root / src / lapack / double / dlaed6.f @ 1
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| 1 | 1 | equemene | SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO ) |
|---|---|---|---|
| 2 | 1 | equemene | * |
| 3 | 1 | equemene | * -- LAPACK routine (version 3.2) -- |
| 4 | 1 | equemene | * -- LAPACK is a software package provided by Univ. of Tennessee, -- |
| 5 | 1 | equemene | * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
| 6 | 1 | equemene | * February 2007 |
| 7 | 1 | equemene | * |
| 8 | 1 | equemene | * .. Scalar Arguments .. |
| 9 | 1 | equemene | LOGICAL ORGATI |
| 10 | 1 | equemene | INTEGER INFO, KNITER |
| 11 | 1 | equemene | DOUBLE PRECISION FINIT, RHO, TAU |
| 12 | 1 | equemene | * .. |
| 13 | 1 | equemene | * .. Array Arguments .. |
| 14 | 1 | equemene | DOUBLE PRECISION D( 3 ), Z( 3 ) |
| 15 | 1 | equemene | * .. |
| 16 | 1 | equemene | * |
| 17 | 1 | equemene | * Purpose |
| 18 | 1 | equemene | * ======= |
| 19 | 1 | equemene | * |
| 20 | 1 | equemene | * DLAED6 computes the positive or negative root (closest to the origin) |
| 21 | 1 | equemene | * of |
| 22 | 1 | equemene | * z(1) z(2) z(3) |
| 23 | 1 | equemene | * f(x) = rho + --------- + ---------- + --------- |
| 24 | 1 | equemene | * d(1)-x d(2)-x d(3)-x |
| 25 | 1 | equemene | * |
| 26 | 1 | equemene | * It is assumed that |
| 27 | 1 | equemene | * |
| 28 | 1 | equemene | * if ORGATI = .true. the root is between d(2) and d(3); |
| 29 | 1 | equemene | * otherwise it is between d(1) and d(2) |
| 30 | 1 | equemene | * |
| 31 | 1 | equemene | * This routine will be called by DLAED4 when necessary. In most cases, |
| 32 | 1 | equemene | * the root sought is the smallest in magnitude, though it might not be |
| 33 | 1 | equemene | * in some extremely rare situations. |
| 34 | 1 | equemene | * |
| 35 | 1 | equemene | * Arguments |
| 36 | 1 | equemene | * ========= |
| 37 | 1 | equemene | * |
| 38 | 1 | equemene | * KNITER (input) INTEGER |
| 39 | 1 | equemene | * Refer to DLAED4 for its significance. |
| 40 | 1 | equemene | * |
| 41 | 1 | equemene | * ORGATI (input) LOGICAL |
| 42 | 1 | equemene | * If ORGATI is true, the needed root is between d(2) and |
| 43 | 1 | equemene | * d(3); otherwise it is between d(1) and d(2). See |
| 44 | 1 | equemene | * DLAED4 for further details. |
| 45 | 1 | equemene | * |
| 46 | 1 | equemene | * RHO (input) DOUBLE PRECISION |
| 47 | 1 | equemene | * Refer to the equation f(x) above. |
| 48 | 1 | equemene | * |
| 49 | 1 | equemene | * D (input) DOUBLE PRECISION array, dimension (3) |
| 50 | 1 | equemene | * D satisfies d(1) < d(2) < d(3). |
| 51 | 1 | equemene | * |
| 52 | 1 | equemene | * Z (input) DOUBLE PRECISION array, dimension (3) |
| 53 | 1 | equemene | * Each of the elements in z must be positive. |
| 54 | 1 | equemene | * |
| 55 | 1 | equemene | * FINIT (input) DOUBLE PRECISION |
| 56 | 1 | equemene | * The value of f at 0. It is more accurate than the one |
| 57 | 1 | equemene | * evaluated inside this routine (if someone wants to do |
| 58 | 1 | equemene | * so). |
| 59 | 1 | equemene | * |
| 60 | 1 | equemene | * TAU (output) DOUBLE PRECISION |
