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mvndst.f
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mvndst.f
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* Note: The test program has been removed and a utlity 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 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 vaules 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 intialization.
*
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