## Sums, products, integrals and similar functions

Although the `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.

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

        GEN (*eval)(void*,GEN)
        void *E;  \\ context: eval(E, x) must evaluate your function at x.

see the Libpari manual for details.

Numerical integration. Starting with version 2.2.9 the "double exponential" univariate integration method is implemented in `intnum` and its variants. Romberg integration is still available under the name `intnumromb`, but superseded.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 non-compact, and the integrand may have reasonable singularities at endpoints. To use `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)-α for some α > -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 abs(x).

See also the discrete summation methods below, sharing the prefix `sum`.

#### asympnum(expr, {k = 20}, {alpha = 1})

Asymptotic expansion of expr, corresponding to a sequence u(n), assuming it has the shape u(n) ~ ∑i ≥ 0 ai n-iα with rational coefficients ai with reasonable height; the algorithm is heuristic and performs repeated calls to limitnum, with `k` and `alpha` are as in `limitnum`

      ? f(n) = n! / (n^n*exp(-n)*sqrt(n));
      ? asympnum(f)
      %2 = []   \\ failure !
      ? l = limitnum(f)
      %3 = 2.5066282746310005024157652848110452530
      ? asympnum(n->f(n)/l) \\ normalize
      %4 = [1, 1/12, 1/288, -139/51840]

and we indeed get a few terms of Stirling's expansion. Note that it helps to normalize with a limit computed to higher accuracy:

      ? \p100
      ? L = limitnum(f)
      ? \p38
      ? asympnum(n->f(n)/L) \\ we get more terms!
      %6 = [1, 1/12, 1/288, -139/51840, -571/2488320, 163879/209018880,\
            5246819/75246796800, -534703531/902961561600]

If `alpha` is not an integer, loss of accuracy is expected, so it should be precomputed to double accuracy, say:

      ? \p38
      ? asympnum(n->-log(1-1/n^Pi),,Pi)
      %1 = [0, 1, 1/2, 1/3]
      ? asympnum(n->-log(1-1/sqrt(n)),,1/2)
      %2 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, \
        1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22]

      ? localprec(100); a = Pi;
      ? asympnum(n->-log(1-1/n^a),,a) \\ better !
      %4 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12]

The library syntax is `asympnum(void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec)`, where `u(E, n, prec)` must return u(n) in precision `prec`. Also available is `GEN asympnum0(GEN u, long muli, GEN alpha, long prec)`, where u must be a vector of sufficient length as above.

#### contfraceval(CF, t, {lim = -1})

Given a continued fraction `CF` output by `contfracinit`, evaluate the first `lim` terms of the continued fraction at `t` (all terms if `lim` is negative or omitted; if positive, `lim` must be less than or equal to the length of `CF`.

The library syntax is `GEN contfraceval(GEN CF, GEN t, long lim)`.

#### contfracinit(M, {lim = -1})

Given M representing the power series S = ∑n ≥ 0 M[n+1]z^n, transform it into a continued fraction; restrict to n ≤ `lim` if latter is non-negative. M can be a vector, a power series, a polynomial, or a rational function. The result is a 2-component vector [A,B] such that S = M[1] / (1+A[1]z+B[1]z^2/(1+A[2]z+B[2]z^2/(1+...1/(1+A[lim/2]z)))). Does not work if any coefficient of M vanishes, nor for series for which certain partial denominators vanish.

The library syntax is `GEN contfracinit(GEN M, long lim)`.

#### derivnum(X = a, expr, {ind = 1})

Numerical derivation of expr with respect to X at X = a. The order of derivation is 1 by default.

      ? derivnum(x=0, sin(exp(x))) - cos(1)
      %1 = 0.E-38

A clumsier approach, which would not work in library mode, is

      ? f(x) = sin(exp(x))
      ? f'(0) - cos(1)
      %2 = 0.E-38

* When a is a numerical type (integer, rational number, real number or `t_COMPLEX` of such), performs numerical derivation.

* When a is a (polynomial, rational function or) power series, compute `derivnum(t = a,f)` as f'(a) = (f(a))'/a':

      ? derivnum(x = 1 + t, sqrt(x))
      %1 = 1/2 - 1/4*t + 3/16*t^2 - 5/32*t^3 + ... + O(t^16)
      ? derivnum(x = 1/(1 + t), sqrt(x))
      %2 = 1/2 + 1/4*t - 1/16*t^2 + 1/32*t^3 + ... + O(t^16)
      ? derivnum(x = 1 + t + O(t^17), sqrt(x))
      %3 = 1/2 - 1/4*t + 3/16*t^2 - 5/32*t^3 + ... + O(t^16)

If the parameter ind is present, it can be

* a non-negative integer m, in which case we return f(m)(x);

* or a vector of orders, in which case we return the vector of derivatives.

      ? derivnum(x = 0, exp(sin(x)), 16) \\ 16-th derivative
      %1 = -52635599.000000000000000000000000000000

      ? round( derivnum(x = 0, exp(sin(x)), [0..13]) )  \\ 0-13-th derivatives
      %2 = [1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, 5973, 309376]

The library syntax is `derivfunk(void *E, GEN (*eval)(void*,GEN), GEN a, GEN ind, long prec)`. Also available is `GEN derivfun(void *E, GEN (*eval)(void *, GEN), GEN a, long prec)`. If a is a numerical type (`t_INT`, `t_FRAC`, `t_REAL` or `t_COMPLEX` of such, we have `GEN derivnumk(void *E, GEN (*eval)(void *, GEN, long), GEN a, GEN ind, long prec)` and `GEN derivnum(void *E, GEN (*eval)(void *, GEN, long prec), GEN a, long prec)`

#### intcirc(X = a, R, expr, {tab})

Numerical integration of (2iπ)-1expr with respect to X on the circle |X-a |= R. In other words, when expr is a meromorphic function, sum of the residues in the corresponding disk; tab is as in `intnum`, except that if computed with `intnuminit` it should be with the endpoints `[-1, 1]`.

      ? \p105
      ? intcirc(s=1, 0.5, zeta(s)) - 1
      time = 496 ms.
      %1 = 1.2883911040127271720 E-101 + 0.E-118*I

The library syntax is `intcirc(void *E, GEN (*eval)(void*,GEN), GEN a,GEN R,GEN tab, long prec)`.

#### intfuncinit(t = a, b, f, {m = 0})

Initialize tables for use with integral transforms (such as Fourier, Laplace or Mellin transforms) in order to compute ∫a^b f(t) k(t,z) dt for some kernel k(t,z). The endpoints a and b are coded as in `intnum`, f is the function to which the integral transform is to be applied and the non-negative integer m is as in `intnum`: multiply the number of sampling points roughly by 2^m, hopefully increasing the accuracy. This function is particularly useful when the function f is hard to compute, such as a gamma product.

Limitation. the endpoints a and b must be at infinity, with the same asymptotic behavior. Oscillating types are not supported. This is easily overcome by integrating vectors of functions, see example below.

Examples.

* numerical Fourier transform F(z) = ∫- oo + oo f(t)e-2iπ z t dt. First the easy case, assume that f decrease exponentially:

         f(t) = exp(-t^2);
         A = [-oo,1];
         B = [+oo,1];
         \p200
         T = intfuncinit(t = A,B , f(t));
         F(z) =
         { my(a = -2*I*Pi*z);
           intnum(t = A,B, exp(a*t), T);
         }
         ? F(1) - sqrt(Pi)*exp(-Pi^2)
         %1 = -1.3... E-212

Now the harder case, f decrease slowly: we must specify the oscillating behavior. Thus, we cannot precompute usefully since everything depends on the point we evaluate at:

         f(t) = 1 / (1+ abs(t));
         \p200
         \\ Fourier cosine transform
         FC(z) =
         { my(a = 2*Pi*z);
           intnum(t = [-oo, a*I], [+oo, a*I], cos(a*t)*f(t));
         }
         FC(1)

* Fourier coefficients: we must integrate over a period, but `intfuncinit` does not support finite endpoints. The solution is to integrate a vector of functions !

      FourierSin(f, T, k) =  \\ first k sine Fourier coeffs
      {
        my (w = 2*Pi/T);
        my (v = vector(k+1));
        intnum(t = -T/2, T/2,
           my (z = exp(I*w*t));
           v[1] = z;
           for (j = 2, k, v[j] = v[j-1]*z);
           f(t) * imag(v)) * 2/T;
      }
      FourierSin(t->sin(2*t), 2*Pi, 10)

The same technique can be used instead of `intfuncinit` to integrate f(t) k(t,z) whenever the list of z-values is known beforehand.