| 61 | 1 | equemene | * The root of the equation f(x). |
| 62 | 1 | equemene | * |
| 63 | 1 | equemene | * INFO (output) INTEGER |
| 64 | 1 | equemene | * = 0: successful exit |
| 65 | 1 | equemene | * > 0: if INFO = 1, failure to converge |
| 66 | 1 | equemene | * |
| 67 | 1 | equemene | * Further Details |
| 68 | 1 | equemene | * =============== |
| 69 | 1 | equemene | * |
| 70 | 1 | equemene | * 30/06/99: Based on contributions by |
| 71 | 1 | equemene | * Ren-Cang Li, Computer Science Division, University of California |
| 72 | 1 | equemene | * at Berkeley, USA |
| 73 | 1 | equemene | * |
| 74 | 1 | equemene | * 10/02/03: This version has a few statements commented out for thread |
| 75 | 1 | equemene | * safety (machine parameters are computed on each entry). SJH. |
| 76 | 1 | equemene | * |
| 77 | 1 | equemene | * 05/10/06: Modified from a new version of Ren-Cang Li, use |
| 78 | 1 | equemene | * Gragg-Thornton-Warner cubic convergent scheme for better stability. |
| 79 | 1 | equemene | * |
| 80 | 1 | equemene | * ===================================================================== |
| 81 | 1 | equemene | * |
| 82 | 1 | equemene | * .. Parameters .. |
| 83 | 1 | equemene | INTEGER MAXIT |
| 84 | 1 | equemene | PARAMETER ( MAXIT = 40 ) |
| 85 | 1 | equemene | DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT |
| 86 | 1 | equemene | PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, |
| 87 | 1 | equemene | $ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 ) |
| 88 | 1 | equemene | * .. |
| 89 | 1 | equemene | * .. External Functions .. |
| 90 | 1 | equemene | DOUBLE PRECISION DLAMCH |
| 91 | 1 | equemene | EXTERNAL DLAMCH |
| 92 | 1 | equemene | * .. |
| 93 | 1 | equemene | * .. Local Arrays .. |
| 94 | 1 | equemene | DOUBLE PRECISION DSCALE( 3 ), ZSCALE( 3 ) |
| 95 | 1 | equemene | * .. |
| 96 | 1 | equemene | * .. Local Scalars .. |
| 97 | 1 | equemene | LOGICAL SCALE |
| 98 | 1 | equemene | INTEGER I, ITER, NITER |
| 99 | 1 | equemene | DOUBLE PRECISION A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F, |
| 100 | 1 | equemene | $ FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1, |
| 101 | 1 | equemene | $ SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4, |
| 102 | 1 | equemene | $ LBD, UBD |
| 103 | 1 | equemene | * .. |
| 104 | 1 | equemene | * .. Intrinsic Functions .. |
| 105 | 1 | equemene | INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT |
| 106 | 1 | equemene | * .. |
| 107 | 1 | equemene | * .. Executable Statements .. |
| 108 | 1 | equemene | * |
| 109 | 1 | equemene | INFO = 0 |
| 110 | 1 | equemene | * |
| 111 | 1 | equemene | IF( ORGATI ) THEN |
| 112 | 1 | equemene | LBD = D(2) |
| 113 | 1 | equemene | UBD = D(3) |
| 114 | 1 | equemene | ELSE |
| 115 | 1 | equemene | LBD = D(1) |
| 116 | 1 | equemene | UBD = D(2) |
| 117 | 1 | equemene | END IF |
| 118 | 1 | equemene | IF( FINIT .LT. ZERO )THEN |
| 119 | 1 | equemene | LBD = ZERO |
| 120 | 1 | equemene | ELSE |
| 121 | 1 | equemene | UBD = ZERO |
| 122 | 1 | equemene | END IF |
| 123 | 1 | equemene | * |
| 124 | 1 | equemene | NITER = 1 |
