root / src / lapack / double / dlasq4.f @ 2
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| 1 | 1 | equemene | SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, |
|---|---|---|---|
| 2 | 1 | equemene | $ DN1, DN2, TAU, TTYPE, G ) |
| 3 | 1 | equemene | * |
| 4 | 1 | equemene | * -- LAPACK routine (version 3.2) -- |
| 5 | 1 | equemene | * |
| 6 | 1 | equemene | * -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
| 7 | 1 | equemene | * -- Laboratory and Beresford Parlett of the Univ. of California at -- |
| 8 | 1 | equemene | * -- Berkeley -- |
| 9 | 1 | equemene | * -- November 2008 -- |
| 10 | 1 | equemene | * |
| 11 | 1 | equemene | * -- LAPACK is a software package provided by Univ. of Tennessee, -- |
| 12 | 1 | equemene | * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
| 13 | 1 | equemene | * |
| 14 | 1 | equemene | * .. Scalar Arguments .. |
| 15 | 1 | equemene | INTEGER I0, N0, N0IN, PP, TTYPE |
| 16 | 1 | equemene | DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU |
| 17 | 1 | equemene | * .. |
| 18 | 1 | equemene | * .. Array Arguments .. |
| 19 | 1 | equemene | DOUBLE PRECISION Z( * ) |
| 20 | 1 | equemene | * .. |
| 21 | 1 | equemene | * |
| 22 | 1 | equemene | * Purpose |
| 23 | 1 | equemene | * ======= |
| 24 | 1 | equemene | * |
| 25 | 1 | equemene | * DLASQ4 computes an approximation TAU to the smallest eigenvalue |
| 26 | 1 | equemene | * using values of d from the previous transform. |
| 27 | 1 | equemene | * |
| 28 | 1 | equemene | * I0 (input) INTEGER |
| 29 | 1 | equemene | * First index. |
| 30 | 1 | equemene | * |
| 31 | 1 | equemene | * N0 (input) INTEGER |
| 32 | 1 | equemene | * Last index. |
| 33 | 1 | equemene | * |
| 34 | 1 | equemene | * Z (input) DOUBLE PRECISION array, dimension ( 4*N ) |
| 35 | 1 | equemene | * Z holds the qd array. |
| 36 | 1 | equemene | * |
| 37 | 1 | equemene | * PP (input) INTEGER |
| 38 | 1 | equemene | * PP=0 for ping, PP=1 for pong. |
| 39 | 1 | equemene | * |
| 40 | 1 | equemene | * NOIN (input) INTEGER |
| 41 | 1 | equemene | * The value of N0 at start of EIGTEST. |
| 42 | 1 | equemene | * |
| 43 | 1 | equemene | * DMIN (input) DOUBLE PRECISION |
| 44 | 1 | equemene | * Minimum value of d. |
| 45 | 1 | equemene | * |
| 46 | 1 | equemene | * DMIN1 (input) DOUBLE PRECISION |
| 47 | 1 | equemene | * Minimum value of d, excluding D( N0 ). |
| 48 | 1 | equemene | * |
| 49 | 1 | equemene | * DMIN2 (input) DOUBLE PRECISION |
| 50 | 1 | equemene | * Minimum value of d, excluding D( N0 ) and D( N0-1 ). |
| 51 | 1 | equemene | * |
| 52 | 1 | equemene | * DN (input) DOUBLE PRECISION |
| 53 | 1 | equemene | * d(N) |
| 54 | 1 | equemene | * |
| 55 | 1 | equemene | * DN1 (input) DOUBLE PRECISION |
| 56 | 1 | equemene | * d(N-1) |
| 57 | 1 | equemene | * |
| 58 | 1 | equemene | * DN2 (input) DOUBLE PRECISION |
| 59 | 1 | equemene | * d(N-2) |
| 60 | 1 | equemene | * |
| 61 | 1 | equemene | * TAU (output) DOUBLE PRECISION |
| 62 | 1 | equemene | * This is the shift. |
| 63 | 1 | equemene | * |
