The Cubature module for Julia

This module provides one- and multi-dimensional adaptive integration routines for the Julia language, including support for vector-valued integrands and facilitation of parallel evaluation of integrands, based on the Cubature Package by Steven G. Johnson.

See also the HCubature package for a pure-Julia implementation of h-adaptive cubature using the same algorithm (which is therefore much more flexible in the types that it can integrate).

h-adaptive versus p-adaptive integration

Adaptive integration works by evaluating the integrand at more and more points until the integrand converges to a specified tolerance (with the error estimated by comparing integral estimates with different numbers of points). The Cubature module implements two schemes for this adaptation: h-adaptivity (routines hquadrature, hcubature, hquadrature_v, and hcubature_v) and p-adaptivity (routines pquadrature, pcubature, pquadrature_v, and pcubature_v). The h- and p-adaptive routines accept the same parameters, so you can use them interchangeably, but they have very different convergence characteristics.

h-adaptive integration works by recursively subdividing the integration domain into smaller and smaller regions, applying the same fixed-order (fixed number of points) integration rule within each sub-region and subdividing a region if its error estimate is too large. (Technically, we use a Gauss-Kronrod rule in 1d and a Genz-Malik rule in higher dimensions.) This is well-suited for functions that have localized sharp features (peaks, kinks, etcetera) in a portion of the domain, because it will adaptively add more points in this region while using a coarser set of points elsewhere. The h-adaptive routines should be your default choice if you know very little about the function you are integrating.

p-adaptive integration works by repeatedly doubling the number of points in the same domain, fitting to higher and higher degree polynomials (in a stable way) until convergence is achieved to the specified tolerance. (Technically, we use Clenshaw-Curtis quadrature rules.) This is best-suited for integrating smooth functions (infinitely differentiable, ideally analytic) in low dimensions (ideally 1 or 2), especially when high accuracy is required.

One technical difference that is sometimes important for functions with singularities at the edges of the integration domain: our h-adaptive algorithm only evaluates the integrand at the interior of the domain (never at the edges), whereas our p-adaptive algorithm also evaluates the integrand at the edges.

(The names "h-adaptive" and "p-adaptive" refer to the fact that the size of the subdomains is often denoted h while the degree of the polynomial fitting is often called p.)

Usage

Before using any of the routines below (and after installing, see above), you should include using Cubature in your code to import the functions from the Cubature module.

One-dimensional integrals of real-valued integrands

The simplest case is to integrate a single real-valued integrand f(x) from xmin to xmax, in which case you can call (similar to Julia's built-in quadgk routine):

(val,err) = hquadrature(f::Function, xmin::Real, xmax::Real;
                        reltol=1e-8, abstol=0, maxevals=0)

for h-adaptive integration, or pquadrature (with the same arguments) for p-adaptive integration. The return value is a tuple of val (the estimated integral) and err (the estimated absolute error in val, usually a conservative upper bound). The required arguments are:

• f is the integrand, a function f(x::Float64) that accepts a real argument (in the integration domain) and returns a real value.

• xmin and xmax are the boundaries of the integration domain. (That is, f is integrated from xmin to xmax.) They must be finite; to compute integrals over infinite or semi-infinite domains, you can use a change of variables.

There are also the following optional keyword arguments:

• reltol is the required relative error tolerance: the adaptive integration will terminate when err ≤ reltol*|val|; the default is 1e-8.

• The optional argument abstol is a required absolute error tolerance: the adaptive integration will terminate when err ≤ abstol. More precisely, the integration will terminate when either the relative- or the absolute-error tolerances are met. abstol defaults to 0, which means that it is ignored, but it can be useful to specify an absoute error tolerance for integrands that may integrate to zero (or nearly zero) because of large cancellations, in which case the problem is ill-conditioned and a small relative error tolerance may be unachievable.

• The optional argument maxevals specifies a (rough) maximum number of function evaluations: the integration will be terminated (and the current estimates returned) if this number is exceeded. The default maxevals is 0, in which case maxevals is ignored (no maximum).

Here is an example that integrates f(x) = x^3 from 0 to 1, printing the x coordinates that are evaluated:

hquadrature(x -> begin println(x); x^3; end, 0,1)

and returning (0.25,2.7755575615628914e-15), which is the correct answer 0.25. If we instead integrate from -1 to 1, the function may never exit: the exact integral is zero, and it is nearly impossible to satisfy the default reltol bound in floating-point arithmetic. In that case, you have to specify an abstol as explained above:

hquadrature(x -> begin println(x); x^3; end, -1,1, abstol=1e-8)

in which case it quickly returns.

