acb_calc_integrate res func param a b rel_goal abs_tol options prec

Computes a rigorous enclosure of the integral

\[`\] \[I = \int_a^b f(t) dt\]

where f is specified by (func, param), following a straight-line path between the complex numbers a and b. For finite results, a, b must be finite and f must be bounded on the path of integration. To compute improper integrals, the user should therefore truncate the path of integration manually (or make a regularizing change of variables, if possible). Returns ARB_CALC_SUCCESS if the integration converged to the target accuracy on all subintervals, and returns ARB_CALC_NO_CONVERGENCE otherwise.

By default, the integrand func will only be called with order = 0 or order = 1; that is, derivatives are not required.

• The integrand will be called with order = 0 to evaluate f normally on the integration path (either at a single point or on a subinterval). In this case, f is treated as a pointwise defined function and can have arbitrary discontinuities.
• The integrand will be called with order = 1 to evaluate f on a domain surrounding a segment of the integration path for the purpose of bounding the error of a quadrature formula. In this case, func must verify that f is holomorphic on this domain (and output a non-finite value if it is not).

The integration algorithm combines direct interval enclosures, Gauss-Legendre quadrature where f is holomorphic, and adaptive subdivision. This strategy supports integrands with discontinuities while providing exponential convergence for typical piecewise holomorphic integrands.

The following parameters control accuracy:

• rel_goal - relative accuracy goal as a number of bits, i.e. target a relative error less than \(\varepsilon_{rel} = 2^{-r}\) where r = rel_goal (note the sign: rel_goal should be nonnegative).
• abs_tol - absolute accuracy goal as a mag_t describing the error tolerance, i.e. target an absolute error less than \(\varepsilon_{abs}\) = abs_tol.
• prec - working precision. This is the working precision used to evaluate the integrand and manipulate interval endpoints. As currently implemented, the algorithm does not attempt to adjust the working precision by itself, and adaptive control of the working precision must be handled by the user.

For typical usage, set rel_goal = prec and abs_tol = \(2^{-prec}\). It usually only makes sense to have rel_goal between 0 and prec.

The algorithm attempts to achieve an error of \(\max(\varepsilon_{abs}, M \varepsilon_{rel})\) on each subinterval, where M is the magnitude of the integral. These parameters are only guidelines; the cumulative error may be larger than both the prescribed absolute and relative error goals, depending on the number of subdivisions, cancellation between segments of the integral, and numerical errors in the evaluation of the integrand.

To compute tiny integrals with high relative accuracy, one should set \(\varepsilon_{abs} \approx M \varepsilon_{rel}\) where M is a known estimate of the magnitude. Setting \(\varepsilon_{abs}\) to 0 is also allowed, forcing use of a relative instead of an absolute tolerance goal. This can be handy for exponentially small or large functions of unknown magnitude. It is recommended to avoid setting \(\varepsilon_{abs}\) very small if possible since the algorithm might need many extra subdivisions to estimate M automatically; if the approximate magnitude can be estimated by some external means (for example if a midpoint-width or endpoint-width estimate is known to be accurate), providing an appropriate \(\varepsilon_{abs} \approx M \varepsilon_{rel}\) will be more efficient.

If the integral has very large magnitude, setting the absolute tolerance to a corresponding large value is recommended for best performance, but it is not necessary for convergence since the absolute tolerance is increased automatically during the execution of the algorithm if the partial integrals are found to have larger error.

Additional options for the integration can be provided via the options parameter (documented below). To use all defaults, NULL can be passed for options.

acb_calc_integrate_opt_init options

Initializes options for use, setting all fields to 0 indicating default values.

acb_calc_integrate_gl_auto_deg res num_eval func param a b tol deg_limit flags prec

Attempts to compute \(I = \int_a^b f(t) dt\) using a single application of Gauss-Legendre quadrature with automatic determination of the quadrature degree so that the error is smaller than tol. Returns ARB_CALC_SUCCESS if the integral has been evaluated successfully or ARB_CALC_NO_CONVERGENCE if the tolerance could not be met. The total number of function evaluations is written to num_eval.

