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This shows that the root α j will be less stable if there are many roots α k close to α j, in the sense that the distance |α j − α k | between them is smaller than |α j |. Example. For the root α 1 = 1, the derivative is equal to 1/19! which is very small; this root is stable even for large changes in t.
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
% The fixed point iteration function is assumed to be input as an % inline function. % This function will calculate and return the fixed point, p, % that makes the expression f(x) = p true to within the desired % tolerance, tol. format compact % This shortens the output. format long % This prints more decimal places. for i = 1: 1000 % get ready ...
α is a root of a polynomial p if and only if α −1 is a root of p ∗. [4] If p(x) ≠ x then p is irreducible if and only if p ∗ is irreducible. [5] p is primitive if and only if p ∗ is primitive. [4] Other properties of reciprocal polynomials may be obtained, for instance:
This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated.
After iteration number 5 we have 4 correct decimals, and the subsequent iteration number 6 confirms that the computed roots are fixed. This general behaviour is characteristic for the method. Also notice that, in this example, the roots are used as soon as they are computed in each iteration.
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MPSolve (Multiprecision Polynomial Solver) is a package for the approximation of the roots of a univariate polynomial. It uses the Aberth method, [1] combined with a careful use of multiprecision. [2] "Mpsolve takes advantage of sparsity, and has special hooks for polynomials that can be evaluated efficiently by straight-line programs" [3]