LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
slag2.f
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00001 *> \brief \b SLAG2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download SLAG2 + dependencies 
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00011 *> [TGZ]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
00022 *                         WR2, WI )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       INTEGER            LDA, LDB
00026 *       REAL               SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       REAL               A( LDA, * ), B( LDB, * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
00039 *> problem  A - w B, with scaling as necessary to avoid over-/underflow.
00040 *>
00041 *> The scaling factor "s" results in a modified eigenvalue equation
00042 *>
00043 *>     s A - w B
00044 *>
00045 *> where  s  is a non-negative scaling factor chosen so that  w,  w B,
00046 *> and  s A  do not overflow and, if possible, do not underflow, either.
00047 *> \endverbatim
00048 *
00049 *  Arguments:
00050 *  ==========
00051 *
00052 *> \param[in] A
00053 *> \verbatim
00054 *>          A is REAL array, dimension (LDA, 2)
00055 *>          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
00056 *>          is less than 1/SAFMIN.  Entries less than
00057 *>          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
00058 *> \endverbatim
00059 *>
00060 *> \param[in] LDA
00061 *> \verbatim
00062 *>          LDA is INTEGER
00063 *>          The leading dimension of the array A.  LDA >= 2.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] B
00067 *> \verbatim
00068 *>          B is REAL array, dimension (LDB, 2)
00069 *>          On entry, the 2 x 2 upper triangular matrix B.  It is
00070 *>          assumed that the one-norm of B is less than 1/SAFMIN.  The
00071 *>          diagonals should be at least sqrt(SAFMIN) times the largest
00072 *>          element of B (in absolute value); if a diagonal is smaller
00073 *>          than that, then  +/- sqrt(SAFMIN) will be used instead of
00074 *>          that diagonal.
00075 *> \endverbatim
00076 *>
00077 *> \param[in] LDB
00078 *> \verbatim
00079 *>          LDB is INTEGER
00080 *>          The leading dimension of the array B.  LDB >= 2.
00081 *> \endverbatim
00082 *>
00083 *> \param[in] SAFMIN
00084 *> \verbatim
00085 *>          SAFMIN is REAL
00086 *>          The smallest positive number s.t. 1/SAFMIN does not
00087 *>          overflow.  (This should always be SLAMCH('S') -- it is an
00088 *>          argument in order to avoid having to call SLAMCH frequently.)
00089 *> \endverbatim
00090 *>
00091 *> \param[out] SCALE1
00092 *> \verbatim
00093 *>          SCALE1 is REAL
00094 *>          A scaling factor used to avoid over-/underflow in the
00095 *>          eigenvalue equation which defines the first eigenvalue.  If
00096 *>          the eigenvalues are complex, then the eigenvalues are
00097 *>          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
00098 *>          exponent range of the machine), SCALE1=SCALE2, and SCALE1
00099 *>          will always be positive.  If the eigenvalues are real, then
00100 *>          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
00101 *>          overflow or underflow, and in fact, SCALE1 may be zero or
00102 *>          less than the underflow threshhold if the exact eigenvalue
00103 *>          is sufficiently large.
00104 *> \endverbatim
00105 *>
00106 *> \param[out] SCALE2
00107 *> \verbatim
00108 *>          SCALE2 is REAL
00109 *>          A scaling factor used to avoid over-/underflow in the
00110 *>          eigenvalue equation which defines the second eigenvalue.  If
00111 *>          the eigenvalues are complex, then SCALE2=SCALE1.  If the
00112 *>          eigenvalues are real, then the second (real) eigenvalue is
00113 *>          WR2 / SCALE2 , but this may overflow or underflow, and in
00114 *>          fact, SCALE2 may be zero or less than the underflow
00115 *>          threshhold if the exact eigenvalue is sufficiently large.
00116 *> \endverbatim
00117 *>
00118 *> \param[out] WR1
00119 *> \verbatim
00120 *>          WR1 is REAL
00121 *>          If the eigenvalue is real, then WR1 is SCALE1 times the
00122 *>          eigenvalue closest to the (2,2) element of A B**(-1).  If the
00123 *>          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
00124 *>          part of the eigenvalues.