| 125 | 1 | equemene | TAU = ZERO |
| 126 | 1 | equemene | IF( KNITER.EQ.2 ) THEN |
| 127 | 1 | equemene | IF( ORGATI ) THEN |
| 128 | 1 | equemene | TEMP = ( D( 3 )-D( 2 ) ) / TWO |
| 129 | 1 | equemene | C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP ) |
| 130 | 1 | equemene | A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 ) |
| 131 | 1 | equemene | B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 ) |
| 132 | 1 | equemene | ELSE |
| 133 | 1 | equemene | TEMP = ( D( 1 )-D( 2 ) ) / TWO |
| 134 | 1 | equemene | C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP ) |
| 135 | 1 | equemene | A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 ) |
| 136 | 1 | equemene | B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 ) |
| 137 | 1 | equemene | END IF |
| 138 | 1 | equemene | TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) ) |
| 139 | 1 | equemene | A = A / TEMP |
| 140 | 1 | equemene | B = B / TEMP |
| 141 | 1 | equemene | C = C / TEMP |
| 142 | 1 | equemene | IF( C.EQ.ZERO ) THEN |
| 143 | 1 | equemene | TAU = B / A |
| 144 | 1 | equemene | ELSE IF( A.LE.ZERO ) THEN |
| 145 | 1 | equemene | TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) |
| 146 | 1 | equemene | ELSE |
| 147 | 1 | equemene | TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) |
| 148 | 1 | equemene | END IF |
| 149 | 1 | equemene | IF( TAU .LT. LBD .OR. TAU .GT. UBD ) |
| 150 | 1 | equemene | $ TAU = ( LBD+UBD )/TWO |
| 151 | 1 | equemene | IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN |
| 152 | 1 | equemene | TAU = ZERO |
| 153 | 1 | equemene | ELSE |
| 154 | 1 | equemene | TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) + |
| 155 | 1 | equemene | $ TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) + |
| 156 | 1 | equemene | $ TAU*Z(3)/( D(3)*( D( 3 )-TAU ) ) |
| 157 | 1 | equemene | IF( TEMP .LE. ZERO )THEN |
| 158 | 1 | equemene | LBD = TAU |
| 159 | 1 | equemene | ELSE |
| 160 | 1 | equemene | UBD = TAU |
| 161 | 1 | equemene | END IF |
| 162 | 1 | equemene | IF( ABS( FINIT ).LE.ABS( TEMP ) ) |
| 163 | 1 | equemene | $ TAU = ZERO |
| 164 | 1 | equemene | END IF |
| 165 | 1 | equemene | END IF |
| 166 | 1 | equemene | * |
| 167 | 1 | equemene | * get machine parameters for possible scaling to avoid overflow |
| 168 | 1 | equemene | * |
| 169 | 1 | equemene | * modified by Sven: parameters SMALL1, SMINV1, SMALL2, |
| 170 | 1 | equemene | * SMINV2, EPS are not SAVEd anymore between one call to the |
| 171 | 1 | equemene | * others but recomputed at each call |
| 172 | 1 | equemene | * |
| 173 | 1 | equemene | EPS = DLAMCH( 'Epsilon' ) |
| 174 | 1 | equemene | BASE = DLAMCH( 'Base' ) |
| 175 | 1 | equemene | SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) / |
| 176 | 1 | equemene | $ THREE ) ) |
| 177 | 1 | equemene | SMINV1 = ONE / SMALL1 |
| 178 | 1 | equemene | SMALL2 = SMALL1*SMALL1 |
| 179 | 1 | equemene | SMINV2 = SMINV1*SMINV1 |
| 180 | 1 | equemene | * |
| 181 | 1 | equemene | * Determine if scaling of inputs necessary to avoid overflow |
| 182 | 1 | equemene | * when computing 1/TEMP**3 |
