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