00125 *> \endverbatim
00126 *>
00127 *> \param[out] WR2
00128 *> \verbatim
00129 *>          WR2 is REAL
00130 *>          If the eigenvalue is real, then WR2 is SCALE2 times the
00131 *>          other eigenvalue.  If the eigenvalue is complex, then
00132 *>          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
00133 *> \endverbatim
00134 *>
00135 *> \param[out] WI
00136 *> \verbatim
00137 *>          WI is REAL
00138 *>          If the eigenvalue is real, then WI is zero.  If the
00139 *>          eigenvalue is complex, then WI is SCALE1 times the imaginary
00140 *>          part of the eigenvalues.  WI will always be non-negative.
00141 *> \endverbatim
00142 *
00143 *  Authors:
00144 *  ========
00145 *
00146 *> \author Univ. of Tennessee 
00147 *> \author Univ. of California Berkeley 
00148 *> \author Univ. of Colorado Denver 
00149 *> \author NAG Ltd. 
00150 *
00151 *> \date November 2011
00152 *
00153 *> \ingroup realOTHERauxiliary
00154 *
00155 *  =====================================================================
00156       SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
00157      $                  WR2, WI )
00158 *
00159 *  -- LAPACK auxiliary routine (version 3.4.0) --
00160 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00161 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00162 *     November 2011
00163 *
00164 *     .. Scalar Arguments ..
00165       INTEGER            LDA, LDB
00166       REAL               SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
00167 *     ..
00168 *     .. Array Arguments ..
00169       REAL               A( LDA, * ), B( LDB, * )
00170 *     ..
00171 *
00172 *  =====================================================================
00173 *
00174 *     .. Parameters ..
00175       REAL               ZERO, ONE, TWO
00176       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
00177       REAL               HALF
00178       PARAMETER          ( HALF = ONE / TWO )
00179       REAL               FUZZY1
00180       PARAMETER          ( FUZZY1 = ONE+1.0E-5 )
00181 *     ..
00182 *     .. Local Scalars ..
00183       REAL               A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
00184      $                   AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
00185      $                   BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
00186      $                   DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
00187      $                   SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
00188      $                   WSCALE, WSIZE, WSMALL
00189 *     ..
00190 *     .. Intrinsic Functions ..
00191       INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
00192 *     ..
00193 *     .. Executable Statements ..
00194 *
00195       RTMIN = SQRT( SAFMIN )
00196       RTMAX = ONE / RTMIN
00197       SAFMAX = ONE / SAFMIN
00198 *
00199 *     Scale A
00200 *
00201       ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
00202      $        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
00203       ASCALE = ONE / ANORM
00204       A11 = ASCALE*A( 1, 1 )
00205       A21 = ASCALE*A( 2, 1 )
00206       A12 = ASCALE*A( 1, 2 )
00207       A22 = ASCALE*A( 2, 2 )
00208 *
00209 *     Perturb B if necessary to insure non-singularity
00210 *
00211       B11 = B( 1, 1 )
00212       B12 = B( 1, 2 )
00213       B22 = B( 2, 2 )
00214       BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
00215       IF( ABS( B11 ).LT.BMIN )
00216      $   B11 = SIGN( BMIN, B11 )
00217       IF( ABS( B22 ).LT.BMIN )
00218      $   B22 = SIGN( BMIN, B22 )
00219 *
00220 *     Scale B
00221 *
00222       BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
00223       BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
00224       BSCALE = ONE / BSIZE
00225       B11 = B11*BSCALE
00226       B12 = B12*BSCALE
00227       B22 = B22*BSCALE
00228 *
00229 *     Compute larger eigenvalue by method described by C. van Loan
00230 *
00231 *     ( AS is A shifted by -SHIFT*B )
00232 *
00233       BINV11 = ONE / B11
00234       BINV22 = ONE / B22
00235       S1 = A11*BINV11
00236       S2 = A22*BINV22
00237       IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
00238          AS12 = A12 - S1*B12
00239          AS22 = A22 - S1*B22
00240          SS = A21*( BINV11*BINV22 )
00241          ABI22 = AS22*BINV22 - SS*B12
00242          PP = HALF*ABI22
00243          SHIFT = S1
00244       ELSE
00245          AS12 = A12 - S2*B12
00246          AS11 = A11 - S2*B11
00247          SS = A21*( BINV11*BINV22 )
00248          ABI22 = -SS*B12
00249          PP = HALF*( AS11*BINV11+ABI22 )
00250          SHIFT = S2
00251       END IF
00252       QQ = SS*AS12
00253       IF( ABS( PP*RTMIN ).GE.ONE ) THEN
00254          DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
00255          R = SQRT( ABS( DISCR ) )*RTMAX
00256       ELSE
00257          IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
00258             DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
00259             R = SQRT( ABS( DISCR ) )*RTMIN
00260          ELSE
00261             DISCR = PP**2 + QQ
00262             R = SQRT( ABS( DISCR ) )
00263          END IF
00264       END IF
00265 *
00266 *     Note: the test of R in the following IF is to cover the case when
00267 *           DISCR is small and negative and is flushed to zero during
00268 *           the calculation of R.  On machines which have a consistent
00269 *           flush-to-zero threshhold and handle numbers above that
00270 *           threshhold correctly, it would not be necessary.
