Abstract
We present an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library. Rapid convergence is ensured for piecewise complex analytic integrals by use of the Petras algorithm, which combines adaptive bisection with adaptive Gaussian quadrature where error bounds are determined via complex magnitudes without evaluating derivatives. The code is general, easy to use, and efficient, often outperforming existing non-rigorous software.
Keywords
- Numerical integration
- Interval arithmetic
- Special functions
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Notes
- 1.
Arb (http://arblib.org) is open source (GNU LGPL) software. For documentation and example code related to this paper, see http://arblib.org/acb_calc.html.
- 2.
Clenshaw-Curtis or double exponential quadrature could be used instead of Gaussian quadrature, but typically require more points for equivalent accuracy. We could also use Taylor series, but this makes supplying f more cumbersome for the user, and computing \(f,f'\ldots ,f^{(n)}\) tends to be more costly than n evaluations of f.
- 3.
In benchmark results, we omit the first-time nodes precomputation overhead.
- 4.
For example, mpmath provides quadgl for Gaussian quadrature, which is 2–3 times faster on some examples, but its precomputations are prohibitive at high precision.
- 5.
An exception is when f has an essential singularity inducing oscillation combined with slow decay. Oscillation with exponential decay is not a problem (as in \(E_4\), \(E_5\)), but integrals like \(\smash {\int _0^1\!\sin (1/x) dx \!=\! \int _1^{\infty } \!\sin (x)/x^2}\) (not benchmarked here) require \(\smash {2^{O(p)}\!}\) work, so we can only hope for 5–10 digits without specialized oscillatory algorithms.
- 6.
As a means to improve performance, we note the standard trick of manually changing variables to turn algebraic growth or decay into exponential decay. Indeed, \(x \rightarrow \sinh (x)\) gives \(E_1 = E_3\). Similarly \(x \rightarrow \tanh (x)\) and \(\smash {x \rightarrow e^{-x}}\) can be used in \(E_0\), \(E_2\).
- 7.
This works for integrating |f| when f is real, but since \(|\cdot |\) on \(\mathbb {C}\) is not holomorphic, integrating |f| for nonreal f must use direct enclosures, with \(2^{O(p)}\) cost. In that case, the user should instead construct complex-extensible real and imaginary parts \(f = g\!+\!h i\) (e.g. via Taylor polynomials if no closed forms exist) and integrate \(\sqrt{g^2 + h^2}\).
References
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Johansson, F. (2018). Numerical Integration in Arbitrary-Precision Ball Arithmetic. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_30
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DOI: https://doi.org/10.1007/978-3-319-96418-8_30
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