Nümerik analiz yöntemleri ve programlama
Numerical methods and programming
- Tez No: 39484
- Danışmanlar: PROF.DR. ALİ NUR GÖNÜLEREN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 127
Özet
general, the coefficients of the terms involving / and in the local truncation error are smaller for the Adams-Moulton method, thus leads to greater stability for the implicit method and smaller rounding errors. In chapter 6, we give some advice how to choose a suitable method for solving a specific problem. Although it is impossible to state universal rules that will guide the user in selecting the best method for a given equation system, a few basic guidelines do exist. These are listed in this chapter. In addition, we discuss some details about the program itself. Moreover, we give some examples of the program in section 6.3, in this chapter. Finally, although the emphasis of the thesis is on algorithms rather than the programming detail in C computer language, we have included the program in the appendix. XI
Özet (Çeviri)
recommended for practical use. It does, however, provide considerable insight into the understanding of several other, more practical methods. The basis of Euler method comes from an application of the initial conditions to a Taylor's series. In «th-order Taylor's expansion, there are the disadvantage of requiring computation and evaluation of the derivatives of f(x,t). This can be very complicated and time-consuming procedure for many problems and, consequently, the Taylor's expansion is seldom used in practice. The Runge-Kutta method provides a set of formulas for selecting the spacing of the internal evaluations required to implement a strategy instead of taking derivatives. Since a number of alternatives exist for the spacing and for the relative weighting to be used for the slopes found, the term Runge-Kutta refers to a large family of methods for handling first-order differential equations. In reality, the Euler method is actually a first-order Runge-Kutta methods. The increased accuracy made possible by the Runge-Kutta technique makes it far more desirable to use then either of the previous method discussed, and more than justifies the additional computational effort needed to use it. Because of this greater accuracy, it is often possible to use a large-sized step-size h. The allowable error at each step will determine the maximum allowable step size that can be used. While using Runge-Kutta methods for the solution of differential equations, information at only previous point (x,-,/,) 's used to evaluate Xi+i at ti+v This is alln“9nt wnen Gto'/o) is used t0 evaluate Xl at f,. However, as the integration of differential equation proceeds, there is an abundance of information available in earlier points which may be used to obtain better accuracy multy-step methods use information at two or more previous points to evaluate %M using the concept of predictor-corrector operations. But in the beginning since only one point is available, other points as required by the multi-step methods have to be evaluated by Runge- Kutta methods. Thus the main drawback of multi-step methods is that they are not self-starting though they offer better accuracy. Muti-step formulas are of two types, open formulas, for example, Adams-Bashforth, and closed formulas, for example, Adams-Moulton. The £th-order Adams-Bashforth method is an explicit Multi-step method obtained by setting p = k-\, a\- üı =- = dk-\ = ^ anc' b-\ = 0 in equation (5.40). In addition, if we set p = k - 2, qx = q2 =...= Qk-2 ~ ® tnus we w'”9e* tne &th-order Adams-Moulton method which is an implicit Multi-step method. It is interesting to compare a Ath-order Adams-Bashforth method to a (&-l)th-order Adams-Moulton method. Both require k evaluations of / per step and both have the same terms in their local truncation errors. Ingeneral, the coefficients of the terms involving / and in the local truncation error are smaller for the Adams-Moulton method, thus leads to greater stability for the implicit method and smaller rounding errors. In chapter 6, we give some advice how to choose a suitable method for solving a specific problem. Although it is impossible to state universal rules that will guide the user in selecting the best method for a given equation system, a few basic guidelines do exist. These are listed in this chapter. In addition, we discuss some details about the program itself. Moreover, we give some examples of the program in section 6.3, in this chapter. Finally, although the emphasis of the thesis is on algorithms rather than the programming detail in C computer language, we have included the program in the appendix. XIrecommended for practical use. It does, however, provide considerable insight into the understanding of several other, more practical methods. The basis of Euler method comes from an application of the initial conditions to a Taylor's series. In «th-order Taylor's expansion, there are the disadvantage of requiring computation and evaluation of the derivatives of f(x,t). This can be very complicated and time-consuming procedure for many problems and, consequently, the Taylor's expansion is seldom used in practice. The Runge-Kutta method provides a set of formulas for selecting the spacing of the internal evaluations required to implement a strategy instead of taking derivatives. Since a number of alternatives exist for the spacing and for the relative weighting to be used for the slopes found, the term Runge-Kutta refers to a large family of methods for handling first-order differential equations. In reality, the Euler method is actually a first-order Runge-Kutta methods. The increased accuracy made possible by the Runge-Kutta technique makes it far more desirable to use then either of the previous method discussed, and more than justifies the additional computational effort needed to use it. Because of this greater accuracy, it is often possible to use a large-sized step-size h. The allowable error at each step