Solving Optimization Problems Involving Polynomial - amazonia.fiocruz.br

Solving Optimization Problems Involving Polynomial Video

In numerical analysisNewton's methodalso known as the Newton—Raphson methodnamed after Isaac Newton and Joseph Raphsonis a root-finding algorithm which produces successively better approximations to the roots or zeroes of a real -valued function.

If the Solving Optimization Problems Involving Polynomial satisfies sufficient assumptions and the initial guess is close, then. The process is repeated as. This algorithm is first in the class of Householder's methodssucceeded by Halley's method. The method can also be extended to complex functions and to systems of equations.

The idea is to start with an initial guess which is reasonably close to the true root, then to approximate the function by its tangent line using calculusand finally to compute the x -intercept of this tangent line by elementary algebra. This x -intercept will typically be a better approximation to the original function's root than the first guess, and the method can be iterated.

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We start the process with some arbitrary initial value x 0. The closer to the zero, the better. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small Proglems by appealing to the intermediate Solving Optimization Problems Involving Polynomial theorem. More details can be found in the analysis section below. Householder's methods are similar but have higher Sooving for even faster convergence. However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if f or its derivatives are computationally expensive to evaluate. The name "Newton's method" is derived from Isaac Newton 's description of a special case of the method in De analysi per aequationes numero terminorum infinitas written inpublished in by William Jones and in De metodis fluxionum et serierum infinitarum written intranslated and published as Method of Fluxions in by John Colson.

However, his method differs substantially from the modern method given above.

Newton applied the method only to polynomials, starting with an initial root estimate and extracting a sequence Invilving error corrections. He used each correction to rewrite the polynomial in terms of the remaining error, and then solved for a new correction by neglecting higher-degree terms. He did not explicitly connect the method with derivatives or present a general formula.

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Newton applied this method to both numerical and algebraic Solving Optimization Problems Involving Polynomial, producing Taylor series in the latter case. Newton may have derived his method from a similar but less precise method by Vieta. A special case of Newton's method for calculating square roots was known since ancient times and is often called the Babylonian method.

This allowed him to derive a reusable iterative expression for each problem. Finally, inThomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero. Arthur Cayley in in The Newton—Fourier imaginary problem was the first to notice the difficulties in see more Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values.

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This opened the way to the study of the theory of iterations of rational functions. Newton's method is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared the number of accurate digits roughly doubles at each step. However, there are some difficulties with the method. Newton's method requires that the derivative can be calculated directly.]