#include "ranlib.h" #include #include #include #define ABS(x) ((x) >= 0 ? (x) : -(x)) #define min(a,b) ((a) <= (b) ? (a) : (b)) #define max(a,b) ((a) >= (b) ? (a) : (b)) void ftnstop(char*); float genbet(float aa,float bb) /* ********************************************************************** float genbet(float aa,float bb) GeNerate BETa random deviate Function Returns a single random deviate from the beta distribution with parameters A and B. The density of the beta is x^(a-1) * (1-x)^(b-1) / B(a,b) for 0 < x < 1 Arguments aa --> First parameter of the beta distribution bb --> Second parameter of the beta distribution Method R. C. H. Cheng Generating Beta Variatew with Nonintegral Shape Parameters Communications of the ACM, 21:317-322 (1978) (Algorithms BB and BC) ********************************************************************** */ { #define expmax 89.0 #define infnty 1.0E38 static float olda = -1.0; static float oldb = -1.0; static float genbet,a,alpha,b,beta,delta,gamma,k1,k2,r,s,t,u1,u2,v,w,y,z; static long qsame; qsame = olda == aa && oldb == bb; if(qsame) goto S20; if(!(aa <= 0.0 || bb <= 0.0)) goto S10; fputs(" AA or BB <= 0 in GENBET - Abort!",stderr); fprintf(stderr," AA: %16.6E BB %16.6E\n",aa,bb); exit(1); S10: olda = aa; oldb = bb; S20: if(!(min(aa,bb) > 1.0)) goto S100; /* Alborithm BB Initialize */ if(qsame) goto S30; a = min(aa,bb); b = max(aa,bb); alpha = a+b; beta = sqrt((alpha-2.0)/(2.0*a*b-alpha)); gamma = a+1.0/beta; S30: S40: u1 = ranf(); /* Step 1 */ u2 = ranf(); v = beta*log(u1/(1.0-u1)); if(!(v > expmax)) goto S50; w = infnty; goto S60; S50: w = a*exp(v); S60: z = pow(u1,2.0)*u2; r = gamma*v-1.3862944; s = a+r-w; /* Step 2 */ if(s+2.609438 >= 5.0*z) goto S70; /* Step 3 */ t = log(z); if(s > t) goto S70; /* Step 4 */ if(r+alpha*log(alpha/(b+w)) < t) goto S40; S70: /* Step 5 */ if(!(aa == a)) goto S80; genbet = w/(b+w); goto S90; S80: genbet = b/(b+w); S90: goto S230; S100: /* Algorithm BC Initialize */ if(qsame) goto S110; a = max(aa,bb); b = min(aa,bb); alpha = a+b; beta = 1.0/b; delta = 1.0+a-b; k1 = delta*(1.38889E-2+4.16667E-2*b)/(a*beta-0.777778); k2 = 0.25+(0.5+0.25/delta)*b; S110: S120: u1 = ranf(); /* Step 1 */ u2 = ranf(); if(u1 >= 0.5) goto S130; /* Step 2 */ y = u1*u2; z = u1*y; if(0.25*u2+z-y >= k1) goto S120; goto S170; S130: /* Step 3 */ z = pow(u1,2.0)*u2; if(!(z <= 0.25)) goto S160; v = beta*log(u1/(1.0-u1)); if(!(v > expmax)) goto S140; w = infnty; goto S150; S140: w = a*exp(v); S150: goto S200; S160: if(z >= k2) goto S120; S170: /* Step 4 Step 5 */ v = beta*log(u1/(1.0-u1)); if(!(v > expmax)) goto S180; w = infnty; goto S190; S180: w = a*exp(v); S190: if(alpha*(log(alpha/(b+w))+v)-1.3862944 < log(z)) goto S120; S200: /* Step 6 */ if(!(a == aa)) goto S210; genbet = w/(b+w); goto S220; S210: genbet = b/(b+w); S230: S220: return genbet; #undef expmax #undef infnty } float genchi(float df) /* ********************************************************************** float genchi(float df) Generate random value of CHIsquare variable Function Generates random deviate from the distribution of a chisquare with DF degrees of freedom random variable. Arguments df --> Degrees of freedom of the chisquare (Must be positive) Method Uses relation between chisquare and gamma. ********************************************************************** */ { static float genchi; if(!(df <= 0.0)) goto S10; fputs("DF <= 0 in GENCHI - ABORT",stderr); fprintf(stderr,"Value of DF: %16.6E\n",df); exit(1); S10: genchi = 2.0*gengam(1.0,df/2.0); return genchi; } float genexp(float av) /* ********************************************************************** float genexp(float av) GENerate EXPonential random deviate Function Generates a single random deviate from an exponential distribution with mean AV. Arguments av --> The mean of the exponential distribution from which a random deviate is to be generated. Method Renames SEXPO from TOMS as slightly modified by BWB to use RANF instead of SUNIF. For details see: Ahrens, J.H. and Dieter, U. Computer Methods for Sampling From the Exponential and Normal Distributions. Comm. ACM, 15,10 (Oct. 1972), 873 - 882. ********************************************************************** */ { static float genexp; genexp = sexpo()*av; return genexp; } float genf(float dfn,float dfd) /* ********************************************************************** float genf(float dfn,float dfd) GENerate random deviate from the F distribution Function Generates a random deviate from the F (variance ratio) distribution with DFN degrees of freedom in the numerator and DFD degrees of freedom in the denominator. Arguments dfn --> Numerator degrees of freedom (Must be positive) dfd --> Denominator degrees of freedom (Must be positive) Method Directly generates ratio of chisquare variates ********************************************************************** */ { static float genf,xden,xnum; if(!(dfn <= 0.0 || dfd <= 0.0)) goto S10; fputs("Degrees of freedom nonpositive in GENF - abort!",stderr); fprintf(stderr,"DFN value: %16.6EDFD value: %16.6E\n",dfn,dfd); exit(1); S10: xnum = genchi(dfn)/dfn; /* GENF = ( GENCHI( DFN ) / DFN ) / ( GENCHI( DFD ) / DFD ) */ xden = genchi(dfd)/dfd; if(!(xden <= 9.999999999998E-39*xnum)) goto S20; fputs(" GENF - generated numbers would cause overflow",stderr); fprintf(stderr," Numerator %16.6E Denominator %16.6E\n",xnum,xden); fputs(" GENF returning 1.0E38",stderr); genf = 1.0E38; goto S30; S20: genf = xnum/xden; S30: return genf; } float gengam(float a,float r) /* ********************************************************************** float gengam(float a,float r) GENerates random deviates from GAMma distribution Function Generates random deviates from the gamma distribution whose density is (A**R)/Gamma(R) * X**(R-1) * Exp(-A*X) Arguments a --> Location parameter of Gamma distribution r --> Shape parameter of Gamma distribution Method Renames SGAMMA from TOMS as slightly modified by BWB to use RANF instead of SUNIF. For details see: (Case R >= 1.0) Ahrens, J.H. and Dieter, U. Generating Gamma Variates by a Modified Rejection Technique. Comm. ACM, 25,1 (Jan. 1982), 47 - 54. Algorithm GD (Case 0.0 <= R <= 1.0) Ahrens, J.H. and Dieter, U. Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions. Computing, 12 (1974), 223-246/ Adapted algorithm GS. ********************************************************************** */ { static float gengam; gengam = sgamma(r); gengam /= a; return gengam; } void genmn(float *parm,float *x,float *work) /* ********************************************************************** void genmn(float *parm,float *x,float *work) GENerate Multivariate Normal random deviate Arguments parm --> Parameters needed to generate multivariate normal deviates (MEANV and Cholesky decomposition of COVM). Set by a previous call to SETGMN. 