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An algorithm is presented that probabilistically computes the exact inverse of a nonsingular n × n integer matrix A using $${({n^3(\log||A||+\log \kappa(A)))}^{1+o(1)}}$$ ( n 3 ( log | | A | | + log κ ( A ) ) ) 1 + o ( 1 ) bit operations. Here, $${||A||= \max_{ij}|A_{ij}|}$$ | | A | | = max i j | A i j | denotes the largest entry in absolute value, ...
The shifted number system is presented: a method for detecting and avoiding error producing carries during approximate computations with truncated expansions of rational numbers. Using the shifted number system the high-order lifting and integrality certification techniques of Storjohann 2003 for polynomial matrices are extended to the integer case. Las Vegas reductions to integer matrix multiplication...
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