The HCubature module is a pure-Julia implementation of multidimensional "h-adaptive" integration. That is, given an n-dimensional integral
then hcubature(f, a, b)
computes the integral, adaptively subdividing
the integration volume into smaller and smaller pieces until convergence
is achieved to the desired tolerance (specified by optional rtol
and
atol
keyword arguments, described in more detail below.
Because hcubature
is written purely in Julia, the integrand f(x)
can return any vector-like object (technically, any type supporting
+
, -
, *
real, and norm
: a Banach space). You can integrate
real, complex, and matrix-valued integrands, for example.
Assuming you've installed the HCubature package (via Pkg.add
) and
loaded it with using HCubature
, you can then use it by calling
the hcubature
function:
hcubature(f, a, b; norm=norm, rtol=sqrt(eps), atol=0, maxevals=typemax(Int))
This computes the n-dimensional integral of f(x), where n == length(a) == length(b)
,
over the hypercube whose corners are given by the vectors (or tuples) a
and b
.
That is, dimension x[i]
is integrated from a[i]
to b[i]
. The
return value of hcubature
is a tuple (I, E)
of the estimated integral
I
and an estimated error E
.
f
should be a function f(x)
that takes an n-dimensional vector x
and returns the integrand at x
. The integrand can be any type that supports
+
, -
, *
real, and norm
functions. For example, the integrand
can be real or complex numbers, vectors, matrices, etcetera.
(For performance, the StaticArrays
package is recommended for use with vector/matrix-valued integrands.)
The integrand f(x)
will be always be passed an SVector{n,T}
,
where SVector
is an efficient vector type defined in the StaticArrays
package and T
is a floating-point type determined by promoting
the endpoint a
and b
coordinates to a floating-point type.
(Your integrand f
should be type-stable: it should always return
a value of the same type, given this type of x
.)
The integrand will never be evaluated exactly at the boundaries of the integration volume. (So, for example, it is possible to have an integrand that blows up at the boundaries, as long as the integral is finite, though such singularities will slow convergence.)
The integration volume is adaptively subdivided, using a cubature
rule due to Genz and Malik (1980), until the estimated error E
satisfies E ≤ max(rtol*norm(I), atol)
, i.e. rtol
and atol
are
the relative and absolute tolerances requested, respectively.
It also stops if the number of f
evaluations exceeds maxevals
.
If neither atol
nor rtol
are specified, the
default rtol
is the square root of the precision eps(T)
of the coordinate type T
described above.
The error is estimated by norm(I - I′)
, where I′
is an alternative
estimated integral (via an "embedded" lower-order cubature rule.)
By default, the norm
function used (for both this and the convergence
test above) is norm
, but you can pass an alternative norm by
the norm
keyword argument. (This is especially useful when f
returns a vector of integrands with different scalings.)
hquadrature(f, a, b; norm=norm, rtol=sqrt(eps), atol=0, maxevals=typemax(Int))
Compute the (1d) integral of f(x) from a
to b
. The
return value of hcubature
is a tuple (I, E)
of the estimated integral
I
and an estimated error E
.
The other parameters are the same as hcubature
(above). hquadrature
is just a convenience wrapper around hcubature
so that you can work
with scalar x
, a
, and b
, rather than 1-component vectors.
Alternatively, for 1d integrals you can import the QuadGK module
and call the quadgk
function, which provides additional flexibility
e.g. in choosing the order of the quadrature rule. (QuadGK
is used
internally anyway by HCubature
to compute the quadrature rule.)
The algorithm of hquadrature
is based on the one 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).
HCubature was written by Steven G. Johnson (SGJ), and is free/open-source software under the MIT/expat license.
SGJ also wrote an earlier C implementation of a similar algorithm that is also callable from Julia via the Cubature.jl package. The HCubature package is a from-scratch re-implementation, not a translation, of this code, both to take advantage of unique features of Julia and to eliminate licensing restrictions arising from the use of C code taken from the HIntLib library. (In both cases, the original DCUHRE Fortran code of Genz was not examined, only the mathematical description in the papers.)