* Note: The test program has been removed and a utility routine mvnun has been * added. RTK 2004-08-10 * * Copyright 2000 by Alan Genz. * Copyright 2004-2005 by Enthought, Inc. * * The subroutine MVNUN is copyrighted by Enthought, Inc. * The rest of the file is copyrighted by Alan Genz and has kindly been offered * to the Scipy project under it's BSD-style license. * * This file contains a short test program and MVNDST, a subroutine * for computing multivariate normal distribution function values. * The file is self contained and should compile without errors on (77) * standard Fortran compilers. The test program demonstrates the use of * MVNDST for computing MVN distribution values for a five dimensional * example problem, with three different integration limit combinations. * * Alan Genz * Department of Mathematics * Washington State University * Pullman, WA 99164-3113 * Email : alangenz@wsu.edu * SUBROUTINE mvnun(d, n, lower, upper, means, covar, maxpts, & abseps, releps, value, inform) * Parameters * * d integer, dimensionality of the data * n integer, the number of data points * lower double(2), the lower integration limits * upper double(2), the upper integration limits * means double(n), the mean of each kernel * covar double(2,2), the covariance matrix * maxpts integer, the maximum number of points to evaluate at * abseps double, absolute error tolerance * releps double, relative error tolerance * value double intent(out), integral value * inform integer intent(out), * if inform == 0: error < eps * elif inform == 1: error > eps, all maxpts used integer n, d, infin(d), maxpts, inform, tmpinf double precision lower(d), upper(d), releps, abseps, & error, value, stdev(d), rho(d*(d-1)/2), & covar(d,d), & nlower(d), nupper(d), means(d,n), tmpval, & inf integer i, j inf = 0d0 inf = 1d0 / inf do i=1,d stdev(i) = dsqrt(covar(i,i)) if (upper(i).eq.inf.and.lower(i).eq.-inf) then infin(i) = -1 else if (lower(i).eq.-inf) then infin(i) = 0 else if (upper(i).eq.inf) then infin(i) = 1 else infin(i) = 2 end if end do do i=1,d do j=1,i-1 rho(j+(i-2)*(i-1)/2) = covar(i,j)/stdev(i)/stdev(j) end do end do value = 0d0 inform = 0 do i=1,n do j=1,d nlower(j) = (lower(j) - means(j,i))/stdev(j) nupper(j) = (upper(j) - means(j,i))/stdev(j) end do call mvndst(d,nlower,nupper,infin,rho,maxpts,abseps,releps, & error,tmpval,tmpinf) value = value + tmpval if (tmpinf .eq. 1) then inform = 1 end if end do value = value / n END SUBROUTINE mvnun_weighted(d, n, lower, upper, means, weights, & covar, maxpts, abseps, releps, & value, inform) * Parameters * * d integer, dimensionality of the data * n integer, the number of data points * lower double(2), the lower integration limits * upper double(2), the upper integration limits * means double(n), the mean of each kernel * weights double(n), the weight of each kernel * covar double(2,2), the covariance matrix * maxpts integer, the maximum number of points to evaluate at * abseps double, absolute error tolerance * releps double, relative error tolerance * value double intent(out), integral value * inform integer intent(out), * if inform == 0: error < eps * elif inform == 1: error > eps, all maxpts used integer n, d, infin(d), maxpts, inform, tmpinf double precision lower(d), upper(d), releps, abseps, & error, value, stdev(d), rho(d*(d-1)/2), & covar(d,d), & nlower(d), nupper(d), means(d,n), tmpval, & inf, weights(n) integer i, j inf = 0d0 inf = 1d0 / inf do i=1,d stdev(i) = dsqrt(covar(i,i)) if (upper(i).eq.inf.and.lower(i).eq.-inf) then infin(i) = -1 else if (lower(i).eq.-inf) then infin(i) = 0 else if (upper(i).eq.inf) then infin(i) = 1 else infin(i) = 2 end if end do do i=1,d do j=1,i-1 rho(j+(i-2)*(i-1)/2) = covar(i,j)/stdev(i)/stdev(j) end do end do value = 0d0 inform = 0 do i=1,n do j=1,d nlower(j) = (lower(j) - means(j,i))/stdev(j) nupper(j) = (upper(j) - means(j,i))/stdev(j) end do call mvndst(d,nlower,nupper,infin,rho,maxpts,abseps,releps, & error,tmpval,tmpinf) value = value + tmpval * weights(i) if (tmpinf .eq. 1) then inform = 1 end if end do END SUBROUTINE MVNDST( N, LOWER, UPPER, INFIN, CORREL, MAXPTS, & ABSEPS, RELEPS, ERROR, VALUE, INFORM ) * * A subroutine for computing multivariate normal probabilities. * This subroutine uses an algorithm given in the paper * "Numerical Computation of Multivariate Normal Probabilities", in * J. of Computational and Graphical Stat., 1(1992), pp. 141-149, by * Alan Genz * Department of Mathematics * Washington State University * Pullman, WA 99164-3113 * Email : AlanGenz@wsu.edu * * Parameters * * N INTEGER, the number of variables. * LOWER REAL, array of lower integration limits. * UPPER REAL, array of upper integration limits. * INFIN INTEGER, array of integration limits flags: * if INFIN(I) < 0, Ith limits are (-infinity, infinity); * if INFIN(I) = 0, Ith limits are (-infinity, UPPER(I)]; * if INFIN(I) = 1, Ith limits are [LOWER(I), infinity); * if INFIN(I) = 2, Ith limits are [LOWER(I), UPPER(I)]. * CORREL REAL, array of correlation coefficients; the correlation * coefficient in row I column J of the correlation matrix * should be stored in CORREL( J + ((I-2)*(I-1))/2 ), for J < I. * THe correlation matrix must be positive semidefinite. * MAXPTS INTEGER, maximum number of function values allowed. This * parameter can be used to limit the time. A sensible * strategy is to start with MAXPTS = 1000*N, and then * increase MAXPTS if ERROR is too large. * ABSEPS REAL absolute error tolerance. * RELEPS REAL relative error tolerance. * ERROR REAL estimated absolute error, with 99% confidence level. * VALUE REAL estimated value for the integral * INFORM INTEGER, termination status parameter: * if INFORM = 0, normal completion with ERROR < EPS; * if INFORM = 1, completion with ERROR > EPS and MAXPTS * function values used; increase MAXPTS to * decrease ERROR; * if INFORM = 2, N > 500 or N < 1. * EXTERNAL MVNDFN INTEGER N, INFIN(*), MAXPTS, INFORM, INFIS, IVLS DOUBLE PRECISION CORREL(*), LOWER(*), UPPER(*), RELEPS, ABSEPS, & ERROR, VALUE, E, D, MVNDNT, MVNDFN COMMON /DKBLCK/IVLS IF ( N .GT. 500 .OR. N .LT. 1 ) THEN INFORM = 2 VALUE = 0 ERROR = 1 ELSE INFORM = MVNDNT(N, CORREL, LOWER, UPPER, INFIN, INFIS, D, E) IF ( N-INFIS .EQ. 0 ) THEN VALUE = 1 ERROR = 0 ELSE IF ( N-INFIS .EQ. 1 ) THEN VALUE = E - D ERROR = 2D-16 ELSE * * Call the lattice rule integration subroutine * IVLS = 0 CALL DKBVRC( N-INFIS-1, IVLS, MAXPTS, MVNDFN, & ABSEPS, RELEPS, ERROR, VALUE, INFORM ) ENDIF ENDIF END DOUBLE PRECISION FUNCTION MVNDFN( N, W ) * * Integrand subroutine * INTEGER N, INFIN(*), INFIS, NL DOUBLE PRECISION W(*), LOWER(*), UPPER(*), CORREL(*), D, E PARAMETER ( NL = 500 ) DOUBLE PRECISION COV(NL*(NL+1)/2), A(NL), B(NL), Y(NL) INTEGER INFI(NL), I, J, IJ, IK, INFA, INFB DOUBLE PRECISION SUM, AI, BI, DI, EI, PHINVS, BVNMVN, MVNDNT SAVE A, B, INFI, COV MVNDFN = 1 INFA = 0 INFB = 0 IK = 1 IJ = 0 DO I = 1, N+1 SUM = 0 DO J = 1, I-1 IJ = IJ + 1 IF ( J .LT. IK ) SUM = SUM + COV(IJ)*Y(J) END DO IF ( INFI(I) .NE. 0 ) THEN IF ( INFA .EQ. 1 ) THEN AI = MAX( AI, A(I) - SUM ) ELSE AI = A(I) - SUM INFA = 1 END IF END IF IF ( INFI(I) .NE. 1 ) THEN IF ( INFB .EQ. 1 ) THEN BI = MIN( BI, B(I) - SUM ) ELSE BI = B(I) - SUM INFB = 1 END IF END IF IJ = IJ + 1 IF ( I .EQ. N+1 .OR. COV(IJ+IK+1) .GT. 0 ) THEN CALL MVNLMS( AI, BI, 2*INFA+INFB-1, DI, EI ) IF ( DI .GE. EI ) THEN MVNDFN = 0 RETURN ELSE MVNDFN = MVNDFN*( EI - DI ) IF ( I .LE. N ) Y(IK) = PHINVS( DI + W(IK)*( EI - DI ) ) IK = IK + 1 INFA = 0 INFB = 0 END IF END IF END DO RETURN * * Entry point for initialization. * ENTRY MVNDNT( N, CORREL, LOWER, UPPER, INFIN, INFIS, D, E ) MVNDNT = 0 * * Initialization and computation of covariance Cholesky factor. * CALL COVSRT( N, LOWER,UPPER,CORREL,INFIN,Y, INFIS,A,B,COV,INFI ) IF ( N - INFIS .EQ. 1 ) THEN CALL MVNLMS( A(1), B(1), INFI(1), D, E ) ELSE IF ( N - INFIS .EQ. 2 ) THEN IF ( ABS( COV(3) ) .GT. 0 ) THEN D = SQRT( 1 + COV(2)**2 ) IF ( INFI(2) .NE. 0 ) A(2) = A(2)/D IF ( INFI(2) .NE. 1 ) B(2) = B(2)/D E = BVNMVN( A, B, INFI, COV(2)/D ) D = 0 ELSE IF ( INFI(1) .NE. 0 ) THEN IF ( INFI(2) .NE. 0 ) A(1) = MAX( A(1), A(2) ) ELSE IF ( INFI(2) .NE. 0 ) A(1) = A(2) END IF IF ( INFI(1) .NE. 1 ) THEN IF ( INFI(2) .NE. 1 ) B(1) = MIN( B(1), B(2) ) ELSE IF ( INFI(2) .NE. 1 ) B(1) = B(2) END IF IF ( INFI(1) .NE. INFI(2) ) INFI(1) = 2 CALL MVNLMS( A(1), B(1), INFI(1), D, E ) END IF INFIS = INFIS + 1 END IF END SUBROUTINE MVNLMS( A, B, INFIN, LOWER, UPPER ) DOUBLE PRECISION A, B, LOWER, UPPER, MVNPHI INTEGER INFIN LOWER = 0 UPPER = 1 IF ( INFIN .GE. 0 ) THEN IF ( INFIN .NE. 0 ) LOWER = MVNPHI(A) IF ( INFIN .NE. 1 ) UPPER = MVNPHI(B) ENDIF UPPER = MAX( UPPER, LOWER ) END SUBROUTINE COVSRT( N, LOWER, UPPER, CORREL, INFIN, Y, & INFIS, A, B, COV, INFI ) * * Subroutine to sort integration limits and determine Cholesky factor. * INTEGER N, INFI(*), INFIN(*), INFIS DOUBLE PRECISION & A(*), B(*), COV(*), LOWER(*), UPPER(*), CORREL(*), Y(*) INTEGER I, J, K, L, M, II, IJ, IL, JMIN DOUBLE PRECISION SUMSQ, AJ, BJ, SUM, SQTWPI, EPS, D, E DOUBLE PRECISION CVDIAG, AMIN, BMIN, DMIN, EMIN, YL, YU