Multi-dimensional integrals of real-valued integrands

The next simplest case is to integrate a single real-valued integrand f(x) over a multidimensional box, with each coordinate x[i] integrated from xmin[i] to xmax[i].

(val,err) = hcubature(f::Function, xmin, xmax;
                      reltol=1e-8, abstol=0, maxevals=0)

for h-adaptive integration, or pcubature (with the same arguments) for p-adaptive integration. The return value is a tuple of val (the estimated integral) and err (the estimated absolute error in val, usually a conservative upper bound). The arguments are:

• f is the integrand, a function f(x::Vector{Float64}) that accepts a vector x (in the integration domain) and returns a real value.

• xmin and xmax are arrays or tuples (or any iterable container) specifying the boundaries xmin[i] and xmax[i] of the integration domain in each coordinate. They must have length(xmin) == length(xmax). (As above, the components must be finite, but you can treat infinite domains via a change of variables).

• The optional keyword arguments reltol, abstol, and maxevals specify termination criteria as for hquadrature above.

Here is the same 1d example as above, integrating f(x) = x^3 from 0 to 1 while the x coordinates that are evaluated:

hcubature(x -> begin println(x[1]); x[1]^3; end, 0,1)

which again returns the correct integral 0.25. The only difference from before is that the argument x of our integrand is now an array, so we must use x[1] to access its value. If we have multiple coordinates, we use x[1], x[2], etcetera, as in this example integrating f(x,y) = x^3 y in the unit box [0,1]x[0,1] (the exact integral is 0.125):

hcubature(x -> begin println(x[1],",",x[2]); x[1]^3*x[2]; end, [0,0],[1,1])

Integrals of vector-valued integrands

In many applications, one wishes to compute integrals of several different integrands over the same domain. Of course, you could simply call hquadrature or hcubature multiple times, once for each integrand. However, in cases where the integrands are closely related functions, it is sometimes much more efficient to compute them together for a given point x than computing them separately. For example, if you have a complex-valued integrand, you could compute two separate integrals of the real and imaginary parts, but it is often more efficient and convenient to compute the real and imaginary parts at the same time.

The Cubature module supports this situation by allowing you to integrate a vector-valued integrand, computing fdim real integrals at once for any given dimension fdim (the dimension of the integrand, which is independent of the dimensionality of the integration domain). This is achieved by calling one of:

(val,err) = hquadrature(fdim::Integer, f::Function, xmin, xmax;
                        reltol=1e-8, abstol=0, maxevals=0,
                        error_norm = Cubature.INDIVIDUAL)
(val,err) = hcubature(fdim::Integer, f::Function, xmin, xmax;
                      reltol=1e-8, abstol=0, maxevals=0,
                      error_norm = Cubature.INDIVIDUAL)

for h-adaptive integration, or pquadrature/pcubature (with the same arguments) for p-adaptive integration. The return value is a tuple of two vectors of length fdim: val (the estimated integrals val[i]) and err (the estimated absolute errors err[i] in val[i]). The arguments are:

• fdim the dimension (number of components) of the integrand, i.e. the number of real-valued integrals to perform simultaneously

• f, the integrand. This is a function f(x, v) of two arguments: the point x in the integration domain (a Float64 for hquadrature and a Vector{Float64} for hcubature), and the vector v::Vector{Float64} of length fdim which is used to output the integrand values. That is, the function f should set v[i] to the value of the i-th integrand upon return. (The return value of f is ignored.) Note: the contents of v must be overwritten in-place by f. If you are not setting v[i] individually, you should do v[:] = ... and not v = ....

• xmin and xmax specify the boundaries of the integration domain, as for hquadrature and hcubature of scalar integrands above.

• The optional keyword arguments reltol, abstol, and maxevals specify termination criteria as for hquadrature above.

• The optional keyword argument error_norm specifies how the convergence criteria for the different integrands are combined. That is, given a vector val of integral estimates and a vector err of error estimates, how do we decide whether to stop? error_norm should be one of the following constants:

  • Cubature.INDIVIDUAL, the default. This terminates the integration when all of the integrals, taken individually, converge. That is, it checks err[i] ≤ reltol*|val[i]| or err[i] ≤ abstol, and only stops when one of these is true for all i.

