HEADER

Function: _header_sums
Class: header
Section: sums
Doc:
 \section{Sums, products, integrals and similar functions}
 \label{se:sums}

 Although the \kbd{gp} calculator is programmable, it is useful to have
 a number of preprogrammed loops, including sums, products, and a certain
 number of recursions. Also, a number of functions from numerical analysis
 like numerical integration and summation of series will be described here.

 One of the parameters in these loops must be the control variable, hence a
 simple variable name. In the descriptions, the letter $X$ will always denote
 any simple variable name, and represents the formal parameter used in the
 function. The expression to be summed, integrated, etc. is any legal PARI
 expression, including of course expressions using loops.

 \misctitle{Library mode}
 Since it is easier to program directly the loops in library mode, these
 functions are mainly useful for GP programming. On the other hand, numerical
 routines code a function (to be integrated, summed, etc.) with two parameters
 named
 \bprog
   GEN (*eval)(void*,GEN)
   void *E;  \\ context: eval(E, x) must evaluate your function at x.
 @eprog\noindent
 see the Libpari manual for details.

 \misctitle{Numerical integration}\sidx{numerical integration}
 Starting with version 2.2.9 the ``double exponential'' univariate
 integration method is implemented in \tet{intnum} and its variants. Romberg
 integration is still available under the name \kbd{intnumromb}, but
 superseded. It is possible to compute numerically integrals to thousands of
 decimal places in reasonable time, as long as the integrand is regular. It is
 also reasonable to compute numerically integrals in several variables,
 although more than two becomes lengthy. The integration domain may be
 noncompact, and the integrand may have reasonable singularities at
 endpoints. To use \kbd{intnum}, you must split the integral into a sum
 of subintegrals where the function has no singularities except at the
 endpoints. Polynomials in logarithms are not considered singular, and
 neglecting these logs, singularities are assumed to be algebraic (asymptotic
 to $C(x-a)^{-\alpha}$ for some $\alpha > -1$ when $x$ is
 close to $a$), or to correspond to simple discontinuities of some (higher)
 derivative of the function. For instance, the point $0$ is a singularity of
 $\text{abs}(x)$.

 See also the discrete summation methods below, sharing the prefix \kbd{sum}.