PARAMETER ( SQTWPI = 2.506628274631001D0, EPS = 1D-10 ) IJ = 0 II = 0 INFIS = 0 DO I = 1, N A(I) = 0 B(I) = 0 INFI(I) = INFIN(I) IF ( INFI(I) .LT. 0 ) THEN INFIS = INFIS + 1 ELSE IF ( INFI(I) .NE. 0 ) A(I) = LOWER(I) IF ( INFI(I) .NE. 1 ) B(I) = UPPER(I) ENDIF DO J = 1, I-1 IJ = IJ + 1 II = II + 1 COV(IJ) = CORREL(II) END DO IJ = IJ + 1 COV(IJ) = 1 END DO * * First move any doubly infinite limits to innermost positions. * IF ( INFIS .LT. N ) THEN DO I = N, N-INFIS+1, -1 IF ( INFI(I) .GE. 0 ) THEN DO J = 1,I-1 IF ( INFI(J) .LT. 0 ) THEN CALL RCSWP( J, I, A, B, INFI, N, COV ) GO TO 10 ENDIF END DO ENDIF 10 END DO * * Sort remaining limits and determine Cholesky factor. * II = 0 DO I = 1, N-INFIS * * Determine the integration limits for variable with minimum * expected probability and interchange that variable with Ith. * DMIN = 0 EMIN = 1 JMIN = I CVDIAG = 0 IJ = II DO J = I, N-INFIS IF ( COV(IJ+J) .GT. EPS ) THEN SUMSQ = SQRT( COV(IJ+J) ) SUM = 0 DO K = 1, I-1 SUM = SUM + COV(IJ+K)*Y(K) END DO AJ = ( A(J) - SUM )/SUMSQ BJ = ( B(J) - SUM )/SUMSQ CALL MVNLMS( AJ, BJ, INFI(J), D, E ) IF ( EMIN + D .GE. E + DMIN ) THEN JMIN = J AMIN = AJ BMIN = BJ DMIN = D EMIN = E CVDIAG = SUMSQ ENDIF ENDIF IJ = IJ + J END DO IF ( JMIN .GT. I ) CALL RCSWP( I, JMIN, A,B, INFI, N, COV ) COV(II+I) = CVDIAG * * Compute Ith column of Cholesky factor. * Compute expected value for Ith integration variable and * scale Ith covariance matrix row and limits. * IF ( CVDIAG .GT. 0 ) THEN IL = II + I DO L = I+1, N-INFIS COV(IL+I) = COV(IL+I)/CVDIAG IJ = II + I DO J = I+1, L COV(IL+J) = COV(IL+J) - COV(IL+I)*COV(IJ+I) IJ = IJ + J END DO IL = IL + L END DO IF ( EMIN .GT. DMIN + EPS ) THEN YL = 0 YU = 0 IF ( INFI(I) .NE. 0 ) YL = -EXP( -AMIN**2/2 )/SQTWPI IF ( INFI(I) .NE. 1 ) YU = -EXP( -BMIN**2/2 )/SQTWPI Y(I) = ( YU - YL )/( EMIN - DMIN ) ELSE IF ( INFI(I) .EQ. 0 ) Y(I) = BMIN IF ( INFI(I) .EQ. 1 ) Y(I) = AMIN IF ( INFI(I) .EQ. 2 ) Y(I) = ( AMIN + BMIN )/2 END IF DO J = 1, I II = II + 1 COV(II) = COV(II)/CVDIAG END DO A(I) = A(I)/CVDIAG B(I) = B(I)/CVDIAG ELSE IL = II + I DO L = I+1, N-INFIS COV(IL+I) = 0 IL = IL + L END DO * * If the covariance matrix diagonal entry is zero, * permute limits and/or rows, if necessary. * * DO J = I-1, 1, -1 IF ( ABS( COV(II+J) ) .GT. EPS ) THEN A(I) = A(I)/COV(II+J) B(I) = B(I)/COV(II+J) IF ( COV(II+J) .LT. 0 ) THEN CALL DKSWAP( A(I), B(I) ) IF ( INFI(I) .NE. 2 ) INFI(I) = 1 - INFI(I) END IF DO L = 1, J COV(II+L) = COV(II+L)/COV(II+J) END DO DO L = J+1, I-1 IF( COV((L-1)*L/2+J+1) .GT. 0 ) THEN IJ = II DO K = I-1, L, -1 DO M = 1, K CALL DKSWAP( COV(IJ-K+M), COV(IJ+M) ) END DO CALL DKSWAP( A(K), A(K+1) ) CALL DKSWAP( B(K), B(K+1) ) M = INFI(K) INFI(K) = INFI(K+1) INFI(K+1) = M IJ = IJ - K END DO GO TO 20 END IF END DO GO TO 20 END IF COV(II+J) = 0 END DO 20 II = II + I Y(I) = 0 END IF END DO ENDIF END * SUBROUTINE DKSWAP( X, Y ) DOUBLE PRECISION X, Y, T T = X X = Y Y = T END * SUBROUTINE RCSWP( P, Q, A, B, INFIN, N, C ) * * Swaps rows and columns P and Q in situ, with P <= Q. * DOUBLE PRECISION A(*), B(*), C(*) INTEGER INFIN(*), P, Q, N, I, J, II, JJ CALL DKSWAP( A(P), A(Q) ) CALL DKSWAP( B(P), B(Q) ) J = INFIN(P) INFIN(P) = INFIN(Q) INFIN(Q) = J JJ = ( P*( P - 1 ) )/2 II = ( Q*( Q - 1 ) )/2 CALL DKSWAP( C(JJ+P), C(II+Q) ) DO J = 1, P-1 CALL DKSWAP( C(JJ+J), C(II+J) ) END DO JJ = JJ + P DO I = P+1, Q-1 CALL DKSWAP( C(JJ+P), C(II+I) ) JJ = JJ + I END DO II = II + Q DO I = Q+1, N CALL DKSWAP( C(II+P), C(II+Q) ) II = II + I END DO END * SUBROUTINE DKBVRC( NDIM, MINVLS, MAXVLS, FUNCTN, ABSEPS, RELEPS, & ABSERR, FINEST, INFORM ) * * Automatic Multidimensional Integration Subroutine * * AUTHOR: Alan Genz * Department of Mathematics * Washington State University * Pulman, WA 99164-3113 * Email: AlanGenz@wsu.edu * * Last Change: 1/15/03 * * KRBVRC computes an approximation to the integral * * 1 1 1 * I I ... I F(X) dx(NDIM)...dx(2)dx(1) * 0 0 0 * * * DKBVRC uses randomized Korobov rules for the first 100 variables. * The primary references are * "Randomization of Number Theoretic Methods for Multiple Integration" * R. Cranley and T.N.L. Patterson, SIAM J Numer Anal, 13, pp. 904-14, * and * "Optimal Parameters for Multidimensional Integration", * P. Keast, SIAM J Numer Anal, 10, pp.831-838. * If there are more than 100 variables, the remaining variables are * integrated using the rules described in the reference * "On a Number-Theoretical Integration Method" * H. Niederreiter, Aequationes Mathematicae, 8(1972), pp. 304-11. * *************** Parameters ******************************************** ****** Input parameters * NDIM Number of variables, must exceed 1, but not exceed 40 * MINVLS Integer minimum number of function evaluations allowed. * MINVLS must not exceed MAXVLS. If MINVLS < 0 then the * routine assumes a previous call has been made with * the same integrand and continues that calculation. * MAXVLS Integer maximum number of function evaluations allowed. * FUNCTN EXTERNALly declared user defined function to be integrated. * It must have parameters (NDIM,Z), where Z is a real array * of dimension NDIM. * * ABSEPS Required absolute accuracy. * RELEPS Required relative accuracy. ****** Output parameters * MINVLS Actual number of function evaluations used. * ABSERR Estimated absolute accuracy of FINEST. * FINEST Estimated value of integral. * INFORM INFORM = 0 for normal exit, when * ABSERR <= MAX(ABSEPS, RELEPS*ABS(FINEST)) * and * INTVLS <= MAXCLS. * INFORM = 1 If MAXVLS was too small to obtain the required * accuracy. In this case a value FINEST is returned with * estimated absolute accuracy ABSERR. ************************************************************************ EXTERNAL FUNCTN INTEGER NDIM, MINVLS, MAXVLS, INFORM, NP, PLIM, NLIM, KLIM, KLIMI, & SAMPLS, I, INTVLS, MINSMP PARAMETER ( PLIM = 28, NLIM = 1000, KLIM = 100, MINSMP = 8 ) INTEGER P(PLIM), C(PLIM,KLIM-1) DOUBLE PRECISION FUNCTN, ABSEPS, RELEPS, FINEST, ABSERR, DIFINT, & FINVAL, VARSQR, VAREST, VARPRD, VALUE DOUBLE PRECISION X(2*NLIM), VK(NLIM), ONE PARAMETER ( ONE = 1 ) SAVE P, C, SAMPLS, NP, VAREST INFORM = 1 INTVLS = 0 KLIMI = KLIM IF ( MINVLS .GE. 0 ) THEN FINEST = 0 VAREST = 0 SAMPLS = MINSMP DO I = MIN( NDIM, 10), PLIM NP = I IF ( MINVLS .LT. 2*SAMPLS*P(I) ) GO TO 10 END DO SAMPLS = MAX( MINSMP, MINVLS/( 2*P(NP) ) ) ENDIF 10 VK(1) = ONE/P(NP) DO I = 2, NDIM IF ( I .LE. KLIM ) THEN VK(I) = MOD( C(NP, MIN(NDIM-1,KLIM-1))*VK(I-1), ONE ) ELSE VK(I) = INT( P(NP)*2**(DBLE(I-KLIM)/(NDIM-KLIM+1)) ) VK(I) = MOD( VK(I)/P(NP), ONE ) END IF END DO FINVAL = 0 VARSQR = 0 DO I = 1, SAMPLS CALL DKSMRC( NDIM, KLIMI, VALUE, P(NP), VK, FUNCTN, X ) DIFINT = ( VALUE - FINVAL )/I FINVAL = FINVAL + DIFINT VARSQR = ( I - 2 )*VARSQR/I + DIFINT**2 END DO INTVLS = INTVLS + 2*SAMPLS*P(NP) VARPRD = VAREST*VARSQR FINEST = FINEST + ( FINVAL - FINEST )/( 1 + VARPRD ) IF ( VARSQR .GT. 0 ) VAREST = ( 1 + VARPRD )/VARSQR ABSERR = 7*SQRT( VARSQR/( 1 + VARPRD ) )/2 IF ( ABSERR .GT. MAX( ABSEPS, ABS(FINEST)*RELEPS ) ) THEN IF ( NP .LT. PLIM ) THEN NP = NP + 1 ELSE SAMPLS = MIN( 3*SAMPLS/2, ( MAXVLS - INTVLS )/( 2*P(NP) ) ) SAMPLS = MAX( MINSMP, SAMPLS ) ENDIF IF ( INTVLS + 2*SAMPLS*P(NP) .LE. MAXVLS ) GO TO 10 ELSE INFORM = 0 ENDIF MINVLS = INTVLS * * Optimal Parameters for Lattice Rules * DATA P( 1),(C( 1,I),I = 1,99)/ 31, 12, 2*9, 13, 8*12, 3*3, 12, & 2*7, 9*12, 3*3, 12, 2*7, 9*12, 3*3, 12, 2*7, 9*12, 3*3, 12, 2*7, & 8*12, 7, 3*3, 3*7, 21*3/ DATA P( 2),(C( 2,I),I = 1,99)/ 47, 13, 11, 17, 10, 6*15, & 22, 2*15, 3*6, 2*15, 9, 13, 3*2, 13, 2*11, 10, 9*15, 3*6, 2*15, & 9, 13, 3*2, 13, 2*11, 10, 9*15, 3*6, 2*15, 9, 13, 3*2, 13, 2*11, & 2*10, 8*15, 6, 2, 3, 2, 3, 12*2/ DATA P( 3),(C( 3,I),I = 1,99)/ 73, 27, 28, 10, 2*11, 20, & 2*11, 28, 2*13, 28, 3*13, 16*14, 2*31, 3*5, 31, 13, 6*11, 7*13, & 16*14, 2*31, 3*5, 11, 13, 7*11, 2*13, 11, 13, 4*5, 14, 13, 8*5/ DATA P( 4),(C( 4,I),I = 1,99)/ 113, 35, 2*27, 36, 22, 2*29, & 20, 45, 3*5, 16*21, 29, 10*17, 12*23, 21, 27, 3*3, 24, 2*27, & 17, 3*29, 17, 4*5, 16*21, 3*17, 6, 2*17, 6, 3, 2*6, 5*3/ DATA P( 5),(C( 5,I),I = 1,99)/ 173, 64, 66, 2*28, 2*44, 55, & 67, 6*10, 2*38, 5*10, 12*49, 2*38, 31, 2*4, 31, 64, 3*4, 64, & 6*45, 19*66, 11, 9*66, 45, 11, 7, 3, 3*2, 27, 5, 2*3, 2*5, 7*2/ DATA P( 6),(C( 6,I),I = 1,99)/ 263, 111, 42, 54, 118, 20, & 2*31, 72, 17, 94, 2*14, 11, 3*14, 94, 4*10, 7*14, 3*11, 7*8, & 5*18, 113, 2*62, 2*45, 17*113, 2*63, 53, 63, 15*67, 5*51, 12, & 51, 12, 51, 5, 2*3, 2*2, 5/ DATA P( 7),(C( 7,I),I = 1,99)/ 397, 163, 154, 83, 43, 82, & 92, 150, 59, 2*76, 47, 2*11, 100, 131, 6*116, 9*138, 21*101, & 6*116, 5*100, 5*138, 19*101, 8*38, 5*3/ DATA P( 8),(C( 8,I),I = 1,99)/ 593, 246, 189, 242, 102, & 2*250, 102, 250, 280, 118, 196, 118, 191, 215, 2*121, & 12*49, 34*171, 8*161, 17*14, 6*10, 103, 4*10, 5/ DATA P( 9),(C( 9,I),I = 1,99)/ 907, 347, 402, 322, 418, & 215, 220, 3*339, 337, 218, 4*315, 4*167, 361, 201, 11*124, & 2*231, 14*90, 4*48, 23*90, 10*243, 9*283, 16, 283, 16, 2*283/ DATA P(10),(C(10,I),I = 1,99)/ 1361, 505, 220, 601, 644, & 612, 160, 3*206, 422, 134, 518, 2*134, 518, 652, 382, & 206, 158, 441, 179, 441, 56, 