Note that the above code includes an unrelated optimization: the sin(j w t) are computed as imaginary parts of exp(i j w t) and the latter by successive multiplications.

* numerical Mellin inversion F(z) = (2iπ)-1 ∫c -i oo c+i oo f(s)z-s ds = (2π)-1 ∫- oo + oo f(c + i t)e-log z(c + it) dt. We take c = 2 in the program below:

         f(s) = gamma(s)^3;  \\ f(c+it) decrease as exp(-3Pi|t|/2)
         c = 2; \\ arbitrary
         A = [-oo,3*Pi/2];
         B = [+oo,3*Pi/2];
         T = intfuncinit(t=A,B, f(c + I*t));
         F(z) =
         { my (a = -log(z));
           intnum(t=A,B, exp(a*I*t), T)*exp(a*c) / (2*Pi);
         }

The library syntax is `intfuncinit(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,long m, long prec)`.

#### intnum(X = a, b, expr, {tab})

Numerical integration of expr on ]a,b[ with respect to X, using the double-exponential method, and thus O(Dlog D) evaluation of the integrand in precision D. The integrand may have values belonging to a vector space over the real numbers; in particular, it can be complex-valued or vector-valued. But it is assumed that the function is regular on ]a,b[. If the endpoints a and b are finite and the function is regular there, the situation is simple:

      ? intnum(x = 0,1, x^2)
      %1 = 0.3333333333333333333333333333
      ? intnum(x = 0,Pi/2, [cos(x), sin(x)])
      %2 = [1.000000000000000000000000000, 1.000000000000000000000000000]

An endpoint equal to ± oo is coded as `+oo` or `-oo`, as expected:

      ? intnum(x = 1,+oo, 1/x^2)
      %3 = 1.000000000000000000000000000

In basic usage, it is assumed that the function does not decrease exponentially fast at infinity:

      ? intnum(x=0,+oo, exp(-x))
        ***   at top-level: intnum(x=0,+oo,exp(-
        ***                 ^ —  —  —  —  —  — --
        *** exp: overflow in expo().

We shall see in a moment how to avoid that last problem, after describing the last optional argument tab.

The tab. argument The routine uses weights wi, which are mostly independent of the function being integrated, evaluated at many sampling points xi and approximates the integral by ∑ wi f(xi). If tab is

* a non-negative integer m, we multiply the number of sampling points by 2^m, hopefully increasing accuracy. Note that the running time increases roughly by a factor 2^m. One may try consecutive values of m until they give the same value up to an accepted error.

* a set of integration tables containing precomputed xi and wi as output by `intnuminit`. This is useful if several integrations of the same type are performed (on the same kind of interval and functions, for a given accuracy): we skip a precomputation of O(Dlog D) elementary functions in accuracy D, whose running time has the same order of magnitude as the evaluation of the integrand. This is in particular useful for multivariate integrals.

Specifying the behavior at endpoints. This is done as follows. An endpoint a is either given as such (a scalar, real or complex, `oo` or `-oo` for ± oo ), or as a two component vector [a,α], to indicate the behavior of the integrand in a neighborhood of a.

If a is finite, the code [a,α] means the function has a singularity of the form (x-a)α, up to logarithms. (If α \ge 0, we only assume the function is regular, which is the default assumption.) If a wrong singularity exponent is used, the result will lose decimals:

      ? c = -9/10;
      ? intnum(x=0, 1, x^c)         \\  assume x-9/10 is regular at 0
      %1 = 9.9999839078827082322596783301939063944
      ? intnum(x=[0,c], 1, x^c)  \\  no, it's not
      %2 = 10.000000000000000000000000000000000000
      ? intnum(x=[0,c/2], 1, x^c) \\  using a wrong exponent is bad
      %3 = 9.9999999997122749095442279375719919769

If a is ± oo , which is coded as `+oo` or `-oo`, the situation is more complicated, and [±`oo`,α] means:

* α = 0 (or no α at all, i.e. simply ±`oo`) assumes that the integrand tends to zero moderately quickly, at least as O(x-2) but not exponentially fast.

* α > 0 assumes that the function tends to zero exponentially fast approximately as exp(-α x). This includes oscillating but quickly decreasing functions such as exp(-x)sin(x).

      ? intnum(x=0, +oo, exp(-2*x))
        ***   at top-level: intnum(x=0,+oo,exp(-
        ***                 ^ —  —  —  —  —  — --
        *** exp: exponent (expo) overflow
      ? intnum(x=0, [+oo, 2], exp(-2*x))  \\  OK!
      %1 = 0.50000000000000000000000000000000000000
      ? intnum(x=0, [+oo, 3], exp(-2*x))  \\  imprecise exponent, still OK !
      %2 = 0.50000000000000000000000000000000000000
      ? intnum(x=0, [+oo, 10], exp(-2*x)) \\  wrong exponent  ==>  disaster
      %3 = 0.49999999999952372962457451698256707393

As the last exemple shows, the exponential decrease rate must be indicated to avoid overflow, but the method is robust enough for a rough guess to be acceptable.

* α < -1 assumes that the function tends to 0 slowly, like xα. Here the algorithm is less robust and it is essential to give a sharp α, unless α ≤ -2 in which case we use the default algorithm as if α were missing (or equal to 0).

      ? intnum(x=1, +oo, x^(-3/2))         \\ default
      %1 = 1.9999999999999999999999999999646391207
      ? intnum(x=1, [+oo,-3/2], x^(-3/2))  \\ precise decrease rate
      %2 = 2.0000000000000000000000000000000000000
      ? intnum(x=1, [+oo,-11/10], x^(-3/2)) \\ worse than default
      %3 = 2.0000000000000000000000000089298011973

The last two codes are reserved for oscillating functions. Let k > 0 real, and g(x) a non-oscillating function tending slowly to 0 (e.g. like a negative power of x), then

* α = k * I assumes that the function behaves like cos(kx)g(x).

* α = -k* I assumes that the function behaves like sin(kx)g(x).

Here it is critical to give the exact value of k. If the oscillating part is not a pure sine or cosine, one must expand it into a Fourier series, use the above codings, and sum the resulting contributions. Otherwise you will get nonsense. Note that cos(kx), and similarly sin(kx), means that very function, and not a translated version such as cos(kx+a).

Note. If f(x) = cos(kx)g(x) where g(x) tends to zero exponentially fast as exp(-α x), it is up to the user to choose between [±`oo`,α] and [±`oo`,k* I], but a good rule of thumb is that if the oscillations are weaker than the exponential decrease, choose [±`oo`,α], otherwise choose [±`oo`,k*I], although the latter can reasonably be used in all cases, while the former cannot. To take a specific example, in most inverse Mellin transforms, the integrand is a product of an exponentially decreasing and an oscillating factor. If we choose the oscillating type of integral we perhaps obtain the best results, at the expense of having to recompute our functions for a different value of the variable z giving the transform, preventing us to use a function such as `intfuncinit`. On the other hand using the exponential type of integral, we obtain less accurate results, but we skip expensive recomputations. See `intfuncinit` for more explanations.

Power series limits. The limits a and b can be power series of non-negative valuation, giving a power series expansion for the integral -- provided it exists.

      ? intnum(t=0,X + O(X^3), exp(t))
      %4 = 1.000...*X - 0.5000...*X^2 + O(X^3)
      ? bestappr( intnum(t=0,X + O(X^17), exp(t)) )- exp(X) + 1
      %5 = O(X^17)

The valuation of the limit cannot be negative since ∫01/X(1+t^2)-1 dt = π/2 - `sign`(X)+O(X^2).

Polynomials and rational functions are also allowed and converted to power series using current `seriesprecision`:

      ? bestappr( intnum(t=1,1+X, 1/t) )
      %6 = X - 1/2*X^2 + 1/3*X^3 - 1/4*X^4 + [...] + 1/15*X^15 + O(X^16)

The function does not work if the integral is singular with the constant coefficient of the series as limit:

      ? intnum(t=X^2+O(X^4),1, 1/sqrt(t))
      %8 = 2.000... - 6.236608109630992528 E28*X^2 + O(X^4)

however you can use

      ? intnum(t=[X^2+O(X^4),-1/2],1, 1/sqrt(t))
      %10 = 2.000000000000000000000000000-2.000000000000000000000000000*X^2+O(X^4)

whis is translated internally to

      ? intnum(t=[0,-1/2],1, 1/sqrt(t))-intnum(t=[0,-1/2],X^2+O(X^4), 1/sqrt(t))

For this form the argument tab can be used only as an integer, not a table precomputed by `intnuminit`.

