3 Zeros of Polynomials and Complex Roots

Polynomials are the functions met most often in practice, and their roots can be found by the methods of Section 2. Two features make them special. They can be evaluated, and differentiated, very cheaply by Horner’s method; and a polynomial with real coefficients may have roots that are not real, which no iteration confined to the real line can find. This section therefore begins with the complex arithmetic the methods need. (The full theory of functions of a complex variable is the subject of the course Theory of Functions of Complex Variables on this site; here we use only what the numerical methods require.)

A polynomial of degree \(n\) is \[P(x) = a_nx^n+a_{n-1}x^{n-1}+\dots +a_1x+a_0,\qquad a_n\neq 0,\] with real or complex coefficients \(a_k\).

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