1 : 1 - size of deviate, P 2 : P + 1 - mean vector P+2 : P*(P+3)/2 + 1 - upper half of cholesky decomposition of cov matrix x <-- Vector deviate generated. work <--> Scratch array Method 1) Generate P independent standard normal deviates - Ei ~ N(0,1) 2) Using Cholesky decomposition find A s.t. trans(A)*A = COVM 3) trans(A)E + MEANV ~ N(MEANV,COVM) ********************************************************************** */ { static long i,icount,j,p,D1,D2,D3,D4; static float ae; p = (long) (*parm); /* Generate P independent normal deviates - WORK ~ N(0,1) */ for(i=1; i<=p; i++) *(work+i-1) = snorm(); for(i=1,D3=1,D4=(p-i+D3)/D3; D4>0; D4--,i+=D3) { /* PARM (P+2 : P*(P+3)/2 + 1) contains A, the Cholesky decomposition of the desired covariance matrix. trans(A)(1,1) = PARM(P+2) trans(A)(2,1) = PARM(P+3) trans(A)(2,2) = PARM(P+2+P) trans(A)(3,1) = PARM(P+4) trans(A)(3,2) = PARM(P+3+P) trans(A)(3,3) = PARM(P+2-1+2P) ... trans(A)*WORK + MEANV ~ N(MEANV,COVM) */ icount = 0; ae = 0.0; for(j=1,D1=1,D2=(i-j+D1)/D1; D2>0; D2--,j+=D1) { icount += (j-1); ae += (*(parm+i+(j-1)*p-icount+p)**(work+j-1)); } *(x+i-1) = ae+*(parm+i); } } void genmul(long n,float *p,long ncat,long *ix) /* ********************************************************************** void genmul(int n,float *p,int ncat,int *ix) GENerate an observation from the MULtinomial distribution Arguments N --> Number of events that will be classified into one of the categories 1..NCAT P --> Vector of probabilities. P(i) is the probability that an event will be classified into category i. Thus, P(i) must be [0,1]. Only the first NCAT-1 P(i) must be defined since P(NCAT) is 1.0 minus the sum of the first NCAT-1 P(i). NCAT --> Number of categories. Length of P and IX. IX <-- Observation from multinomial distribution. All IX(i) will be nonnegative and their sum will be N. Method Algorithm from page 559 of Devroye, Luc Non-Uniform Random Variate Generation. Springer-Verlag, New York, 1986. ********************************************************************** */ { static float prob,ptot,sum; static long i,icat,ntot; if(n < 0) ftnstop("N < 0 in GENMUL"); if(ncat <= 1) ftnstop("NCAT <= 1 in GENMUL"); ptot = 0.0F; for(i=0; i 1.0F) ftnstop("Some P(i) > 1 in GENMUL"); ptot += *(p+i); } if(ptot > 0.99999F) ftnstop("Sum of P(i) > 1 in GENMUL"); /* Initialize variables */ ntot = n; sum = 1.0F; for(i=0; i Degrees of freedom of the chisquare (Must be > 1.0) xnonc --> Noncentrality parameter of the chisquare (Must be >= 0.0) Method Uses fact that noncentral chisquare is the sum of a chisquare deviate with DF-1 degrees of freedom plus the square of a normal deviate with mean XNONC and standard deviation 1. ********************************************************************** */ { static float gennch; if(!(df <= 1.0 || xnonc < 0.0)) goto S10; fputs("DF <= 1 or XNONC < 0 in GENNCH - ABORT",stderr); fprintf(stderr,"Value of DF: %16.6E Value of XNONC%16.6E\n",df,xnonc); exit(1); S10: gennch = genchi(df-1.0)+pow(gennor(sqrt(xnonc),1.0),2.0); return gennch; } float gennf(float dfn,float dfd,float xnonc) /* ********************************************************************** float gennf(float dfn,float dfd,float xnonc) GENerate random deviate from the Noncentral F distribution Function Generates a random deviate from the noncentral F (variance ratio) distribution with DFN degrees of freedom in the numerator, and DFD degrees of freedom in the denominator, and noncentrality parameter XNONC. Arguments dfn --> Numerator degrees of freedom (Must be >= 1.0) dfd --> Denominator degrees of freedom (Must be positive) xnonc --> Noncentrality parameter (Must be nonnegative) Method Directly generates ratio of noncentral numerator chisquare variate to central denominator chisquare variate. ********************************************************************** */ { static float gennf,xden,xnum; static long qcond; qcond = dfn <= 1.0 || dfd <= 0.0 || xnonc < 0.0; if(!qcond) goto S10; fputs("In GENNF - Either (1) Numerator DF <= 1.0 or",stderr); fputs("(2) Denominator DF < 0.0 or ",stderr); fputs("(3) Noncentrality parameter < 0.0",stderr); fprintf(stderr, "DFN value: %16.6EDFD value: %16.6EXNONC value: \n%16.6E\n",dfn,dfd, xnonc); exit(1); S10: xnum = gennch(dfn,xnonc)/dfn; /* GENNF = ( GENNCH( DFN, XNONC ) / DFN ) / ( GENCHI( DFD ) / DFD ) */ xden = genchi(dfd)/dfd; if(!(xden <= 9.999999999998E-39*xnum)) goto S20; fputs(" GENNF - generated numbers would cause overflow",stderr); fprintf(stderr," Numerator %16.6E Denominator %16.6E\n",xnum,xden); fputs(" GENNF returning 1.0E38",stderr); gennf = 1.0E38; goto S30; S20: gennf = xnum/xden; S30: return gennf; } float gennor(float av,float sd) /* ********************************************************************** float gennor(float av,float sd) GENerate random deviate from a NORmal distribution Function Generates a single random deviate from a normal distribution with mean, AV, and standard deviation, SD. Arguments av --> Mean of the normal distribution. sd --> Standard deviation of the normal distribution. Method Renames SNORM from TOMS as slightly modified by BWB to use RANF instead of SUNIF. For details see: Ahrens, J.H. and Dieter, U. Extensions of Forsythe's Method for Random Sampling from the Normal Distribution. Math. Comput., 27,124 (Oct. 1973), 927 - 937. ********************************************************************** */ { static float gennor; gennor = sd*snorm()+av; return gennor; } void genprm(long *iarray,int larray) /* ********************************************************************** void genprm(long *iarray,int larray) GENerate random PeRMutation of iarray Arguments iarray <--> On output IARRAY is a random permutation of its value on input larray <--> Length of IARRAY ********************************************************************** */ { static long i,itmp,iwhich,D1,D2; for(i=1,D1=1,D2=(larray-i+D1)/D1; D2>0; D2--,i+=D1) { iwhich = ignuin(i,larray); itmp = *(iarray+iwhich-1); *(iarray+iwhich-1) = *(iarray+i-1); *(iarray+i-1) = itmp; } } float genunf(float low,float high) /* ********************************************************************** float genunf(float low,float high) GeNerate Uniform Real between LOW and HIGH Function Generates a real uniformly distributed between LOW and HIGH. Arguments low --> Low bound (exclusive) on real value to be generated high --> High bound (exclusive) on real value to be generated ********************************************************************** */ { static float genunf; if(!