1 \documentclass[11pt,a4paper]{article}
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7 \title{Paper1_kahina}
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9 \section{Root finding problem}
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10 we consider a polynomial of degree \textit{n} having coefficients
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11 in the complex \textit{C} and zeros $\alpha
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12 _{i},\textit{i=1,...,n}$. \\
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14 {\Large$p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),
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18 the root finding problem consist to find
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19 all n root of \textit{p(x)}. the problem of finding a root is
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20 equivalent to the problem of finding a fixed-point. To see this
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21 consider the fixed-point problem of finding the n-dimensional
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26 where $g : C^{n}\longrightarrow C^{n}$. Note that we can easily
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27 rewrite this fixed-point problem as a root-finding problem by
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28 setting $f (x) = x-g(x)$ and likewise we can recast the
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29 root-finding problem into a fixed-point problem by setting
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33 Often it will not be possible to solve such nonlinear equation
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34 root-finding problems analytically. When this occurs we turn to
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35 numerical methods to approximate the solution. Generally speaking,
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36 algorithms for solving problems numerically can be divided into
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37 two main groups: direct methods and iterative methods.
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39 Direct methods exist only for $n \leqslant4$,solved in closed form by G. Cardano
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40 in the mid-16th century. However, N.H. Abel in the early 19th
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41 century showed that polynomials of degree five or more could not
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42 be solved by directs methods. Since then researchers have
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43 concentrated on numerical (iterative) methods such as the famous
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44 Newton s method, Bernoulli s method of the 18th, and Graeffe s.
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45 With the advent of electronic computers, different methods has
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46 been developed such as the Jenkins-Traub method, Larkin s method,
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47 Muller s method, and several methods for simultaneous
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48 approximation of all the roots, starting with the Durand-Kerner
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52 $ Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})} $
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55 This formula is mentioned for the first time from Weiestrass [12]
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56 as part of the fundamental theorem of Algebra and is rediscovered
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57 from Ilieff~\cite{Ilief50} [2], Docev [3], Durand [4], Kerner [5].
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58 Another method discovered from Borsch-Supan [6] and also described
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59 and brought in the following form from Ehrlich [7] and
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60 Aberth~\cite{Aberth73}.
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63 $ Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})}
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64 {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}} $
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67 Aberth, Ehrlich and Farmer-Loizou [10] have proved that the above
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68 method has cubic order of convergence for simple roots.
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71 Iterative methods raise several problem when implemented e.g.
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72 specific sizes of numbers must be used to deal with this
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73 difficulty.Moreover,the convergence time of iterative methods
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74 drastically increase like the degrees of high polynomials. The
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75 parallelization of these algorithms will improve the convergence
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78 Many authors have treated the problem of parallelization of
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79 simultaneous methods. Freeman [13] has tested the DK method, EA
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80 method and another method of the fourth order proposed from Farmer
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81 and Loizou [10],on a 8- processor linear chain, for polynomial of
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82 degree up to 8. The third method often diverges, but the first two
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83 methods have speed-up 5.5 (speed-up=(Time on one processor)/(Time
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84 on p processors)). Later Freeman and Bane [14] consider
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85 asynchronous algorithms, in which each processor continues to
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86 update its approximations even although the latest values of other
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87 $z_i((k))$ have not received from the other processors, in
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88 difference with the synchronous version where it would wait. in
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89 [15]proposed two methods of parallelization for architecture with
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90 shared memory and distributed memory,it able to compute the root
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91 of polynomial degree 10000 on 430 s with only 8 pc and 2
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92 communications per iteration. Compare to the sequential it take
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93 3300 s to obtain the same results.
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95 After this few works discuses this problem until the apparition of
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96 the Compute Unified Device Architecture (CUDA) [19],a parallel
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97 computing platform and a programming model invented by NVIDIA. the
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98 computing ability of GPU has exceeded the counterpart of CPU. It
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99 is a waste of resource to be just a graphics card for GPU. CUDA
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100 adopts a totally new computing architecture to use the hardware
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101 resources provided by GPU in order to offer a stronger computing
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102 ability to the massive data computing.
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105 Indeed [16]proposed the implementation of the Durand-Kerner method
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106 on GPU (Graphics Processing Unit). The main result prove that a
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107 parallel implementation is 10 times as fast as the sequential
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108 implementation on a single CPU for high degree polynomials that is
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109 greater than about 48000.
