Algebraic Algorithms

This is a preliminary version of a Chapter on Algebraic Algorithms in the upcoming Computing Handbook Set Computer Science (Volume I), CRCPress/Taylor and Francis Group. Algebraic algorithms deal with numbers, vectors, matrices, polynomials, formal power series, exponential and differential polynomials, rational functions, algebraic sets, curves and surfaces. In this vast area, manipulation with matrices and polynomials is most fundamental for modern computations in Sciences, Engineering, and Signal and Image Processing. They include the solution of a polynomial equation and linear and polynomial systems of equations, univariate and multivariate polynomial evaluation, interpolation, factorization and decompositions, rational interpolation, computing matrix factorization and decompositions (which in turn include various triangular and orthogonal factorizations such as LU, PLU, QR, QRP, QLP, CS, LR, Cholesky factorizations and eigenvalue and singular value decompositions), computation of the matrix characteristic and minimal polynomials, determinants, Smith and Frobenius normal forms, ranks, and (generalized) inverses, univariate and multivariate polynomial resultants, Newton’s polytopes, greatest common divisors, and least common multiples as well as manipulation with truncated series and algebraic sets. Such problems can be solved based on the error-free symbolic computations with infinite precision. The computer library GMP and computer algebra