2*559, 14*56, 2*101, 56, & 8*101, 7*193, 21*101, 17*122, 4*101/ DATA P(11),(C(11,I),I = 1,99)/ 2053, 794, 325, 960, 528, & 2*247, 338, 366, 847, 2*753, 236, 2*334, 461, 711, 652, & 3*381, 652, 7*381, 226, 7*326, 126, 10*326, 2*195, 19*55, & 7*195, 11*132, 13*387/ DATA P(12),(C(12,I),I = 1,99)/ 3079, 1189, 888, 259, 1082, 725, & 811, 636, 965, 2*497, 2*1490, 392, 1291, 2*508, 2*1291, 508, & 1291, 2*508, 4*867, 934, 7*867, 9*1284, 4*563, 3*1010, 208, & 838, 3*563, 2*759, 564, 2*759, 4*801, 5*759, 8*563, 22*226/ DATA P(13),(C(13,I),I = 1,99)/ 4621, 1763, 1018, 1500, 432, & 1332, 2203, 126, 2240, 1719, 1284, 878, 1983, 4*266, & 2*747, 2*127, 2074, 127, 2074, 1400, 10*1383, 1400, 7*1383, & 507, 4*1073, 5*1990, 9*507, 17*1073, 6*22, 1073, 6*452, 318, & 4*301, 2*86, 15/ DATA P(14),(C(14,I),I = 1,99)/ 6947, 2872, 3233, 1534, 2941, & 2910, 393, 1796, 919, 446, 2*919, 1117, 7*103, 2311, 3117, 1101, & 2*3117, 5*1101, 8*2503, 7*429, 3*1702, 5*184, 34*105, 13*784/ DATA P(15),(C(15,I),I = 1,99)/ 10427, 4309, 3758, 4034, 1963, & 730, 642, 1502, 2246, 3834, 1511, 2*1102, 2*1522, 2*3427, & 3928, 2*915, 4*3818, 3*4782, 3818, 4782, 2*3818, 7*1327, 9*1387, & 13*2339, 18*3148, 3*1776, 3*3354, 925, 2*3354, 5*925, 8*2133/ DATA P(16),(C(16,I),I = 1,99)/ 15641, 6610, 6977, 1686, 3819, & 2314, 5647, 3953, 3614, 5115, 2*423, 5408, 7426, 2*423, & 487, 6227, 2660, 6227, 1221, 3811, 197, 4367, 351, & 1281, 1221, 3*351, 7245, 1984, 6*2999, 3995, 4*2063, 1644, & 2063, 2077, 3*2512, 4*2077, 19*754, 2*1097, 4*754, 248, 754, & 4*1097, 4*222, 754,11*1982/ DATA P(17),(C(17,I),I = 1,99)/ 23473, 9861, 3647, 4073, 2535, & 3430, 9865, 2830, 9328, 4320, 5913, 10365, 8272, 3706, 6186, & 3*7806, 8610, 2563, 2*11558, 9421, 1181, 9421, 3*1181, 9421, & 2*1181, 2*10574, 5*3534, 3*2898, 3450, 7*2141, 15*7055, 2831, & 24*8204, 3*4688, 8*2831/ DATA P(18),(C(18,I),I = 1,99)/ 35221, 10327, 7582, 7124, 8214, & 9600, 10271, 10193, 10800, 9086, 2365, 4409, 13812, & 5661, 2*9344, 10362, 2*9344, 8585, 11114, 3*13080, 6949, & 3*3436, 13213, 2*6130, 2*8159, 11595, 8159, 3436, 18*7096, & 4377, 7096, 5*4377, 2*5410, 32*4377, 2*440, 3*1199/ DATA P(19),(C(19,I),I = 1,99)/ 52837, 19540, 19926, 11582, & 11113, 24585, 8726, 17218, 419, 3*4918, 15701, 17710, & 2*4037, 15808, 11401, 19398, 2*25950, 4454, 24987, 11719, & 8697, 5*1452, 2*8697, 6436, 21475, 6436, 22913, 6434, 18497, & 4*11089, 2*3036, 4*14208, 8*12906, 4*7614, 6*5021, 24*10145, & 6*4544, 4*8394/ DATA P(20),(C(20,I),I = 1,99)/ 79259, 34566, 9579, 12654, & 26856, 37873, 38806, 29501, 17271, 3663, 10763, 18955, & 1298, 26560, 2*17132, 2*4753, 8713, 18624, 13082, 6791, & 1122, 19363, 34695, 4*18770, 15628, 4*18770, 33766, 6*20837, & 5*6545, 14*12138, 5*30483, 19*12138, 9305, 13*11107, 2*9305/ DATA P(21),(C(21,I),I = 1,99)/118891, 31929, 49367, 10982, 3527, & 27066, 13226, 56010, 18911, 40574, 2*20767, 9686, 2*47603, & 2*11736, 41601, 12888, 32948, 30801, 44243, 2*53351, 16016, & 2*35086, 32581, 2*2464, 49554, 2*2464, 2*49554, 2464, 81, 27260, & 10681, 7*2185, 5*18086, 2*17631, 3*18086, 37335, 3*37774, & 13*26401, 12982, 6*40398, 3*3518, 9*37799, 4*4721, 4*7067/ DATA P(22),(C(22,I),I = 1,99)/178349, 40701, 69087, 77576, 64590, & 39397, 33179, 10858, 38935, 43129, 2*35468, 5279, 2*61518, 27945, & 2*70975, 2*86478, 2*20514, 2*73178, 2*43098, 4701, & 2*59979, 58556, 69916, 2*15170, 2*4832, 43064, 71685, 4832, & 3*15170, 3*27679, 2*60826, 2*6187, 5*4264, 45567, 4*32269, & 9*62060, 13*1803, 12*51108, 2*55315, 5*54140, 13134/ DATA P(23),(C(23,I),I = 1,99)/267523, 103650, 125480, 59978, & 46875, 77172, 83021, 126904, 14541, 56299, 43636, 11655, & 52680, 88549, 29804, 101894, 113675, 48040, 113675, & 34987, 48308, 97926, 5475, 49449, 6850, 2*62545, 9440, & 33242, 9440, 33242, 9440, 33242, 9440, 62850, 3*9440, & 3*90308, 9*47904, 7*41143, 5*36114, 24997, 14*65162, 7*47650, & 7*40586, 4*38725, 5*88329/ DATA P(24),(C(24,I),I = 1,99)/401287, 165843, 90647, 59925, & 189541, 67647, 74795, 68365, 167485, 143918, 74912, & 167289, 75517, 8148, 172106, 126159,3*35867, 121694, & 52171, 95354, 2*113969, 76304, 2*123709, 144615, 123709, & 2*64958, 32377, 2*193002, 25023, 40017, 141605, 2*189165, & 141605, 2*189165, 3*141605, 189165, 20*127047, 10*127785, & 6*80822, 16*131661, 7114, 131661/ DATA P(25),(C(25,I),I = 1,99)/601942, 130365, 236711, 110235, & 125699, 56483, 93735, 234469, 60549, 1291, 93937, & 245291, 196061, 258647, 162489, 176631, 204895, 73353, & 172319, 28881, 136787,2*122081, 275993, 64673, 3*211587, & 2*282859, 211587, 242821, 3*256865, 122203, 291915, 122203, & 