acb_calc.h – calculus with complex-valued functions

This module provides functions for operations of calculus over the complex numbers (intended to include root-finding, integration, and so on). The numerical integration code is described in [Joh2018a].

Types, macros and constants

acb_calc_func_t

Typedef for a pointer to a function with signature:

int func(acb_ptr out, const acb_t inp, void * param, slong order, slong prec)

implementing a univariate complex function \(f(z)\). The param argument may be used to pass through additional parameters to the function. The return value is reserved for future use as an error code. It can be assumed that out and inp are not aliased.

When called with order = 0, func should write to out the value of \(f(z)\) at the point inp, evaluated at a precision of prec bits. In this case, f can be an arbitrary complex function, which may have branch cuts or even be non-holomorphic.

When called with order = n for \(n \ge 1\), func should write to out the first n coefficients in the Taylor series expansion of \(f(z)\) at the point inp, evaluated at a precision of prec bits. In this case, the implementation of func must verify that f is holomorphic on the complex interval defined by z, and set the coefficients in out to non-finite values otherwise.

For algorithms that do not rely on derivatives, func will always get called with order = 0 or order = 1, in which case the user only needs to implement evaluation of the direct function value \(f(z)\) (without derivatives). With order = 1, func must verify holomorphicity (unlike the order = 0 case).

If f is built from field operations and meromorphic functions, then no special action is necessary when order is positive since division by zero or evaluation of builtin functions at poles automatically produces infinite enclosures. However, manual action is needed for bounded functions with branch cuts. For example, when evaluating \(\sqrt{z}\), the output must be set to an non-finite value if \(z\) overlaps with the branch cut \([-\infty,0]\). The easiest way to accomplish this is to use versions of basic functions (sqrt, log, pow, etc.) that test holomorphicity of their arguments individually.

Some functions with branch cut detection are available as builtins: see acb_sqrt_analytic(), acb_rsqrt_analytic(), acb_log_analytic(), acb_pow_analytic(). It is not difficult to write custom functions of this type, using the following pattern:

/* Square root function on C with detection of the branch cut. */
void sqrt_analytic(acb_t res, const acb_t z, int analytic, slong prec)
{
    if (analytic &&
        arb_contains_zero(acb_imagref(z)) &&
        arb_contains_nonpositive(acb_realref(z))))
    {
        acb_indeterminate(res);
    }
    else
    {
        acb_sqrt(res, z, prec);
    }
}

The built-in methods acb_real_abs(), acb_real_sgn(), acb_real_heaviside(), acb_real_floor(), acb_real_ceil(), acb_real_max(), acb_real_min() provide piecewise holomorphic functions that are useful for integrating piecewise-defined real functions.

For example, here we define a piecewise holomorphic extension of the function \(f(z) = \sqrt{\lfloor z \rfloor}\) (for simplicity, without implementing derivatives):

int func(acb_ptr out, const acb_t inp, void * param, slong order, slong prec)
{
    if (order > 1) flint_abort();  /* derivatives not implemented */

    acb_real_floor(out, inp, order != 0, prec);
    acb_sqrt_analytic(out, out, order != 0, prec);
    return 0;
}

(Here, acb_real_sqrtpos() may be slightly better if it is known that z will be nonnegative on the path.)

See the demo program examples/integrals.c for more examples.

Integration

int acb_calc_integrate(acb_t res, acb_calc_func_t func, void * param, const acb_t a, const acb_t b, slong rel_goal, const mag_t abs_tol, const acb_calc_integrate_opt_t options, slong 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.

Options for integration

acb_calc_integrate_opt_struct
acb_calc_integrate_opt_t

This structure contains several fields, explained below. An acb_calc_integrate_opt_t is defined as an array of acb_calc_integrate_opt_struct of length 1, permitting it to be passed by reference. An acb_calc_integrate_opt_t must be initialized before use, which sets all fields to 0 or NULL. For fields that have not been set to other values, the integration algorithm will choose defaults automatically (based on the precision and accuracy goals). This structure will most likely be extended in the future to accommodate more options.

slong deg_limit

Maximum quadrature degree for each subinterval. If a zero or negative value is provided, the limit is set to a default value which currently equals \(0.5 \cdot \min(prec, rel\_goal) + 60\) for Gauss-Legendre quadrature. A higher quadrature degree can be beneficial for functions that are holomorphic on a large domain around the integration path and yet behave irregularly, such as oscillatory entire functions. The drawback of increasing the degree is that the precomputation time for quadrature nodes increases.

slong eval_limit

Maximum number of function evaluations. If a zero or negative value is provided, the limit is set to a default value which currently equals \(1000 \cdot prec + prec^2\). This is the main parameter used to limit the amount of work before aborting due to possible slow convergence or non-convergence. A lower limit allows aborting faster. A higher limit may be needed for integrands with many discontinuities or many singularities close to the integration path. This limit is only taken as a rough guideline, and the actual number of function evaluations may be slightly higher depending on the actual subdivisions.

slong depth_limit

Maximum search depth for adaptive subdivision. Technically, this is not the limit on the local bisection depth but the limit on the number of simultaneously queued subintervals. If a zero or negative value is provided, the limit is set to the default value \(2 \cdot \text{prec}\). Warning: memory usage may increase in proportion to this limit.

int use_heap

By default (if set to 0), new subintervals generated by adaptive bisection will be appended to the top of a stack. If set to 1, a binary heap will be used to maintain a priority queue where the subintervals with larger error have higher priority. This sometimes gives better results in case of convergence failure, but can lead to a much larger array of subintervals (requiring a higher depth_limit) when many global bisections are needed.

int verbose

If set to 1, some information about the overall integration process is printed to standard output. If set to 2, information about each subinterval is printed.

void acb_calc_integrate_opt_init(acb_calc_integrate_opt_t options)

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

Local integration algorithms

int acb_calc_integrate_gl_auto_deg(acb_t res, slong * num_eval, acb_calc_func_t func, void * param, const acb_t a, const acb_t b, const mag_t tol, slong deg_limit, int flags, slong 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.

Integration (old)

void acb_calc_cauchy_bound(arb_t bound, acb_calc_func_t func, void * param, const acb_t x, const arb_t radius, slong maxdepth, slong 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.

int acb_calc_integrate_taylor(acb_t res, acb_calc_func_t func, void * param, const acb_t a, const acb_t b, const arf_t inner_radius, const arf_t outer_radius, slong accuracy_goal, slong 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.