| 183 | 1 | equemene | * |
| 184 | 1 | equemene | IF( ORGATI ) THEN |
| 185 | 1 | equemene | TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) ) |
| 186 | 1 | equemene | ELSE |
| 187 | 1 | equemene | TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) ) |
| 188 | 1 | equemene | END IF |
| 189 | 1 | equemene | SCALE = .FALSE. |
| 190 | 1 | equemene | IF( TEMP.LE.SMALL1 ) THEN |
| 191 | 1 | equemene | SCALE = .TRUE. |
| 192 | 1 | equemene | IF( TEMP.LE.SMALL2 ) THEN |
| 193 | 1 | equemene | * |
| 194 | 1 | equemene | * Scale up by power of radix nearest 1/SAFMIN**(2/3) |
| 195 | 1 | equemene | * |
| 196 | 1 | equemene | SCLFAC = SMINV2 |
| 197 | 1 | equemene | SCLINV = SMALL2 |
| 198 | 1 | equemene | ELSE |
| 199 | 1 | equemene | * |
| 200 | 1 | equemene | * Scale up by power of radix nearest 1/SAFMIN**(1/3) |
| 201 | 1 | equemene | * |
| 202 | 1 | equemene | SCLFAC = SMINV1 |
| 203 | 1 | equemene | SCLINV = SMALL1 |
| 204 | 1 | equemene | END IF |
| 205 | 1 | equemene | * |
| 206 | 1 | equemene | * Scaling up safe because D, Z, TAU scaled elsewhere to be O(1) |
| 207 | 1 | equemene | * |
| 208 | 1 | equemene | DO 10 I = 1, 3 |
| 209 | 1 | equemene | DSCALE( I ) = D( I )*SCLFAC |
| 210 | 1 | equemene | ZSCALE( I ) = Z( I )*SCLFAC |
| 211 | 1 | equemene | 10 CONTINUE |
| 212 | 1 | equemene | TAU = TAU*SCLFAC |
| 213 | 1 | equemene | LBD = LBD*SCLFAC |
| 214 | 1 | equemene | UBD = UBD*SCLFAC |
| 215 | 1 | equemene | ELSE |
| 216 | 1 | equemene | * |
| 217 | 1 | equemene | * Copy D and Z to DSCALE and ZSCALE |
| 218 | 1 | equemene | * |
| 219 | 1 | equemene | DO 20 I = 1, 3 |
| 220 | 1 | equemene | DSCALE( I ) = D( I ) |
| 221 | 1 | equemene | ZSCALE( I ) = Z( I ) |
| 222 | 1 | equemene | 20 CONTINUE |
| 223 | 1 | equemene | END IF |
| 224 | 1 | equemene | * |
| 225 | 1 | equemene | FC = ZERO |
| 226 | 1 | equemene | DF = ZERO |
| 227 | 1 | equemene | DDF = ZERO |
| 228 | 1 | equemene | DO 30 I = 1, 3 |
| 229 | 1 | equemene | TEMP = ONE / ( DSCALE( I )-TAU ) |
| 230 | 1 | equemene | TEMP1 = ZSCALE( I )*TEMP |
| 231 | 1 | equemene | TEMP2 = TEMP1*TEMP |
| 232 | 1 | equemene | TEMP3 = TEMP2*TEMP |
| 233 | 1 | equemene | FC = FC + TEMP1 / DSCALE( I ) |
| 234 | 1 | equemene | DF = DF + TEMP2 |
| 235 | 1 | equemene | DDF = DDF + TEMP3 |
| 236 | 1 | equemene | 30 CONTINUE |
| 237 | 1 | equemene | F = FINIT + TAU*FC |
| 238 | 1 | equemene | * |
| 239 | 1 | equemene | IF( ABS( F ).LE.ZERO ) |
| 240 | 1 | equemene | $ GO TO 60 |
| 241 | 1 | equemene | IF( F .LE. ZERO )THEN |
| 242 | 1 | equemene | LBD = TAU |
| 243 | 1 | equemene | ELSE |
| 244 | 1 | equemene | UBD = TAU |
| 245 | 1 | equemene | END IF |
| 246 | 1 | equemene | * |
| 247 | 1 | equemene | * Iteration begins -- Use Gragg-Thornton-Warner cubic convergent |
| 248 | 1 | equemene | * scheme |
| 249 | 1 | equemene | * |
| 250 | 1 | equemene | * It is not hard to see that |
| 251 | 1 | equemene | * |
| 252 | 1 | equemene | * 1) Iterations will go up monotonically |
| 253 | 1 | equemene | * if FINIT < 0; |
| 254 | 1 | equemene | * |
| 255 | 1 | equemene | * 2) Iterations will go down monotonically |