2*291915, 122203, 2*25639, 291803, 245397, 284047, & 7*245397, 94241, 2*66575, 19*217673, 10*210249, 15*94453/ DATA P(26),(C(26,I),I = 1,99)/902933, 333459, 375354, 102417, & 383544, 292630, 41147, 374614, 48032, 435453, 281493, 358168, & 114121, 346892, 238990, 317313, 164158, 35497, 2*70530, 434839, & 3*24754, 393656, 2*118711, 148227, 271087, 355831, 91034, & 2*417029, 2*91034, 417029, 91034, 2*299843, 2*413548, 308300, & 3*413548, 3*308300, 413548, 5*308300, 4*15311, 2*176255, 6*23613, & 172210, 4* 204328, 5*121626, 5*200187, 2*121551, 12*248492, & 5*13942/ DATA P(27), (C(27,I), I = 1,99)/ 1354471, 500884, 566009, 399251, & 652979, 355008, 430235, 328722, 670680, 2*405585, 424646, & 2*670180, 641587, 215580, 59048, 633320, 81010, 20789, 2*389250, & 2*638764, 2*389250, 398094, 80846, 2*147776, 296177, 2*398094, & 2*147776, 396313, 3*578233, 19482, 620706, 187095, 620706, & 187095, 126467, 12*241663, 321632, 2*23210, 3*394484, 3*78101, & 19*542095, 3*277743, 12*457259/ DATA P(28), (C(28,I), I = 1, 99)/ 2031713, 858339, 918142, 501970, & 234813, 460565, 31996, 753018, 256150, 199809, 993599, 245149, & 794183, 121349, 150619, 376952, 2*809123, 804319, 67352, 969594, & 434796, 969594, 804319, 391368, 761041, 754049, 466264, 2*754049, & 466264, 2*754049, 282852, 429907, 390017, 276645, 994856, 250142, & 144595, 907454, 689648, 4*687580, 978368, 687580, 552742, 105195, & 942843, 768249, 4*307142, 7*880619, 11*117185, 11*60731, & 4*178309, 8*74373, 3*214965/ * END * SUBROUTINE DKSMRC( NDIM, KLIM, SUMKRO, PRIME, VK, FUNCTN, X ) EXTERNAL FUNCTN INTEGER NDIM, NK, KLIM, PRIME, K, J, JP DOUBLE PRECISION SUMKRO, VK(*), FUNCTN, X(*), ONE, XT, MVNUNI PARAMETER ( ONE = 1 ) SUMKRO = 0 NK = MIN( NDIM, KLIM ) DO J = 1, NK - 1 JP = J + MVNUNI()*( NK + 1 - J ) XT = VK(J) VK(J) = VK(JP) VK(JP) = XT END DO DO J = 1, NDIM X(NDIM+J) = MVNUNI() END DO DO K = 1, PRIME DO J = 1, NDIM X(J) = ABS( 2*MOD( K*VK(J) + X(NDIM+J), ONE ) - 1 ) END DO SUMKRO = SUMKRO + ( FUNCTN(NDIM,X) - SUMKRO )/( 2*K - 1 ) DO J = 1, NDIM X(J) = 1 - X(J) END DO SUMKRO = SUMKRO + ( FUNCTN(NDIM,X) - SUMKRO )/( 2*K ) END DO END * DOUBLE PRECISION FUNCTION MVNPHI( Z ) * * Normal distribution probabilities accurate to 1.e-15. * Z = no. of standard deviations from the mean. * * Based upon algorithm 5666 for the error function, from: * Hart, J.F. et al, 'Computer Approximations', Wiley 1968 * * Programmer: Alan Miller * * Latest revision - 30 March 1986 * DOUBLE PRECISION P0, P1, P2, P3, P4, P5, P6, * Q0, Q1, Q2, Q3, Q4, Q5, Q6, Q7, * Z, P, EXPNTL, CUTOFF, ROOTPI, ZABS PARAMETER( * P0 = 220.20 68679 12376 1D0, * P1 = 221.21 35961 69931 1D0, * P2 = 112.07 92914 97870 9D0, * P3 = 33.912 86607 83830 0D0, * P4 = 6.3739 62203 53165 0D0, * P5 = .70038 30644 43688 1D0, * P6 = .035262 49659 98910 9D0 ) PARAMETER( * Q0 = 440.41 37358 24752 2D0, * Q1 = 793.82 65125 19948 4D0, * Q2 = 637.33 36333 78831 1D0, * Q3 = 296.56 42487 79673 7D0, * Q4 = 86.780 73220 29460 8D0, * Q5 = 16.064 17757 92069 5D0, * Q6 = 1.7556 67163 18264 2D0, * Q7 = .088388 34764 83184 4D0 ) PARAMETER( ROOTPI = 2.5066 28274 63100 1D0 ) PARAMETER( CUTOFF = 7.0710 67811 86547 5D0 ) * ZABS = ABS(Z) * * |Z| > 37 * IF ( ZABS .GT. 37 ) THEN P = 0 ELSE * * |Z| <= 37 * EXPNTL = EXP( -ZABS**2/2 ) * * |Z| < CUTOFF = 10/SQRT(2) * IF ( ZABS .LT. CUTOFF ) THEN P = EXPNTL*( (((((P6*ZABS + P5)*ZABS + P4)*ZABS + P3)*ZABS * + P2)*ZABS + P1)*ZABS + P0)/(((((((Q7*ZABS + Q6)*ZABS * + Q5)*ZABS + Q4)*ZABS + Q3)*ZABS + Q2)*ZABS + Q1)*ZABS * + Q0 ) * * |Z| >= CUTOFF. * ELSE P = EXPNTL/( ZABS + 1/( ZABS + 2/( ZABS + 3/( ZABS * + 4/( ZABS + 0.65D0 ) ) ) ) )/ROOTPI END IF END IF IF ( Z .GT. 0 ) P = 1 - P MVNPHI = P END DOUBLE PRECISION FUNCTION PHINVS(P) * * ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3 * * Produces the normal deviate Z corresponding to a given lower * tail area of P. * * The hash sums below are the sums of the mantissas of the * coefficients. They are included for use in checking * transcription. * DOUBLE PRECISION SPLIT1, SPLIT2, CONST1, CONST2, * A0, A1, A2, A3, A4, A5, A6, A7, B1, B2, B3, B4, B5, B6, B7, * C0, C1, C2, C3, C4, C5, C6, C7, D1, D2, D3, D4, D5, D6, D7, * E0, E1, E2, E3, E4, E5, E6, E7, F1, F2, F3, F4, F5, F6, F7, * P, Q, R PARAMETER ( SPLIT1 = 0.425, SPLIT2 = 5, * CONST1 = 0.180625D0, CONST2 = 1.6D0 ) * * Coefficients for P close to 0.5 * PARAMETER ( * A0 = 3.38713 28727 96366 6080D0, * A1 = 1.33141 66789 17843 7745D+2, * A2 = 1.97159 09503 06551 4427D+3, * A3 = 1.37316 93765 50946 1125D+4, * A4 = 4.59219 53931 54987 1457D+4, * A5 = 6.72657 70927 00870 0853D+4, * A6 = 3.34305 75583 58812 8105D+4, * A7 = 2.50908 09287 30122 6727D+3, * B1 = 4.23133 30701 60091 1252D+1, * B2 = 6.87187 00749 20579 0830D+2, * B3 = 5.39419 60214 24751 1077D+3, * B4 = 2.12137 94301 58659 5867D+4, * B5 = 3.93078 95800 09271 0610D+4, * B6 = 2.87290 85735 72194 2674D+4, * B7 = 5.22649 52788 52854 5610D+3 ) * HASH SUM AB 55.88319 28806 14901 4439 * * Coefficients for P not close to 0, 0.5 or 1. * PARAMETER ( * C0 = 1.42343 71107 49683 57734D0, * C1 = 4.63033 78461 56545 29590D0, * C2 = 5.76949 72214 60691 40550D0, * C3 = 3.64784 83247 63204 60504D0, * C4 = 1.27045 82524 52368 38258D0, * C5 = 2.41780 72517 74506 11770D-1, * C6 = 2.27238 44989 26918 45833D-2, * C7 = 7.74545 01427 83414 07640D-4, * D1 = 2.05319 16266 37758 82187D0, * D2 = 1.67638 48301 83803 84940D0, * D3 = 6.89767 33498 51000 04550D-1, * D4 = 1.48103 97642 74800 74590D-1, * D5 = 1.51986 66563 61645 71966D-2, * D6 = 5.47593 80849 95344 94600D-4, * D7 = 1.05075 00716 44416 84324D-9 ) * HASH SUM CD 49.33206 50330 16102 89036 * * Coefficients for P near 0 or 1. * PARAMETER ( * E0 = 6.65790 46435 01103 77720D0, * E1 = 5.46378 49111 64114 36990D0, * E2 = 1.78482 65399 17291 33580D0, * E3 = 2.96560 57182 85048 91230D-1, * E4 = 2.65321 89526 57612 30930D-2, * E5 = 1.24266 09473 88078 43860D-3, * E6 = 2.71155 55687 43487 57815D-5, * E7 = 2.01033 43992 92288 13265D-7, * F1 = 5.99832 20655 58879 37690D-1, * F2 = 1.36929 88092 27358 05310D-1, * F3 = 1.48753 61290 85061 48525D-2, * F4 = 7.86869 13114 56132 59100D-4, * F5 = 1.84631 83175 10054 68180D-5, * F6 = 1.42151 17583 16445 88870D-7, * F7 = 2.04426 31033 89939 78564D-15 ) * HASH SUM EF 47.52583 31754 92896 71629 * Q = ( 2*P - 1 )/2 IF ( ABS(Q) .LE. SPLIT1 ) THEN R = CONST1 - Q*Q PHINVS = Q*( ( ( ((((A7*R + A6)*R + A5)*R + A4)*R + A3) * *R + A2 )*R + A1 )*R + A0 ) * /( ( ( ((((B7*R + B6)*R + B5)*R + B4)*R + B3) * *R + B2 )*R + B1 )*R + 1 ) ELSE R = MIN( P, 1 - P ) IF ( R .GT. 0 ) THEN R = SQRT( -LOG(R) ) IF ( R .LE. SPLIT2 ) THEN R = R - CONST2 PHINVS = ( ( ( ((((C7*R + C6)*R + C5)*R + C4)*R + C3) * *R + C2 )*R + C1 )*R + C0 ) * /( ( ( ((((D7*R + D6)*R + D5)*R + D4)*R + D3) * *R + D2 )*R + D1 )*R + 1 ) ELSE R = R - SPLIT2 PHINVS = ( ( ( ((((E7*R + E6)*R + E5)*R + E4)*R + E3) * *R + E2 )*R + E1 )*R + E0 ) * /( ( ( ((((F7*R + F6)*R + F5)*R + F4)*R + F3) * *R + F2 )*R + F1 )*R + 1 ) END IF ELSE PHINVS = 9 END IF IF ( Q .LT. 0 ) PHINVS = - PHINVS END IF END DOUBLE PRECISION FUNCTION BVNMVN( LOWER, UPPER, INFIN, CORREL ) * * A function for computing bivariate normal probabilities. * * Parameters * * LOWER REAL, array of lower integration limits. * UPPER REAL, array of upper integration limits. * INFIN INTEGER, array of integration limits flags: * if INFIN(I) = 0, Ith limits are (-infinity, UPPER(I)]; * if INFIN(I) = 1, Ith limits are [LOWER(I), infinity); * if INFIN(I) = 2, Ith limits are [LOWER(I), UPPER(I)]. * CORREL REAL, correlation coefficient. * DOUBLE PRECISION LOWER(*), UPPER(*), CORREL, BVU INTEGER INFIN(*) IF ( INFIN(1) .EQ. 2 .AND. INFIN(2) .EQ. 2 ) THEN BVNMVN = BVU ( LOWER(1), LOWER(2), CORREL ) + - BVU ( UPPER(1), LOWER(2), CORREL ) + - BVU ( LOWER(1), UPPER(2), CORREL ) + + BVU ( UPPER(1), UPPER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 2 .AND. INFIN(2) .EQ. 1 ) THEN BVNMVN = BVU ( LOWER(1), LOWER(2), CORREL ) + - BVU ( UPPER(1), LOWER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 1 .AND. INFIN(2) .EQ. 2 ) THEN BVNMVN = BVU ( LOWER(1), LOWER(2), CORREL ) + - BVU ( LOWER(1), UPPER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 2 .AND. INFIN(2) .EQ. 0 ) THEN BVNMVN = BVU ( -UPPER(1), -UPPER(2), CORREL ) + - BVU ( -LOWER(1), -UPPER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 0 .AND. INFIN(2) .EQ. 2 ) THEN BVNMVN = BVU ( -UPPER(1), -UPPER(2), CORREL ) + - BVU ( -UPPER(1), -LOWER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 1 .AND. INFIN(2) .EQ. 0 ) THEN BVNMVN = BVU ( LOWER(1), -UPPER(2), -CORREL ) ELSE IF ( INFIN(1) .EQ. 0 .AND. INFIN(2) .EQ. 1 ) THEN BVNMVN = BVU ( -UPPER(1), LOWER(2), -CORREL ) ELSE IF ( INFIN(1) .EQ. 1 .AND. INFIN(2) .EQ. 1 ) THEN BVNMVN = BVU ( LOWER(1), LOWER(2), CORREL ) ELSE IF ( INFIN(1) .EQ. 0 .AND. INFIN(2) .EQ. 0 ) THEN BVNMVN = BVU ( -UPPER(1), -UPPER(2), CORREL ) END IF END DOUBLE PRECISION FUNCTION BVU( SH, SK, R ) * * A function for computing bivariate normal probabilities. * * Yihong Ge * Department of Computer Science and Electrical Engineering * Washington State University * Pullman, WA 99164-2752 * and * Alan Genz * Department of Mathematics * Washington State University * Pullman, WA 99164-3113 * Email : alangenz@wsu.edu * * BVN - calculate the probability that X is larger than SH and Y is * larger than SK. * * Parameters * * SH REAL, integration limit * SK REAL, integration limit * R REAL, correlation coefficient * LG INTEGER, number of Gauss Rule Points and Weights * DOUBLE PRECISION BVN, SH, SK, R, ZERO, TWOPI INTEGER I, LG, NG PARAMETER ( ZERO = 0, TWOPI = 6.283185307179586D0 ) DOUBLE PRECISION X(10,3), W(10,3), AS, A, B, C, D, RS, XS DOUBLE PRECISION MVNPHI, SN, ASR, H, K, BS, HS, HK SAVE X, W * Gauss Legendre Points and Weights, N = 6 DATA ( W(I,1), X(I,1), I = 1,3) / * 0.1713244923791705D+00,-0.9324695142031522D+00, * 0.3607615730481384D+00,-0.6612093864662647D+00, * 0.4679139345726904D+00,-0.2386191860831970D+00/ * Gauss Legendre Points and Weights, N = 12 DATA ( W(I,2), X(I,2), I = 1,6) / * 0.4717533638651177D-01,-0.9815606342467191D+00, * 0.1069393259953183D+00,-0.9041172563704750D+00, * 0.1600783285433464D+00,-0.7699026741943050D+00, * 0.2031674267230659D+00,-0.5873179542866171D+00, * 0.2334925365383547D+00,-0.3678314989981802D+00, * 0.2491470458134029D+00,-0.1252334085114692D+00/ * Gauss Legendre Points and Weights, N = 20 DATA ( W(I,3), X(I,3), I = 1,10) / * 0.1761400713915212D-01,-0.9931285991850949D+00, * 0.4060142980038694D-01,-0.9639719272779138D+00, * 0.6267204833410906D-01,-0.9122344282513259D+00, * 0.8327674157670475D-01,-0.8391169718222188D+00, * 0.1019301198172404D+00,-0.7463319064601508D+00, * 0.1181945319615184D+00,-0.6360536807265150D+00, * 0.1316886384491766D+00,-0.5108670019508271D+00, * 0.1420961093183821D+00,-0.3737060887154196D+00, * 0.1491729864726037D+00,-0.2277858511416451D+00, * 0.1527533871307259D+00,-0.7652652113349733D-01/ IF ( ABS(R) .LT. 0.3 ) THEN NG = 1 LG = 3 ELSE IF ( ABS(R) .LT. 0.75 ) THEN NG = 2 LG = 6 ELSE NG = 3 LG = 10 ENDIF H = SH K = SK HK = H*K BVN = 0 IF ( ABS(R) .LT. 0.925 ) THEN HS = ( H*H + K*K )/2 ASR = ASIN(R) DO I = 1, LG SN = SIN(ASR*( X(I,NG)+1 )/2) BVN = BVN + W(I,NG)*EXP( ( SN*HK - HS )/( 1 - SN*SN ) ) SN = SIN(ASR*(-X(I,NG)+1 )/2) BVN = BVN + W(I,NG)*EXP( ( SN*HK - HS )/( 1 - SN*SN ) ) END DO BVN = BVN*ASR/(2*TWOPI) + MVNPHI(-H)*MVNPHI(-K) ELSE IF ( R .LT. 0 ) THEN K = -K HK = -HK ENDIF IF ( ABS(R) .LT. 1 ) THEN AS = ( 1 - R )*( 1 + R ) A = SQRT(AS) BS = ( H - K )**2 C = ( 4 - HK )/8 D = ( 12 - HK )/16 BVN = A*EXP( -(BS/AS + HK)/2 ) + *( 1 - C*(BS - AS)*(1 - D*BS/5)/3 + C*D*AS*AS/5 ) IF ( HK .GT. -160 ) THEN B = SQRT(BS) BVN = BVN - EXP(-HK/2)*SQRT(TWOPI)*MVNPHI(-B/A)*B + *( 1 - C*BS*( 1 - D*BS/5 )/3 ) ENDIF A = A/2 DO I = 1, LG XS = ( A*(X(I,NG)+1) )**2 RS = SQRT( 1 - XS ) BVN = BVN + A*W(I,NG)* + ( EXP( -BS/(2*XS) - HK/(1+RS) )/RS + - EXP( -(BS/XS+HK)/2 )*( 1 + C*XS*( 1 + D*XS ) ) ) XS = AS*(-X(I,NG)+1)**2/4 RS = SQRT( 1 - XS ) BVN = BVN + A*W(I,NG)*EXP( -(BS/XS + HK)/2 ) + *( EXP( -HK*(1-RS)/(2*(1+RS)) )/RS + - ( 1 + C*XS*( 1 + D*XS ) ) ) END DO BVN = -BVN/TWOPI ENDIF IF ( R .GT. 0 ) BVN = BVN + MVNPHI( -MAX( H, K ) ) IF ( R .LT. 0 ) BVN = -BVN + MAX( ZERO, MVNPHI(-H)-MVNPHI(-K) ) ENDIF BVU = BVN END DOUBLE PRECISION FUNCTION MVNUNI() * * Uniform (0,1) random number generator * * Reference: * L'Ecuyer, Pierre (1996), * "Combined Multiple Recursive Random Number Generators" * Operations Research 44, pp. 816-822. * * INTEGER A12, A13, A21, A23, P12, P13, P21, P23 INTEGER Q12, Q13, Q21, Q23, R12, R13, R21, R23 INTEGER X10, X11, X12, X20, X21, X22, Z, M1, M2, H DOUBLE PRECISION INVMP1 PARAMETER ( M1 = 2147483647, M2 = 2145483479 ) PARAMETER ( A12 = 63308, Q12 = 33921, R12 = 12979 ) PARAMETER ( A13 = -183326, Q13 = 11714, R13 = 2883 ) PARAMETER ( A21 = 86098, Q21 = 24919, R21 = 7417 ) PARAMETER ( A23 = -539608, Q23 = 3976, R23 = 2071 ) PARAMETER ( INVMP1 = 4.656612873077392578125D-10 ) * INVMP1 = 1/(M1+1) SAVE X10, X11, X12, X20, X21, X22 DATA X10, X11, X12, X20, X21, X22 & / 15485857, 17329489, 36312197, 55911127, 75906931, 96210113 / * * Component 1 * H = X10/Q13 P13 = -A13*( X10 - H*Q13 ) - H*R13 H = X11/Q12 P12 = A12*( X11 - H*Q12 ) - H*R12 IF ( P13 .LT. 0 ) P13 = P13 + M1 IF ( P12 .LT. 0 ) P12 = P12 + M1 X10 = X11 X11 = X12 X12 = P12 - P13 IF ( X12 .LT. 0 ) X12 = X12 + M1 * * Component 2 * H = X20/Q23 P23 = -A23*( X20 - H*Q23 ) - H*R23 H = X22/Q21 P21 = A21*( X22 - H*Q21 ) - H*R21 IF ( P23 .LT. 0 ) P23 = P23 + M2 IF ( P21 .LT. 0 ) P21 = P21 + M2 X20 = X21 X21 = X22 X22 = P21 - P23 IF ( X22 .LT. 0 ) X22 = X22 + M2 * * Combination * Z = X12 - X22 IF ( Z .LE. 0 ) Z = Z + M1 MVNUNI = Z*INVMP1 END