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On the other side if the Newton's method will converge then it is quite rapid with convergence of order p = 2, quadratic convergence. + +Newton's method is also effective if the function derrivative if far from zero, so the slope of the function is steep, conversely if the derrivative is close the zero the method is not recommended. \subsection{Problem} @@ -115,6 +125,69 @@ So our polynomial looks like this: Using the M{\"u}ller's method. We have to implement both MM1 and MM2 versions. We also need to find real roots using the Newton's method and compare these results with what we got from MM2 version of the M{\"u}ller's method. \section{Theoretical Introduction} +M{\"u}ller's method revoles around the idea of approximating the polynomial locally close to the root by a quadratic function. Based on three different points we can use quadratic interpolation and develop our method. This means that we can treat it as a generalization of secant method. That being said wwe can also realize it in an efficient way if we use just one point. We can use for this case values of polynomial, and its first and second derrivative at current point. + +There are two versions of M{\"u}ller's method: \textbf{MM1} and \textbf{MM2}. + +\subsection{MM1} +Given three points: $x_0; x_1; x_2$ and their polynomial values: $f(x_0), f(x_1), f(x_2)$ we construct a (quadratic) function passing through these points. Then we find roots of this parabola and we choose one of these rots for the approximation of the result. + +For example: +Assume that $x_2$ is the approximation of the root. +Let's introduce variable $z$ such that: +\[ z = x - x_2 \] +And differences: +\[ z_0 = x_0 - x_2 \] +\[ z_1 = x_1 - x_2 \] + +We have quadratic function: +\[ y(z) = az^2 + bz + c \] + +Using three points from above we get: +\[ y(z_0) = az_0^2 + bz_0 + c = f(x_0) \] +\[ y(z_1) = az_1^2 + bz_1 + c = f(x_1) \] +\[ y(z_2) = c = f(x_2) \] + +And then we get system of equation that we can solve to find $a$ and $b$: +\[ az_0^2 + bz_0 = f(x_0) - f(x_2) \] +\[ az_1^2 + bz_1 = f(x_1) - f(x_2) \] + +Roots are equal to: +\[ z_+ = \frac{-2c}{b+\sqrt{b^2 - 4ac}} \] +\[ z_- = \frac{-2c}{b-\sqrt{b^2 - 4ac}} \] + +We choose a root with smaller absolute value for next iteration: +\[ z_min = \min{|z_+, z_-|} \] +\[ x_3 = x_2 + z_min \] + +Then we choose new point $x_3$ and two selected from $x_0, x_1, x_2$ which were closer to $x_3$. + +This method should also work for $\delta < 0 $ + + +\subsection{MM2} +This method being numerically more effective is usually recommended. +We calculate values of a polynomial and its first and second derrivatives at one point. + +from definition of quadratic function: +\[ y(z) = az^2 + bz + c \] +we can get: +\[ z = x - x_k \] +If $z = 0$ then: +\[ y(0) = c = f(x_k) \] +\[ y^{'}(0) = b = f^{'}(x_k) \] +\[ y^{''}(0) = 2a = f^{''}(x_k) \] + +We can derive from that formula for roots: + +\[ z_{\pm} = \frac{-2f(x_k)}{f^{'}(x_k) \pm \sqrt{ (f^{'}(x_k))^2 - 2f(x_k)f^{''}(x_k)}}\] + +Then we choose root with smaller absolute value for next iteration: +\[ x_{k+1} = x_k + z_{min} \] + +Again this method should be implemented in complexd number arithmetic. +This method is locally convergent with order of convergence equal to 1.84. It is locally more effective that secant method and it is almost as fast as Newton's method while being capable of finding complex roots. It can be used to find roots of polynomials or another nonlinear functions. + \section{Results} \subsection{Comparison of results between MM1 and MM2} @@ -129,6 +202,14 @@ Using the Laguerre's method. Then we should compare those results with the MM2 v \section{Theoretical Introduction} +Laguerre's method is defined by a single formula: + +\[ x_{k+1} = x_k - \frac{nf(x_k)}{f^{'}(x_k) \pm \sqrt{(n-1)[(n-1)( (f^{'}(x_k))^2 - nf(x_k)f^{''}(x_k) )]}} \] +Where: +$n$ - order of the polynomial + +This formula is similar to the one from MM2 but also takes order of the polynomial into consideration. In general this method is better. For polynomials with real roots it is globally convergent. It does not have formal analysis for complex roots but it usually shows good numerical properties, although divergence may happen. + \section{Results} \subsection{Comparison of results between MM1 and MM2} diff --git a/ENUME/projectB/report.toc b/ENUME/projectB/report.toc index 77534fd7..0577e97b 100644 --- a/ENUME/projectB/report.toc +++ b/ENUME/projectB/report.toc @@ -11,13 +11,15 @@ \contentsline {subsection}{\numberline {1.2.3}Results}{4}{subsection.1.2.3}% \contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{5}{chapter.2}% \contentsline {section}{\numberline {2.1}Problem}{5}{section.2.1}% -\contentsline {section}{\numberline {2.2}Theoretical Introduction}{6}{section.2.2}% -\contentsline {section}{\numberline {2.3}Results}{6}{section.2.3}% -\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{6}{subsection.2.3.1}% -\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{6}{subsection.2.3.2}% -\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{7}{chapter.3}% -\contentsline {section}{\numberline {3.1}Problem}{7}{section.3.1}% -\contentsline {section}{\numberline {3.2}Theoretical Introduction}{7}{section.3.2}% -\contentsline {section}{\numberline {3.3}Results}{7}{section.3.3}% -\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{7}{subsection.3.3.1}% -\contentsline {chapter}{\numberline {4}Code appendix}{8}{chapter.4}% +\contentsline {section}{\numberline {2.2}Theoretical Introduction}{5}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}MM1}{6}{subsection.2.2.1}% +\contentsline {subsection}{\numberline {2.2.2}MM2}{7}{subsection.2.2.2}% +\contentsline {section}{\numberline {2.3}Results}{8}{section.2.3}% +\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{8}{subsection.2.3.1}% +\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{8}{subsection.2.3.2}% +\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{9}{chapter.3}% +\contentsline {section}{\numberline {3.1}Problem}{9}{section.3.1}% +\contentsline {section}{\numberline {3.2}Theoretical Introduction}{9}{section.3.2}% +\contentsline {section}{\numberline {3.3}Results}{10}{section.3.3}% +\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{10}{subsection.3.3.1}% +\contentsline {chapter}{\numberline {4}Code appendix}{11}{chapter.4}%