The PairwiseSummation command performs a summation using a pairwise algorithm, known to substantially reduce the accumulated round-off error when adding floating-point numbers if compared with adding them one at a time. Although there exist other techniques that can have even smaller round-off errors, pairwise summation provides sufficient advantage while having a much lower computational cost than slightly better algorithms.

The first argument, U, is expected to be either a one-dimensional Array or a procedure of one single argument - say $j$ - such that $U\left(j\right)$ gives the $j^{\mathrm{th}}$ term to be added. When U is an Array, if the (optional) summation limits m and n where not indicated, the Array dimensions (as returned by the ArrayDims command) are taken for summation limits. If however U is a procedure, the values of m and n should be indicated.

When the value of Digits is smaller or equal to the value of evalhf(Digits) and provided that the procedure or Array U satisfy required conditions (see evalhf), the pairwise summation is performed under evalhf, that is using the floating-point hardware of the underlying system and done in double precision. This can significantly speed up the summation process. Otherwise, the summation is performed anyway using option hfloat.

Initialization: Load the package and set the display of special functions in output to typeset mathematical notation (textbook notation):

$\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right)\;\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right)\:$

$\left\{\mathrm{Add}\,\mathrm{Evalb}\,\mathrm{Zoom}\,\mathrm{QuadrantNumbers}\,\mathrm{Singularities}\,\mathrm{GenerateRecurrence}\,\mathrm{PairwiseSummation}\right\}$

$\mathrm{Digits}≔15$

$\mathrm{Digits}≔15$

$A ≔ \mathrm{QuadrantNumbers}\left(\mathrm{around} \= \mathrm{map}\left(u \to \left(u\/10000\right)\, \left[\$\left(1 .. 5000\right)\right]\right)\right)$

$B≔\mathrm{ArrayTools}:-\mathrm{Concatenate}\left(2\,A_1\,A_2\,A_3\,A_4\right)$

$\mathrm{\_t0}≔\mathrm{time}\left(\right)\:$$\mathrm{add}\left(B\right)\;$$\mathrm{`time\; consumed`}=\mathrm{time}\left(\right)-\mathrm{\_t0}$

$8.63950358284711-38.2988008918779\mathrm{I}$

$\mathrm{time\; consumed}=0.093$

$\mathrm{\_t0}≔\mathrm{time}\left(\right)\:$$\mathrm{PairwiseSummation}\left(B\right)\;$$\mathrm{`time\; consumed`}=\mathrm{time}\left(\right)-\mathrm{\_t0}$

$8.63950358331158-38.2988008918301\mathrm{I}$

$\mathrm{time\; consumed}=0.047$

$\mathrm{evalf}\left[15\right]\left(\mathrm{evalf}\left[20\right]\left(\mathrm{add}\left(B\right)\right)\right)$

$8.63950358331122-38.2988008918305\mathrm{I}$

$\mathrm{evalf}\left[15\right]\left(\mathrm{evalf}\left[20\right]\left(\mathrm{PairwiseSummation}\left(B\right)\right)\right)$

$8.63950358331123-38.2988008918305\mathrm{I}$

[1] Wikipedia, "Pairwise summation". http://en.wikipedia.org/wiki/Pairwise_summation

The MathematicalFunctions[Evalf][PairwiseSummation] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.