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Kompleks katsayılı çokterimlilerin köklerinin bilgisayar programı ile hesaplanması

Overviev of this study

  1. Tez No: 39836
  2. Yazar: AYHAN YALKUT
  3. Danışmanlar: PROF.DR. METİN DEMİRALP
  4. Tez Türü: Yüksek Lisans
  5. Konular: Mühendislik Bilimleri, Engineering Sciences
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1994
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 45

Özet

ÖZET Günümüzde, Sayısal Hesap problemlerini çözmek için bir çok modüler program yazılmıştır. Ancak, bu programlar problemlere genel bir çözüm yöntemiyle yaklaşmakta, özel durumlara sahip problemler için sonuçlar yeterli duyarlıkta elde edilemeyebilmektedir. Bu tez'in amacı, bir çokterimlinin köklerinin kompleks düzlem üzerinde bulunduğu bölgeleri Cauchy Simitleri ve Daraltılabilen Çemberler yöntemleri ile belirlemek ve bu bölgelerdeki köklere Newton-Raphson yöntemi ile yeterli duyarlıkta yaklaşabilmektir. Program Borland C++ programlama dilinde yazılmıştır. Kompleks sayılar üzerinde matematiksel işlemlerin çok kolay gerçekleştirilebilmesi ve istenilen duyarlıkta sayılarla işlem yapılabilmesi nedeni ile C++ programlama dili seçilmiştir. Program bir ana program ve dokuz alt-programdan oluşmaktadır. Verilen bir çokterimlinin kompleks katsayıları bir bağlı listede tutulmaktadır. Cauchy Simitleri ve köklerin kompleks düzlem üzerindeki bölgeleri grafik ekranda gösterilmektedir. IV

Özet (Çeviri)

The aim of this study, when a polynomial is given, is to determine the zone of the root on complex plane with Cauchy rings and Shrinkable rings methods and to find out the root of this zone with Newton-Raphson methods. We are now going to establish general methods for computing a root of the equation j{X)= u, where j{X) can be algebraic or transcendental function. We intend to start with a trial value and then construct better and better approximation. If we represent the function y = f(x) graphically, the problem can be so formulated that we are looking for the intersection between the curve and the x-axis. The basic idea is now to replace the curve by a suitable straight line whose intersection with the x-axis can easily be computed. The line passes through the last approximation point v>,vV> but lis direction can be chosen in many different ways. One can be prescribe either a fixed direction, eg, parallel to the secant through the first points, [x0j>q) and [x^j, or to the tangent through the first point, or such a variable direction that the line simply coincides with the secant through yxn_byn_,j and vw> or with the tangent through \x“y”). If the slope of the line is denoted by k, we have the following four possibilities: 1. k =(y, - y0)/(x} - x0) fixed secant 2. k = f'(x0) fixed tangent 3. k =(yn - y“_i)f(x”- *“_,) variable secant 4. k = /'(*”) variable tangent In all cases we obtain an iteration formula of the following form: We shall now examine the convergence of the different methods. Let § be the exact value of the simple root we are looking for: j(E,)= u, j(%)* u. Further we set xn = Ş + zn, and for the first two cases we gety-yn = k(x-xn) and, putting y = 0, v- -, v- = Xn~yn X ^ k Hence % + e“+1 = § + e”- M + e»)/* = ? + e“ - [./©+ 8,,//5 m 2 where the plus sign should be chosen. Thus we have found 8«+i = Ke» m The method with variable secant which has just been examined is known as Regula falsi. In case 4 we find; VIor e”/'(l + eJ-/Q; + E“) '/.(i+«g E”/'(§Ke“2/”(§H-/(g)-B“/'(l)-(Bn2/2y^)--. 6,1+1 rrç + e ^ Hence the leading term is E«+l ”~ ' 2/'(|)“ and the convergence is said to be quadratic. This fact also means that the number of correct decimals is approximately doubled at every iteration, at least if the factor j 'X^pzj \§; is not too large. The process which has just been described is known as the Newton-Raphson method. The iteration formula has the form Xn+\ Xn As has already been mentioned, the geometrical interpretation defines xn+i as the intersection between the tangent in \xnyn) and x-axis. Newton-Raphson's formula can also be obtained analytically from the condition flx”+ A)- /(*>///'(*>^/“(*>= 0, where xn is an approximation of the root. If all terms of second and higher degree in h are neglected, we get h_ /(*”) and x ~x J^L exactly. The method can be used for both algebraic and transcendental equations, and it also works when coefficients or roots are complex. It should be noted, however, that in the case of algebraic equation with real coefficients, a complex root cannot be reached with a real string value. Cauchy rings is improved by Cauchy. It is defined a zone between two rings with the same centre. It is computed as follows: VIIfor I -rj > R > 0 l^)|>HîKi-fctJ-l^-Jal} for |r| > R, P(z) is always positive ! j_k^_h^__N=0 With the same way, \an\rn + \an_x\r" 1 ++|a,|r + |a0| = 0 The program consist of a main and ten sub-function. Main program gets the degree of the polynomial from user and calls the INPUT function. INPUT function gets complex coefficients of the polynomial from user and adds this complex coefficients to a linked list. When user enters last coefficients of polynomial, it assigns NULL variable to the end of linked list and return to main function. Main function calls Cauchy rings function. Cauchy rings function computes the radius of Cauchy rings by using linked list that has the coefficients of polynomial. It computes the average of radius and returns to the main program. Main program calls Shrinkable rings function with the average radius that Cauchy rings had computed. Shrinkable rings function finds minimum radius for a lot of points on the average radius that is passed by the main program. It calls TestCember function to test whether this rings with minimum radius is intersecting each other. If the rings have no intersection, each rings has a different root. The coordinate of the centre of rings and radius of the rings are added to a linked list. It returns to main program. Main program calls Graphics function. VIIIGraphic function draws the Cauchy rings and rings, that is constructed in Shrinkable rings function, on a graphical screen and returns to main program. Main program branches to Newton-Raphson function for each coordinate that has been computed in the Shrinkable rings function. IX

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