00271 *
00272       IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
00273          SUM = PP + SIGN( R, PP )
00274          DIFF = PP - SIGN( R, PP )
00275          WBIG = SHIFT + SUM
00276 *
00277 *        Compute smaller eigenvalue
00278 *
00279          WSMALL = SHIFT + DIFF
00280          IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
00281             WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
00282             WSMALL = WDET / WBIG
00283          END IF
00284 *
00285 *        Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
00286 *        for WR1.
00287 *
00288          IF( PP.GT.ABI22 ) THEN
00289             WR1 = MIN( WBIG, WSMALL )
00290             WR2 = MAX( WBIG, WSMALL )
00291          ELSE
00292             WR1 = MAX( WBIG, WSMALL )
00293             WR2 = MIN( WBIG, WSMALL )
00294          END IF
00295          WI = ZERO
00296       ELSE
00297 *
00298 *        Complex eigenvalues
00299 *
00300          WR1 = SHIFT + PP
00301          WR2 = WR1
00302          WI = R
00303       END IF
00304 *
00305 *     Further scaling to avoid underflow and overflow in computing
00306 *     SCALE1 and overflow in computing w*B.
00307 *
00308 *     This scale factor (WSCALE) is bounded from above using C1 and C2,
00309 *     and from below using C3 and C4.
00310 *        C1 implements the condition  s A  must never overflow.
00311 *        C2 implements the condition  w B  must never overflow.
00312 *        C3, with C2,
00313 *           implement the condition that s A - w B must never overflow.
00314 *        C4 implements the condition  s    should not underflow.
00315 *        C5 implements the condition  max(s,|w|) should be at least 2.
00316 *
00317       C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
00318       C2 = SAFMIN*MAX( ONE, BNORM )
00319       C3 = BSIZE*SAFMIN
00320       IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
00321          C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
00322       ELSE
00323          C4 = ONE
00324       END IF
00325       IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
00326          C5 = MIN( ONE, ASCALE*BSIZE )
00327       ELSE
00328          C5 = ONE
00329       END IF
00330 *
00331 *     Scale first eigenvalue
00332 *
00333       WABS = ABS( WR1 ) + ABS( WI )
00334       WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
00335      $        MIN( C4, HALF*MAX( WABS, C5 ) ) )
00336       IF( WSIZE.NE.ONE ) THEN
00337          WSCALE = ONE / WSIZE
00338          IF( WSIZE.GT.ONE ) THEN
00339             SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
00340      $               MIN( ASCALE, BSIZE )
00341          ELSE
00342             SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
00343      $               MAX( ASCALE, BSIZE )
00344          END IF
00345          WR1 = WR1*WSCALE
00346          IF( WI.NE.ZERO ) THEN
00347             WI = WI*WSCALE
00348             WR2 = WR1
00349             SCALE2 = SCALE1
00350          END IF
00351       ELSE
00352          SCALE1 = ASCALE*BSIZE
00353          SCALE2 = SCALE1
00354       END IF
00355 *
00356 *     Scale second eigenvalue (if real)
00357 *
00358       IF( WI.EQ.ZERO ) THEN
00359          WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
00360      $           MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
00361          IF( WSIZE.NE.ONE ) THEN
00362             WSCALE = ONE / WSIZE
00363             IF( WSIZE.GT.ONE ) THEN
00364                SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
00365      $                  MIN( ASCALE, BSIZE )
00366             ELSE
00367                SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
00368      $                  MAX( ASCALE, BSIZE )
00369             END IF
00370             WR2 = WR2*WSCALE
00371          ELSE
00372             SCALE2 = ASCALE*BSIZE
00373          END IF
00374       END IF
00375 *
00376 *     End of SLAG2
00377 *
00378       RETURN
00379       END
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