| 64 | 1 | equemene | * TTYPE (output) INTEGER |
| 65 | 1 | equemene | * Shift type. |
| 66 | 1 | equemene | * |
| 67 | 1 | equemene | * G (input/output) REAL |
| 68 | 1 | equemene | * G is passed as an argument in order to save its value between |
| 69 | 1 | equemene | * calls to DLASQ4. |
| 70 | 1 | equemene | * |
| 71 | 1 | equemene | * Further Details |
| 72 | 1 | equemene | * =============== |
| 73 | 1 | equemene | * CNST1 = 9/16 |
| 74 | 1 | equemene | * |
| 75 | 1 | equemene | * ===================================================================== |
| 76 | 1 | equemene | * |
| 77 | 1 | equemene | * .. Parameters .. |
| 78 | 1 | equemene | DOUBLE PRECISION CNST1, CNST2, CNST3 |
| 79 | 1 | equemene | PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0, |
| 80 | 1 | equemene | $ CNST3 = 1.050D0 ) |
| 81 | 1 | equemene | DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD |
| 82 | 1 | equemene | PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0, |
| 83 | 1 | equemene | $ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0, |
| 84 | 1 | equemene | $ TWO = 2.0D0, HUNDRD = 100.0D0 ) |
| 85 | 1 | equemene | * .. |
| 86 | 1 | equemene | * .. Local Scalars .. |
| 87 | 1 | equemene | INTEGER I4, NN, NP |
| 88 | 1 | equemene | DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S |
| 89 | 1 | equemene | * .. |
| 90 | 1 | equemene | * .. Intrinsic Functions .. |
| 91 | 1 | equemene | INTRINSIC MAX, MIN, SQRT |
| 92 | 1 | equemene | * .. |
| 93 | 1 | equemene | * .. Executable Statements .. |
| 94 | 1 | equemene | * |
| 95 | 1 | equemene | * A negative DMIN forces the shift to take that absolute value |
| 96 | 1 | equemene | * TTYPE records the type of shift. |
| 97 | 1 | equemene | * |
| 98 | 1 | equemene | IF( DMIN.LE.ZERO ) THEN |
| 99 | 1 | equemene | TAU = -DMIN |
| 100 | 1 | equemene | TTYPE = -1 |
| 101 | 1 | equemene | RETURN |
| 102 | 1 | equemene | END IF |
| 103 | 1 | equemene | * |
| 104 | 1 | equemene | NN = 4*N0 + PP |
| 105 | 1 | equemene | IF( N0IN.EQ.N0 ) THEN |
| 106 | 1 | equemene | * |
| 107 | 1 | equemene | * No eigenvalues deflated. |
| 108 | 1 | equemene | * |
| 109 | 1 | equemene | IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN |
| 110 | 1 | equemene | * |
| 111 | 1 | equemene | B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) ) |
| 112 | 1 | equemene | B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) ) |
| 113 | 1 | equemene | A2 = Z( NN-7 ) + Z( NN-5 ) |
| 114 | 1 | equemene | * |
| 115 | 1 | equemene | * Cases 2 and 3. |
| 116 | 1 | equemene | * |
| 117 | 1 | equemene | IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN |
| 118 | 1 | equemene | GAP2 = DMIN2 - A2 - DMIN2*QURTR |
| 119 | 1 | equemene | IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN |
| 120 | 1 | equemene | GAP1 = A2 - DN - ( B2 / GAP2 )*B2 |
| 121 | 1 | equemene | ELSE |
| 122 | 1 | equemene | GAP1 = A2 - DN - ( B1+B2 ) |
| 123 | 1 | equemene | END IF |
| 124 | 1 | equemene | IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN |
