001:       SUBROUTINE DLARUV( ISEED, N, X )
002: *
003: *  -- LAPACK auxiliary routine (version 3.2) --
004: *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
005: *     November 2006
006: *
007: *     .. Scalar Arguments ..
008:       INTEGER            N
009: *     ..
010: *     .. Array Arguments ..
011:       INTEGER            ISEED( 4 )
012:       DOUBLE PRECISION   X( N )
013: *     ..
014: *
015: *  Purpose
016: *  =======
017: *
018: *  DLARUV returns a vector of n random real numbers from a uniform (0,1)
019: *  distribution (n <= 128).
020: *
021: *  This is an auxiliary routine called by DLARNV and ZLARNV.
022: *
023: *  Arguments
024: *  =========
025: *
026: *  ISEED   (input/output) INTEGER array, dimension (4)
027: *          On entry, the seed of the random number generator; the array
028: *          elements must be between 0 and 4095, and ISEED(4) must be
029: *          odd.
030: *          On exit, the seed is updated.
031: *
032: *  N       (input) INTEGER
033: *          The number of random numbers to be generated. N <= 128.
034: *
035: *  X       (output) DOUBLE PRECISION array, dimension (N)
036: *          The generated random numbers.
037: *
038: *  Further Details
039: *  ===============
040: *
041: *  This routine uses a multiplicative congruential method with modulus
042: *  2**48 and multiplier 33952834046453 (see G.S.Fishman,
043: *  'Multiplicative congruential random number generators with modulus
044: *  2**b: an exhaustive analysis for b = 32 and a partial analysis for
045: *  b = 48', Math. Comp. 189, pp 331-344, 1990).
046: *
047: *  48-bit integers are stored in 4 integer array elements with 12 bits
048: *  per element. Hence the routine is portable across machines with
049: *  integers of 32 bits or more.
050: *
051: *  =====================================================================
052: *
053: *     .. Parameters ..
054:       DOUBLE PRECISION   ONE
055:       PARAMETER          ( ONE = 1.0D0 )
056:       INTEGER            LV, IPW2
057:       DOUBLE PRECISION   R
058:       PARAMETER          ( LV = 128, IPW2 = 4096, R = ONE / IPW2 )
059: *     ..
060: *     .. Local Scalars ..
061:       INTEGER            I, I1, I2, I3, I4, IT1, IT2, IT3, IT4, J
062: *     ..
063: *     .. Local Arrays ..
064:       INTEGER            MM( LV, 4 )
065: *     ..
066: *     .. Intrinsic Functions ..
067:       INTRINSIC          DBLE, MIN, MOD
068: *     ..
069: *     .. Data statements ..
070:       DATA               ( MM( 1, J ), J = 1, 4 ) / 494, 322, 2508,
071:      $                   2549 /
072:       DATA               ( MM( 2, J ), J = 1, 4 ) / 2637, 789, 3754,
073:      $                   1145 /
074:       DATA               ( MM( 3, J ), J = 1, 4 ) / 255, 1440, 1766,
075:      $                   2253 /
076:       DATA               ( MM( 4, J ), J = 1, 4 ) / 2008, 752, 3572,
077:      $                   305 /
078:       DATA               ( MM( 5, J ), J = 1, 4 ) / 1253, 2859, 2893,
079:      $                   3301 /
080:       DATA               ( MM( 6, J ), J = 1, 4 ) / 3344, 123, 307,
081:      $                   1065 /
082:       DATA               ( MM( 7, J ), J = 1, 4 ) / 4084, 1848, 1297,
083:      $                   3133 /
084:       DATA               ( MM( 8, J ), J = 1, 4 ) / 1739, 643, 3966,
085:      $                   2913 /
086:       DATA               ( MM( 9, J ), J = 1, 4 ) / 3143, 2405, 758,
087:      $                   3285 /
088:       DATA               ( MM( 10, J ), J = 1, 4 ) / 3468, 2638, 2598,
089:      $                   1241 /
090:       DATA               ( MM( 11, J ), J = 1, 4 ) / 688, 2344, 3406,
091:      $                   1197 /
092:       DATA               ( MM( 12, J ), J = 1, 4 ) / 1657, 46, 2922,
