Cubature (Multidimensional integration)
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Cubature
Steven G. Johnson has written a simple C package for adaptive multidimensional integration (cubature) of vectorvalued integrands over hypercubes, i.e. to compute integrals of the form:
(Of course, it can handle scalar integrands as the special case where is a onedimensional vector: the dimensionalities of and are independent.) The integrand can be evaluated for an array of points at once to enable easy parallelization. The code, which is distributed as free software under the terms of the GNU General Public License (v2 or later), implements two algorithms for adaptive integration.
The first, hadaptive integration (recursively partitioning the integration domain into smaller subdomains, applying the same integration rule to each, until convergence is achieved), is based on the algorithms described in:
 A. C. Genz and A. A. Malik, "An adaptive algorithm for numeric integration over an Ndimensional rectangular region," J. Comput. Appl. Math. 6 (4), 295–302 (1980).
 J. Berntsen, T. O. Espelid, and A. Genz, "An adaptive algorithm for the approximate calculation of multiple integrals," ACM Trans. Math. Soft. 17 (4), 437–451 (1991).
This algorithm is best suited for a moderate number of dimensions (say, < 7), and is superseded for highdimensional integrals by other methods (e.g. Monte Carlo variants or sparse grids).
(Note that we do not use any of the original DCUHRE code by Genz, which is not under a free/opensource license.) Our code is based in part on code borrowed from the HIntLib numericintegration library by Rudolf Schürer and from code for GaussKronrod quadrature (for 1d integrals) from the GNU Scientific Library, both of which are free software under the GNU GPL. (Another freesoftware multidimensional integration library, unrelated to our code here but also implementing the Genz–Malik algorithm among other techniques, is Cuba.)
The second, padaptive integration (repeatedly doubling the degree of the quadrature rules until convergence is achieved), is based on a tensor product of Clenshaw–Curtis quadrature rules. This algorithm is often superior to hadaptive integration for smooth integrands in a few (≤3) dimensions, but is a poor choice in higher dimensions or for nonsmooth integrands.
For the most part, the padaptive routines below are dropin replacements for the hadaptive routines, with the same arguments etcetera, so you can experiment to see which one works best for your problem. One difference: the hadaptive routines do *not* evaluate the integrand on the boundaries of the integration volume, whereas the padaptive routines *do* evaluate the integrand at the boundaries. This means that the p adaptive routines require more care in cases where there are singularities at the boundaries.
I am also grateful to Dmitry Turbiner (dturbiner ατ alum.mit.edu), who implemented an initial prototype of the "vectorized" functionality (see below) for evaluating an array of points in a single call, which facilitates parallelization of the integrand evaluation.
Download
The current version of the code can be downloaded from:
a gzipped tar file. This unpacks to a directory containing a README
file with instructions and a standalone hcubature.c
or pcubature.c
file (along with a couple of private header files) that you can compile and link into your program for hadaptive and padaptive integration, respectively, and a header file cubature.h
that you #include
.
The test.c
file contains a little test program which is produced if you compile that file with DHCUBATURE
or DPCUBATURE
and link with hcubature.c
or pcubature.c
, respectively, as described below.
B. Narasimhan wrote a GNU R interface for an earlier version these routines, which can be downloaded here: http://cran.rproject.org/web/packages/cubature/index.html. The most recent version of the cubature code with the old interface (hadaptive only) may be downloaded from cubature20101018.tgz.
A Julia interface can be obtained from Cubature.jl. A Python cubature.py interface written by Saullo Castro is also available.
Usage
You should compile hcubature.c
and/or pcubature.c
and link it with your program, and #include
the header file cubature.h
.
The central subroutine you will be calling for hadaptive cubature is:
int hcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err);
or pcubature
(same arguments) for padaptive cubature. (See also the vectorized interface below.)
This integrates a function F(x), returning a vector of FDIM integrands, where x is a DIMdimensional vector ranging from XMIN to XMAX (i.e. in a hypercube XMIN_{i} ≤ x_{i} ≤ XMAX_{i}).
MAXEVAL specifies a maximum number of function evaluations (0 for no limit). [Note: the actual number of evaluations may somewhat exceed MAXEVAL: MAXEVAL is rounded up to an integer number of subregion evaluations.] Otherwise, the integration stops when the estimated error is less than REQABSERROR (the absolute error requested) or when the estimated error is less than REQRELERROR × integral value (the relative error requested). (Either of the error tolerances can be set to zero to ignore it.)