| 125 | 1 | equemene | S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN ) |
| 126 | 1 | equemene | TTYPE = -2 |
| 127 | 1 | equemene | ELSE |
| 128 | 1 | equemene | S = ZERO |
| 129 | 1 | equemene | IF( DN.GT.B1 ) |
| 130 | 1 | equemene | $ S = DN - B1 |
| 131 | 1 | equemene | IF( A2.GT.( B1+B2 ) ) |
| 132 | 1 | equemene | $ S = MIN( S, A2-( B1+B2 ) ) |
| 133 | 1 | equemene | S = MAX( S, THIRD*DMIN ) |
| 134 | 1 | equemene | TTYPE = -3 |
| 135 | 1 | equemene | END IF |
| 136 | 1 | equemene | ELSE |
| 137 | 1 | equemene | * |
| 138 | 1 | equemene | * Case 4. |
| 139 | 1 | equemene | * |
| 140 | 1 | equemene | TTYPE = -4 |
| 141 | 1 | equemene | S = QURTR*DMIN |
| 142 | 1 | equemene | IF( DMIN.EQ.DN ) THEN |
| 143 | 1 | equemene | GAM = DN |
| 144 | 1 | equemene | A2 = ZERO |
| 145 | 1 | equemene | IF( Z( NN-5 ) .GT. Z( NN-7 ) ) |
| 146 | 1 | equemene | $ RETURN |
| 147 | 1 | equemene | B2 = Z( NN-5 ) / Z( NN-7 ) |
| 148 | 1 | equemene | NP = NN - 9 |
| 149 | 1 | equemene | ELSE |
| 150 | 1 | equemene | NP = NN - 2*PP |
| 151 | 1 | equemene | B2 = Z( NP-2 ) |
| 152 | 1 | equemene | GAM = DN1 |
| 153 | 1 | equemene | IF( Z( NP-4 ) .GT. Z( NP-2 ) ) |
| 154 | 1 | equemene | $ RETURN |
| 155 | 1 | equemene | A2 = Z( NP-4 ) / Z( NP-2 ) |
| 156 | 1 | equemene | IF( Z( NN-9 ) .GT. Z( NN-11 ) ) |
| 157 | 1 | equemene | $ RETURN |
| 158 | 1 | equemene | B2 = Z( NN-9 ) / Z( NN-11 ) |
| 159 | 1 | equemene | NP = NN - 13 |
| 160 | 1 | equemene | END IF |
| 161 | 1 | equemene | * |
| 162 | 1 | equemene | * Approximate contribution to norm squared from I < NN-1. |
| 163 | 1 | equemene | * |
| 164 | 1 | equemene | A2 = A2 + B2 |
| 165 | 1 | equemene | DO 10 I4 = NP, 4*I0 - 1 + PP, -4 |
| 166 | 1 | equemene | IF( B2.EQ.ZERO ) |
| 167 | 1 | equemene | $ GO TO 20 |
| 168 | 1 | equemene | B1 = B2 |
| 169 | 1 | equemene | IF( Z( I4 ) .GT. Z( I4-2 ) ) |
| 170 | 1 | equemene | $ RETURN |
| 171 | 1 | equemene | B2 = B2*( Z( I4 ) / Z( I4-2 ) ) |
| 172 | 1 | equemene | A2 = A2 + B2 |
| 173 | 1 | equemene | IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) |
| 174 | 1 | equemene | $ GO TO 20 |
| 175 | 1 | equemene | 10 CONTINUE |
| 176 | 1 | equemene | 20 CONTINUE |
| 177 | 1 | equemene | A2 = CNST3*A2 |
| 178 | 1 | equemene | * |
| 179 | 1 | equemene | * Rayleigh quotient residual bound. |
| 180 | 1 | equemene | * |
| 181 | 1 | equemene | IF( A2.LT.CNST1 ) |
| 182 | 1 | equemene | $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) |
| 183 | 1 | equemene | END IF |
| 184 | 1 | equemene | ELSE IF( DMIN.EQ.DN2 ) THEN |
| 185 | 1 | equemene | * |
| 186 | 1 | equemene | * Case 5. |
| 187 | 1 | equemene | * |
| 188 | 1 | equemene | TTYPE = -5 |
| 189 | 1 | equemene | S = QURTR*DMIN |
| 190 | 1 | equemene | * |
| 191 | 1 | equemene | * Compute contribution to norm squared from I > NN-2. |
| 192 | 1 | equemene | * |
| 193 | 1 | equemene | NP = NN - 2*PP |
| 194 | 1 | equemene | B1 = Z( NP-2 ) |
| 195 | 1 | equemene | B2 = Z( NP-6 ) |
| 196 | 1 | equemene | GAM = DN2 |