| 256 | 1 | equemene | * if FINIT > 0. |
| 257 | 1 | equemene | * |
| 258 | 1 | equemene | ITER = NITER + 1 |
| 259 | 1 | equemene | * |
| 260 | 1 | equemene | DO 50 NITER = ITER, MAXIT |
| 261 | 1 | equemene | * |
| 262 | 1 | equemene | IF( ORGATI ) THEN |
| 263 | 1 | equemene | TEMP1 = DSCALE( 2 ) - TAU |
| 264 | 1 | equemene | TEMP2 = DSCALE( 3 ) - TAU |
| 265 | 1 | equemene | ELSE |
| 266 | 1 | equemene | TEMP1 = DSCALE( 1 ) - TAU |
| 267 | 1 | equemene | TEMP2 = DSCALE( 2 ) - TAU |
| 268 | 1 | equemene | END IF |
| 269 | 1 | equemene | A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF |
| 270 | 1 | equemene | B = TEMP1*TEMP2*F |
| 271 | 1 | equemene | C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF |
| 272 | 1 | equemene | TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) ) |
| 273 | 1 | equemene | A = A / TEMP |
| 274 | 1 | equemene | B = B / TEMP |
| 275 | 1 | equemene | C = C / TEMP |
| 276 | 1 | equemene | IF( C.EQ.ZERO ) THEN |
| 277 | 1 | equemene | ETA = B / A |
| 278 | 1 | equemene | ELSE IF( A.LE.ZERO ) THEN |
| 279 | 1 | equemene | ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) |
| 280 | 1 | equemene | ELSE |
| 281 | 1 | equemene | ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) |
| 282 | 1 | equemene | END IF |
| 283 | 1 | equemene | IF( F*ETA.GE.ZERO ) THEN |
| 284 | 1 | equemene | ETA = -F / DF |
| 285 | 1 | equemene | END IF |
| 286 | 1 | equemene | * |
| 287 | 1 | equemene | TAU = TAU + ETA |
| 288 | 1 | equemene | IF( TAU .LT. LBD .OR. TAU .GT. UBD ) |
| 289 | 1 | equemene | $ TAU = ( LBD + UBD )/TWO |
| 290 | 1 | equemene | * |
| 291 | 1 | equemene | FC = ZERO |
| 292 | 1 | equemene | ERRETM = ZERO |
| 293 | 1 | equemene | DF = ZERO |
| 294 | 1 | equemene | DDF = ZERO |
| 295 | 1 | equemene | DO 40 I = 1, 3 |
| 296 | 1 | equemene | TEMP = ONE / ( DSCALE( I )-TAU ) |
| 297 | 1 | equemene | TEMP1 = ZSCALE( I )*TEMP |
| 298 | 1 | equemene | TEMP2 = TEMP1*TEMP |
| 299 | 1 | equemene | TEMP3 = TEMP2*TEMP |
| 300 | 1 | equemene | TEMP4 = TEMP1 / DSCALE( I ) |
| 301 | 1 | equemene | FC = FC + TEMP4 |
| 302 | 1 | equemene | ERRETM = ERRETM + ABS( TEMP4 ) |
| 303 | 1 | equemene | DF = DF + TEMP2 |
| 304 | 1 | equemene | DDF = DDF + TEMP3 |
| 305 | 1 | equemene | 40 CONTINUE |
| 306 | 1 | equemene | F = FINIT + TAU*FC |
| 307 | 1 | equemene | ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) + |
| 308 | 1 | equemene | $ ABS( TAU )*DF |
| 309 | 1 | equemene | IF( ABS( F ).LE.EPS*ERRETM ) |
| 310 | 1 | equemene | $ GO TO 60 |
| 311 | 1 | equemene | IF( F .LE. ZERO )THEN |
| 312 | 1 | equemene | LBD = TAU |
| 313 | 1 | equemene | ELSE |
| 314 | 1 | equemene | UBD = TAU |
| 315 | 1 | equemene | END IF |
| 316 | 1 | equemene | 50 CONTINUE |
| 317 | 1 | equemene | INFO = 1 |
| 318 | 1 | equemene | 60 CONTINUE |
| 319 | 1 | equemene | * |
| 320 | 1 | equemene | * Undo scaling |
| 321 | 1 | equemene | * |
| 322 | 1 | equemene | IF( SCALE ) |
| 323 | 1 | equemene | $ TAU = TAU*SCLINV |
| 324 | 1 | equemene | RETURN |
| 325 | 1 | equemene | * |
| 326 | 1 | equemene | * End of DLAED6 |
| 327 | 1 | equemene | * |
| 328 | 1 | equemene | END |