For vectorvalued integrands (FDIM > 1), NORM specifies the norm that is used to measure the error and determine convergence properties. (The NORM argument is irrelevant for FDIM ≤ 1 and is ignored.) Given vectors v and e of estimated integrals and errors therein, respectively, the NORM argument takes on one of the following enumerated constant values:

ERROR_L1
,ERROR_L2
,ERROR_LINF
: the absolute error is measured as e and the relative error as e/v, where ... is the L_{1}, L_{2}, or L_{∞} norm, respectively. (x in the L_{1} norm is the sum of the absolute values of the components, in the L_{2} norm is the root mean square of the components, and in the L_{∞} norm is the maximum absolute value of the components)

ERROR_INDIVIDUAL
: Convergence is achieved only when each integrand (each component of v and e) individually satisfies the requested error tolerances.

ERROR_PAIRED
: LikeERROR_INDIVIDUAL
, except that the integrands are grouped into consecutive pairs, with the error tolerance applied in an L2 sense to each pair. This option is mainly useful for integrating vectors of complex numbers, where each consecutive pair
of real integrands is the real and imaginary parts of a single complex integrand, and you only care about the error in the complex plane rather than the error in the real and imaginary parts separately.
VAL
and ERR
are arrays of length FDIM
, which upon return are the computed integral values and estimated errors, respectively. (The estimated errors are based on an embedded cubature rule of lower order; for smooth functions, this estimate is usually conservative.)
The return value of hcubature
and pcubature
is 0 on success and nonzero if there was an error (mainly only outofmemory situations or if the integrand signals an error). For a nonzero return value, the contents of the VAL
and ERR
arrays are undefined.
The integrand function F
should be a function of the form:
int f(unsigned ndim, const double *x, void *fdata, unsigned fdim, double *fval);
Here, the input is an array X
of length NDIM
(the point to be evaluated), the output is an array FVAL
of length FDIM
(the vector of function values at the point X
). he return value should be 0 on success or a nonzero value if an error occurred and the integration is to be terminated immediately (hcubature
will then return a nonzero error code).
The FDATA
argument of F
is equal to the FDATA
argument passed to hcubature
—this can be used by the caller to pass any additional information through to F
as needed (rather than using global variables, which are not reentrant). If F
does not need any additional data, you can just pass FDATA
= NULL
and ignore the FDATA
argument to F
.
"Vectorized" interface
These integration algorithms actually evaluate the integrand in "batches" of several points at a time. It is often useful to have access to this information so that your integrand function is not called for one point at a time, but rather for a whole "vector" of many points at once. For example, you may want to evaluate the integrand in parallel at different points. This functionality is available by calling:
int hcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, unsigned maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err);
(and similarly for pcubature_v
). All of the arguments and the return value are identical to hcubature
, above, except that now the integrand F
is of type integrand_v
, corresponding to a function of a different form. The integrand function F
should now be a function of the form:
int f(unsigned ndim, unsigned npts, const double *x, void *fdata, unsigned fdim, double *fval);
Now, X
is not a single point, but an array of NPTS
points (length NPTS
×NDIM
), and upon return the values of all FDIM
integrands at all NPTS
points should be stored in FVAL
(length NPTS
×FDIM
). In particular, x[i*ndim + j]
is the j
th coordinate of the i
th point (i
<npts
and j
<ndim
), and the k
th function evaluation (k
<fdim
) for the i
th point is returned in fval[i*fdim + k]
. (Note: the fval
indexing is changed compared to the adapt_integrate_v
interface in previous versions.)
Again, the return value should be 0 on success or nonzero to terminate the integration immediately (e.g. if an error occurred).
The size of NPTS
will vary with the dimensionality of the problem; higherdimensional problems will have (exponentially) larger NPTS, allowing for the possibility of more parallelism. Currently, for hcubature_v
, NPTS
starts at 15 in 1d, 17 in 2d, and 33 in 3d, but as adapt_integrate_v
calls your integrand more and more times the value of NPTS will grow. e.g. if you end up requiring several thousand points in total, NPTS
may grow to several hundred. We utilize an algorithm from:
 I. Gladwell, "Vectorization of one dimensional quadrature codes," pp. 230–238 in Numerical Integration. Recent Developments, Software and Applications, G. Fairweather and P. M. Keast, eds., NATO ASI Series C203, Dordrecht (1987).
as described in the article "Parallel globally adaptive algorithms for multidimensional integration" by Bull and Freeman (1994).
