
00046 *  In IEEE arithmetic, the code works correctly if one matrix element is
00047 *  infinite.
00048 *
00049 *  Overflow will not occur unless the largest singular value itself
00050 *  overflows, or is within a few ulps of overflow. (On machines with
00051 *  partial overflow, like the Cray, overflow may occur if the largest
00052 *  singular value is within a factor of 2 of overflow.)
00053 *
00054 *  Underflow is harmless if underflow is gradual. Otherwise, results
00055 *  may correspond to a matrix modified by perturbations of size near
00056 *  the underflow threshold.
00057 *
00058 *  ====================================================================
00059 *
00060 *     .. Parameters ..
00061       REAL               ZERO
00062       PARAMETER          ( ZERO = 0.0E0 )
00063       REAL               ONE
00064       PARAMETER          ( ONE = 1.0E0 )
00065       REAL               TWO
00066       PARAMETER          ( TWO = 2.0E0 )
00067 *     ..
00068 *     .. Local Scalars ..
00069       REAL               AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
00070 *     ..
00071 *     .. Intrinsic Functions ..
00072       INTRINSIC          ABS, MAX, MIN, SQRT
00073 *     ..
00074 *     .. Executable Statements ..
00075 *
00076       FA = ABS( F )
00077       GA = ABS( G )
00078       HA = ABS( H )
00079       FHMN = MIN( FA, HA )
00080       FHMX = MAX( FA, HA )
00081       IF( FHMN.EQ.ZERO ) THEN
00082          SSMIN = ZERO
00083          IF( FHMX.EQ.ZERO ) THEN
00084             SSMAX = GA
00085          ELSE
00086             SSMAX = MAX( FHMX, GA )*SQRT( ONE+
00087      $              ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
00088          END IF
00089       ELSE
00090          IF( GA.LT.FHMX ) THEN
00091             AS = ONE + FHMN / FHMX
00092             AT = ( FHMX-FHMN ) / FHMX
00093             AU = ( GA / FHMX )**2
00094             C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
00095             SSMIN = FHMN*C
00096             SSMAX = FHMX / C
00097          ELSE
00098             AU = FHMX / GA
00099             IF( AU.EQ.ZERO ) THEN
00100 *
00101 *              Avoid possible harmful underflow if exponent range
00102 *              asymmetric (true SSMIN may not underflow even if
00103 *              AU underflows)
00104 *
00105                SSMIN = ( FHMN*FHMX ) / GA
00106                SSMAX = GA
00107             ELSE
00108                AS = ONE + FHMN / FHMX
00109                AT = ( FHMX-FHMN ) / FHMX
00110                C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
00111      $             SQRT( ONE+( AT*AU )**2 ) )
00112                SSMIN = ( FHMN*C )*AU
00113                SSMIN = SSMIN + SSMIN
00114                SSMAX = GA / ( C+C )
00115             END IF
00116          END IF
00117       END IF
00118       RETURN
00119 *
00120 *     End of SLAS2
00121 *
00122       END
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