093:      $                   3729 /
094:       DATA               ( MM( 13, J ), J = 1, 4 ) / 1238, 3814, 1038,
095:      $                   2501 /
096:       DATA               ( MM( 14, J ), J = 1, 4 ) / 3166, 913, 2934,
097:      $                   1673 /
098:       DATA               ( MM( 15, J ), J = 1, 4 ) / 1292, 3649, 2091,
099:      $                   541 /
100:       DATA               ( MM( 16, J ), J = 1, 4 ) / 3422, 339, 2451,
101:      $                   2753 /
102:       DATA               ( MM( 17, J ), J = 1, 4 ) / 1270, 3808, 1580,
103:      $                   949 /
104:       DATA               ( MM( 18, J ), J = 1, 4 ) / 2016, 822, 1958,
105:      $                   2361 /
106:       DATA               ( MM( 19, J ), J = 1, 4 ) / 154, 2832, 2055,
107:      $                   1165 /
108:       DATA               ( MM( 20, J ), J = 1, 4 ) / 2862, 3078, 1507,
109:      $                   4081 /
110:       DATA               ( MM( 21, J ), J = 1, 4 ) / 697, 3633, 1078,
111:      $                   2725 /
112:       DATA               ( MM( 22, J ), J = 1, 4 ) / 1706, 2970, 3273,
113:      $                   3305 /
114:       DATA               ( MM( 23, J ), J = 1, 4 ) / 491, 637, 17,
115:      $                   3069 /
116:       DATA               ( MM( 24, J ), J = 1, 4 ) / 931, 2249, 854,
117:      $                   3617 /
118:       DATA               ( MM( 25, J ), J = 1, 4 ) / 1444, 2081, 2916,
119:      $                   3733 /
120:       DATA               ( MM( 26, J ), J = 1, 4 ) / 444, 4019, 3971,
121:      $                   409 /
122:       DATA               ( MM( 27, J ), J = 1, 4 ) / 3577, 1478, 2889,
123:      $                   2157 /
124:       DATA               ( MM( 28, J ), J = 1, 4 ) / 3944, 242, 3831,
125:      $                   1361 /
126:       DATA               ( MM( 29, J ), J = 1, 4 ) / 2184, 481, 2621,
127:      $                   3973 /
128:       DATA               ( MM( 30, J ), J = 1, 4 ) / 1661, 2075, 1541,
129:      $                   1865 /
130:       DATA               ( MM( 31, J ), J = 1, 4 ) / 3482, 4058, 893,
131:      $                   2525 /
132:       DATA               ( MM( 32, J ), J = 1, 4 ) / 657, 622, 736,
133:      $                   1409 /
134:       DATA               ( MM( 33, J ), J = 1, 4 ) / 3023, 3376, 3992,
135:      $                   3445 /
136:       DATA               ( MM( 34, J ), J = 1, 4 ) / 3618, 812, 787,
137:      $                   3577 /
138:       DATA               ( MM( 35, J ), J = 1, 4 ) / 1267, 234, 2125,
139:      $                   77 /
140:       DATA               ( MM( 36, J ), J = 1, 4 ) / 1828, 641, 2364,
141:      $                   3761 /
142:       DATA               ( MM( 37, J ), J = 1, 4 ) / 164, 4005, 2460,
143:      $                   2149 /
144:       DATA               ( MM( 38, J ), J = 1, 4 ) / 3798, 1122, 257,
145:      $                   1449 /
146:       DATA               ( MM( 39, J ), J = 1, 4 ) / 3087, 3135, 1574,
147:      $                   3005 /
148:       DATA               ( MM( 40, J ), J = 1, 4 ) / 2400, 2640, 3912,
149:      $                   225 /
150:       DATA               ( MM( 41, J ), J = 1, 4 ) / 2870, 2302, 1216,
151:      $                   85 /
152:       DATA               ( MM( 42, J ), J = 1, 4 ) / 3876, 40, 3248,
153:      $                   3673 /
154:       DATA               ( MM( 43, J ), J = 1, 4 ) / 1905, 1832, 3401,
155:      $                   3117 /
156:       DATA               ( MM( 44, J ), J = 1, 4 ) / 1593, 2247, 2124,
157:      $                   3089 /
158:       DATA               ( MM( 45, J ), J = 1, 4 ) / 1797, 2034, 2762,
159:      $                   1349 /
160:       DATA               ( MM( 46, J ), J = 1, 4 ) / 1234, 2637, 149,
