Algebraic and Numerical Algorithms

Arithmetic manipulation with matrices and polynomials is a common subject for algebraic (or symbolic) and numerical computing. Typical computational problems in these areas 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, including 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 inverses, determinants, Smith and Frobenius normal forms, ranks, characteristic and minimal polynomials, univariate and multivariate polynomial resultants, Newton’s polytopes, and 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 algebraic (symbolic) computations with infinite precision. This demanding task is achieved in the present day advanced computer library GMP and computer algebra systems such as Maple and Mathematica by employing various nontrivial computational techniques such as the Euclidean algorithm and continuous fraction approximation, Hensel’s and Newton’s lifting, Chinese Remainder algorithm, elimination and resultant methods, and Gröbner bases computation. The price for the achieved accuracy is the increase of the memory space and computer time supporting the computations.