(low > high)) goto S10; fprintf(stderr,"LOW > HIGH in GENUNF: LOW %16.6E HIGH: %16.6E\n",low,high); fputs("Abort",stderr); exit(1); S10: genunf = low+(high-low)*ranf(); return genunf; } void gscgn(long getset,long *g) /* ********************************************************************** void gscgn(long getset,long *g) Get/Set GeNerator Gets or returns in G the number of the current generator Arguments getset --> 0 Get 1 Set g <-- Number of the current random number generator (1..32) ********************************************************************** */ { #define numg 32L static long curntg = 1; if(getset == 0) *g = curntg; else { if(*g < 0 || *g > numg) { fputs(" Generator number out of range in GSCGN",stderr); exit(0); } curntg = *g; } #undef numg } void gsrgs(long getset,long *qvalue) /* ********************************************************************** void gsrgs(long getset,long *qvalue) Get/Set Random Generators Set Gets or sets whether random generators set (initialized). Initially (data statement) state is not set If getset is 1 state is set to qvalue If getset is 0 state returned in qvalue ********************************************************************** */ { static long qinit = 0; if(getset == 0) *qvalue = qinit; else qinit = *qvalue; } void gssst(long getset,long *qset) /* ********************************************************************** void gssst(long getset,long *qset) Get or Set whether Seed is Set Initialize to Seed not Set If getset is 1 sets state to Seed Set If getset is 0 returns T in qset if Seed Set Else returns F in qset ********************************************************************** */ { static long qstate = 0; if(getset != 0) qstate = 1; else *qset = qstate; } long ignbin(long n,float pp) /* ********************************************************************** long ignbin(long n,float pp) GENerate BINomial random deviate Function Generates a single random deviate from a binomial distribution whose number of trials is N and whose probability of an event in each trial is P. Arguments n --> The number of trials in the binomial distribution from which a random deviate is to be generated. p --> The probability of an event in each trial of the binomial distribution from which a random deviate is to be generated. ignbin <-- A random deviate yielding the number of events from N independent trials, each of which has a probability of event P. Method This is algorithm BTPE from: Kachitvichyanukul, V. and Schmeiser, B. W. Binomial Random Variate Generation. Communications of the ACM, 31, 2 (February, 1988) 216. ********************************************************************** SUBROUTINE BTPEC(N,PP,ISEED,JX) BINOMIAL RANDOM VARIATE GENERATOR MEAN .LT. 30 -- INVERSE CDF MEAN .GE. 30 -- ALGORITHM BTPE: ACCEPTANCE-REJECTION VIA FOUR REGION COMPOSITION. THE FOUR REGIONS ARE A TRIANGLE (SYMMETRIC IN THE CENTER), A PAIR OF PARALLELOGRAMS (ABOVE THE TRIANGLE), AND EXPONENTIAL LEFT AND RIGHT TAILS. BTPE REFERS TO BINOMIAL-TRIANGLE-PARALLELOGRAM-EXPONENTIAL. BTPEC REFERS TO BTPE AND "COMBINED." THUS BTPE IS THE RESEARCH AND BTPEC IS THE IMPLEMENTATION OF A COMPLETE USABLE ALGORITHM. REFERENCE: VORATAS KACHITVICHYANUKUL AND BRUCE SCHMEISER, "BINOMIAL RANDOM VARIATE GENERATION," COMMUNICATIONS OF THE ACM, FORTHCOMING WRITTEN: SEPTEMBER 1980. LAST REVISED: MAY 1985, JULY 1987 REQUIRED SUBPROGRAM: RAND() -- A UNIFORM (0,1) RANDOM NUMBER GENERATOR ARGUMENTS N : NUMBER OF BERNOULLI TRIALS (INPUT) PP : PROBABILITY OF SUCCESS IN EACH TRIAL (INPUT) ISEED: RANDOM NUMBER SEED (INPUT AND OUTPUT) JX: RANDOMLY GENERATED OBSERVATION (OUTPUT) VARIABLES PSAVE: VALUE OF PP FROM THE LAST CALL TO BTPEC NSAVE: VALUE OF N FROM THE LAST CALL TO BTPEC XNP: VALUE OF THE MEAN FROM THE LAST CALL TO BTPEC P: PROBABILITY USED IN THE GENERATION PHASE OF BTPEC FFM: TEMPORARY VARIABLE EQUAL TO XNP + P M: INTEGER VALUE OF THE CURRENT MODE FM: FLOATING POINT VALUE OF THE CURRENT MODE XNPQ: TEMPORARY VARIABLE USED IN SETUP AND SQUEEZING STEPS P1: AREA OF THE TRIANGLE C: HEIGHT OF THE PARALLELOGRAMS XM: CENTER OF THE TRIANGLE XL: LEFT END OF THE TRIANGLE XR: RIGHT END OF THE TRIANGLE AL: TEMPORARY VARIABLE XLL: RATE FOR THE LEFT EXPONENTIAL TAIL XLR: RATE FOR THE RIGHT EXPONENTIAL TAIL P2: AREA OF THE PARALLELOGRAMS P3: AREA OF THE LEFT EXPONENTIAL TAIL P4: AREA OF THE RIGHT EXPONENTIAL TAIL U: A U(0,P4) RANDOM VARIATE USED FIRST TO SELECT ONE OF THE FOUR REGIONS AND THEN CONDITIONALLY TO GENERATE A VALUE FROM THE REGION V: A U(0,1) RANDOM NUMBER USED TO GENERATE THE RANDOM VALUE (REGION 1) OR TRANSFORMED INTO THE VARIATE TO ACCEPT OR REJECT THE CANDIDATE VALUE IX: INTEGER CANDIDATE VALUE X: PRELIMINARY CONTINUOUS CANDIDATE VALUE IN REGION 2 LOGIC AND A FLOATING POINT IX IN THE ACCEPT/REJECT LOGIC K: ABSOLUTE VALUE OF (IX-M) F: THE HEIGHT OF THE SCALED DENSITY FUNCTION USED IN THE ACCEPT/REJECT DECISION WHEN BOTH M AND IX ARE SMALL ALSO USED IN THE INVERSE TRANSFORMATION R: THE RATIO P/Q G: CONSTANT USED IN CALCULATION OF PROBABILITY MP: MODE PLUS ONE, THE LOWER INDEX FOR EXPLICIT CALCULATION OF F WHEN IX IS GREATER THAN M IX1: CANDIDATE VALUE PLUS ONE, THE LOWER INDEX FOR EXPLICIT CALCULATION OF F WHEN IX IS LESS THAN M I: INDEX FOR EXPLICIT CALCULATION OF F FOR BTPE AMAXP: MAXIMUM ERROR OF THE LOGARITHM OF NORMAL BOUND YNORM: LOGARITHM OF NORMAL BOUND ALV: NATURAL LOGARITHM OF THE ACCEPT/REJECT VARIATE V X1,F1,Z,W,Z2,X2,F2, AND W2 ARE TEMPORARY VARIABLES TO BE USED IN THE FINAL ACCEPT/REJECT TEST QN: PROBABILITY OF NO SUCCESS IN N TRIALS REMARK IX AND JX COULD LOGICALLY BE THE SAME VARIABLE, WHICH WOULD SAVE A MEMORY POSITION AND A LINE OF CODE. HOWEVER, SOME COMPILERS (E.G.,CDC MNF) OPTIMIZE BETTER WHEN THE ARGUMENTS ARE NOT INVOLVED. ISEED NEEDS TO BE DOUBLE PRECISION IF THE IMSL ROUTINE GGUBFS IS USED TO GENERATE UNIFORM RANDOM NUMBER, OTHERWISE TYPE OF ISEED SHOULD BE DICTATED BY THE UNIFORM GENERATOR ********************************************************************** *****DETERMINE APPROPRIATE ALGORITHM AND WHETHER SETUP IS NECESSARY */ { static float psave = -1.0; static long nsave = -1; static long ignbin,i,ix,ix1,k,m,mp,T1; static float al,alv,amaxp,c,f,f1,f2,ffm,fm,g,p,p1,p2,p3,p4,q,qn,r,u,v,w,w2,x,x1, x2,xl,xll,xlr,xm,xnp,xnpq,xr,ynorm,z,z2; if(pp != psave) goto S10; if(n != nsave) goto S20; if(xnp < 30.0) goto S150; goto S30; S10: /* *****SETUP, PERFORM ONLY WHEN PARAMETERS CHANGE */ psave = pp; p = min(psave,1.0-psave); q = 1.0-p; S20: xnp = n*p; nsave = n; if(xnp < 30.0) goto S140; ffm = xnp+p; m = ffm; fm = m; xnpq = xnp*q; p1 = (long) (2.195*sqrt(xnpq)-4.6*q)+0.5; xm = fm+0.5; xl = xm-p1; xr = xm+p1; c = 0.134+20.5/(15.3+fm); al = (ffm-xl)/(ffm-xl*p); xll = al*(1.0+0.5*al); al = (xr-ffm)/(xr*q); xlr = al*(1.0+0.5*al); p2 = p1*(1.0+c+c); p3 = p2+c/xll; p4 = p3+c/xlr; S30: /* *****GENERATE VARIATE */ u = ranf()*p4; v = ranf(); /* TRIANGULAR REGION */ if(u > p1) goto S40; ix = xm-p1*v+u; goto S170; S40: /* PARALLELOGRAM REGION */ if(u > p2) goto S50; x = xl+(u-p1)/c; v = v*c+1.0-ABS(xm-x)/p1; if(v > 1.0 || v <= 0.0) goto S30; ix = x; goto S70; S50: /* LEFT TAIL */ if(u > p3) goto S60; ix = xl+log(v)/xll; if(ix < 0) goto S30; v *= ((u-p2)*xll); goto S70; S60: /* RIGHT TAIL */ ix = xr-log(v)/xlr; if(ix > n) goto S30; v *= ((u-p3)*xlr); S70: /* *****DETERMINE APPROPRIATE WAY TO PERFORM ACCEPT/REJECT TEST */ k = ABS(ix-m); if(k > 20 && k < xnpq/2-1) goto S130; /* EXPLICIT EVALUATION */ f = 1.0; r = p/q; g = (n+1)*r; T1 = m-ix; if(T1 < 0) goto S80; else if(T1 == 0) goto S120; else goto S100; S80: mp = m+1; for(i=mp; i<=ix; i++) f *= (g/i-r); goto S120; S100: ix1 = ix+1; for(i=ix1; i<=m; i++) f /= (g/i-r); S120: if(v <= f) goto S170; goto S30; S130: /* SQUEEZING USING UPPER AND LOWER BOUNDS ON ALOG(F(X)) */ amaxp = k/xnpq*((k*(k/3.0+0.625)+0.1666666666666)/xnpq+0.5); ynorm = -(k*k/(2.0*xnpq)); alv = log(v); if(alv < ynorm-amaxp) goto S170; if(alv > ynorm+amaxp) goto S30; /* STIRLING'S FORMULA TO MACHINE ACCURACY FOR THE FINAL ACCEPTANCE/REJECTION TEST */ x1 = ix+1.0; f1 = fm+1.0; z = n+1.0-fm; w = n-ix+1.0; z2 = z*z; x2 = x1*x1; f2 = f1*f1; w2 = w*w; if(alv <= xm*log(f1/x1)+(n-m+0.5)*log(z/w)+(ix-m)*log(w*p/(x1*q))+(13860.0- (462.0-(132.0-(99.0-140.0/f2)/f2)/f2)/f2)/f1/166320.0+(13860.0-(462.0- (132.0-(99.0-140.0/z2)/z2)/z2)/z2)/z/166320.0+(13860.0-(462.0-(132.0- (99.0-140.0/x2)/x2)/x2)/x2)/x1/166320.0+(13860.0-(462.0-(132.0-(99.0 -140.0/w2)/w2)/w2)/w2)/w/166320.0) goto S170; goto S30; S140: /* INVERSE CDF LOGIC FOR MEAN LESS THAN 30 */ qn = pow(q,(double)n); r = p/q; g = r*(n+1); S150: ix = 0; f = qn; u = ranf(); S160: if(u < f) goto S170; if(ix > 110) goto S150; u -= f; ix += 1; f *= (g/ix-r); goto S160; S170: if(psave > 0.5) ix = n-ix; ignbin = ix; return ignbin; } long ignnbn(long n,float p) /* ********************************************************************** long ignnbn(long n,float p) GENerate Negative BiNomial random deviate Function Generates a single random deviate from a negative binomial distribution. Arguments N --> The number of trials in the negative binomial distribution from which a random deviate is to be generated. P --> The probability of an event. Method Algorithm from page 480 of Devroye, Luc Non-Uniform Random Variate Generation. Springer-Verlag, New York, 1986. ********************************************************************** */ { static long ignnbn; static float y,a,r; /* .. .. Executable Statements .. */ /* Check Arguments */ if(n < 0) ftnstop("N < 0 in IGNNBN"); if(p <= 0.0F) ftnstop("P <= 0 in IGNNBN"); if(p >= 1.0F) ftnstop("P >= 1 in IGNNBN"); /* Generate Y, a random gamma (n,(1-p)/p) variable */ r = (float)n; a = p/(1.0F-p); y = gengam(a,r); /* Generate a random Poisson(y) variable */ ignnbn = ignpoi(y); return ignnbn; } long ignpoi(float mu) /* ********************************************************************** long ignpoi(float mu) GENerate POIsson random deviate Function Generates a single random deviate from a Poisson distribution with mean AV. Arguments av --> The mean of the Poisson distribution from which a random deviate is to be generated. genexp <-- The random deviate. Method Renames KPOIS from TOMS as slightly modified by BWB to use RANF instead of SUNIF. For details see: Ahrens, J.H. and Dieter, U. Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2 (June 1982),163-179 ********************************************************************** ********************************************************************** P O I S S O N DISTRIBUTION ********************************************************************** ********************************************************************** FOR DETAILS SEE: AHRENS, J.H. AND DIETER, U. COMPUTER GENERATION OF POISSON DEVIATES FROM MODIFIED NORMAL DISTRIBUTIONS. ACM TRANS. MATH. SOFTWARE, 8,2 (JUNE 1982), 163 - 179. (SLIGHTLY MODIFIED VERSION OF THE PROGRAM IN THE ABOVE ARTICLE) ********************************************************************** INTEGER FUNCTION IGNPOI(IR,MU) INPUT: IR=CURRENT STATE OF BASIC RANDOM NUMBER GENERATOR MU=MEAN MU OF THE POISSON DISTRIBUTION OUTPUT: IGNPOI=SAMPLE FROM THE POISSON-(MU)-DISTRIBUTION MUPREV=PREVIOUS MU, MUOLD=MU AT LAST EXECUTION OF STEP P OR B. TABLES: COEFFICIENTS A0-A7 FOR STEP F. FACTORIALS FACT COEFFICIENTS A(K) - FOR PX = FK*V*V*SUM(A(K)*V**K)-DEL SEPARATION OF CASES A AND B */ { extern float fsign( float num, float sign ); static float a0 = -0.5; static float a1 = 0.3333333; static float a2 = -0.2500068; static float a3 = 0.2000118; static float a4 = -0.1661269; static float a5 = 0.1421878; static float a6 = -0.1384794; static float a7 = 0.125006; static float muold = 0.0; static float muprev = 0.0; static float fact[10] = { 1.0,1.0,2.0,6.0,24.0,120.0,720.0,5040.0,40320.0,362880.0 }; static long ignpoi,j,k,kflag,l,m; static float b1,b2,c,c0,c1,c2,c3,d,del,difmuk,e,fk,fx,fy,g,omega,p,p0,px,py,q,s, t,u,v,x,xx,pp[35]; if(mu == muprev) goto S10; if(mu < 10.0) goto S120; /* C A S E A. (RECALCULATION OF S,D,L IF MU HAS CHANGED) */ muprev = mu; s = sqrt(mu); d = 6.0*mu*mu; /* THE POISSON PROBABILITIES PK EXCEED THE DISCRETE NORMAL PROBABILITIES FK WHENEVER K >= M(MU). L=IFIX(MU-1.1484) IS AN UPPER BOUND TO M(MU) FOR ALL MU >= 10 . */ l = (long) (mu-1.1484); S10: /* STEP N. NORMAL SAMPLE - SNORM(IR) FOR STANDARD NORMAL DEVIATE */ g = mu+s*snorm(); if(g < 0.0) goto S20; ignpoi = (long) (g); /* STEP I. IMMEDIATE ACCEPTANCE IF IGNPOI IS LARGE ENOUGH */ if(ignpoi >= l) return ignpoi; /* STEP S. SQUEEZE ACCEPTANCE - SUNIF(IR) FOR (0,1)-SAMPLE U */ fk = (float)ignpoi; difmuk = mu-fk; u = ranf(); if(d*u >= difmuk*difmuk*difmuk) return ignpoi; S20: /* STEP P. PREPARATIONS FOR STEPS Q AND H. (RECALCULATIONS OF PARAMETERS IF NECESSARY) .3989423=(2*PI)**(-.5) .416667E-1=1./24. .1428571=1./7. THE QUANTITIES B1, B2, C3, C2, C1, C0 ARE FOR THE HERMITE