We shall now see many examples to get a feeling for what the various parameters achieve. All examples below assume precision is set to 115 decimal digits. We first type

      ? \p 115

Apparent singularities. In many cases, apparent singularities can be ignored. For instance, if f(x) = 1 /(exp(x)-1) - exp(-x)/x, then ∫0^ oo f(x)dx = γ, Euler's constant `Euler`. But

      ? f(x) = 1/(exp(x)-1) - exp(-x)/x
      ? intnum(x = 0, [oo,1],  f(x)) - Euler
      %1 = 0.E-115

But close to 0 the function f is computed with an enormous loss of accuracy, and we are in fact lucky that it get multiplied by weights which are sufficiently close to 0 to hide this:

      ? f(1e-200)
      %2 = -3.885337784451458142 E84

A more robust solution is to define the function differently near special points, e.g. by a Taylor expansion

      ? F = truncate( f(t + O(t^10)) ); \\  expansion around t = 0
      ? poldegree(F)
      %4 = 7
      ? g(x) = if (x > 1e-18, f(x), subst(F,t,x)); \\  note that 7.18 > 105
      ? intnum(x = 0, [oo,1],  g(x)) - Euler
      %2 = 0.E-115

It is up to the user to determine constants such as the 10-18 and 10 used above.

True singularities. With true singularities the result is worse. For instance

      ? intnum(x = 0, 1,  x^(-1/2)) - 2
      %1 = -3.5... E-68 \\  only 68 correct decimals

      ? intnum(x = [0,-1/2], 1,  x^(-1/2)) - 2
      %2 = 0.E-114 \\  better

Oscillating functions.

      ? intnum(x = 0, oo, sin(x) / x) - Pi/2
      %1 = 16.19.. \\  nonsense
      ? intnum(x = 0, [oo,1], sin(x)/x) - Pi/2
      %2 = -0.006.. \\  bad
      ? intnum(x = 0, [oo,-I], sin(x)/x) - Pi/2
      %3 = 0.E-115 \\  perfect
      ? intnum(x = 0, [oo,-I], sin(2*x)/x) - Pi/2  \\  oops, wrong k
      %4 = 0.06...
      ? intnum(x = 0, [oo,-2*I], sin(2*x)/x) - Pi/2
      %5 = 0.E-115 \\  perfect

      ? intnum(x = 0, [oo,-I], sin(x)^3/x) - Pi/4
      %6 = -0.0008... \\  bad
      ? sin(x)^3 - (3*sin(x)-sin(3*x))/4
      %7 = O(x^17)

We may use the above linearization and compute two oscillating integrals with endpoints `[oo, -I]` and `[oo, -3*I]` respectively, or notice the obvious change of variable, and reduce to the single integral (1/2)∫0^ oo sin(x)/xdx. We finish with some more complicated examples:

      ? intnum(x = 0, [oo,-I], (1-cos(x))/x^2) - Pi/2
      %1 = -0.0003... \\  bad
      ? intnum(x = 0, 1, (1-cos(x))/x^2) \
      + intnum(x = 1, oo, 1/x^2) - intnum(x = 1, [oo,I], cos(x)/x^2) - Pi/2
      %2 = 0.E-115 \\  perfect

      ? intnum(x = 0, [oo, 1], sin(x)^3*exp(-x)) - 0.3
      %3 = -7.34... E-55 \\  bad
      ? intnum(x = 0, [oo,-I], sin(x)^3*exp(-x)) - 0.3
      %4 = 8.9... E-103 \\  better. Try higher m
      ? tab = intnuminit(0,[oo,-I], 1); \\  double number of sampling points
      ? intnum(x = 0, oo, sin(x)^3*exp(-x), tab) - 0.3
      %6 = 0.E-115 \\  perfect

Warning. Like `sumalt`, `intnum` often assigns a reasonable value to diverging integrals. Use these values at your own risk! For example:

      ? intnum(x = 0, [oo, -I], x^2*sin(x))
      %1 = -2.0000000000...

Note the formula ∫0^ oo sin(x)/x^sdx = cos(π s/2) Γ(1-s) , a priori valid only for 0 < Re(s) < 2, but the right hand side provides an analytic continuation which may be evaluated at s = -2...

Multivariate integration. Using successive univariate integration with respect to different formal parameters, it is immediate to do naive multivariate integration. But it is important to use a suitable `intnuminit` to precompute data for the internal integrations at least!

For example, to compute the double integral on the unit disc x^2+y^2 ≤ 1 of the function x^2+y^2, we can write

      ? tab = intnuminit(-1,1);
      ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab),tab) - Pi/2
      %2 = -7.1... E-115 \\  OK

The first tab is essential, the second optional. Compare:

      ? tab = intnuminit(-1,1);
      time = 4 ms.
      ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2));
      time = 3,092 ms. \\  slow
      ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab), tab);
      time = 252 ms.  \\  faster
      ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab));
      time = 261 ms.  \\  the internal integral matters most

The library syntax is `intnum(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,GEN tab, long prec)`, where an omitted tab is coded as `NULL`.

#### intnumgauss(X = a, b, expr, {tab})

Numerical integration of expr on the compact interval [a,b] with respect to X using Gauss-Legendre quadrature; `tab` is either omitted or precomputed with `intnumgaussinit`. As a convenience, it can be an integer n in which case we call `intnumgaussinit`(n) and use n-point quadrature.

      ? test(n, b = 1) = T=intnumgaussinit(n);\
          intnumgauss(x=-b,b, 1/(1+x^2),T) - 2*atan(b);
      ? test(0) \\ default
      %1 = -9.490148553624725335 E-22
      ? test(40)
      %2 = -6.186629001816965717 E-31
      ? test(50)
      %3 = -1.1754943508222875080 E-38
      ? test(50, 2) \\ double interval length
      %4 = -4.891779568527713636 E-21
      ? test(90, 2) \\ n must almost be doubled as well!
      %5 = -9.403954806578300064 E-38

On the other hand, we recommend to split the integral and change variables rather than increasing n too much:

      ? f(x) = 1/(1+x^2);
      ? b = 100;
      ? intnumgauss(x=0,1, f(x)) + intnumgauss(x=1,1/b, f(1/x)*(-1/x^2)) - atan(b)
      %3 = -1.0579449157400587572 E-37

The library syntax is `GEN intnumgauss0(GEN X, GEN b, GEN expr, GEN tab = NULL, long prec)`.

#### intnumgaussinit({n})

Initialize tables for n-point Gauss-Legendre integration of a smooth function f lon a compact interval [a,b] at current `realprecision`. If n is omitted, make a default choice n ~ `realprecision`, suitable for analytic functions on [-1,1]. The error is bounded by

((b-a)2n+1 (n!)^4)/((2n+1)[(2n)!]^3) f(2n) (ξ) , a < ξ < b

so, if the interval length increases, n should be increased as well.

      ? T = intnumgaussinit();
      ? intnumgauss(t=-1,1,exp(t), T) - exp(1)+exp(-1)
      %1 = -5.877471754111437540 E-39
      ? intnumgauss(t=-10,10,exp(t), T) - exp(10)+exp(-10)
      %2 = -8.358367809712546836 E-35
      ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
      %3 = -9.490148553624725335 E-22

      ? T = intnumgaussinit(50);
      ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
      %5 = -1.1754943508222875080 E-38
      ? intnumgauss(t=-5,5,1/(1+t^2), T) - 2*atan(5)
      %6 = -1.2[...]E-8

On the other hand, we recommend to split the integral and change variables rather than increasing n too much, see `intnumgauss`.

The library syntax is `GEN intnumgaussinit(long n, long prec)`.