will determine the maximum allowable step size that can be used. While using Runge-Kutta methods for the solution of differential equations, information at only previous point (x,-,/,) 's used to evaluate Xi+i at ti+v This is alln“9nt wnen Gto'/o) is used t0 evaluate Xl at f,. However, as the integration of differential equation proceeds, there is an abundance of information available in earlier points which may be used to obtain better accuracy multy-step methods use information at two or more previous points to evaluate %M using the concept of predictor-corrector operations. But in the beginning since only one point is available, other points as required by the multi-step methods have to be evaluated by Runge- Kutta methods. Thus the main drawback of multi-step methods is that they are not self-starting though they offer better accuracy. Muti-step formulas are of two types, open formulas, for example, Adams-Bashforth, and closed formulas, for example, Adams-Moulton. The £th-order Adams-Bashforth method is an explicit Multi-step method obtained by setting p = k-\, a\- üı =- = dk-\ = ^ anc' b-\ = 0 in equation (5.40). In addition, if we set p = k - 2, qx = q2 =...= Qk-2 ~ ® tnus we w'”9e* tne &th-order Adams-Moulton method which is an implicit Multi-step method. It is interesting to compare a Ath-order Adams-Bashforth method to a (&-l)th-order Adams-Moulton method. Both require k evaluations of / per step and both have the same terms in their local truncation errors. Ingeneral, the coefficients of the terms involving / and in the local truncation error are smaller for the Adams-Moulton method, thus leads to greater stability for the implicit method and smaller rounding errors. In chapter 6, we give some advice how to choose a suitable method for solving a specific problem. Although it is impossible to state universal rules that will guide the user in selecting the best method for a given equation system, a few basic guidelines do exist. These are listed in this chapter. In addition, we discuss some details about the program itself. Moreover, we give some examples of the program in section 6.3, in this chapter. Finally, although the emphasis of the thesis is on algorithms rather than the programming detail in C computer language, we have included the program in the appendix. XIrecommended for practical use. It does, however, provide considerable insight into the understanding of several other, more practical methods. The basis of Euler method comes from an application of the initial conditions to a Taylor's series. In «th-order Taylor's expansion, there are the disadvantage of requiring computation and evaluation of the derivatives of f(x,t). This can be very complicated and time-consuming procedure for many problems and, consequently, the Taylor's expansion is seldom used in practice. The Runge-Kutta method provides a set of formulas for selecting the spacing of the internal evaluations required to implement a strategy instead of taking derivatives. Since a number of alternatives exist for the spacing and for the relative weighting to be used for the slopes found, the term Runge-Kutta refers to a large family of methods for handling first-order differential equations. In reality, the Euler method is actually a first-order Runge-Kutta methods. The increased accuracy made possible by the Runge-Kutta technique makes it far more desirable to use then either of the previous method discussed, and more than justifies the additional computational effort needed to use it. Because of this greater accuracy, it is often possible to use a large-sized step-size h. The allowable error at each step will determine the maximum allowable step size that can be used. While using Runge-Kutta methods for the solution of differential equations, information at only previous point (x,-,/,) 's used to evaluate Xi+i at ti+v This is alln“9nt wnen Gto'/o) is used t0 evaluate Xl at f,. However, as the integration of differential equation proceeds, there is an abundance of information available in earlier points which may be used to obtain better accuracy multy-step methods use information at two or more previous points to evaluate %M using the concept of predictor-corrector operations. But in the beginning since only one point is available, other points as required by the multi-step methods have to be evaluated by Runge- Kutta methods. Thus the main drawback of multi-step methods is that they are not self-starting though they offer better accuracy. Muti-step formulas are of two types, open formulas, for example, Adams-Bashforth, and closed formulas, for example, Adams-Moulton. The £th-order Adams-Bashforth method is an explicit Multi-step method obtained by setting p = k-\, a\- üı =- = dk-\ = ^ anc' b-\ = 0 in equation (5.40). In addition, if we set p = k - 2, qx = q2 =...= Qk-2 ~ ® tnus we w'”9e* tne &th-order Adams-Moulton method which is an implicit Multi-step method. It is interesting to compare a Ath-order Adams-Bashforth method to a (&-l)th-order Adams-Moulton method. Both require k evaluations of / per step and both have the same terms in their local truncation errors. Ingeneral, the coefficients of the terms involving / and in the local truncation error are smaller for the Adams-Moulton method, thus leads to greater stability for the implicit method and smaller rounding errors. In chapter 6, we give some advice how to choose a suitable method for solving a specific problem. Although it is impossible to state universal rules that will guide the user in selecting the best method for a given equation system, a few basic guidelines do exist. These are listed in this chapter. In addition, we discuss some details about the program itself. Moreover, we give some examples of the program in section 6.3, in this chapter. Finally, although the emphasis of the thesis is on algorithms rather than the programming detail in C computer language, we have included the program in the appendix. XI
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