GRAFFE ROOT SQUARING METHOD PDF

A root -finding method which was among the most popular methods for finding roots of univariate polynomials in the 19th and 20th centuries. Graeffe's method has a number of drawbacks, among which are that its usual formulation leads to exponents exceeding the maximum allowed by floating-point arithmetic and also that it can map well-conditioned polynomials into ill-conditioned ones. However, these limitations are avoided in an efficient implementation by Malajovich and Zubelli The method proceeds by multiplying a polynomial by and noting that.

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A root -finding method which was among the most popular methods for finding roots of univariate polynomials in the 19th and 20th centuries. Graeffe's method has a number of drawbacks, among which are that its usual formulation leads to exponents exceeding the maximum allowed by floating-point arithmetic and also that it can map well-conditioned polynomials into ill-conditioned ones.

However, these limitations are avoided in an efficient implementation by Malajovich and Zubelli The method proceeds by multiplying a polynomial by and noting that.

Since the coefficients are given by Vieta's formulas. Bini, D. Complexity 12 , , Brodetsky, S. Cambridge Philos. Cajori, F. New York: Chelsea, p. Dedieu, J.

I Math , , Grau, A. Householder, A. Monthly 66 , , Jana, P. Von and Biot, M. New York: Mcgraw-Hill, pp. Malajovich, G. Complexity 17 , , Ostrowski, A. Pan, V. Runge, C. Berlin and Leipzig, Germany: de Gruyter, pp. Whittaker, E. New York: Dover, pp. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

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Graeffe's Method

In mathematics , Graeffe's method or Dandelin—Lobachesky—Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in and Lobachevsky in This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Let p x be a polynomial of degree n. Graeffe observed that if one separates p x into its odd and even parts:. This expression involves the squaring of two polynomials of only half the degree, and is therefore used in most implementations of the method.

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