On the complexity of computing determinants

Abstract.We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in $$\left( {n^{3.2} \log \left\| A \right\|} \right)^{1 + o(1)} $$ and $$\left( {n^{2.697263} \log \left\| A \right\|} \right)^{1 + o(1)} $$ bit operations; here $$\left\| A \right\|$$ denotes the largest entry in absolute value and the exponent adjustment by “+o(1)” captures additional factors $$C_1 (\log n)^{C_2 } \left( {\log \log \left\| A \right\|} \right)^{C_3 } $$ for positive real constants C1, C2, C3. The bit complexity $$\left( {n^{3.2} \log \left\| A \right\|} \right)^{1 + o(1)} $$ results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n3.2+o(1) and O(n2.697263) ring additions, subtractions and multiplications.