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Integration of functions are approximations of the area that the functions cover. Matrices are similar to some boxes being stacked together by rows and columns, but the objects they contain are the numbers. Our protagonists are Riemann surfaces, which have the shape of donuts with one hole or even more holes. Our job is to compute period matrices, which are one category of matrices, of Riemann surfaces, by integrating some special functions on the surface. We provide the proven error bound of period matrices, since each number of period matrices are transcendental, which means that they cannot be compared easily just like pi. We adapt several methods, including Taylor's theorem with the Cauchy form of the error bound and other theorems to derive an error bound. We adapt the existing work to SageMath and obtain two algorithms. We expect the new methods to significantly reduce the running time and be convenient for researchers in need.
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