1      SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
2     $                  WR2, WI )
3*
4*  -- LAPACK auxiliary routine (version 3.0) --
5*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
6*     Courant Institute, Argonne National Lab, and Rice University
7*     March 31, 1993
8*
9*     .. Scalar Arguments ..
10      INTEGER            LDA, LDB
11      DOUBLE PRECISION   SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
12*     ..
13*     .. Array Arguments ..
14      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
15*     ..
16*
17*  Purpose
18*  =======
19*
20*  DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
21*  problem  A - w B, with scaling as necessary to avoid over-/underflow.
22*
23*  The scaling factor "s" results in a modified eigenvalue equation
24*
25*      s A - w B
26*
27*  where  s  is a non-negative scaling factor chosen so that  w,  w B,
28*  and  s A  do not overflow and, if possible, do not underflow, either.
29*
30*  Arguments
31*  =========
32*
33*  A       (input) DOUBLE PRECISION array, dimension (LDA, 2)
34*          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
35*          is less than 1/SAFMIN.  Entries less than
36*          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
37*
38*  LDA     (input) INTEGER
39*          The leading dimension of the array A.  LDA >= 2.
40*
41*  B       (input) DOUBLE PRECISION array, dimension (LDB, 2)
42*          On entry, the 2 x 2 upper triangular matrix B.  It is
43*          assumed that the one-norm of B is less than 1/SAFMIN.  The
44*          diagonals should be at least sqrt(SAFMIN) times the largest
45*          element of B (in absolute value); if a diagonal is smaller
46*          than that, then  +/- sqrt(SAFMIN) will be used instead of
47*          that diagonal.
48*
49*  LDB     (input) INTEGER
50*          The leading dimension of the array B.  LDB >= 2.
51*
52*  SAFMIN  (input) DOUBLE PRECISION
53*          The smallest positive number s.t. 1/SAFMIN does not
54*          overflow.  (This should always be DLAMCH('S') -- it is an
55*          argument in order to avoid having to call DLAMCH frequently.)
56*
57*  SCALE1  (output) DOUBLE PRECISION
58*          A scaling factor used to avoid over-/underflow in the
59*          eigenvalue equation which defines the first eigenvalue.  If
60*          the eigenvalues are complex, then the eigenvalues are
61*          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
62*          exponent range of the machine), SCALE1=SCALE2, and SCALE1
63*          will always be positive.  If the eigenvalues are real, then
64*          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
65*          overflow or underflow, and in fact, SCALE1 may be zero or
66*          less than the underflow threshhold if the exact eigenvalue
67*          is sufficiently large.
68*
69*  SCALE2  (output) DOUBLE PRECISION
70*          A scaling factor used to avoid over-/underflow in the
71*          eigenvalue equation which defines the second eigenvalue.  If
72*          the eigenvalues are complex, then SCALE2=SCALE1.  If the
73*          eigenvalues are real, then the second (real) eigenvalue is
74*          WR2 / SCALE2 , but this may overflow or underflow, and in
75*          fact, SCALE2 may be zero or less than the underflow
76*          threshhold if the exact eigenvalue is sufficiently large.
77*
78*  WR1     (output) DOUBLE PRECISION
79*          If the eigenvalue is real, then WR1 is SCALE1 times the
80*          eigenvalue closest to the (2,2) element of A B**(-1).  If the
81*          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
82*          part of the eigenvalues.
83*
84*  WR2     (output) DOUBLE PRECISION
85*          If the eigenvalue is real, then WR2 is SCALE2 times the
86*          other eigenvalue.  If the eigenvalue is complex, then
87*          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
88*
89*  WI      (output) DOUBLE PRECISION
90*          If the eigenvalue is real, then WI is zero.  If the
91*          eigenvalue is complex, then WI is SCALE1 times the imaginary
92*          part of the eigenvalues.  WI will always be non-negative.
93*
94*  =====================================================================
95*
96*     .. Parameters ..
97      DOUBLE PRECISION   ZERO, ONE, TWO
98      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
99      DOUBLE PRECISION   HALF
100      PARAMETER          ( HALF = ONE / TWO )
101      DOUBLE PRECISION   FUZZY1
102      PARAMETER          ( FUZZY1 = ONE+1.0D-5 )
103*     ..
104*     .. Local Scalars ..
105      DOUBLE PRECISION   A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
106     $                   AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
107     $                   BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
108     $                   DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
109     $                   SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
110     $                   WSCALE, WSIZE, WSMALL
111*     ..
112*     .. Intrinsic Functions ..
113      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
114*     ..
115*     .. Executable Statements ..
