README

It is likely that some of the comments in this file are out of date.

Possible sources of nan (will be fixed of course):
----------------------------------------------------

    * out of range errors in precomputed tables

    * out of range errors in data that is precomputed for J_Integrals and IC7 (shouldn't even happen if the data is created correctly)
    
    * When K is large or epsilon is very small, compute_exponential_sums_via_euler_maclaurin() might end with nans or infs from out
        of range errors on the correction terms.
    * H_method1() can end up returning nan if j is very large from out of range errors. When this is compiled
        with -ffast-math, these errors are much more likely, because -ffast-math doesn't always work correctly with
        the full exponent range for complex numbers. For this reason, we don't compile this file with -ffast-math.


Hard-coded parameters that (maybe) shouldn't:
----------------------------------------------------

    * j = 18 in stage3.cc

    * epsilon = exp(-20) in stage3.cc

    * number 1200 in stage_3_start in main_sum.cc

    * dimension of array Z[30] in stage3.cc (but there's little need to change it since
      it's unlikely we'll used an array of a larger size ever)

    * dimension of array Z[30] in main_sum.cc

    * dimension of array Z[30] in main_sum.h

    * Only the first 23 coefficients of array Z[30] are computed in stage3.cc


Possible cosmetic changes:
----------------------------------------------------

    * compute_exponential_sums function takes in the end-point of the block,
      rather than its length, which is a little bit at odds with all other
      functions.


Files structure:
----------------------------------------------------

    * It's reasonable to remove all the functions above "the mark" from main_sum.h 
      since we're only concerned with the _main sum_ here, and not yet with computing 
      zeta. These functions (or whichever ones survive in a final release) should be
      in a seperate "compute_zeta" files & directory.


Redundancies:
----------------------------------------------------

    * none, of course


General remarks:
----------------------------------------------------

    * The algorithm computes the sum:

            sum_{n = 1}^N exp(i*t*log(n)) / sqrt(n) = sum_{n = 1}^N n^{i*t - 0.5}

      rather than its conjugate. 

    * The comments try to make a distinction between a "chunk", which is the basic 
      unit fed to zeta_block_stage_x, and a "block", which is the basic unit fed to
      zeta_block_stage_x_basic. A "chunk" is relatively large piece of the main sum
      and is computed in "blocks", which are smaller pieces where certain taylor
      expansions are applicable.