161:      $                   2057 /
162:       DATA               ( MM( 47, J ), J = 1, 4 ) / 3460, 1287, 2245,
163:      $                   413 /
164:       DATA               ( MM( 48, J ), J = 1, 4 ) / 328, 1691, 166,
165:      $                   65 /
166:       DATA               ( MM( 49, J ), J = 1, 4 ) / 2861, 496, 466,
167:      $                   1845 /
168:       DATA               ( MM( 50, J ), J = 1, 4 ) / 1950, 1597, 4018,
169:      $                   697 /
170:       DATA               ( MM( 51, J ), J = 1, 4 ) / 617, 2394, 1399,
171:      $                   3085 /
172:       DATA               ( MM( 52, J ), J = 1, 4 ) / 2070, 2584, 190,
173:      $                   3441 /
174:       DATA               ( MM( 53, J ), J = 1, 4 ) / 3331, 1843, 2879,
175:      $                   1573 /
176:       DATA               ( MM( 54, J ), J = 1, 4 ) / 769, 336, 153,
177:      $                   3689 /
178:       DATA               ( MM( 55, J ), J = 1, 4 ) / 1558, 1472, 2320,
179:      $                   2941 /
180:       DATA               ( MM( 56, J ), J = 1, 4 ) / 2412, 2407, 18,
181:      $                   929 /
182:       DATA               ( MM( 57, J ), J = 1, 4 ) / 2800, 433, 712,
183:      $                   533 /
184:       DATA               ( MM( 58, J ), J = 1, 4 ) / 189, 2096, 2159,
185:      $                   2841 /
186:       DATA               ( MM( 59, J ), J = 1, 4 ) / 287, 1761, 2318,
187:      $                   4077 /
188:       DATA               ( MM( 60, J ), J = 1, 4 ) / 2045, 2810, 2091,
189:      $                   721 /
190:       DATA               ( MM( 61, J ), J = 1, 4 ) / 1227, 566, 3443,
191:      $                   2821 /
192:       DATA               ( MM( 62, J ), J = 1, 4 ) / 2838, 442, 1510,
193:      $                   2249 /
194:       DATA               ( MM( 63, J ), J = 1, 4 ) / 209, 41, 449,
195:      $                   2397 /
196:       DATA               ( MM( 64, J ), J = 1, 4 ) / 2770, 1238, 1956,
197:      $                   2817 /
198:       DATA               ( MM( 65, J ), J = 1, 4 ) / 3654, 1086, 2201,
199:      $                   245 /
200:       DATA               ( MM( 66, J ), J = 1, 4 ) / 3993, 603, 3137,
201:      $                   1913 /
202:       DATA               ( MM( 67, J ), J = 1, 4 ) / 192, 840, 3399,
203:      $                   1997 /
204:       DATA               ( MM( 68, J ), J = 1, 4 ) / 2253, 3168, 1321,
205:      $                   3121 /
206:       DATA               ( MM( 69, J ), J = 1, 4 ) / 3491, 1499, 2271,
207:      $                   997 /
208:       DATA               ( MM( 70, J ), J = 1, 4 ) / 2889, 1084, 3667,
209:      $                   1833 /
210:       DATA               ( MM( 71, J ), J = 1, 4 ) / 2857, 3438, 2703,
211:      $                   2877 /
212:       DATA               ( MM( 72, J ), J = 1, 4 ) / 2094, 2408, 629,
213:      $                   1633 /
214:       DATA               ( MM( 73, J ), J = 1, 4 ) / 1818, 1589, 2365,
215:      $                   981 /
216:       DATA               ( MM( 74, J ), J = 1, 4 ) / 688, 2391, 2431,
217:      $                   2009 /
218:       DATA               ( MM( 75, J ), J = 1, 4 ) / 1407, 288, 1113,
219:      $                   941 /
220:       DATA               ( MM( 76, J ), J = 1, 4 ) / 634, 26, 3922,
221:      $                   2449 /
222:       DATA               ( MM( 77, J ), J = 1, 4 ) / 3231, 512, 2554,
223:      $                   197 /
224:       DATA               ( MM( 78, J ), J = 1, 4 ) / 815, 1456, 184,
225:      $                   2441 /
226:       DATA               ( MM( 79, J ), J = 1, 4 ) / 3524, 171, 2099,
227:      $                   285 /