For the interval \([-1,1]\), the error of the n-point Gauss-Legendre rule is bounded by

\[`\] \[\left| I - \sum_{k=0}^{n-1} w_k f(x_k) \right| \le \frac{64 M}{15 (\rho-1) \rho^{2n-1}}\]

if \(f\) is holomorphic with \(|f(z)| \le M\) inside the ellipse E with foci \(\pm 1\) and semiaxes \(X\) and \(Y = \sqrt{X^2 - 1}\) such that \(\rho = X + Y\) with \(\rho > 1\) [Tre2008].

For an arbitrary interval, we use \(\int_a^b f(t) dt = \int_{-1}^1 g(t) dt\) where \(g(t) = \Delta f(\Delta t + m)\), \(\Delta = \tfrac{1}{2}(b-a)\), \(m = \tfrac{1}{2}(a+b)\). With \(I = [\pm X] + [\pm Y]i\), this means that we evaluate \(\Delta f(\Delta I + m)\) to get the bound \(M\). (An improvement would be to reduce the wrapping effect of rotating the ellipse when the path is not rectilinear).

We search for an \(X\) that makes the error small by trying steps \(2^{2^k}\). Larger \(X\) will give smaller \(1 / \rho^{2n-1}\) but larger \(M\). If we try successive larger values of \(k\), we can abort when \(M = \infty\) since this either means that we have hit a singularity or a branch cut or that overestimation in the evaluation of \(f\) is becoming too severe.

acb_calc_cauchy_bound bound func param x radius maxdepth prec

Sets bound to a ball containing the value of the integral

\[`\] \[C(x,r) = \frac{1}{2 \pi r} \oint_{|z-x| = r} |f(z)| dz = \int_0^1 |f(x+re^{2\pi i t})| dt\]

where f is specified by (func, param) and r is given by radius. The integral is computed using a simple step sum. The integration range is subdivided until the order of magnitude of b can be determined (i.e. its error bound is smaller than its midpoint), or until the step length has been cut in half maxdepth times. This function is currently implemented completely naively, and repeatedly subdivides the whole integration range instead of performing adaptive subdivisions.

acb_calc_integrate_taylor res func param a b inner_radius outer_radius accuracy_goal prec

Computes the integral

\[`\] \[I = \int_a^b f(t) dt\]

where f is specified by (func, param), following a straight-line path between the complex numbers a and b which both must be finite.

The integral is approximated by piecewise centered Taylor polynomials. Rigorous truncation error bounds are calculated using the Cauchy integral formula. More precisely, if the Taylor series of f centered at the point m is \(f(m+x) = \sum_{n=0}^{\infty} a_n x^n\), then

\[`\] \[\int f(m+x) = \left( \sum_{n=0}^{N-1} a_n \frac{x^{n+1}}{n+1} \right) + \left( \sum_{n=N}^{\infty} a_n \frac{x^{n+1}}{n+1} \right).\]

For sufficiently small x, the second series converges and its absolute value is bounded by

\[`\] \[\sum_{n=N}^{\infty} \frac{C(m,R)}{R^n} \frac{|x|^{n+1}}{N+1} = \frac{C(m,R) R x}{(R-x)(N+1)} \left( \frac{x}{R} \right)^N.\]

It is required that any singularities of f are isolated from the path of integration by a distance strictly greater than the positive value outer_radius (which is the integration radius used for the Cauchy bound). Taylor series step lengths are chosen so as not to exceed inner_radius, which must be strictly smaller than outer_radius for convergence. A smaller inner_radius gives more rapid convergence of each Taylor series but means that more series might have to be used. A reasonable choice might be to set inner_radius to half the value of outer_radius, giving roughly one accurate bit per term.

The truncation point of each Taylor series is chosen so that the absolute truncation error is roughly \(2^{-p}\) where p is given by accuracy_goal (in the future, this might change to a relative accuracy). Arithmetic operations and function evaluations are performed at a precision of prec bits. Note that due to accumulation of numerical errors, both values may have to be set higher (and the endpoints may have to be computed more accurately) to achieve a desired accuracy.

This function chooses the evaluation points uniformly rather than implementing adaptive subdivision.