#### intnuminit(a, b, {m = 0})

Initialize tables for integration from a to b, where a and b are coded as in `intnum`. Only the compactness, the possible existence of singularities, the speed of decrease or the oscillations at infinity are taken into account, and not the values. For instance intnuminit(-1,1) is equivalent to intnuminit(0,Pi), and intnuminit([0,-1/2],oo) is equivalent to intnuminit([-1,-1/2], -oo); on the other hand, the order matters and intnuminit([0,-1/2], [1,-1/3]) is not equivalent to intnuminit([0,-1/3], [1,-1/2]) !

If m is present, it must be non-negative and we multiply the default number of sampling points by 2^m (increasing the running time by a similar factor).

The result is technical and liable to change in the future, but we document it here for completeness. Let x = φ(t), t ∈ ]- oo , oo [ be an internally chosen change of variable, achieving double exponential decrease of the integrand at infinity. The integrator `intnum` will compute h ∑|n| < N φ'(nh) F(φ(nh)) for some integration step h and truncation parameter N. In basic use, let

      [h, x0, w0, xp, wp, xm, wm] = intnuminit(a,b);

* h is the integration step

* x0 = φ(0) and w0 = φ'(0),

* xp contains the φ(nh), 0 < n < N,

* xm contains the φ(nh), 0 < -n < N, or is empty.

* wp contains the φ'(nh), 0 < n < N,

* wm contains the φ'(nh), 0 < -n < N, or is empty.

The arrays xm and wm are left empty when φ is an odd function. In complicated situations when non-default behavior is specified at end points, `intnuminit` may return up to 3 such arrays, corresponding to a splitting of up to 3 integrals of basic type.

If the functions to be integrated later are of the form F = f(t) k(t,z) for some kernel k (e.g. Fourier, Laplace, Mellin,...), it is useful to also precompute the values of f(φ(nh)), which is accomplished by `intfuncinit`. The hard part is to determine the behavior of F at endpoints, depending on z.

The library syntax is `GEN intnuminit(GEN a, GEN b, long m, long prec)`.

#### intnumromb(X = a, b, expr, {flag = 0})

Numerical integration of expr (smooth in ]a,b[), with respect to X. Suitable for low accuracy; if expr is very regular (e.g. analytic in a large region) and high accuracy is desired, try `intnum` first.

Set flag = 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 flag = 1, uses a general driver routine for doing numerical integration, making no particular assumption (slow).

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

If flag = 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: `realprecision` about 30 decimal digits (`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 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 ∫0^ oo e-x^2dx to 28 decimal digits, then one can set infinity equal to 10 for example, and certainly not to `1e1000`.

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

#### laurentseries(f, {M = seriesprecision}, {x = 'x})

Expand f as a Laurent series around x = 0 to order M. This function computes f(x + O(x^n)) until n is large enough so it must be possible to evaluate f on a power series with 0 constant term.

      ? laurentseries(t->sin(t)/(1-cos(t)), 5)
      %1 = 2*x^-1 - 1/6*x - 1/360*x^3 - 1/15120*x^5 + O(x^6)
      ? laurentseries(log)
        ***   at top-level: laurentseries(log)
        ***                 ^ —  —  —  —  —  —
        ***   in function laurentseries: log
        ***                              ^ —
        *** log: domain error in log: series valuation != 0

Note that individual Laurent coefficients of order ≤ M can be retrieved from s = `laurentseries`(f,M) via `polcoeff(s,i)` for any i ≤ M. The series s may occasionally be more precise that the required O(xM+1).

With respect to successive calls to `derivnum`, `laurentseries` is both faster and more precise:

      ? laurentseries(t->log(3+t),1)
      %1 = 1.0986122886681096913952452369225257047 + 1/3*x - 1/18*x^2 + O(x^3)
      ? derivnum(t=0,log(3+t),1)
      %2 = 0.33333333333333333333333333333333333333
      ? derivnum(t=0,log(3+t),2)
      %3 = -0.11111111111111111111111111111111111111

      ? f = x->sin(exp(x));
      ? polcoeff(laurentseries(x->f(x+2), 1), 1)
      %5 = 3.3129294231043339804683687620360224365
      ? exp(2) * cos(exp(2));
      %6 = 3.3129294231043339804683687620360224365
      ? derivnum(x = 2, f(x))
      %7 = 3.3129294231043339804683687620360224364 \\ 1 ulp off

      ? default(realprecision,115);
      ? for(i=1,10^4, laurentseries(x->f(x+2),1))
      time = 279 ms.
      ? for(i=1,10^4, derivnum(x=2,f(x)))  \\ ... and slower
      time = 1,134 ms.

The library syntax is `laurentseries(void *E, GEN (*f)(void*,GEN,long), long M, long v, long prec)`.

#### limitnum(expr, {k = 20}, {alpha = 1})

Lagrange-Zagier numerical extrapolation of expr, corresponding to a sequence un, either given by a closure `n- > u(n)` or by a vector of values I.e., assuming that un tends to a finite limit ℓ, try to determine ℓ. This routine is purely numerical and heuristic, thus may or may not work on your examples; k is ignored if u is given by a vector, and otherwise is a multiplier such that we extrapolate from u(kn).

Assume that un has an asymptotic expansion in n-α : un = ℓ + ∑i ≥ 1 ai n-iα for some ai.

      ? limitnum(n -> n*sin(1/n))
      %1 = 1.0000000000000000000000000000000000000

      ? limitnum(n -> (1+1/n)^n) - exp(1)
      %2 = 0.E-37

      ? limitnum(n -> 2^(4*n+1)*(n!)^4 / (2*n)! /(2*n+1)! )
      %3 = 3.1415926535897932384626433832795028842
      ? Pi
      %4 = 3.1415926535897932384626433832795028842

If un is given by a vector, it must be long enough for the extrapolation to make sense: at least k times the current `realprecision`. The preferred format is thus a closure, although it becomes inconvenient when un cannot be directly computed in time polynomial in log n, for instance if it is defined as a sum or by induction. In that case, passing a vector of values is the best option. It usually pays off to interpolate u(kn) for some k > 1:

      ? limitnum(vector(10,n,(1+1/n)^n))
       ***                 ^ —  —  —  —  —  — --
       *** limitnum: non-existent component in limitnum: index < 20
      \\ at this accuracy, we must have at least 20 values
      ? limitnum(vector(20,n,(1+1/n)^n)) - exp(1)
      %5 = -2.05... E-20
      ? limitnum(vector(20,n, m=10*n;(1+1/m)^m)) - exp(1) \\ better accuracy
      %6 = 0.E-37

      ? v = vector(20); s = 0;
      ? for(i=1,#v, s += 1/i; v[i]= s - log(i));
      ? limitnum(v) - Euler
      %9 = -1.6... E-19

      ? V = vector(200); s = 0;
      ? for(i=1,#V, s += 1/i; V[i]= s);
      ? v = vector(#V \ 10, i, V[10*i] - log(10*i));
      ? limitnum(v) - Euler
      %13 = 6.43... E-29

The library syntax is `limitnum(void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec)`, where `u(E, n, prec)` must return u(n) in precision `prec`. Also available is `GEN limitnum0(GEN u, long muli, GEN alpha, long prec)`, where u must be a vector of sufficient length as above.

#### prod(X = a, b, expr, {x = 1})

Product of expression expr, initialized at x, the formal parameter X going from a to b. As for `sum`, the main purpose of the initialization parameter x is to force the type of the operations being performed. For example if it is set equal to the integer 1, operations will start being done exactly. If it is set equal to the real 1., they will be done using real numbers having the default precision. If it is set equal to the power series 1+O(X^k) for a certain k, they will be done using power series of precision at most k. These are the three most common initializations.

As an extreme example, compare

      ? prod(i=1, 100, 1 - X^i);  \\  this has degree 5050 !!
      time = 128 ms.
      ? prod(i=1, 100, 1 - X^i, 1 + O(X^101))
      time = 8 ms.
      %2 = 1 - X - X^2 + X^5 + X^7 - X^12 - X^15 + X^22 + X^26 - X^35 - X^40 + \
      X^51 + X^57 - X^70 - X^77 + X^92 + X^100 + O(X^101)

Of course, in this specific case, it is faster to use `eta`, which is computed using Euler's formula.

      ? prod(i=1, 1000, 1 - X^i, 1 + O(X^1001));
      time = 589 ms.
      ? \ps1000
      seriesprecision = 1000 significant terms
      ? eta(X) - %
      time = 8ms.
      %4 = O(X^1001)

The library syntax is `produit(GEN a, GEN b, char *expr, GEN x)`.