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111 The mean part of our work is to implement the Aberth method on GPU
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112 and compare it with the Durand Kerner
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113 implementation.................To be continued..................
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116 \section{Aberth method and difficulties}
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117 A cubically convergent iteration method for finding zeros of
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118 polynomials was proposed by O.Aberth[?].The Aberth method is a
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119 purely algebraic derivation.To illustrate the derivation, we let
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120 $w_{i}(z)$ be the product of linear factor $ w_{i}(z)=\prod_{j=1,j
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121 \neq i}^{n} (z-x_{j})$
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123 and rational function $R_{i}(z)$ be the correction term of
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124 Weistrass method (?)
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125 $$R_{i}(z)=\dfrac{p(z)}{w_{i}(Z)} , i=1,2,...,n. $$
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127 Differentiating the rational function $R_{i}(z)$ and applying the
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128 Newton method, we have
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129 $$\dfrac{R_{i}(z)}{R_{i}^{'}(z)}= \dfrac{p(z)}{p^{'}(z)-p(z)\dfrac{w_{i}(z)}{w_{i}^{'}(z)}}= \dfrac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\dfrac{1}{z-x_{i}}}, i=1,2,...,n $$
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130 Substituting $x_{j}$ for z we obtain the Aberth iteration method
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132 Let present the means stages of Aberth's method.
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134 \subsection{Polynomials Initialization}
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135 The initialization of polynomial P(z) with complex coefficients
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137 $$ p(z)=\sum{a_{i}z^{n-i}}. where a_{n} \neq 0,a_{0}=1, a_{i}\subset C $$
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140 \subsection{Vector $Z^{0)}$ Initialization}
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142 The choice of the initial points $z^{(0)}_{i} , i = 1, . . . , n,$
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143 from which starting the iteration (2) or (3), is rather delicate
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144 since the number of steps needed by the iterative method to reach
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145 a given approximation strongly depends on it. In [1] the Aberth
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146 iteration is started by selecting n equispaced points on a circle
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147 of center 0 and radius r, where r is an upper bound to the moduli
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148 of the zeros. After[18] performs this choice by selecting complex
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149 numbers along different circles and relies on the result of [19].
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151 $$\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}}; v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};u_{i}=2.|a_{i}|^{\frac{1}{i}}; v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2} $$
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153 \subsection{Iterative Function Hi}
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154 The operator used with Aberth's method is corresponding to the
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155 following equation which will enable the convergence towards
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156 polynomial solutions, provided all the roots are distinct.
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158 $$ H_{i}(z)=z_{i}-\frac{1}{\frac{P^{'}(z_{i})}{P(z_{i})}-\sum_{j\neq i}{\frac{1}{z_{i}-z_{j}}}} $$
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160 \subsection{Convergence condition}
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161 determines the success of the termination. It consists in stopping
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162 the iterative function $H_{i}(z)$ when the are stable,the method
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163 converge sufficiently:
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164 $$ \forall i \in [1,n]; \frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}< \xi$$
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166 \section{Difficulties and amelioration}
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167 the Aberth method implementation suffer of overflow problems. This
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168 situation occurs, for instance, in the case where a polynomial
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169 having positive coefficients and large degree is computed at a
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170 point $\xi$ where $|\xi| > 1$.Indeed the limited number in the
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171 mantissa of floating takings the computation of P(z) wrong when z
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172 is large. for example $(10^{50}) +1+ (- 10_{50})$ will give result
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173 0 instead of 1 in reality.consequently we can't compute the roots
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174 for large polynomial's degree. This problem was discuss in [17]
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175 for the Durand-Kerner method, the authors propose to use the
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176 logratihm and the exponential of a complex:
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178 $$ \forall(x,y)\in R^{*2}; \ln (x+i.y)=\ln(x^{2}+y^{2}) 2+i.\arcsin(y\sqrt{x^{2}+y^{2}})_{\left] -\pi, \pi\right[ } $$
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179 $$ \forall(x,y)\in R^{*2}; \exp(x+i.y)= \exp(x).\exp(i.y)$$
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180 $$ =\exp(x).\cos(y)+i.\exp(x).\sin(y)$$
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183 The application of logarithm can replace any multiplications and
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184 divisions with additions and subtractions; consequently it
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185 manipulates lower absolute values and can be compute the roots for
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186 large polynomial's degree exceed 1000[17].
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188 Applying this solution for the Aberth method we obtain the
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189 iteration function with logarithm:
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190 %%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
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192 $$ H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left( p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln\left(1- \exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left( \sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right) \right) \right)$$
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196 this solution is applying when it is necessary
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198 \section{The implementation of simultaneous methods in a parallel computer}
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201 \bibliographystyle{plain}
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202 \bibliography{biblio}
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203 %% \begin{thebibliography}{2}
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205 %% \bibitem [1] {1} O. Aberth, Iteration Methods for Finding
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208 %% all Zeros of a Polynomial Simultaneously, Math. Comput. 27, 122
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209 %% (1973) 339
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211 %% \bibitem [2] {2} Ilieff, L. (1948-50), On the approximations of Newton, Annual
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212 %% Sofia Univ. 46, 167-171.
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214 %% \bibitem [3] {3} Docev, K. (1962), An alternative method of Newton for
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215 %% simultaneous calculation of all the roots of a given algebraic
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216 %% equation, Phys. Math. J., Bulg. Acad. Sci. 5, 136-139.
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218 %% \bibitem [4]{4} Durand, E. (1960), Solution Numerique des Equations
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219 %% Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une
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220 %% Polynome. Masson, Paris.
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222 %% \bibitem [4] {4} Aberth, O. (1973), Iterative methods for finding all zeros of
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223 %% a polynomial simultaneously, Math. Comp. 27, 339-344.
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225 %% \bibitem [5] {5} Kerner, I.O. (1966), Ein Gesamtschritteverfahren zur
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226 %% Berechnung der Nullstellen von Polynomen, Numer. Math. 8, 290-294.
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228 %% \bibitem [6]{6} Borch-Supan, W. (1963), A posteriori error for the zeros of
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229 %% polynomials, Numer. Math. 5, 380-398.
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231 %% \bibitem [7] {7} Ehrlich, L. W. (1967), A modified Newton method for
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232 %% polynomials, Comm. Ass. Comput. Mach. 10, 107-108.
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238 %% Math. 14, 45-58.
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240 %% \bibitem [11]{11} E. Durand, Solutions numŽeriques des Žequations algŽebriques,
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241 %% Tome 1: Equations du type F(X) = 0; Racines d
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242 %% Masson, Paris 1960.
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244 %% \bibitem [12] {12} Weierstrass, K. (1903), Neuer Beweis des Satzes, dass
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245 %% jede ganze rationale function einer veranderlichen dagestellt
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246 %% werden kann als ein product aus linearen functionen derselben
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247 %% veranderlichen, Ges. Werke 3, 251-269.
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248 %% \bibitem [13] {13} Freeman, T. L. (1989), Calculating polynomial zeros on a
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249 %% local memory parallel computer, Parallel Computing 12, 351-358.
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251 %% \bibitem [14] {14} Freeman, T. L., Brankin, R. K. (1990), Asynchronous
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252 %% polynomial zero-finding algorithms, Parallel Computing 17,
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255 %% \bibitem [15] {15} Raphaël,C. François,S. (2001), Extraction de racines dans des
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256 %% polynômes creux de degré élevé. RSRCP (Réseaux et Systèmes
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258 %% Algorithmes itératifs parallèles et distribués, 13(1):67--81.
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260 %% \bibitem [16]{16} Kahina, G. Raphaël, C. Abderrahmane, S. A
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261 %% parallel implementation of the Durand-Kerner algorithm for
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262 %% polynomial root-finding on GPU. In INDS 2014, Int. Conf. on
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263 %% advanced Networking, Distributed Systems and Applications, Bejaia,
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264 %% Algeria, pages 53--57, June 2014. IEEE
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266 %% \bibitem [17] {17} Karim Rhofir, François Spies, and Jean-Claude Miellou.
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267 %%Perfectionnements de la méthode asynchrone de Durand-Kerner pour
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268 %%les polynômes complexes. Calculateurs Parallèles, 10(4):449-- 458,
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270 %% \bibitem [18] {18} Bini, D. A. Numerical computation of polynomial zeros by
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271 %%means of Aberth
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274 %%Bull. A.M.S., 47 (1941) 742
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275 %% \end{thebibliography}
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