Function: intnumromb
Section: sums
C-Name: intnumromb0_bitprec
Prototype: V=GGED0,L,b
Help: intnumromb(X=a,b,expr,{flag=0}): numerical integration of expr (smooth in
 ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.
 expr can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be
 plus or minus infinity (chosen suitably), but of same sign; 3: expr has only
 limits at a or b.
Wrapper: (,,G)
Description:
  (gen,gen,gen,?small):gen:prec intnumromb_bitprec(${3 cookie}, ${3 wrapper}, $1, $2, $4, $bitprec)
Doc: numerical integration of \var{expr} (smooth in $]a,b[$), with respect to
 $X$. Suitable for low accuracy; if \var{expr} is very regular (e.g. analytic
 in a large region) and high accuracy is desired, try \tet{intnum} first.

 Set $\fl=0$ (or omit it altogether) when $a$ and $b$ are not too large, the
 function is smooth, and can be evaluated exactly everywhere on the interval
 $[a,b]$.

 If $\fl=1$, uses a general driver routine for doing numerical integration,
 making no particular assumption (slow).

 $\fl=2$ is tailored for being used when $a$ or $b$ are infinite using the
 change of variable $t = 1/X$. One \emph{must} have $ab>0$, and in fact if
 for example $b=+\infty$, then it is preferable to have $a$ as large as
 possible, at least $a\ge1$.

 If $\fl=3$, the function is allowed to be undefined
 at $a$ (but right continuous) or $b$ (left continuous),
 for example the function $\sin(x)/x$ between $x=0$ and $1$.

 The user should not require too much accuracy: \tet{realprecision} about
 30 decimal digits (\tet{realbitprecision} about 100 bits) is OK,
 but not much more. In addition, analytical cleanup of the integral must have
 been done: there must be no singularities in the interval or at the
 boundaries. In practice this can be accomplished with a change of
 variable. Furthermore, for improper integrals, where one or both of the
 limits of integration are plus or minus infinity, the function must decrease
 sufficiently rapidly at infinity, which can often be accomplished through
 integration by parts. Finally, the function to be integrated should not be
 very small (compared to the current precision) on the entire interval. This
 can of course be accomplished by just multiplying by an appropriate constant.

 Note that \idx{infinity} can be represented with essentially no loss of
 accuracy by an appropriate huge number. However beware of real underflow
 when dealing with rapidly decreasing functions. For example, in order to
 compute the $\int_0^\infty e^{-x^2}\,dx$ to 28 decimal digits, then one can
 set infinity equal to 10 for example, and certainly not to \kbd{1e1000}.

 \synt{intnumromb_bitprec}{void *E, GEN (*eval)(void*,GEN), GEN a, GEN b, long flag, long bitprec},
 where $\kbd{eval}(x, E)$ returns the value of the function at $x$.
 You may store any additional information required by \kbd{eval} in $E$, or set
 it to \kbd{NULL}. The historical variant
 \synt{intnumromb}{\dots, long prec}, where \kbd{prec} is expressed in words,
 not bits, is obsolete and should no longer be used.
