verified almost all relations by GiNaC [@GiNac] with an error bound $<10^{-90}$.

This work was started while I was visiting Chern Institute of Mathematics at Nankai University and the Morningside Center of Mathematics at Beijing, China in the summer of 2007. I would like to thank both institutions and my hosts Chengming Bai and Fei Xu for their hospitality and the ideal working environment. I also want to thank Jens Vollinga for answering some of my questions regarding the numerical computation of the multiple polylog values. The paper was revised later while I was visiting the Institute for Advanced Study and thanks are due to Prof. Deligne for his patient explanation of [@DG] and many insightful remarks on the paper. This work was partially supported by a faculty development fund from Eckerd College.

The double shuffle relations and the algebra $\fA$ {#sec:RDSoverC}
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It is Kontsevich [@K1] who first noticed that MZVs can be represented by iterated integrals (cf. [@Rac]). We now extend this to MPVs. Set $$a=\frac{dt}{t},\qquad b_i=\frac{\mu^i dt}{1-\mu^i t}\quad
\text{ for } i=0,1,\dots,N-1.$$ For every positive integer $n$ define $$y_{n,i}:=a^{n-1} b_i.$$ Then it is straight-forward to verify using that if $(s_1,\mu^{i_1})\ne (1,1)$ then (cf. [@Rac (2.5)]) $$\label{equ:mzv}
L_N(s_1,\dots,s_n|i_1,i_2,\dots,i_n)=\int_0^1
