fredrikj.net / blog /

# Numerical multidimensional infinite series

*February 12, 2010*

Another new feature in mpmath: support for multidimensional series in `nsum`. Some very simple examples (finite and infinite summations can be combined in any desired order):

>>> nsum(lambda i,j: 2**(-i-j), [0,inf], [0,inf]) 4.0 >>> nsum(lambda i,j,k: 2**(-i-j-k), [0,inf], [0,inf], [0,inf]) 8.0 >>> nsum(lambda i,j,k,l: i/2**j*(i+l)/2**k, [1,2], [0,inf], [1,inf], [2,3]) 50.0 >>> nsum((lambda i,j,k,l: 1/(2**(i**2+j**2+k**2+l**2))), *([[-inf,inf]]*4)) 20.5423960756379 >>> jtheta(3,0,0.5)**4 20.5423960756379

One could of course also make nested calls to `nsum`, but having a direct syntax is much more convenient. Nested evaluation is also usually inefficient for convergence acceleration, so this is not what `nsum` does internally. Instead, it combines all the infinite summations to a single summation over growing hypercubes. The distinction is very important for conditionally convergent series.

For example, `nsum` can now directly evaluate the Madelung constant in three dimensions (ignore=True is to ignore the singular term at the origin):

>>> nsum(lambda i,j,k: (-1)**(i+j+k)/(i**2+j**2+k**2)**0.5, ... [-inf,inf], [-inf,inf], [-inf,inf], ignore=True) -1.74756459463318

While this evaluation takes several seconds, it is somewhat remarkable that a precise value can be obtained at all. A better way to compute the Madelung constant is using the following rapidly convergent 2D series:

>>> f = lambda i,j: -12*pi*sech(0.5*pi*sqrt((2*i+1)**2+(2*j+1)**2))**2 >>> nsum(f, [0,inf], [0,inf]) -1.74756459463318 >>> mp.dps = 100 >>> nsum(f, [0,inf], [0,inf]) -1.74756459463318219063621203554439740348516143662474175815282535076504062353276 117989075836269460789

Another nice application is to evaluate Eisenstein series directly from the definition:

>>> tau = 1j >>> q = qfrom(tau=tau) >>> nsum(lambda m,n: (m+n*tau)**(-4), [-inf,inf], [-inf,inf], ignore=True) (3.1512120021539 + 0.0j) >>> 0.5*sum(jtheta(n,0,q)**8 for n in [2,3,4])*(2*zeta(4)) (3.1512120021539 + 0.0j)

fredrikj.net | Blog index | RSS feed | Follow me on Mastodon | Become a sponsor