TOMS725
Evaluate Multivariate Normal Integrals

TOMS725 is a FORTRAN77 library which evaluates multivariate normal integrals associated with the computation of cumulative probabilities associated with a multidimensional variable governed by a normal probability density function with a known correlation matrix, by Zvi Drezner.

In particular, we wish to compute the probability P(H,R) that a sample vector X will satisfy

        x(i) <= h(i) for 1 <= i <= m
      

        1/sqrt(2^m*pi^m*det(R)) * 
        integral (-oo < x(m) < h(m) ) * 
        ...
        integral (-oo < x(2) < h(2) ) *
        integral (-oo < x(1) < h(1) )
        exp ( -0.5 * x' * inverse(R) * x ) dx1 dx2 ... dxm
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TOMS725 is available in a FORTRAN77 version.

Related Data and Programs:

HERMITE_POLYNOMIAL, a FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

PROB, a FORTRAN77 library which evaluates, samples, inverts, and characterizes a number of Probability Density Functions (PDF's) and Cumulative Density Functions (CDF's), including anglit, arcsin, benford, birthday, bernoulli, beta_binomial, beta, binomial, bradford, burr, cardiod, cauchy, chi, chi squared, circular, cosine, deranged, dipole, dirichlet mixture, discrete, empirical, english sentence and word length, error, exponential, extreme values, f, fisk, folded normal, frechet, gamma, generalized logistic, geometric, gompertz, gumbel, half normal, hypergeometric, inverse gaussian, laplace, levy, logistic, log normal, log series, log uniform, lorentz, maxwell, multinomial, nakagami, negative binomial, normal, pareto, planck, poisson, power, quasigeometric, rayleigh, reciprocal, runs, sech, semicircular, student t, triangle, uniform, von mises, weibull, zipf.

Reference:

1. Zvi Drezner,
   Algorithm 725: Computation of the Multivariate Normal Integral,
   Transactions on Mathematical Software,
   Volume 18, Number 4, December 1992, pages 470-480.

Source Code:

• toms725.f, the source code.
• toms725.sh, BASH commands to compile the source code.

Examples and Tests:

• toms725_prb.f, a sample calling program.
• toms725_prb.sh, BASH commands to compile and run the sample program.
• toms725_prb_output.txt, the output file.

List of Routines:

• DMV CALCULATES THE MULTIVARIATE NORMAL INTEGRAL.
• DMV1 CALCULATES THE PROBABILITY FOR A GIVEN K.
• SOLVE calculates probability for a given vector H and value K.
• DPOFA factors a double precision symmetric positive definite matrix.
• DPODI computes determinant or inverse of symmetric positive definite matrices.
• DAXPY computes a constant times a vector plus a vector.
• DSCAL scales a vector by a constant.
• DDOT forms the dot product of two vectors.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.