#### prodeuler(X = a, b, expr)

Product of expression expr, initialized at 1. (i.e. to a real number equal to 1 to the current `realprecision`), the formal parameter X ranging over the prime numbers between a and b.

The library syntax is `prodeuler(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, long prec)`.

#### prodeulerrat(F, {s = 1}, {a = 2})

∏p ≥ a, p primeF(p^s), where F is a rational function.

      ? prodeulerrat(1+1/q^3,1)
      %1 = 1.1815649490102569125693997341604542605
      ? zeta(3)/zeta(6)
      %2 = 1.1815649490102569125693997341604542606

The library syntax is `GEN prodeulerrat(GEN F, GEN s = NULL, long a, long prec)`.

#### prodinf(X = a, expr, {flag = 0})

infinite product of expression expr, the formal parameter X starting at a. The evaluation stops when the relative error of the expression minus 1 is less than the default precision. In particular, non-convergent products result in infinite loops. The expressions must always evaluate to an element of ℂ.

If flag = 1, do the product of the (1+expr) instead.

The library syntax is `prodinf(void *E, GEN (*eval)(void*,GEN), GEN a, long prec)` (flag = 0), or `prodinf1` with the same arguments (flag = 1).

#### prodnumrat(F, a)

∏n ≥ aF(n), where F-1 is a rational function of degree less than or equal to -2.

      ? prodnumrat(1+1/x^2,1)
      %1 = 3.6760779103749777206956974920282606665

The library syntax is `GEN prodnumrat(GEN F, long a, long prec)`.

#### solve(X = a, b, expr)

Find a real root of expression expr between a and b, under the condition expr(X = a) * expr(X = b) ≤ 0. (You will get an error message `roots must be bracketed in solve` if this does not hold.) This routine uses Brent's method and can fail miserably if expr is not defined in the whole of [a,b] (try `solve(x = 1, 2, tan(x))`).

The library syntax is `zbrent(void *E,GEN (*eval)(void*,GEN),GEN a,GEN b,long prec)`.

#### solvestep(X = a, b, step, expr, {flag = 0})

Find zeros of a continuous function in the real interval [a,b] by naive interval splitting. This function is heuristic and may or may not find the intended zeros. Binary digits of flag mean

* 1: return as soon as one zero is found, otherwise return all zeros found;

* 2: refine the splitting until at least one zero is found (may loop indefinitely if there are no zeros);

* 4: do a multiplicative search (we must have a > 0 and step > 1), otherwise an additive search; step is the multiplicative or additive step.

* 8: refine the splitting until at least one zero is very close to an integer.

      ? solvestep(X=0,10,1,sin(X^2),1)
      %1 = 1.7724538509055160272981674833411451828
      ? solvestep(X=1,12,2,besselj(4,X),4)
      %2 = [7.588342434..., 11.064709488...]

The library syntax is `solvestep(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, GEN step,long flag,long prec)`.

#### sum(X = a, b, expr, {x = 0})

Sum of expression expr, initialized at x, the formal parameter going from a to b. As for `prod`, the initialization parameter x may be given to force the type of the operations being performed.

As an extreme example, compare

      ? sum(i=1, 10^4, 1/i); \\  rational number: denominator has 4345 digits.
      time = 236 ms.
      ? sum(i=1, 5000, 1/i, 0.)
      time = 8 ms.
      %2 = 9.787606036044382264178477904

#### sumalt(X = a, expr, {flag = 0})

Numerical summation of the series expr, which should be an alternating series (-1)^k ak, the formal variable X starting at a. Use an algorithm of Cohen, Villegas and Zagier (Experiment. Math. 9 (2000), no. 1, 3--12).

If flag = 0, assuming that the ak are the moments of a positive measure on [0,1], the relative error is O(3+sqrt8)-n after using ak for k ≤ n. If `realprecision` is p, we thus set n = log(10)p/log(3+sqrt8) ~ 1.3 p; besides the time needed to compute the ak, k ≤ n, the algorithm overhead is negligible: time O(p^2) and space O(p).

If flag = 1, use a variant with more complicated polynomials, see `polzagier`. If the ak are the moments of w(x)dx where w (or only xw(x^2)) is a smooth function extending analytically to the whole complex plane, convergence is in O(14.4-n). If xw(x^2) extends analytically to a smaller region, we still have exponential convergence, with worse constants. Usually faster when the computation of ak is expensive. If `realprecision` is p, we thus set n = log(10)p/log(14.4) ~ 0.86 p; besides the time needed to compute the ak, k ≤ n, the algorithm overhead is not negligible: time O(p^3) and space O(p^2). Thus, even if the analytic conditions for rigorous use are met, this variant is only worthwile if the ak are hard to compute, at least O(p^2) individually on average: otherwise we gain a small constant factor (1.5, say) in the number of needed ak at the expense of a large overhead.

The conditions for rigorous use are hard to check but the routine is best used heuristically: even divergent alternating series can sometimes be summed by this method, as well as series which are not exactly alternating (see for example Section se:user_defined). It should be used to try and guess the value of an infinite sum. (However, see the example at the end of Section se:userfundef.)

If the series already converges geometrically, `suminf` is often a better choice:

      ? \p28
      ? sumalt(i = 1, -(-1)^i / i)  - log(2)
      time = 0 ms.
      %1 = -2.524354897 E-29
      ? suminf(i = 1, -(-1)^i / i)   \\  Had to hit C-C
        ***   at top-level: suminf(i=1,-(-1)^i/i)
        ***                                ^ —  —
        *** suminf: user interrupt after 10min, 20,100 ms.
      ? \p1000
      ? sumalt(i = 1, -(-1)^i / i)  - log(2)
      time = 90 ms.
      %2 = 4.459597722 E-1002

      ? sumalt(i = 0, (-1)^i / i!) - exp(-1)
      time = 670 ms.
      %3 = -4.03698781490633483156497361352190615794353338591897830587 E-944
      ? suminf(i = 0, (-1)^i / i!) - exp(-1)
      time = 110 ms.
      %4 = -8.39147638 E-1000   \\   faster and more accurate

The library syntax is `sumalt(void *E, GEN (*eval)(void*,GEN),GEN a,long prec)`. Also available is `sumalt2` with the same arguments (flag = 1).

#### sumdiv(n, X, expr)

Sum of expression expr over the positive divisors of n. This function is a trivial wrapper essentially equivalent to

        D = divisors(n);
        for (i = 1, #D, X = D[i]; eval(expr))

(except that `X` is lexically scoped to the `sumdiv` loop). If expr is a multiplicative function, use `sumdivmult`.

#### sumdivmult(n, d, expr)

Sum of multiplicative expression expr over the positive divisors d of n. Assume that expr evaluates to f(d) where f is multiplicative: f(1) = 1 and f(ab) = f(a)f(b) for coprime a and b.

#### sumeulerrat(F, {s = 1}, {a = 2})

∑p ≥ a, p primeF(p^s), where F is a rational function.

      ? sumeulerrat(1/q)
      %1 = 0.45224742004106549850654336483224793418

The library syntax is `GEN sumeulerrat(GEN F, GEN s = NULL, long a, long prec)`.

#### suminf(X = a, expr)

infinite sum of expression expr, the formal parameter X starting at a. The evaluation stops when the relative error of the expression is less than the default precision for 3 consecutive evaluations. The expressions must always evaluate to a complex number.

If the series converges slowly, make sure `realprecision` is low (even 28 digits may be too much). In this case, if the series is alternating or the terms have a constant sign, `sumalt` and `sumpos` should be used instead.

      ? \p28
      ? suminf(i = 1, -(-1)^i / i)   \\  Had to hit C-C
        ***   at top-level: suminf(i=1,-(-1)^i/i)
        ***                                ^ —  —
        *** suminf: user interrupt after 10min, 20,100 ms.
      ? sumalt(i = 1, -(-1)^i / i) - log(2)
      time = 0 ms.
      %1 = -2.524354897 E-29

The library syntax is `suminf(void *E, GEN (*eval)(void*,GEN), GEN a, long prec)`.

#### sumnum(n = a, f, {tab})

Numerical summation of f(n) at high accuracy using Euler-MacLaurin, the variable n taking values from a to + oo , where f is assumed to have positive values and is a C^ oo function; `a` must be an integer and `tab`, if given, is the output of `sumnuminit`. The latter precomputes abscissas and weights, speeding up the computation; it also allows to specify the behavior at infinity via `sumnuminit([+oo, asymp])`.