APPROXIMATIONS TO THE DISCRETE NORMAL PROBABILITIES FK. C=.1069/MU GUARANTEES MAJORIZATION BY THE 'HAT'-FUNCTION. */ if(mu == muold) goto S30; muold = mu; omega = 0.3989423/s; b1 = 4.166667E-2/mu; b2 = 0.3*b1*b1; c3 = 0.1428571*b1*b2; c2 = b2-15.0*c3; c1 = b1-6.0*b2+45.0*c3; c0 = 1.0-b1+3.0*b2-15.0*c3; c = 0.1069/mu; S30: if(g < 0.0) goto S50; /* 'SUBROUTINE' F IS CALLED (KFLAG=0 FOR CORRECT RETURN) */ kflag = 0; goto S70; S40: /* STEP Q. QUOTIENT ACCEPTANCE (RARE CASE) */ if(fy-u*fy <= py*exp(px-fx)) return ignpoi; S50: /* STEP E. EXPONENTIAL SAMPLE - SEXPO(IR) FOR STANDARD EXPONENTIAL DEVIATE E AND SAMPLE T FROM THE LAPLACE 'HAT' (IF T <= -.6744 THEN PK < FK FOR ALL MU >= 10.) */ e = sexpo(); u = ranf(); u += (u-1.0); t = 1.8+fsign(e,u); if(t <= -0.6744) goto S50; ignpoi = (long) (mu+s*t); fk = (float)ignpoi; difmuk = mu-fk; /* 'SUBROUTINE' F IS CALLED (KFLAG=1 FOR CORRECT RETURN) */ kflag = 1; goto S70; S60: /* STEP H. HAT ACCEPTANCE (E IS REPEATED ON REJECTION) */ if(c*fabs(u) > py*exp(px+e)-fy*exp(fx+e)) goto S50; return ignpoi; S70: /* STEP F. 'SUBROUTINE' F. CALCULATION OF PX,PY,FX,FY. CASE IGNPOI .LT. 10 USES FACTORIALS FROM TABLE FACT */ if(ignpoi >= 10) goto S80; px = -mu; py = pow(mu,(double)ignpoi)/ *(fact+ignpoi); goto S110; S80: /* CASE IGNPOI .GE. 10 USES POLYNOMIAL APPROXIMATION A0-A7 FOR ACCURACY WHEN ADVISABLE .8333333E-1=1./12. .3989423=(2*PI)**(-.5) */ del = 8.333333E-2/fk; del -= (4.8*del*del*del); v = difmuk/fk; if(fabs(v) <= 0.25) goto S90; px = fk*log(1.0+v)-difmuk-del; goto S100; S90: px = fk*v*v*(((((((a7*v+a6)*v+a5)*v+a4)*v+a3)*v+a2)*v+a1)*v+a0)-del; S100: py = 0.3989423/sqrt(fk); S110: x = (0.5-difmuk)/s; xx = x*x; fx = -0.5*xx; fy = omega*(((c3*xx+c2)*xx+c1)*xx+c0); if(kflag <= 0) goto S40; goto S60; S120: /* C A S E B. (START NEW TABLE AND CALCULATE P0 IF NECESSARY) */ muprev = 0.0; if(mu == muold) goto S130; muold = mu; m = max(1L,(long) (mu)); l = 0; p = exp(-mu); q = p0 = p; S130: /* STEP U. UNIFORM SAMPLE FOR INVERSION METHOD */ u = ranf(); ignpoi = 0; if(u <= p0) return ignpoi; /* STEP T. TABLE COMPARISON UNTIL THE END PP(L) OF THE PP-TABLE OF CUMULATIVE POISSON PROBABILITIES (0.458=PP(9) FOR MU=10) */ if(l == 0) goto S150; j = 1; if(u > 0.458) j = min(l,m); for(k=j; k<=l; k++) { if(u <= *(pp+k-1)) goto S180; } if(l == 35) goto S130; S150: /* STEP C. CREATION OF NEW POISSON PROBABILITIES P AND THEIR CUMULATIVES Q=PP(K) */ l += 1; for(k=l; k<=35; k++) { p = p*mu/(float)k; q += p; *(pp+k-1) = q; if(u <= q) goto S170; } l = 35; goto S130; S170: l = k; S180: ignpoi = k; return ignpoi; } long ignuin(long low,long high) /* ********************************************************************** long ignuin(long low,long high) GeNerate Uniform INteger Function Generates an integer uniformly distributed between LOW and HIGH. Arguments low --> Low bound (inclusive) on integer value to be generated high --> High bound (inclusive) on integer value to be generated Note If (HIGH-LOW) > 2,147,483,561 prints error message on * unit and stops the program. ********************************************************************** IGNLGI generates integers between 1 and 2147483562 MAXNUM is 1 less than maximum generable value */ { #define maxnum 2147483561L static long ignuin,ign,maxnow,range,ranp1; if(!(low > high)) goto S10; fputs(" low > high in ignuin - ABORT",stderr); exit(1); S10: range = high-low; if(!(range > maxnum)) goto S20; fputs(" high - low too large in ignuin - ABORT",stderr); exit(1); S20: if(!(low == high)) goto S30; ignuin = low; return ignuin; S30: /* Number to be generated should be in range 0..RANGE Set MAXNOW so that the number of integers in 0..MAXNOW is an integral multiple of the number in 0..RANGE */ ranp1 = range+1; maxnow = maxnum/ranp1*ranp1; S40: ign = ignlgi()-1; if(!(ign <= maxnow)) goto S50; ignuin = low+ign%ranp1; return ignuin; S50: goto S40; #undef maxnum #undef err1 #undef err2 } long lennob( char *str ) /* Returns the length of str ignoring trailing blanks but not other white space. */ { long i, i_nb; for (i=0, i_nb= -1L; *(str+i); i++) if ( *(str+i) != ' ' ) i_nb = i; return (i_nb+1); } long mltmod(long a,long s,long m) /* ********************************************************************** long mltmod(long a,long s,long m) Returns (A*S) MOD M This is a transcription from Pascal to Fortran of routine MULtMod_Decompos from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments a, s, m --> ********************************************************************** */ { #define h 32768L static long mltmod,a0,a1,k,p,q,qh,rh; /* H = 2**((b-2)/2) where b = 32 because we are using a 32 bit machine. On a different machine recompute H */ if(!(a <= 0 || a >= m || s <= 0 || s >= m)) goto S10; fputs(" a, m, s out of order in mltmod - ABORT!",stderr); fprintf(stderr," a = %12ld s = %12ld m = %12ld\n",a,s,m); fputs(" mltmod requires: 0 < a < m; 0 < s < m",stderr); exit(1); S10: if(!(a < h)) goto S20; a0 = a; p = 0; goto S120; S20: a1 = a/h; a0 = a-h*a1; qh = m/h; rh = m-h*qh; if(!(a1 >= h)) goto S50; a1 -= h; k = s/qh; p = h*(s-k*qh)-k*rh; S30: if(!(p < 0)) goto S40; p += m; goto S30; S40: goto S60; S50: p = 0; S60: /* P = (A2*S*H)MOD M */ if(!(a1 != 0)) goto S90; q = m/a1; k = s/q; p -= (k*(m-a1*q)); if(p > 0) p -= m; p += (a1*(s-k*q)); S70: if(!(p < 0)) goto S80; p += m; goto S70; S90: S80: k = p/qh; /* P = ((A2*H + A1)*S)MOD M */ p = h*(p-k*qh)-k*rh; S100: if(!(p < 0)) goto S110; p += m; goto S100; S120: S110: if(!(a0 != 0)) goto S150; /* P = ((A2*H + A1)*H*S)MOD M */ q = m/a0; k = s/q; p -= (k*(m-a0*q)); if(p > 0) p -= m; p += (a0*(s-k*q)); S130: if(!(p < 0)) goto S140; p += m; goto S130; S150: S140: mltmod = p; return mltmod; #undef h } void phrtsd(char* phrase,long *seed1,long *seed2) /* ********************************************************************** void phrtsd(char* phrase,long *seed1,long *seed2) PHRase To SeeDs Function Uses a phrase (character string) to generate two seeds for the RGN random number generator. Arguments phrase --> Phrase to be used for random number generation seed1 <-- First seed for generator seed2 <-- Second seed for generator Note Trailing blanks are eliminated before the seeds are generated. Generated seed values will fall in the range 1..2^30 (1..1,073,741,824) ********************************************************************** */ { static char table[] = "abcdefghijklmnopqrstuvwxyz\ ABCDEFGHIJKLMNOPQRSTUVWXYZ\ 0123456789\ !