Numerical Polynomial Algebra
In many important areas of scientific computing, polynomials in one or more variables are employed in the mathematical modeling of real-life phenomena; yet most of classical computer algebra assumes exact rational data. This book is the first comprehensive treatment of numerical polynomial algebra, an emerging area that falls between classical numerical analysis and classical computer algebra, and which has received surprisingly little attention so far. The author introduces a conceptual framework that permits the meaningful solution of various algebraic problems with multivariate polynomial equations whose coefficients have some indeterminacy; for this purpose, he combines approaches of both numerical linear algebra and commutative algebra. For the application scientist, this book provides both a survey of polynomial problems in scientific computing that may be solved numerically and a guide to their numerical treatment. In addition, the book provides both introductory sections and novel extensions, making it more easily accessible.
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Representations of Polynomial Ideals
Polynomials with Coefficients of Limited Accuracy
Approximate Numerical Computation
Various Tasks with Empirical Univariate Polynomials
One Multivariate Polynomial
0-dimensional ideal affine varieties algebraic problem algorithm analysis approximate zero associated assume backward error border basis elements cluster zeros codimension complete intersection complex components computation consider data space defined Definition degree determination differential dimension divisor dual space eigenvalues eigenvectors empirical polynomial empirical system equations evaluation exact Example factors function Groebner basis implies indetermination invariant subspace isolated zeros joint eigenvectors leading monomials linear algebra linear system m-fold zero matrix minimal monomial basis multiple zero multiplication matrices Aa multiplicative structure multivariate polynomial neighborhood nontrivial rows norm normal form normal set representation Ns(p numerical numerical analysis obtain parameter perturbation polynomial algebra polynomial ideal polynomial p e polynomial system potential Proof Proposition quadratic quotient ring residual resp S-polynomials satisfy situation solution span specified subset subspace Sylvester matrix syzygies term order Theorem Ti[I tolerance Ts(m univariate polynomials values vanish variables vector space yields zero manifold zero set