      ? \p500
      ? z3 = zeta(3);
      ? sumpos(n = 1, n^-3) - z3
      time = 2,332 ms.
      %2 = 2.438468843 E-501
      ? sumnum(n = 1, n^-3) - z3 \\ here slower than sumpos
      time = 2,752 ms.
      %3 = 0.E-500

Complexity. The function f will be evaluated at O(D log D) real arguments, where D ~ `realprecision`.log(10). The routine is geared towards slowly decreasing functions: if f decreases exponentially fast, then one of `suminf` or `sumpos` should be preferred. If f satisfies the stronger hypotheses required for Monien summation, i.e. if f(1/z) is holomorphic in a complex neighbourhood of [0,1], then `sumnummonien` will be faster since it only requires O(D/log D) evaluations:

      ? sumnummonien(n = 1, 1/n^3) - z3
      time = 1,985 ms.
      %3 = 0.E-500

The `tab` argument precomputes technical data not depending on the expression being summed and valid for a given accuracy, speeding up immensely later calls:

      ? tab = sumnuminit();
      time = 2,709 ms.
      ? sumnum(n = 1, 1/n^3, tab) - z3 \\ now much faster than sumpos
      time = 40 ms.
      %5 = 0.E-500

      ? tabmon = sumnummonieninit(); \\ Monien summation allows precomputations too
      time = 1,781 ms.
      ? sumnummonien(n = 1, 1/n^3, tabmon) - z3
      time = 2 ms.
      %7 = 0.E-500

The speedup due to precomputations becomes less impressive when the function f is expensive to evaluate, though:

      ? sumnum(n = 1, lngamma(1+1/n)/n, tab);
      time = 14,180 ms.

      ? sumnummonien(n = 1, lngamma(1+1/n)/n, tabmon); \\ fewer evaluations
      time = 717 ms.

Behaviour at infinity. By default, `sumnum` assumes that expr decreases slowly at infinity, but at least like O(n-2). If the function decreases like nα for some -2 < α < -1, then it must be indicated via

        tab = sumnuminit([+oo, alpha]); /* alpha < 0 slow decrease */

otherwise loss of accuracy is expected. If the functions decreases quickly, like exp(-α n) for some α > 0, then it must be indicated via

        tab = sumnuminit([+oo, alpha]); /* alpha  > 0 exponential decrease */

otherwise exponent overflow will occur.

      ? sumnum(n=1,2^-n)
       ***   at top-level: sumnum(n=1,2^-n)
       ***                             ^ — -
       *** _^_: overflow in expo().
      ? tab = sumnuminit([+oo,log(2)]); sumnum(n=1,2^-n, tab)
      %1 = 1.000[...]

As a shortcut, one can also input

        sumnum(n = [a, asymp], f)

instead of

        tab = sumnuminit(asymp);
        sumnum(n = a, f, tab)

Further examples.

      ? \p200
      ? sumnum(n = 1, n^(-2)) - zeta(2) \\ accurate, fast
      time = 200 ms.
      %1 = -2.376364457868949779 E-212
      ? sumpos(n = 1, n^(-2)) - zeta(2)  \\ even faster
      time = 96 ms.
      %2 = 0.E-211
      ? sumpos(n=1,n^(-4/3)) - zeta(4/3)   \\ now much slower
      time = 13,045 ms.
      %3 = -9.980730723049589073 E-210
      ? sumnum(n=1,n^(-4/3)) - zeta(4/3)  \\ fast but inaccurate
      time = 365 ms.
      %4 = -9.85[...]E-85
      ? sumnum(n=[1,-4/3],n^(-4/3)) - zeta(4/3) \\ with decrease rate, now accurate
      time = 416 ms.
      %5 = -4.134874156691972616 E-210

      ? tab = sumnuminit([+oo,-4/3]);
      time = 196 ms.
      ? sumnum(n=1, n^(-4/3), tab) - zeta(4/3) \\ faster with precomputations
      time = 216 ms.
      %5 = -4.134874156691972616 E-210
      ? sumnum(n=1,-log(n)*n^(-4/3), tab) - zeta'(4/3)
      time = 321 ms.
      %7 = 7.224147951921607329 E-210

Note that in the case of slow decrease (α < 0), the exact decrease rate must be indicated, while in the case of exponential decrease, a rough value will do. In fact, for exponentially decreasing functions, `sumnum` is given for completeness and comparison purposes only: one of `suminf` or `sumpos` should always be preferred.

      ? sumnum(n=[1, 1], 2^-n) \\ pretend we decrease as exp(-n)
      time = 240 ms.
      %8 = 1.000[...] \\ perfect
      ? sumpos(n=1, 2^-n)
      %9 = 1.000[...] \\ perfect and instantaneous

Beware cancellation. The function f(n) is evaluated for huge values of n, so beware of cancellation in the evaluation:

      ? f(n) = 2 - 1/n - 2*n*log(1+1/n); \\ result is O(1/n^2)
      ? z = -2 + log(2*Pi) - Euler;
      ? sumnummonien(n=1, f(n)) - z
      time = 149 ms.
      %12 = 0.E-212  \\ perfect
      ? sumnum(n=1, f(n)) - z
      time = 116 ms.
      %13 = -948.216[...] \\ junk

As `sumnum(n = 1, print(n))` shows, we evaluate f(n) for n > 1e233 and our implementation of f suffers from massive cancellation since we are summing two terms of the order of O(1) for a result in O(1/n^2). You can either rewrite your sum so that individual terms are evaluated without cancellation or locally replace f(n) by an accurate asymptotic expansion:

      ? F = truncate( f(1/x + O(x^30)) );
      ? sumnum(n=1, if(n > 1e7, subst(F,x,1/n), f(n))) - z
      %15 = 1.1 E-212 \\ now perfect

The library syntax is `sumnum((void *E, GEN (*eval)(void*, GEN), GEN a, GEN tab, long prec))` where an omitted tab is coded as `NULL`.

#### sumnumap(n = a, f, {tab})

Numerical summation of f(n) at high accuracy using Abel-Plana, the variable n taking values from a to + oo , where f is holomorphic in the right half-place Re(z) > a; `a` must be an integer and `tab`, if given, is the output of `sumnumapinit`. The latter precomputes abscissas and weights, speeding up the computation; it also allows to specify the behavior at infinity via `sumnumapinit([+oo, asymp])`.

      ? \p500
      ? z3 = zeta(3);
      ? sumpos(n = 1, n^-3) - z3
      time = 2,332 ms.
      %2 = 2.438468843 E-501
      ? sumnumap(n = 1, n^-3) - z3 \\ here slower than sumpos
      time = 2,565 ms.
      %3 = 0.E-500

Complexity. The function f will be evaluated at O(D log D) real arguments and O(D) complex arguments, where D ~ `realprecision`.log(10). The routine is geared towards slowly decreasing functions: if f decreases exponentially fast, then one of `suminf` or `sumpos` should be preferred. The default algorithm `sumnum` is usually a little slower than `sumnumap` but its initialization function `sumnuminit` becomes much faster as `realprecision` increases.

If f satisfies the stronger hypotheses required for Monien summation, i.e. if f(1/z) is holomorphic in a complex neighbourhood of [0,1], then `sumnummonien` will be faster since it only requires O(D/log D) evaluations:

      ? sumnummonien(n = 1, 1/n^3) - z3
      time = 1,128 ms.
      %3 = 0.E-500

The `tab` argument precomputes technical data not depending on the expression being summed and valid for a given accuracy, speeding up immensely later calls:

      ? tab = sumnumapinit();
      time = 2,567 ms.
      ? sumnumap(n = 1, 1/n^3, tab) - z3 \\ now much faster than sumpos
      time = 39 ms.
      %5 = 0.E-500

      ? tabmon = sumnummonieninit(); \\ Monien summation allows precomputations too
      time = 1,125 ms.
      ? sumnummonien(n = 1, 1/n^3, tabmon) - z3
      time = 2 ms.
      %7 = 0.E-500

The speedup due to precomputations becomes less impressive when the function f is expensive to evaluate, though:

      ? sumnumap(n = 1, lngamma(1+1/n)/n, tab);
      time = 10,762 ms.