@#$%^&*()_+[];:'\\\"<>?,./"; long ix; static long twop30 = 1073741824L; static long shift[5] = { 1L,64L,4096L,262144L,16777216L }; static long i,ichr,j,lphr,values[5]; extern long lennob(char *str); *seed1 = 1234567890L; *seed2 = 123456789L; lphr = lennob(phrase); if(lphr < 1) return; for(i=0; i<=(lphr-1); i++) { for (ix=0; table[ix]; ix++) if (*(phrase+i) == table[ix]) break; if (!table[ix]) ix = 0; ichr = ix % 64; if(ichr == 0) ichr = 63; for(j=1; j<=5; j++) { *(values+j-1) = ichr-j; if(*(values+j-1) < 1) *(values+j-1) += 63; } for(j=1; j<=5; j++) { *seed1 = ( *seed1+*(shift+j-1)**(values+j-1) ) % twop30; *seed2 = ( *seed2+*(shift+j-1)**(values+6-j-1) ) % twop30; } } #undef twop30 } float ranf(void) /* ********************************************************************** float ranf(void) RANDom number generator as a Function Returns a random floating point number from a uniform distribution over 0 - 1 (endpoints of this interval are not returned) using the current generator This is a transcription from Pascal to Fortran of routine Uniform_01 from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) ********************************************************************** */ { static float ranf; /* 4.656613057E-10 is 1/M1 M1 is set in a data statement in IGNLGI and is currently 2147483563. If M1 changes, change this also. */ ranf = ignlgi()*4.656613057E-10; return ranf; } float sexpo(void) /* ********************************************************************** (STANDARD-) E X P O N E N T I A L DISTRIBUTION ********************************************************************** ********************************************************************** FOR DETAILS SEE: AHRENS, J.H. AND DIETER, U. COMPUTER METHODS FOR SAMPLING FROM THE EXPONENTIAL AND NORMAL DISTRIBUTIONS. COMM. ACM, 15,10 (OCT. 1972), 873 - 882. ALL STATEMENT NUMBERS CORRESPOND TO THE STEPS OF ALGORITHM 'SA' IN THE ABOVE PAPER (SLIGHTLY MODIFIED IMPLEMENTATION) Modified by Barry W. Brown, Feb 3, 1988 to use RANF instead of SUNIF. The argument IR thus goes away. ********************************************************************** Q(N) = SUM(ALOG(2.0)**K/K!) K=1,..,N , THE HIGHEST N (HERE 8) IS DETERMINED BY Q(N)=1.0 WITHIN STANDARD PRECISION */ { static float q[8] = { 0.6931472,0.9333737,0.9888778,0.9984959,0.9998293,0.9999833,0.9999986,1.0 }; static long i; static float sexpo,a,u,ustar,umin; static float *q1 = q; a = 0.0; u = ranf(); goto S30; S20: a += *q1; S30: u += u; if(u <= 1.0) goto S20; u -= 1.0; if(u > *q1) goto S60; sexpo = a+u; return sexpo; S60: i = 1; ustar = ranf(); umin = ustar; S70: ustar = ranf(); if(ustar < umin) umin = ustar; i += 1; if(u > *(q+i-1)) goto S70; sexpo = a+umin**q1; return sexpo; } float sgamma(float a) /* ********************************************************************** (STANDARD-) G A M M A DISTRIBUTION ********************************************************************** ********************************************************************** PARAMETER A >= 1.0 ! ********************************************************************** FOR DETAILS SEE: AHRENS, J.H. AND DIETER, U. GENERATING GAMMA VARIATES BY A MODIFIED REJECTION TECHNIQUE. COMM. ACM, 25,1 (JAN. 1982), 47 - 54. STEP NUMBERS CORRESPOND TO ALGORITHM 'GD' IN THE ABOVE PAPER (STRAIGHTFORWARD IMPLEMENTATION) Modified by Barry W. Brown, Feb 3, 1988 to use RANF instead of SUNIF. The argument IR thus goes away. ********************************************************************** PARAMETER 0.0 < A < 1.0 ! ********************************************************************** FOR DETAILS SEE: AHRENS, J.H. AND DIETER, U. COMPUTER METHODS FOR SAMPLING FROM GAMMA, BETA, POISSON AND BINOMIAL DISTRIBUTIONS. COMPUTING, 12 (1974), 223 - 246. (ADAPTED IMPLEMENTATION OF ALGORITHM 'GS' IN THE ABOVE PAPER) ********************************************************************** INPUT: A =PARAMETER (MEAN) OF THE STANDARD GAMMA DISTRIBUTION OUTPUT: SGAMMA = SAMPLE FROM THE GAMMA-(A)-DISTRIBUTION COEFFICIENTS Q(K) - FOR Q0 = SUM(Q(K)*A**(-K)) COEFFICIENTS A(K) - FOR Q = Q0+(T*T/2)*SUM(A(K)*V**K) COEFFICIENTS E(K) - FOR EXP(Q)-1 = SUM(E(K)*Q**K) PREVIOUS A PRE-SET TO ZERO - AA IS A', AAA IS A" SQRT32 IS THE SQUAREROOT OF 32 = 5.656854249492380 */ { extern float fsign( float num, float sign ); static float q1 = 4.166669E-2; static float q2 = 2.083148E-2; static float q3 = 8.01191E-3; static float q4 = 1.44121E-3; static float q5 = -7.388E-5; static float q6 = 2.4511E-4; static float q7 = 2.424E-4; static float a1 = 0.3333333; static float a2 = -0.250003; static float a3 = 0.2000062; static float a4 = -0.1662921; static float a5 = 0.1423657; static float a6 = -0.1367177; static float a7 = 0.1233795; static float e1 = 1.0; static float e2 = 0.4999897; static float e3 = 0.166829; static float e4 = 4.07753E-2; static float e5 = 1.0293E-2; static float aa = 0.0; static float aaa = 0.0; static float sqrt32 = 5.656854; static float sgamma,s2,s,d,t,x,u,r,q0,b,si,c,v,q,e,w,p; if(a == aa) goto S10; if(a < 1.0) goto S120; /* STEP 1: RECALCULATIONS OF S2,S,D IF A HAS CHANGED */ aa = a; s2 = a-0.5; s = sqrt(s2); d = sqrt32-12.0*s; S10: /* STEP 2: T=STANDARD NORMAL DEVIATE, X=(S,1/2)-NORMAL DEVIATE. IMMEDIATE ACCEPTANCE (I) */ t = snorm(); x = s+0.5*t; sgamma = x*x; if(t >= 0.0) return sgamma; /* STEP 3: U= 0,1 -UNIFORM SAMPLE. SQUEEZE ACCEPTANCE (S) */ u = ranf(); if(d*u <= t*t*t) return sgamma; /* STEP 4: RECALCULATIONS OF Q0,B,SI,C IF NECESSARY */ if(a == aaa) goto S40; aaa = a; r = 1.0/ a; q0 = ((((((q7*r+q6)*r+q5)*r+q4)*r+q3)*r+q2)*r+q1)*r; /* APPROXIMATION DEPENDING ON SIZE OF PARAMETER A THE CONSTANTS IN THE EXPRESSIONS FOR B, SI AND C WERE ESTABLISHED BY NUMERICAL EXPERIMENTS */ if(a <= 3.686) goto S30; if(a <= 13.022) goto S20; /* CASE 3: A .GT. 13.022 */ b = 1.77; si = 0.75; c = 0.1515/s; goto S40; S20: /* CASE 2: 3.686 .LT. A .LE. 13.022 */ b = 1.654+7.6E-3*s2; si = 1.68/s+0.275; c = 6.2E-2/s+2.4E-2; goto S40; S30: /* CASE 1: A .LE. 3.686 */ b = 0.463+s+0.178*s2; si = 1.235; c = 0.195/s-7.9E-2+1.6E-1*s; S40: /* STEP 5: NO QUOTIENT TEST IF X NOT POSITIVE */ if(x <= 0.0) goto S70; /* STEP 6: CALCULATION OF V AND QUOTIENT Q */ v = t/(s+s); if(fabs(v) <= 0.25) goto S50; q = q0-s*t+0.25*t*t+(s2+s2)*log(1.0+v); goto S60; S50: q = q0+0.5*t*t*((((((a7*v+a6)*v+a5)*v+a4)*v+a3)*v+a2)*v+a1)*v; S60: /* STEP 7: QUOTIENT ACCEPTANCE (Q) */ if(log(1.0-u) <= q) return sgamma; S70: /* STEP 8: E=STANDARD EXPONENTIAL DEVIATE U= 0,1 -UNIFORM DEVIATE T=(B,SI)-DOUBLE EXPONENTIAL (LAPLACE) SAMPLE */ e = sexpo(); u = ranf(); u += (u-1.0); t = b+fsign(si*e,u); /* STEP 9: REJECTION IF T .LT. TAU(1) = -.71874483771719 */ if(t < -0.7187449) goto