228:       DATA               ( MM( 80, J ), J = 1, 4 ) / 1914, 1677, 3228,
229:      $                   1473 /
230:       DATA               ( MM( 81, J ), J = 1, 4 ) / 516, 2657, 4012,
231:      $                   2741 /
232:       DATA               ( MM( 82, J ), J = 1, 4 ) / 164, 2270, 1921,
233:      $                   3129 /
234:       DATA               ( MM( 83, J ), J = 1, 4 ) / 303, 2587, 3452,
235:      $                   909 /
236:       DATA               ( MM( 84, J ), J = 1, 4 ) / 2144, 2961, 3901,
237:      $                   2801 /
238:       DATA               ( MM( 85, J ), J = 1, 4 ) / 3480, 1970, 572,
239:      $                   421 /
240:       DATA               ( MM( 86, J ), J = 1, 4 ) / 119, 1817, 3309,
241:      $                   4073 /
242:       DATA               ( MM( 87, J ), J = 1, 4 ) / 3357, 676, 3171,
243:      $                   2813 /
244:       DATA               ( MM( 88, J ), J = 1, 4 ) / 837, 1410, 817,
245:      $                   2337 /
246:       DATA               ( MM( 89, J ), J = 1, 4 ) / 2826, 3723, 3039,
247:      $                   1429 /
248:       DATA               ( MM( 90, J ), J = 1, 4 ) / 2332, 2803, 1696,
249:      $                   1177 /
250:       DATA               ( MM( 91, J ), J = 1, 4 ) / 2089, 3185, 1256,
251:      $                   1901 /
252:       DATA               ( MM( 92, J ), J = 1, 4 ) / 3780, 184, 3715,
253:      $                   81 /
254:       DATA               ( MM( 93, J ), J = 1, 4 ) / 1700, 663, 2077,
255:      $                   1669 /
256:       DATA               ( MM( 94, J ), J = 1, 4 ) / 3712, 499, 3019,
257:      $                   2633 /
258:       DATA               ( MM( 95, J ), J = 1, 4 ) / 150, 3784, 1497,
259:      $                   2269 /
260:       DATA               ( MM( 96, J ), J = 1, 4 ) / 2000, 1631, 1101,
261:      $                   129 /
262:       DATA               ( MM( 97, J ), J = 1, 4 ) / 3375, 1925, 717,
263:      $                   1141 /
264:       DATA               ( MM( 98, J ), J = 1, 4 ) / 1621, 3912, 51,
265:      $                   249 /
266:       DATA               ( MM( 99, J ), J = 1, 4 ) / 3090, 1398, 981,
267:      $                   3917 /
268:       DATA               ( MM( 100, J ), J = 1, 4 ) / 3765, 1349, 1978,
269:      $                   2481 /
270:       DATA               ( MM( 101, J ), J = 1, 4 ) / 1149, 1441, 1813,
271:      $                   3941 /
272:       DATA               ( MM( 102, J ), J = 1, 4 ) / 3146, 2224, 3881,
273:      $                   2217 /
274:       DATA               ( MM( 103, J ), J = 1, 4 ) / 33, 2411, 76,
275:      $                   2749 /
276:       DATA               ( MM( 104, J ), J = 1, 4 ) / 3082, 1907, 3846,
277:      $                   3041 /
278:       DATA               ( MM( 105, J ), J = 1, 4 ) / 2741, 3192, 3694,
279:      $                   1877 /
280:       DATA               ( MM( 106, J ), J = 1, 4 ) / 359, 2786, 1682,
281:      $                   345 /
282:       DATA               ( MM( 107, J ), J = 1, 4 ) / 3316, 382, 124,
283:      $                   2861 /
284:       DATA               ( MM( 108, J ), J = 1, 4 ) / 1749, 37, 1660,
285:      $                   1809 /
286:       DATA               ( MM( 109, J ), J = 1, 4 ) / 185, 759, 3997,
287:      $                   3141 /
288:       DATA               ( MM( 110, J ), J = 1, 4 ) / 2784, 2948, 479,
289:      $                   2825 /
290:       DATA               ( MM( 111, J ), J = 1, 4 ) / 2202, 1862, 1141,
291:      $                   157 /
292:       DATA               ( MM( 112, J ), J = 1, 4 ) / 2199, 3802, 886,
293:      $                   2881 /
294:       DATA               ( MM( 113, J ), J = 1, 4 ) / 1364, 2423, 3514,
295:      $                   3637 /