| 197 | 1 | equemene | IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 ) |
| 198 | 1 | equemene | $ RETURN |
| 199 | 1 | equemene | A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 ) |
| 200 | 1 | equemene | * |
| 201 | 1 | equemene | * Approximate contribution to norm squared from I < NN-2. |
| 202 | 1 | equemene | * |
| 203 | 1 | equemene | IF( N0-I0.GT.2 ) THEN |
| 204 | 1 | equemene | B2 = Z( NN-13 ) / Z( NN-15 ) |
| 205 | 1 | equemene | A2 = A2 + B2 |
| 206 | 1 | equemene | DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4 |
| 207 | 1 | equemene | IF( B2.EQ.ZERO ) |
| 208 | 1 | equemene | $ GO TO 40 |
| 209 | 1 | equemene | B1 = B2 |
| 210 | 1 | equemene | IF( Z( I4 ) .GT. Z( I4-2 ) ) |
| 211 | 1 | equemene | $ RETURN |
| 212 | 1 | equemene | B2 = B2*( Z( I4 ) / Z( I4-2 ) ) |
| 213 | 1 | equemene | A2 = A2 + B2 |
| 214 | 1 | equemene | IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) |
| 215 | 1 | equemene | $ GO TO 40 |
| 216 | 1 | equemene | 30 CONTINUE |
| 217 | 1 | equemene | 40 CONTINUE |
| 218 | 1 | equemene | A2 = CNST3*A2 |
| 219 | 1 | equemene | END IF |
| 220 | 1 | equemene | * |
| 221 | 1 | equemene | IF( A2.LT.CNST1 ) |
| 222 | 1 | equemene | $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) |
| 223 | 1 | equemene | ELSE |
| 224 | 1 | equemene | * |
| 225 | 1 | equemene | * Case 6, no information to guide us. |
| 226 | 1 | equemene | * |
| 227 | 1 | equemene | IF( TTYPE.EQ.-6 ) THEN |
| 228 | 1 | equemene | G = G + THIRD*( ONE-G ) |
| 229 | 1 | equemene | ELSE IF( TTYPE.EQ.-18 ) THEN |
| 230 | 1 | equemene | G = QURTR*THIRD |
| 231 | 1 | equemene | ELSE |
| 232 | 1 | equemene | G = QURTR |
| 233 | 1 | equemene | END IF |
| 234 | 1 | equemene | S = G*DMIN |
| 235 | 1 | equemene | TTYPE = -6 |
| 236 | 1 | equemene | END IF |
| 237 | 1 | equemene | * |
| 238 | 1 | equemene | ELSE IF( N0IN.EQ.( N0+1 ) ) THEN |
| 239 | 1 | equemene | * |
| 240 | 1 | equemene | * One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN. |
| 241 | 1 | equemene | * |
| 242 | 1 | equemene | IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN |
| 243 | 1 | equemene | * |
| 244 | 1 | equemene | * Cases 7 and 8. |
| 245 | 1 | equemene | * |
| 246 | 1 | equemene | TTYPE = -7 |
| 247 | 1 | equemene | S = THIRD*DMIN1 |
| 248 | 1 | equemene | IF( Z( NN-5 ).GT.Z( NN-7 ) ) |
| 249 | 1 | equemene | $ RETURN |
| 250 | 1 | equemene | B1 = Z( NN-5 ) / Z( NN-7 ) |
| 251 | 1 | equemene | B2 = B1 |
| 252 | 1 | equemene | IF( B2.EQ.ZERO ) |
| 253 | 1 | equemene | $ GO TO 60 |
| 254 | 1 | equemene | DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 |
| 255 | 1 | equemene | A2 = B1 |
| 256 | 1 | equemene | IF( Z( I4 ).GT.Z( I4-2 ) ) |
| 257 | 1 | equemene | $ RETURN |
| 258 | 1 | equemene | B1 = B1*( Z( I4 ) / Z( I4-2 ) ) |
| 259 | 1 | equemene | B2 = B2 + B1 |
| 260 | 1 | equemene | IF( HUNDRD*MAX( B1, A2 ).LT.B2 ) |
| 261 | 1 | equemene | $ GO TO 60 |
| 262 | 1 | equemene | 50 CONTINUE |
| 263 | 1 | equemene | 60 CONTINUE |
| 264 | 1 | equemene | B2 = SQRT( CNST3*B2 ) |