      ? sumnummonien(n = 1, lngamma(1+1/n)/n, tabmon); \\ fewer evaluations
      time = 205 ms.

Behaviour at infinity. By default, `sumnumap` assumes that expr decreases slowly at infinity, but at least like O(n-2). If the function decreases like nα for some -2 < α < -1, then it must be indicated via

        tab = sumnumapinit([+oo, alpha]); /* alpha < 0 slow decrease */

otherwise loss of accuracy is expected. If the functions decreases quickly, like exp(-α n) for some α > 0, then it must be indicated via

        tab = sumnumapinit([+oo, alpha]); /* alpha  > 0 exponential decrease */

otherwise exponent overflow will occur.

      ? sumnumap(n=1,2^-n)
       ***   at top-level: sumnumap(n=1,2^-n)
       ***                             ^ — -
       *** _^_: overflow in expo().
      ? tab = sumnumapinit([+oo,log(2)]); sumnumap(n=1,2^-n, tab)
      %1 = 1.000[...]

As a shortcut, one can also input

        sumnumap(n = [a, asymp], f)

instead of

        tab = sumnumapinit(asymp);
        sumnumap(n = a, f, tab)

Further examples.

      ? \p200
      ? sumnumap(n = 1, n^(-2)) - zeta(2) \\ accurate, fast
      time = 169 ms.
      %1 = -4.752728915737899559 E-212
      ? sumpos(n = 1, n^(-2)) - zeta(2)  \\ even faster
      time = 79 ms.
      %2 = 0.E-211
      ? sumpos(n=1,n^(-4/3)) - zeta(4/3)   \\ now much slower
      time = 10,518 ms.
      %3 = -9.980730723049589073 E-210
      ? sumnumap(n=1,n^(-4/3)) - zeta(4/3)  \\ fast but inaccurate
      time = 309 ms.
      %4 = -2.57[...]E-78
      ? sumnumap(n=[1,-4/3],n^(-4/3)) - zeta(4/3) \\ decrease rate: now accurate
      time = 329 ms.
      %6 = -5.418110963941205497 E-210

      ? tab = sumnumapinit([+oo,-4/3]);
      time = 160 ms.
      ? sumnumap(n=1, n^(-4/3), tab) - zeta(4/3) \\ faster with precomputations
      time = 175 ms.
      %5 = -5.418110963941205497 E-210
      ? sumnumap(n=1,-log(n)*n^(-4/3), tab) - zeta'(4/3)
      time = 258 ms.
      %7 = 9.125239518216767153 E-210

Note that in the case of slow decrease (α < 0), the exact decrease rate must be indicated, while in the case of exponential decrease, a rough value will do. In fact, for exponentially decreasing functions, `sumnumap` is given for completeness and comparison purposes only: one of `suminf` or `sumpos` should always be preferred.

      ? sumnumap(n=[1, 1], 2^-n) \\ pretend we decrease as exp(-n)
      time = 240 ms.
      %8 = 1.000[...] \\ perfect
      ? sumpos(n=1, 2^-n)
      %9 = 1.000[...] \\ perfect and instantaneous

The library syntax is `sumnumap((void *E, GEN (*eval)(void*,GEN), GEN a, GEN tab, long prec))` where an omitted tab is coded as `NULL`.

#### sumnumapinit({asymp})

Initialize tables for Abel-Plana summation of a series ∑ f(n), where f is holomorphic in a right half-plane. If given, `asymp` is of the form [`+oo`, α], as in `intnum` and indicates the decrease rate at infinity of functions to be summed. A positive α > 0 encodes an exponential decrease of type exp(-α n) and a negative -2 < α < -1 encodes a slow polynomial decrease of type nα.

      ? \p200
      ? sumnumap(n=1, n^-2);
      time = 163 ms.
      ? tab = sumnumapinit();
      time = 160 ms.
      ? sumnum(n=1, n^-2, tab); \\ faster
      time = 7 ms.

      ? tab = sumnumapinit([+oo, log(2)]); \\ decrease like 2^-n
      time = 164 ms.
      ? sumnumap(n=1, 2^-n, tab) - 1
      time = 36 ms.
      %5 = 3.0127431466707723218 E-282

      ? tab = sumnumapinit([+oo, -4/3]); \\ decrease like n^(-4/3)
      time = 166 ms.
      ? sumnumap(n=1, n^(-4/3), tab);
      time = 181 ms.

The library syntax is `GEN sumnumapinit(GEN asymp = NULL, long prec)`.

#### sumnuminit({asymp})

Initialize tables for Euler-MacLaurin delta summation of a series with positive terms. If given, `asymp` is of the form [`+oo`, α], as in `intnum` and indicates the decrease rate at infinity of functions to be summed. A positive α > 0 encodes an exponential decrease of type exp(-α n) and a negative -2 < α < -1 encodes a slow polynomial decrease of type nα.

      ? \p200
      ? sumnum(n=1, n^-2);
      time = 200 ms.
      ? tab = sumnuminit();
      time = 188 ms.
      ? sumnum(n=1, n^-2, tab); \\ faster
      time = 8 ms.

      ? tab = sumnuminit([+oo, log(2)]); \\ decrease like 2^-n
      time = 200 ms.
      ? sumnum(n=1, 2^-n, tab)
      time = 44 ms.

      ? tab = sumnuminit([+oo, -4/3]); \\ decrease like n^(-4/3)
      time = 200 ms.
      ? sumnum(n=1, n^(-4/3), tab);
      time = 221 ms.

The library syntax is `GEN sumnuminit(GEN asymp = NULL, long prec)`.

#### sumnumlagrange(n = a, f, {tab})

Numerical summation of f(n) from n = a to + oo using Lagrange summation; a must be an integer, and the optional argument `tab` is the output of `sumnumlagrangeinit`. By default, the program assumes that the Nth remainder has an asymptotic expansion in integral powers of 1/N. If not, initialize `tab` using `sumnumlagrangeinit(al)`, where the asymptotic expansion of the remainder is integral powers of 1/Nal; al can be equal to 1 (default), 1/2, 1/3, or 1/4, and also equal to 2, but in this latter case it is the Nth remainder minus one half of the last summand which has an asymptotic expansion in integral powers of 1/N^2.

      ? \p1000
      ? z3 = zeta(3);
      ? sumpos(n = 1, n^-3) - z3
      time = 8,088 ms.
      %2 = -2.08[...] E-1001
      ? sumnumlagrange(n = 1, n^-3) - z3 \\ much faster than sumpos
      time = 40 ms.
      %3 = 0.E-1001
      ? tab = sumnumlagrangeinit(2);
      time = 20 ms.
      ? sumnumlagrange(n = 1, n^-3, tab) - z3
      time = 4 ms. /* even faster */
      %5 = 0.E-1001

      ? \p115
      ? tab = sumnumlagrangeinit([1/3,1/3]);
      time = 316 ms.
      ? sumnumlagrange(n = 1, n^-(7/3), tab) - zeta(7/3)
      time = 24 ms.
      %7 = 0.E-115
      ? sumnumlagrange(n = 1, n^(-2/3) - 3*(n^(1/3)-(n-1)^(1/3)), tab) - zeta(2/3)
      time = 32 ms.
      %8 = 1.0151767349262596893 E-115

Complexity. The function f is evaluated at O(D) integer arguments, where D ~ `realprecision`.log(10).

The library syntax is `sumnumlagrange((void *E, GEN (*eval)(void*, GEN), GEN a, GEN tab, long prec))` where an omitted tab is coded as `NULL`.

#### sumnumlagrangeinit({asymp}, {c1})

Initialize tables for Lagrange summation of a series. By default, assume that the remainder R(n) = ∑m ≥ n f(m) has an asymptotic expansion R(n) = ∑m ≥ n f(n) ~ ∑i ≥ 1 ai / n^i at infinity. The argument `asymp` allows to specify different expansions:

* a real number β means R(n) = n-β ∑i ≥ 1 ai / n^i

* a `t_CLOSURE` g means R(n) = g(n) ∑i ≥ 1 ai / n^i (The preceding case corresponds to g(n) = n-β.)

* a pair [α,β] where β is as above and α ∈ {2, 1, 1/2, 1/3, 1/4}. We let R2(n) = R(n) - f(n)/2 and R_α(n) = R(n) for α != 2. Then R_α(n) = g(n) ∑i ≥ 1 ai / niα Note that the initialization times increase considerable for the α is this list (1/4 being the slowest).

