Smale’s Fundamental Theorem of Algebra Reconsidered

Smale's Fundamental Theorem of Algebra Reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Our’s is polynomial in the Bézout number and the di...

Fundamental Theorem of Algebra

The following propositions are true: (1) For all natural numbers n, m such that n 6= 0 and m 6= 0 holds (n ·m− n − m) + 1 ­ 0. (2) For all real numbers x, y such that y > 0 holds min(x,y) max(x,y) ¬ 1. (3) For all real numbers x, y such that for every real number c such that c > 0 and c < 1 holds c · x ­ y holds y ¬ 0. (4) Let p be a finite sequence of elements of R. Suppose that for every natu...

Fundamental theorem of algebra

In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solving polynomial equations with real or complex coefficie...

The Fundamental Theorem of Linear Algebra

The SVD-Fundamental Theorem of Linear Algebra

Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and A T ; a 1 through a n and h 1 through h m are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N (A) is the orthogonal complement of R(A T). These four s...