| 265 | 1 | equemene | A2 = DMIN1 / ( ONE+B2**2 ) |
| 266 | 1 | equemene | GAP2 = HALF*DMIN2 - A2 |
| 267 | 1 | equemene | IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN |
| 268 | 1 | equemene | S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) |
| 269 | 1 | equemene | ELSE |
| 270 | 1 | equemene | S = MAX( S, A2*( ONE-CNST2*B2 ) ) |
| 271 | 1 | equemene | TTYPE = -8 |
| 272 | 1 | equemene | END IF |
| 273 | 1 | equemene | ELSE |
| 274 | 1 | equemene | * |
| 275 | 1 | equemene | * Case 9. |
| 276 | 1 | equemene | * |
| 277 | 1 | equemene | S = QURTR*DMIN1 |
| 278 | 1 | equemene | IF( DMIN1.EQ.DN1 ) |
| 279 | 1 | equemene | $ S = HALF*DMIN1 |
| 280 | 1 | equemene | TTYPE = -9 |
| 281 | 1 | equemene | END IF |
| 282 | 1 | equemene | * |
| 283 | 1 | equemene | ELSE IF( N0IN.EQ.( N0+2 ) ) THEN |
| 284 | 1 | equemene | * |
| 285 | 1 | equemene | * Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN. |
| 286 | 1 | equemene | * |
| 287 | 1 | equemene | * Cases 10 and 11. |
| 288 | 1 | equemene | * |
| 289 | 1 | equemene | IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN |
| 290 | 1 | equemene | TTYPE = -10 |
| 291 | 1 | equemene | S = THIRD*DMIN2 |
| 292 | 1 | equemene | IF( Z( NN-5 ).GT.Z( NN-7 ) ) |
| 293 | 1 | equemene | $ RETURN |
| 294 | 1 | equemene | B1 = Z( NN-5 ) / Z( NN-7 ) |
| 295 | 1 | equemene | B2 = B1 |
| 296 | 1 | equemene | IF( B2.EQ.ZERO ) |
| 297 | 1 | equemene | $ GO TO 80 |
| 298 | 1 | equemene | DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 |
| 299 | 1 | equemene | IF( Z( I4 ).GT.Z( I4-2 ) ) |
| 300 | 1 | equemene | $ RETURN |
| 301 | 1 | equemene | B1 = B1*( Z( I4 ) / Z( I4-2 ) ) |
| 302 | 1 | equemene | B2 = B2 + B1 |
| 303 | 1 | equemene | IF( HUNDRD*B1.LT.B2 ) |
| 304 | 1 | equemene | $ GO TO 80 |
| 305 | 1 | equemene | 70 CONTINUE |
| 306 | 1 | equemene | 80 CONTINUE |
| 307 | 1 | equemene | B2 = SQRT( CNST3*B2 ) |
| 308 | 1 | equemene | A2 = DMIN2 / ( ONE+B2**2 ) |
| 309 | 1 | equemene | GAP2 = Z( NN-7 ) + Z( NN-9 ) - |
| 310 | 1 | equemene | $ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2 |
| 311 | 1 | equemene | IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN |
| 312 | 1 | equemene | S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) |
| 313 | 1 | equemene | ELSE |
| 314 | 1 | equemene | S = MAX( S, A2*( ONE-CNST2*B2 ) ) |
| 315 | 1 | equemene | END IF |
| 316 | 1 | equemene | ELSE |
| 317 | 1 | equemene | S = QURTR*DMIN2 |
| 318 | 1 | equemene | TTYPE = -11 |
| 319 | 1 | equemene | END IF |
| 320 | 1 | equemene | ELSE IF( N0IN.GT.( N0+2 ) ) THEN |
| 321 | 1 | equemene | * |
| 322 | 1 | equemene | * Case 12, more than two eigenvalues deflated. No information. |
| 323 | 1 | equemene | * |
| 324 | 1 | equemene | S = ZERO |
| 325 | 1 | equemene | TTYPE = -12 |
| 326 | 1 | equemene | END IF |
| 327 | 1 | equemene | * |
| 328 | 1 | equemene | TAU = S |
| 329 | 1 | equemene | RETURN |
| 330 | 1 | equemene | * |
| 331 | 1 | equemene | * End of DLASQ4 |
| 332 | 1 | equemene | * |
| 333 | 1 | equemene | END |