The constant c1 is technical and computed by the program, but can be set by the user: the number of interpolation steps will be chosen close to c1.B, where B is the bit accuracy.

      ? \p2000
      ? sumnumlagrange(n=1, n^-2);
      time = 173 ms.
      ? tab = sumnumlagrangeinit();
      time = 172 ms.
      ? sumnumlagrange(n=1, n^-2, tab);
      time = 4 ms.

      ? \p115
      ? sumnumlagrange(n=1, n^(-4/3)) - zeta(4/3);
      %1 = -0.1093[...] \\ junk: expansion in n^(1/3)
      time = 84 ms.
      ? tab = sumnumlagrangeinit([1/3,0]); \\ alpha = 1/3
      time = 336 ms.
      ? sumnumlagrange(n=1, n^(-4/3), tab) - zeta(4/3)
      time = 84 ms.
      %3 = 1.0151767349262596893 E-115 \\ now OK

      ? tab = sumnumlagrangeinit(1/3); \\ alpha = 1, beta = 1/3: much faster
      time = 3ms
      ? sumnumlagrange(n=1, n^(-4/3), tab) - zeta(4/3) \\ ... but wrong
      %5 = -0.273825[...]   \\ junk !
      ? tab = sumnumlagrangeinit(-2/3); \\ alpha = 1, beta = -2/3
      time = 3ms
      ? sumnumlagrange(n=1, n^(-4/3), tab) - zeta(4/3)
      %6 = 2.030353469852519379 E-115 \\ now OK

in The final example with ζ(4/3), the remainder R1(n) is of the form n-1/3 ∑i ≥ 0 ai / n^i, i.e. n2/3 ∑i ≥ 1 ai / n^i. The explains the wrong result for β = 1/3 and the correction with β = -2/3.

The library syntax is `GEN sumnumlagrangeinit(GEN asymp = NULL, GEN c1 = NULL, long prec)`.

#### sumnummonien(n = a, f, {tab})

Numerical summation ∑n ≥ a f(n) at high accuracy, the variable n taking values from the integer a to + oo using Monien summation, which assumes that f(1/z) has a complex analytic continuation in a (complex) neighbourhood of the segment [0,1].

The function f is evaluated at O(D / log D) real arguments, where D ~ `realprecision`.log(10). By default, assume that f(n) = O(n-2) and has a non-zero asymptotic expansion f(n) = ∑i ≥ 2 ai n-i at infinity. To handle more complicated behaviors and allow time-saving precomputations (for a given `realprecision`), see `sumnummonieninit`.

The library syntax is `GEN sumnummonien0(GEN n, GEN f, GEN tab = NULL, long prec)`.

#### sumnummonieninit({asymp}, {w}, {n0 = 1})

Initialize tables for Monien summation of a series ∑n ≥ n0 f(n) where f(1/z) has a complex analytic continuation in a (complex) neighbourhood of the segment [0,1].

By default, assume that f(n) = O(n-2) and has a non-zero asymptotic expansion f(n) = ∑i ≥ 2 ai / n^i at infinity. Note that the sum starts at i = 2! The argument `asymp` allows to specify different expansions:

* a real number β > 0 means f(n) = ∑i ≥ 1 ai / ni + β (Now the summation starts at 1.)

* a vector [α,β] of reals, where we must have α > 0 and α + β > 1 to ensure convergence, means that f(n) = ∑i ≥ 1 ai / nα i + β Note that `asymp` = [1, β] is equivalent to `asymp` = β.

      ? \p57
      ? s = sumnum(n = 1, sin(1/sqrt(n)) / n); \\ reference point

      ? \p38
      ? sumnummonien(n = 1, sin(1/sqrt(n)) / n) - s
      %2 = -0.001[...] \\ completely wrong

      ? t = sumnummonieninit(1/2);  \\ f(n) = sumi 1 / n^(i+1/2)
      ? sumnummonien(n = 1, sin(1/sqrt(n)) / n, t) - s
      %3 = 0.E-37 \\ now correct

(As a matter of fact, in the above summation, the result given by `sumnum` at `\p38` is slighly incorrect, so we had to increase the accuracy to `\p57`.)

The argument w is used to sum expressions of the form ∑n ≥ n0 f(n) w(n), for varying f as above, and fixed weight function w, where we further assume that the auxiliary sums gw(m) = ∑n ≥ n0 w(n) / nα m + β converge for all m ≥ 1. Note that for non-negative integers k, and weight w(n) = (log n)^k, the function gw(m) = ζ(k)(α m + β) has a simple expression; for general weights, gw is computed using `sumnum`. The following variants are available

* an integer k ≥ 0, to code w(n) = (log n)^k;

* a `t_CLOSURE` computing the values w(n), where we assume that w(n) = O(n^ε) for all ε > 0;

* a vector [w, `fast`], where w is a closure as above and `fast` is a scalar; we assume that w(n) = O(n`fast`+ε); note that `w` = [w, 0] is equivalent to `w` = w. Note that if w decreases exponentially, `suminf` should be used instead.

The subsequent calls to `sumnummonien` must use the same value of n0 as was used here.

      ? \p300
      ? sumnummonien(n = 1, n^-2*log(n)) + zeta'(2)
      time = 328 ms.
      %1 = -1.323[...]E-6 \\ completely wrong, f does not satisfy hypotheses !
      ? tab = sumnummonieninit(, 1); \\ codes w(n) = log(n)
      time = 3,993 ms.
      ? sumnummonien(n = 1, n^-2, tab) + zeta'(2)
      time = 41 ms.
      %3 = -5.562684646268003458 E-309  \\ now perfect

      ? tab = sumnummonieninit(, n->log(n)); \\ generic, slower
      time = 9,808 ms.
      ? sumnummonien(n = 1, n^-2, tab) + zeta'(2)
      time = 40 ms.
      %5 = -5.562684646268003458 E-309  \\ identical result

The library syntax is `GEN sumnummonieninit(GEN asymp = NULL, GEN w = NULL, GEN n0 = NULL, long prec)`.

#### sumnumrat(F, a)

∑n ≥ aF(n), where F is a rational function of degree less than or equal to -2 and where poles of F at integers ≥ a are omitted from the summation. The argument a must be a `t_INT` or `-oo`.

      ? sumnumrat(1/(x^2+1)^2,0)
      %1 = 1.3068369754229086939178621382829073480
      ? sumnumrat(1/x^2, -oo) \\ value at x=0 is discarded
      %2 = 3.2898681336964528729448303332920503784
      ? 2*zeta(2)
      %3 = 3.2898681336964528729448303332920503784

When deg F = -1, we define ∑- oo oo F(n) := ∑n ≥ 0 (F(n) + F(-1-n)):

      ? sumnumrat(1/x, -oo)
      %4 = 0.E-38

The library syntax is `GEN sumnumrat(GEN F, GEN a, long prec)`.

#### sumpos(X = a, expr, {flag = 0})

Numerical summation of the series expr, which must be a series of terms having the same sign, the formal variable X starting at a. The algorithm used is Van Wijngaarden's trick for converting such a series into an alternating one, then we use `sumalt`. For regular functions, the function `sumnum` is in general much faster once the initializations have been made using `sumnuminit`.

The routine is heuristic and assumes that expr is more or less a decreasing function of X. In particular, the result will be completely wrong if expr is 0 too often. We do not check either that all terms have the same sign. As `sumalt`, this function should be used to try and guess the value of an infinite sum.

If flag = 1, use `sumalt`(,1) instead of `sumalt`(,0), see Section se:sumalt. Requiring more stringent analytic properties for rigorous use, but allowing to compute fewer series terms.

To reach accuracy 10-p, both algorithms require O(p^2) space; furthermore, assuming the terms decrease polynomially (in O(n-C)), both need to compute O(p^2) terms. The `sumpos`(,1) variant has a smaller implied constant (roughly 1.5 times smaller). Since the `sumalt`(,1) overhead is now small compared to the time needed to compute series terms, this last variant should be about 1.5 faster. On the other hand, the achieved accuracy may be much worse: as for `sumalt`, since conditions for rigorous use are hard to check, the routine is best used heuristically.

The library syntax is `sumpos(void *E, GEN (*eval)(void*,GEN),GEN a,long prec)`. Also available is `sumpos2` with the same arguments (flag = 1).