116*
117      RTMIN = SQRT( SAFMIN )
118      RTMAX = ONE / RTMIN
119      SAFMAX = ONE / SAFMIN
120*
121*     Scale A
122*
123      ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
124     $        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
125      ASCALE = ONE / ANORM
126      A11 = ASCALE*A( 1, 1 )
127      A21 = ASCALE*A( 2, 1 )
128      A12 = ASCALE*A( 1, 2 )
129      A22 = ASCALE*A( 2, 2 )
130*
131*     Perturb B if necessary to insure non-singularity
132*
133      B11 = B( 1, 1 )
134      B12 = B( 1, 2 )
135      B22 = B( 2, 2 )
136      BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
137      IF( ABS( B11 ).LT.BMIN )
138     $   B11 = SIGN( BMIN, B11 )
139      IF( ABS( B22 ).LT.BMIN )
140     $   B22 = SIGN( BMIN, B22 )
141*
142*     Scale B
143*
144      BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
145      BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
146      BSCALE = ONE / BSIZE
147      B11 = B11*BSCALE
148      B12 = B12*BSCALE
149      B22 = B22*BSCALE
150*
151*     Compute larger eigenvalue by method described by C. van Loan
152*
153*     ( AS is A shifted by -SHIFT*B )
154*
155      BINV11 = ONE / B11
156      BINV22 = ONE / B22
157      S1 = A11*BINV11
158      S2 = A22*BINV22
159      IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
160         AS12 = A12 - S1*B12
161         AS22 = A22 - S1*B22
162         SS = A21*( BINV11*BINV22 )
163         ABI22 = AS22*BINV22 - SS*B12
164         PP = HALF*ABI22
165         SHIFT = S1
166      ELSE
167         AS12 = A12 - S2*B12
168         AS11 = A11 - S2*B11
169         SS = A21*( BINV11*BINV22 )
170         ABI22 = -SS*B12
171         PP = HALF*( AS11*BINV11+ABI22 )
172         SHIFT = S2
173      END IF
174      QQ = SS*AS12
175      IF( ABS( PP*RTMIN ).GE.ONE ) THEN
176         DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
177         R = SQRT( ABS( DISCR ) )*RTMAX
178      ELSE
179         IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
180            DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
181            R = SQRT( ABS( DISCR ) )*RTMIN
182         ELSE
183            DISCR = PP**2 + QQ
184            R = SQRT( ABS( DISCR ) )
185         END IF
186      END IF
187*
188*     Note: the test of R in the following IF is to cover the case when
189*           DISCR is small and negative and is flushed to zero during
190*           the calculation of R.  On machines which have a consistent
191*           flush-to-zero threshhold and handle numbers above that
192*           threshhold correctly, it would not be necessary.
193*
194      IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
195         SUM = PP + SIGN( R, PP )
196         DIFF = PP - SIGN( R, PP )
197         WBIG = SHIFT + SUM
198*
199*        Compute smaller eigenvalue
200*
201         WSMALL = SHIFT + DIFF
202         IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
203            WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
204            WSMALL = WDET / WBIG
205         END IF
206*
207*        Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
208*        for WR1.
209*
210         IF( PP.GT.ABI22 ) THEN
211            WR1 = MIN( WBIG, WSMALL )
212            WR2 = MAX( WBIG, WSMALL )
213         ELSE
214            WR1 = MAX( WBIG, WSMALL )
215            WR2 = MIN( WBIG, WSMALL )
216         END IF
217         WI = ZERO
218      ELSE
219*
220*        Complex eigenvalues
221*
222         WR1 = SHIFT + PP
223         WR2 = WR1
224         WI = R
225      END IF
226*
227*     Further scaling to avoid underflow and overflow in computing
228*     SCALE1 and overflow in computing w*B.
229*
230*     This scale factor (WSCALE) is bounded from above using C1 and C2,
231*     and from below using C3 and C4.
232*        C1 implements the condition  s A  must never overflow.
233*        C2 implements the condition  w B  must never overflow.
234*        C3, with C2,
235*           implement the condition that s A - w B must never overflow.
236*        C4 implements the condition  s    should not underflow.
237*        C5 implements the condition  max(s,|w|) should be at least 2.
238*
239      C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
240      C2 = SAFMIN*MAX( ONE, BNORM )
241      C3 = BSIZE*SAFMIN
242      IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
243         C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
244      ELSE
245         C4 = ONE
246      END IF
247      IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
248         C5 = MIN( ONE, ASCALE*BSIZE )
249      ELSE
250         C5 = ONE
251      END IF
252*
253*     Scale first eigenvalue
254*
255      WABS = ABS( WR1 ) + ABS( WI )
256      WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
257     $        MIN( C4, HALF*MAX( WABS, C5 ) ) )
258      IF( WSIZE.NE.ONE ) THEN
259         WSCALE = ONE / WSIZE
260         IF( WSIZE.GT.ONE ) THEN
261            SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
262     $               MIN( ASCALE, BSIZE )
263         ELSE
264            SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
265     $               MAX( ASCALE, BSIZE )
266         END IF
267         WR1 = WR1*WSCALE
268         IF( WI.NE.ZERO ) THEN
269            WI = WI*WSCALE
270            WR2 = WR1
271            SCALE2 = SCALE1
272         END IF
273      ELSE
274         SCALE1 = ASCALE*BSIZE
275         SCALE2 = SCALE1
276      END IF
277*
278*     Scale second eigenvalue (if real)
279*
280      IF( WI.EQ.ZERO ) THEN
281         WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
282     $           MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
283         IF( WSIZE.NE.ONE ) THEN
284            WSCALE = ONE / WSIZE
285            IF( WSIZE.GT.ONE ) THEN
286               SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
287     $                  MIN( ASCALE, BSIZE )
288            ELSE
289               SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
290     $                  MAX( ASCALE, BSIZE )
291            END IF
292            WR2 = WR2*WSCALE
293         ELSE
294            SCALE2 = ASCALE*BSIZE
295         END IF
296      END IF
297*
298*     End of DLAG2
299*
300      RETURN
301      END
302