  • Cubature.PAIRED. This is like Cubature.INDIVIDUAL, but applies the convergence criteria to consecutive pairs of integrands, as if these integrands were real and imaginary parts of complex numbers. (This is mainly useful for integrating complex functions in cases where you only care about error in the complex plane as opposed to error in the real and imaginary parts taken individually.)

  • Cubature.L1, Cubature.L2, or Cubature.LINF. These terminate the integration when |err| ≤ reltol*|val| or |err| ≤ abstol, where |...| denotes a norm applied to the whole vector of errors or integrals. In particular, the L1 norm (sum of absolute values), the L2 norm (the root-mean-square value), or the L-infinity norm (the maximum absolute value), respectively. These are useful if you only care about the error in the vector of integrals taken as a whole in some norm, rather than the relative error in the components taken individually (which could be large if some of the components integrate almost to zero). We provide three different norms for completeness, but probably the choice of norm doesn't matter too much; pick Cubature.L1 if you aren't sure.

Here is an example, similar to above, which integrates a vector of three integrands (x, x^2, x^3) from 0 to 1:

hquadrature(3, (x,v) -> v[:] = x.^[1:3], 0,1)

returning ([0.5, 0.333333, 0.25],[5.55112e-15, 3.70074e-15, 2.77556e-15]), which are of course the correct integrals.

Parallelizing the integrand evaluation

These numerical integration algorithms actually call your integrand function for batches of points at a time, not just point-by-point. It is useful to expose this information for parellelization: your code may be able to evaluate the integrand in parallel for multiple points.

This is provided by a "vectorized" interface to the Cubature module: functions hquadrature_v, pquadrature_v, hcubature_v, and pcubature_v, which have exactly the same arguments as the functions described in the previous sections, except that the integrand function f must accept different arguments.

In particular, for the _v integration routines, the integrand must be a function f(x,v) where x is an array of n points to evaluate and v is an array in which to store the values of the integrands at those points. n is determined at runtime and varies between calls to f. The shape of the arrays depends upon which routine is called:

• For hquadrature_v and pquadrature_v with real-valued integrands (no fdim argument), x and v are both 1d Float64 arrays of length n of the points (input) and values (output), respectively.

• For hcubature_v and pcubature_v with real-valued integrands (no fdim argument) in d integration dimensions, x is a 2d Float64 array of size d×n holding the points x[:,i] at which to evaluate the integrand, and v is a 1d Float64 array of length n in which to store the resulting integrand values.

• For hquadrature_v and pquadrature_v with vector-valued integrands (an fdim argument), x is a 1d Float64 array of length n of points at which to evaluate the integrands, and v is a 2d Float64 array of size fdim×n in which to store the values v[:,i] at these points.

• For hcubature_v and pcubature_v with vector-valued integrands (an fdim argument) in d integration dimensions, x is a 2d Float64 array of length d×n of points x[:,i] at which to evaluate the integrands, and v is a 2d Float64 array of size fdim×n in which to store the values v[:,i] at these points.

Technical Algorithms and References

The h-adaptive integration routines are based on those described in:

• A. C. Genz and A. A. Malik, "An adaptive algorithm for numeric integration over an N-dimensional rectangular region," J. Comput. Appl. Math., vol. 6 (no. 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., vol. 17 (no. 4), 437-451 (1991).

which we implemented in a C library, the Cubature Package, that is called from Julia.

Note that we do ''not'' use any of the original DCUHRE code by Genz, which is not under a free/open-source license.) Our code is based in part on code borrowed from the HIntLib numeric-integration library by Rudolf Schürer and from code for Gauss-Kronrod quadrature (for 1d integrals) from the GNU Scientific Library, both of which are free software under the GNU GPL. (Another free-software multi-dimensional integration library, unrelated to our code here but also implementing the Genz-Malik algorithm among other techniques, is Cuba.)

The hcubature_v technique is adapted 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 J. M. Bull and T. L. Freeman, "Parallel Globally Adaptive Algorithms for Multi-dimensional Integration," http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.6638 (1994).

The p-adaptive integration algorithm is simply a tensor product of nested Clenshaw-Curtis quadrature rules for power-of-two sizes, using a pre-computed table of points and weights up to order 2^20.

Author

This module was written by Steven G. Johnson.