296:       DATA               ( MM( 114, J ), J = 1, 4 ) / 1244, 2051, 1301,
297:      $                   1465 /
298:       DATA               ( MM( 115, J ), J = 1, 4 ) / 2020, 2295, 3604,
299:      $                   2829 /
300:       DATA               ( MM( 116, J ), J = 1, 4 ) / 3160, 1332, 1888,
301:      $                   2161 /
302:       DATA               ( MM( 117, J ), J = 1, 4 ) / 2785, 1832, 1836,
303:      $                   3365 /
304:       DATA               ( MM( 118, J ), J = 1, 4 ) / 2772, 2405, 1990,
305:      $                   361 /
306:       DATA               ( MM( 119, J ), J = 1, 4 ) / 1217, 3638, 2058,
307:      $                   2685 /
308:       DATA               ( MM( 120, J ), J = 1, 4 ) / 1822, 3661, 692,
309:      $                   3745 /
310:       DATA               ( MM( 121, J ), J = 1, 4 ) / 1245, 327, 1194,
311:      $                   2325 /
312:       DATA               ( MM( 122, J ), J = 1, 4 ) / 2252, 3660, 20,
313:      $                   3609 /
314:       DATA               ( MM( 123, J ), J = 1, 4 ) / 3904, 716, 3285,
315:      $                   3821 /
316:       DATA               ( MM( 124, J ), J = 1, 4 ) / 2774, 1842, 2046,
317:      $                   3537 /
318:       DATA               ( MM( 125, J ), J = 1, 4 ) / 997, 3987, 2107,
319:      $                   517 /
320:       DATA               ( MM( 126, J ), J = 1, 4 ) / 2573, 1368, 3508,
321:      $                   3017 /
322:       DATA               ( MM( 127, J ), J = 1, 4 ) / 1148, 1848, 3525,
323:      $                   2141 /
324:       DATA               ( MM( 128, J ), J = 1, 4 ) / 545, 2366, 3801,
325:      $                   1537 /
326: *     ..
327: *     .. Executable Statements ..
328: *
329:       I1 = ISEED( 1 )
330:       I2 = ISEED( 2 )
331:       I3 = ISEED( 3 )
332:       I4 = ISEED( 4 )
333: *
334:       DO 10 I = 1, MIN( N, LV )
335: *         
336:   20     CONTINUE
337: *
338: *        Multiply the seed by i-th power of the multiplier modulo 2**48
339: *
340:          IT4 = I4*MM( I, 4 )
341:          IT3 = IT4 / IPW2
342:          IT4 = IT4 - IPW2*IT3
343:          IT3 = IT3 + I3*MM( I, 4 ) + I4*MM( I, 3 )
344:          IT2 = IT3 / IPW2
345:          IT3 = IT3 - IPW2*IT2
346:          IT2 = IT2 + I2*MM( I, 4 ) + I3*MM( I, 3 ) + I4*MM( I, 2 )
347:          IT1 = IT2 / IPW2
348:          IT2 = IT2 - IPW2*IT1
349:          IT1 = IT1 + I1*MM( I, 4 ) + I2*MM( I, 3 ) + I3*MM( I, 2 ) +
350:      $         I4*MM( I, 1 )
351:          IT1 = MOD( IT1, IPW2 )
352: *
353: *        Convert 48-bit integer to a real number in the interval (0,1)
354: *
355:          X( I ) = R*( DBLE( IT1 )+R*( DBLE( IT2 )+R*( DBLE( IT3 )+R*
356:      $            DBLE( IT4 ) ) ) )
357: *
358:          IF (X( I ).EQ.1.0D0) THEN
359: *           If a real number has n bits of precision, and the first
360: *           n bits of the 48-bit integer above happen to be all 1 (which
361: *           will occur about once every 2**n calls), then X( I ) will
362: *           be rounded to exactly 1.0. 
363: *           Since X( I ) is not supposed to return exactly 0.0 or 1.0,
364: *           the statistically correct thing to do in this situation is
365: *           simply to iterate again.
366: *           N.B. the case X( I ) = 0.0 should not be possible.  
367:             I1 = I1 + 2
368:             I2 = I2 + 2
369:             I3 = I3 + 2
370:             I4 = I4 + 2
371:             GOTO 20
372:          END IF
373: *
374:    10 CONTINUE
375: *
376: *     Return final value of seed
377: *
378:       ISEED( 1 ) = IT1
379:       ISEED( 2 ) = IT2
380:       ISEED( 3 ) = IT3
381:       ISEED( 4 ) = IT4
382:       RETURN
383: *
384: *     End of DLARUV
385: *
386:       END
387: