A Sequence of Nearest Polynomials with Given Factors

Hiroshi Sekigawa ()
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Hiroshi Sekigawa: Tokyo University of Science

A chapter in Computer Mathematics, 2014, pp 141-145 from Springer

Abstract: Abstract Let $$K$$ K be either $$\mathbb {R}$$ R or $$\mathbb {C}$$ C , and $$p$$ p and $$f_0$$ f 0 be polynomials in $$K[x_1,\ldots ,x_s]$$ K [ x 1 , … , x s ] such that $$p\ne 0$$ p ≠ 0 , $$\Vert f_0\Vert =1$$ ‖ f 0 ‖ = 1 , where $$\Vert f_0\Vert $$ ‖ f 0 ‖ is the Euclidean norm of $$f_0$$ f 0 , and the coefficient of $$f_0$$ f 0 with the maximal absolute value is a positive real number. For $$j=1$$ j = 1 , $$2$$ 2 , ..., let $$p_{2j-1}=f_{j-1}g_j$$ p 2 j - 1 = f j - 1 g j be the nearest polynomial to $$p$$ p such that $$f_{j-1}|p_{2j-1}$$ f j - 1 | p 2 j - 1 and $$\deg (p_{2j-1})\le \deg (p)$$ deg ( p 2 j - 1 ) ≤ deg ( p ) , where $$\deg $$ deg is the total degree, and $$p_{2j}=c_j f_j g_j$$ p 2 j = c j f j g j be the nearest polynomial to $$p$$ p such that $$c_j\in K$$ c j ∈ K , $$g_j|p_{2j}$$ g j | p 2 j , $$\deg (p_{2j})\le \deg (p)$$ deg ( p 2 j ) ≤ deg ( p ) , $$\Vert f_j\Vert = 1$$ ‖ f j ‖ = 1 , and the coefficient of $$f_j$$ f j with the maximal absolute value is a positive real number. We investigate the behavior of the sequences $$\{\,p_j\,\}$$ { p j } , $$\{\,f_j\,\}$$ { f j } , $$\{\,g_j\,\}$$ { g j } , and $$\{\,c_j\,\}$$ { c j } .

Keywords: Maximum Absolute Value; Positive Real Number; Total Degree; Euclidean Norm; Monotone Nonincreasing (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-43799-5_12

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DOI: 10.1007/978-3-662-43799-5_12

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