S70; /* STEP 10: CALCULATION OF V AND QUOTIENT Q */ v = t/(s+s); if(fabs(v) <= 0.25) goto S80; q = q0-s*t+0.25*t*t+(s2+s2)*log(1.0+v); goto S90; S80: q = q0+0.5*t*t*((((((a7*v+a6)*v+a5)*v+a4)*v+a3)*v+a2)*v+a1)*v; S90: /* STEP 11: HAT ACCEPTANCE (H) (IF Q NOT POSITIVE GO TO STEP 8) */ if(q <= 0.0) goto S70; if(q <= 0.5) goto S100; w = exp(q)-1.0; goto S110; S100: w = ((((e5*q+e4)*q+e3)*q+e2)*q+e1)*q; S110: /* IF T IS REJECTED, SAMPLE AGAIN AT STEP 8 */ if(c*fabs(u) > w*exp(e-0.5*t*t)) goto S70; x = s+0.5*t; sgamma = x*x; return sgamma; S120: /* ALTERNATE METHOD FOR PARAMETERS A BELOW 1 (.3678794=EXP(-1.)) */ aa = 0.0; b = 1.0+0.3678794*a; S130: p = b*ranf(); if(p >= 1.0) goto S140; sgamma = exp(log(p)/ a); if(sexpo() < sgamma) goto S130; return sgamma; S140: sgamma = -log((b-p)/ a); if(sexpo() < (1.0-a)*log(sgamma)) goto S130; return sgamma; } float snorm(void) /* ********************************************************************** (STANDARD-) N O R M A L DISTRIBUTION ********************************************************************** ********************************************************************** FOR DETAILS SEE: AHRENS, J.H. AND DIETER, U. EXTENSIONS OF FORSYTHE'S METHOD FOR RANDOM SAMPLING FROM THE NORMAL DISTRIBUTION. MATH. COMPUT., 27,124 (OCT. 1973), 927 - 937. ALL STATEMENT NUMBERS CORRESPOND TO THE STEPS OF ALGORITHM 'FL' (M=5) IN THE ABOVE PAPER (SLIGHTLY MODIFIED IMPLEMENTATION) Modified by Barry W. Brown, Feb 3, 1988 to use RANF instead of SUNIF. The argument IR thus goes away. ********************************************************************** THE DEFINITIONS OF THE CONSTANTS A(K), D(K), T(K) AND H(K) ARE ACCORDING TO THE ABOVEMENTIONED ARTICLE */ { static float a[32] = { 0.0,3.917609E-2,7.841241E-2,0.11777,0.1573107,0.1970991,0.2372021,0.2776904, 0.3186394,0.36013,0.4022501,0.4450965,0.4887764,0.5334097,0.5791322, 0.626099,0.6744898,0.7245144,0.7764218,0.8305109,0.8871466,0.9467818, 1.00999,1.077516,1.150349,1.229859,1.318011,1.417797,1.534121,1.67594, 1.862732,2.153875 }; static float d[31] = { 0.0,0.0,0.0,0.0,0.0,0.2636843,0.2425085,0.2255674,0.2116342,0.1999243, 0.1899108,0.1812252,0.1736014,0.1668419,0.1607967,0.1553497,0.1504094, 0.1459026,0.14177,0.1379632,0.1344418,0.1311722,0.128126,0.1252791, 0.1226109,0.1201036,0.1177417,0.1155119,0.1134023,0.1114027,0.1095039 }; static float t[31] = { 7.673828E-4,2.30687E-3,3.860618E-3,5.438454E-3,7.0507E-3,8.708396E-3, 1.042357E-2,1.220953E-2,1.408125E-2,1.605579E-2,1.81529E-2,2.039573E-2, 2.281177E-2,2.543407E-2,2.830296E-2,3.146822E-2,3.499233E-2,3.895483E-2, 4.345878E-2,4.864035E-2,5.468334E-2,6.184222E-2,7.047983E-2,8.113195E-2, 9.462444E-2,0.1123001,0.136498,0.1716886,0.2276241,0.330498,0.5847031 }; static float h[31] = { 3.920617E-2,3.932705E-2,3.951E-2,3.975703E-2,4.007093E-2,4.045533E-2, 4.091481E-2,4.145507E-2,4.208311E-2,4.280748E-2,4.363863E-2,4.458932E-2, 4.567523E-2,4.691571E-2,4.833487E-2,4.996298E-2,5.183859E-2,5.401138E-2, 5.654656E-2,5.95313E-2,6.308489E-2,6.737503E-2,7.264544E-2,7.926471E-2, 8.781922E-2,9.930398E-2,0.11556,0.1404344,0.1836142,0.2790016,0.7010474 }; static long i; static float snorm,u,s,ustar,aa,w,y,tt; u = ranf(); s = 0.0; if(u > 0.5) s = 1.0; u += (u-s); u = 32.0*u; i = (long) (u); if(i == 32) i = 31; if(i == 0) goto S100; /* START CENTER */ ustar = u-(float)i; aa = *(a+i-1); S40: if(ustar <= *(t+i-1)) goto S60; w = (ustar-*(t+i-1))**(h+i-1); S50: /* EXIT (BOTH CASES) */ y = aa+w; snorm = y; if(s == 1.0) snorm = -y; return snorm; S60: /* CENTER CONTINUED */ u = ranf(); w = u*(*(a+i)-aa); tt = (0.5*w+aa)*w; goto S80; S70: tt = u; ustar = ranf(); S80: if(ustar > tt) goto S50; u = ranf(); if(ustar >= u) goto S70; ustar = ranf(); goto S40; S100: /* START TAIL */ i = 6; aa = *(a+31); goto S120; S110: aa += *(d+i-1); i += 1; S120: u += u; if(u < 1.0) goto S110; u -= 1.0; S140: w = u**(d+i-1); tt = (0.5*w+aa)*w; goto S160; S150: tt = u; S160: ustar = ranf(); if(ustar > tt) goto S50; u = ranf(); if(ustar >= u) goto S150; u = ranf(); goto S140; } float fsign( float num, float sign ) /* Transfers sign of argument sign to argument num */ { if ( ( sign>0.0f && num<0.0f ) || ( sign<0.0f && num>0.0f ) ) return -num; else return num; } /************************************************************************ FTNSTOP: Prints msg to standard error and then exits ************************************************************************/ void ftnstop(char* msg) /* msg - error message */ { if (msg != NULL) fprintf(stderr,"%s\n",msg); exit(0); } void advnst(long k) /* ********************************************************************** void advnst(long k) ADV-a-N-ce ST-ate Advances the state of the current generator by 2^K values and resets the initial seed to that value. This is a transcription from Pascal to Fortran of routine Advance_State from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments k -> The generator is advanced by2^K values ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gscgn(long getset,long *g); extern long Xm1,Xm2,Xa1,Xa2,Xcg1[],Xcg2[]; static long g,i,ib1,ib2; static long qrgnin; /* Abort unless random number generator initialized */ gsrgs(0L,&qrgnin); if(qrgnin) goto S10; fputs(" ADVNST called before random generator initialized - ABORT",stderr); exit(1); S10: gscgn(0L,&g); ib1 = Xa1; ib2 = Xa2; for(i=1; i<=k; i++) { ib1 = mltmod(ib1,ib1,Xm1); ib2 = mltmod(ib2,ib2,Xm2); } setsd(mltmod(ib1,*(Xcg1+g-1),Xm1),mltmod(ib2,*(Xcg2+g-1),Xm2)); /* NOW, IB1 = A1**K AND IB2 = A2**K */ #undef numg } void getsd(long *iseed1,long *iseed2) /* ********************************************************************** void getsd(long *iseed1,long *iseed2) GET SeeD Returns the value of two integer seeds of the current generator This is a transcription from Pascal to Fortran of routine Get_State from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments iseed1 <- First integer seed of generator G iseed2 <- Second integer seed of generator G ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gscgn(long getset,long *g); extern long Xcg1[],Xcg2[]; static long g; static long qrgnin; /* Abort unless random number generator initialized */ gsrgs(0L,&qrgnin); if(qrgnin) goto S10; fprintf(stderr,"%s\n", " GETSD called before random number generator initialized -- abort!"); exit(0); S10: gscgn(0L,&g); *iseed1 = *(Xcg1+g-1); *iseed2 = *(Xcg2+g-1); #undef numg } long ignlgi(void) /* ********************************************************************** long ignlgi(void) GeNerate LarGe Integer Returns a random integer following