Example
As a simple example, consider the Gaussian integral of the scalar function over the hypercube [ − 2,2]^{3} in 3 dimensions. You could compute this integral via code that looks like:
#include <stdio.h> #include <math.h> #include "cubature.h" int f(unsigned ndim, const double *x, void *fdata, unsigned fdim, double *fval) { double sigma = *((double *) fdata); // we can pass σ via fdata argument double sum = 0; unsigned i; for (i = 0; i < ndim; ++i) sum += x[i] * x[i]; // compute the output value: note that fdim should == 1 from below fval[0] = exp(sigma * sum); return 0; // success }
then, later in the program where we call hcubature
:
{ double xmin[3] = {2,2,2}, xmax[3] = {2,2,2}, sigma = 0.5, val, err; hcubature(1, f, &sigma, 3, xmin, xmax, 0, 0, 1e4, ERROR_INDIVIDUAL, &val, &err); printf("Computed integral = %0.10g +/ %g\n", val, err); }
Here, we have specified a relative error tolerance of 10^{ − 4} (and no absolute error tolerance or maximum number of function evaluations). Note also that, to demonstrate the fdata
parameter, we have used it to pass the σ value through to our function (rather than hardcoding the value of σ in f
or using a global variable).
The output should be:
Computed integral = 13.69609043 +/ 0.00136919
Note that the estimated relative error is 0.00136919/13.69609043 = 9.9969×10^{–5}, within our requested tolerance of 10^{ − 4}. The actual error in the integral value, as can be determined e.g. by running the integration with a much lower tolerance, is much smaller: the integral is too small by about 0.00002, for an actual relative error of about 1.4×10^{–6}. As mentioned above, for smooth integrands the estimated error is almost always conservative (which means, unfortunately, that the integrator usually does more function evaluations than it needs to).
With the vectorized interface hcubature_v
, one would instead use:
int f(unsigned ndim, unsigned npts, const double *x, void *fdata, unsigned fdim, double *fval) { double sigma = *((double *) fdata); unsigned i, j; for (j = 0; j < npts; ++j) { // evaluate the integrand for npts points double sum = 0; for (i = 0; i < ndim; ++i) sum += x[j*ndim+i] * x[j*ndim+i]; fval[j] = exp(sigma * sum); } return 0; // success }
Infinite intervals
Integrals over infinite or semiinfinite intervals is possible by a change of variables. This is best illustrated in one dimension.
To compute an integral over a semiinfinite interval, you can perform the change of variables x=a+t/(1t):
For an infinite interval, you can perform the change of variables x=t/(1t^{2}):
Note the Jacobian factors multiplying f(⋅⋅⋅) in both integrals, and also that the limits of the t integrals are different in the two cases.
In multiple dimensions, one simply performs this change of variables on each dimension separately, as desired, multiplying the integrand by the corresponding Jacobian factor for each dimension being transformed.
The Jacobian factors diverge as the endpoints are approached. However, if f(x) goes to zero at least as fast as 1/x^{2}, then the limit of the integrand (including the Jacobian factor) is finite at the endpoints. If your f(x) vanishes more slowly than 1/x^{2} but still faster than 1/x, then the integrand blows up at the endpoints but the integral is still finite (it is an integrable singularity), so the code will work (although it may take many function evaluations to converge). If your f(x) vanishes only as 1/x, then it is not absolutely convergent and much more care is required even to define what you are trying to compute. (In any case, the hadaptive quadrature/cubature rules currently employed in cubature.c
do not evaluate the integrand at the endpoints, so you need not implement special handling for t=1.)
Test program
To compile a test program, just compile cubature.c
while #defining TEST_INTEGRATOR
, e.g. (on Unix or GNU/Linux) via:
cc DTEST_INTEGRATOR o cubature_test cubature.c lm
The usage is then:
./cubature_test <dim> <tol> <integrand> <maxeval>
where <dim> = # dimensions, <tol> = relative tolerance, <integrand> is 0–7 for one of eight possible test integrands (see below) and <maxeval> is the maximum number of function evaluations (0 for none, the default).
The different test integrands are:
 0: a product of cosine functions
 1: a Gaussian integral of exp(x^2), remapped to [0,infinity) limits
 2: volume of a hypersphere (integrating a discontinuous function!)
 3: a simple polynomial (product of coordinates)
 4: a Gaussian centered in the middle of the integration volume
 5: a sum of two Gaussians
 6: an example function by Tsuda, a product of terms with near poles
 7: a test integrand by Morokoff and Caflisch, a simple product of dimth roots of the coordinates (weakly singular at the boundary)
For example:
./cubature_test 3 1e5 4
integrates the Gaussian function (4) to a desired relative error tolerance of 10^{–5} in 3 dimensions. The output is:
3dim integral, tolerance = 1e05 integrand 4: integral = 1, est err = 9.99952e06, true err = 2.54397e08 #evals = 82203
Notice that it finds the integral after 82203 function evaluations with an estimated error of about 10^{–5}, but the true error (compared to the exact result) is much smaller (2.5×10^{–8}): the error estimation is typically conservative when applied to smooth functions like this.