a uniform distribution over (1, 2147483562) using the current generator. This is a transcription from Pascal to Fortran of routine Random from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gssst(long getset,long *qset); extern void gscgn(long getset,long *g); extern void inrgcm(void); extern long Xm1,Xm2,Xa1,Xa2,Xcg1[],Xcg2[]; extern long Xqanti[]; static long ignlgi,curntg,k,s1,s2,z; static long qqssd,qrgnin; /* IF THE RANDOM NUMBER PACKAGE HAS NOT BEEN INITIALIZED YET, DO SO. IT CAN BE INITIALIZED IN ONE OF TWO WAYS : 1) THE FIRST CALL TO THIS ROUTINE 2) A CALL TO SETALL. */ gsrgs(0L,&qrgnin); if(!qrgnin) inrgcm(); gssst(0,&qqssd); if(!qqssd) setall(1234567890L,123456789L); /* Get Current Generator */ gscgn(0L,&curntg); s1 = *(Xcg1+curntg-1); s2 = *(Xcg2+curntg-1); k = s1/53668L; s1 = Xa1*(s1-k*53668L)-k*12211; if(s1 < 0) s1 += Xm1; k = s2/52774L; s2 = Xa2*(s2-k*52774L)-k*3791; if(s2 < 0) s2 += Xm2; *(Xcg1+curntg-1) = s1; *(Xcg2+curntg-1) = s2; z = s1-s2; if(z < 1) z += (Xm1-1); if(*(Xqanti+curntg-1)) z = Xm1-z; ignlgi = z; return ignlgi; #undef numg } void initgn(long isdtyp) /* ********************************************************************** void initgn(long isdtyp) INIT-ialize current G-e-N-erator Reinitializes the state of the current generator This is a transcription from Pascal to Fortran of routine Init_Generator from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments isdtyp -> The state to which the generator is to be set isdtyp = -1 => sets the seeds to their initial value isdtyp = 0 => sets the seeds to the first value of the current block isdtyp = 1 => sets the seeds to the first value of the next block ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gscgn(long getset,long *g); extern long Xm1,Xm2,Xa1w,Xa2w,Xig1[],Xig2[],Xlg1[],Xlg2[],Xcg1[],Xcg2[]; static long g; static long qrgnin; /* Abort unless random number generator initialized */ gsrgs(0L,&qrgnin); if(qrgnin) goto S10; fprintf(stderr,"%s\n", " INITGN called before random number generator initialized -- abort!"); exit(1); S10: gscgn(0L,&g); if(-1 != isdtyp) goto S20; *(Xlg1+g-1) = *(Xig1+g-1); *(Xlg2+g-1) = *(Xig2+g-1); goto S50; S20: if(0 != isdtyp) goto S30; goto S50; S30: /* do nothing */ if(1 != isdtyp) goto S40; *(Xlg1+g-1) = mltmod(Xa1w,*(Xlg1+g-1),Xm1); *(Xlg2+g-1) = mltmod(Xa2w,*(Xlg2+g-1),Xm2); goto S50; S40: fprintf(stderr,"%s\n","isdtyp not in range in INITGN"); exit(1); S50: *(Xcg1+g-1) = *(Xlg1+g-1); *(Xcg2+g-1) = *(Xlg2+g-1); #undef numg } void inrgcm(void) /* ********************************************************************** void inrgcm(void) INitialize Random number Generator CoMmon Function Initializes common area for random number generator. This saves the nuisance of a BLOCK DATA routine and the difficulty of assuring that the routine is loaded with the other routines. ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern long Xm1,Xm2,Xa1,Xa2,Xa1w,Xa2w,Xa1vw,Xa2vw; extern long Xqanti[]; static long T1; static long i; /* V=20; W=30; A1W = MOD(A1**(2**W),M1) A2W = MOD(A2**(2**W),M2) A1VW = MOD(A1**(2**(V+W)),M1) A2VW = MOD(A2**(2**(V+W)),M2) If V or W is changed A1W, A2W, A1VW, and A2VW need to be recomputed. An efficient way to precompute a**(2*j) MOD m is to start with a and square it j times modulo m using the function MLTMOD. */ Xm1 = 2147483563L; Xm2 = 2147483399L; Xa1 = 40014L; Xa2 = 40692L; Xa1w = 1033780774L; Xa2w = 1494757890L; Xa1vw = 2082007225L; Xa2vw = 784306273L; for(i=0; i First of two integer seeds iseed2 -> Second of two integer seeds ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gssst(long getset,long *qset); extern void gscgn(long getset,long *g); extern long Xm1,Xm2,Xa1vw,Xa2vw,Xig1[],Xig2[]; static long T1; static long g,ocgn; static long qrgnin; T1 = 1; /* TELL IGNLGI, THE ACTUAL NUMBER GENERATOR, THAT THIS ROUTINE HAS BEEN CALLED. */ gssst(1,&T1); gscgn(0L,&ocgn); /* Initialize Common Block if Necessary */ gsrgs(0L,&qrgnin); if(!qrgnin) inrgcm(); *Xig1 = iseed1; *Xig2 = iseed2; initgn(-1L); for(g=2; g<=numg; g++) { *(Xig1+g-1) = mltmod(Xa1vw,*(Xig1+g-2),Xm1); *(Xig2+g-1) = mltmod(Xa2vw,*(Xig2+g-2),Xm2); gscgn(1L,&g); initgn(-1L); } gscgn(1L,&ocgn); #undef numg } void setant(long qvalue) /* ********************************************************************** void setant(long qvalue) SET ANTithetic Sets whether the current generator produces antithetic values. If X is the value normally returned from a uniform [0,1] random number generator then 1 - X is the antithetic value. If X is the value normally returned from a uniform [0,N] random number generator then N - 1 - X is the antithetic value. All generators are initialized to NOT generate antithetic values. This is a transcription from Pascal to Fortran of routine Set_Antithetic from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments qvalue -> nonzero if generator G is to generating antithetic values, otherwise zero ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gscgn(long getset,long *g); extern long Xqanti[]; static long g; static long qrgnin; /* Abort unless random number generator initialized */ gsrgs(0L,&qrgnin); if(qrgnin) goto S10; fprintf(stderr,"%s\n", " SETANT called before random number generator initialized -- abort!"); exit(1); S10: gscgn(0L,&g); Xqanti[g-1] = qvalue; #undef numg } void setsd(long iseed1,long iseed2) /* ********************************************************************** void setsd(long iseed1,long iseed2) SET S-ee-D of current generator Resets the initial seed of the current generator to ISEED1 and ISEED2. The seeds of the other generators remain unchanged. This is a transcription from Pascal to Fortran of routine Set_Seed from the paper L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package with Splitting Facilities." ACM Transactions on Mathematical Software, 17:98-111 (1991) Arguments iseed1 -> First integer seed iseed2 -> Second integer seed ********************************************************************** */ { #define numg 32L extern void gsrgs(long getset,long *qvalue); extern void gscgn(long getset,long *g); extern long Xig1[],Xig2[]; static long g; static long qrgnin; /* Abort unless random number generator initialized */ gsrgs(0L,&qrgnin); if(qrgnin) goto S10; fprintf(stderr,"%s\n", " SETSD called before random number generator initialized -- abort!"); exit(1); S10: gscgn(0L,&g); *(Xig1+g-1) = iseed1; *(Xig2+g-1) = iseed2; initgn(-1L); #undef numg } long Xm1,Xm2,Xa1,Xa2,Xcg1[32],Xcg2[32],Xa1w,Xa2w,Xig1[32],Xig2[32],Xlg1[32], Xlg2[32],Xa1vw,Xa2vw; long Xqanti[32];