научная статья по теме PROPERTIES OF SUMS OF SOME ELEMENTARY FUNCTIONS AND THEIR APPLICATION TO COMPUTATIONAL AND MODELING PROBLEMS Математика

Текст научной статьи на тему «PROPERTIES OF SUMS OF SOME ELEMENTARY FUNCTIONS AND THEIR APPLICATION TO COMPUTATIONAL AND MODELING PROBLEMS»

ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ, 2011, том 51, № 5, с. 748-761

УДК 519.615.5

PROPERTIES OF SUMS OF SOME ELEMENTARY FUNCTIONS AND THEIR APPLICATION TO COMPUTATIONAL AND MODELING PROBLEMS

© 2011 Yu. K. Shestopaloff

(142 Kennard Ave Toronto, Ontario M3H4M5, Canada) e-mail: shes169@yahoo.ca Received April 27, 2010; in final form September 27, 2010

The article proposes methods for creating corresponding sums of functions and sums of corresponding series, whose appropriate equations have equal number of solutions. We consider equations composed of power functions (generalized polynomials) and sums of exponential functions. Using the concept of corresponding functions, we prove the relationships between the properties of polynomial, power and sums of exponential functions. One of the results is generalization of Descartes Rule of Signs for sums of other than polynomial functions and sums of their series. Obtained results are applied to two practical problems. One is the finding of an adequate description of transition electrical signals. Secondly, the proved theorems are applied to the problem of finding the initial value for iterative algorithms used to solve one particular case of IRR (internal rate of return) equation for mortgage calculations. Overall, the results are proved to be beneficial for theoretical and practical applications in industry and in different areas of science and technology.

Key words: polynomials; Descartes rule of signs, power function; exponential function; logarithmic functions; real solutions; corresponding equations, transition process, IRR equation.

Ю.К. Шестопалов. "Свойства сумм некоторых элементарных функций и использование этих свойств для моделирования и вычислительных задач". Предлагается метод, с помощью которого суммам функций одного типа можно сопоставить сумму функций другого типа, при этом соответствующие уравнения будут иметь одинаковое число решений. Подход не ограничивается суммами с конечным числом членов, но применим также к рядам. Рассмотрены обобщенные полиномиальные уравнения, у членов которых степень действительное число, а также сумма конечного числа экспоненциальных функций и сумма ряда; одним из следствий явилось подтверждение правила Декарта о максимально возможном числе решений соответствующих уравнений для сумм названных функций. Полученные результаты были использованы для моделирования суммарного воздействия переходных электрических процессов. Экспериментальная проверка подтвердила адекватность модели и ее существенно более высокую точность по сравнению с ранее используемым подходом; было правильно предсказано максимальное число осцилляций сигнала и амплитуда. Анализ уравнения внутренней ставки доходности, широко используемого в финансовой математике, позволил показать, что в случае выплаты регулярных платежей это уравнение имеет два решения. Предложен метод для нахождения начального значения для итерационных процедур, применяемых для численного решения этого уравнения, которое обеспечивает быструю сходимость к правильному решению. Библ. 8. Фиг. 4.

Ключевые слова: обобщенные полиномы, степенные функции, экспоненциальные функции, ряды, число решений, уравнение внутренней ставки доходности, начальное значение, сходимость.

1. FORMULATION OF THE PROBLEM

All dynamical systems and devices, such as mechanical and electrical, include, in one form or another, certain transition processes. The performance of a majority of electronic devices, both analog and digital, is affected by the permanent presence of transition processes arising from fast changes of voltage and current, which is especially the case with high frequency digital devices. Correct evaluation of characteristics of transition processes and understanding of their nature is important for design of reliable and high performance electronic devices. Usually, in electrical devices with a large number of components, the oscil-

lating nature of transition processes is well expressed. This fact gives the rise to the approach of modeling these processes by the attenuating harmonic signals of the form

P(t) = A exp(-bt) sin(rot + ф). (1)

Here, P(t) is a transition process; t is time; A and b are real constants, ю and ф are angle frequency and phase shift accordingly.

When such processes attenuate slowly, there is a chance that the sum of such transition processes can produce a voltage spike (when the crests of different sinusoids coincide) that can damage semiconductor electrical components or significantly disturb a device's performance. On the other hand, in many instances, despite the presence of oscillations shown by direct measurements, the analysis of electrical schemes does not reveal sources that could produce oscillating signals of type (1). When analyzing such a problem, we made the assumption that the observed oscillations are produced by a sum of attenuating exponential signals. This can be presented as follows

P(t) = £ Aj exp(-bjt), (2)

i

where real values Aj and bj relate to one exponential signal.

The sources of transition signals that are represented by summands in (2) are usually easily identifiable; in most instances, they are associated with elementary electrical components, such as series RC circuits. Beside the straightforward identification of sources of such transition signals, another advantage of approach (2) is that in this case the chances of creating a dangerous voltage spike are much lower, in practical terms, actually zero. So, the designer of electrical devices for which (2) holds true does not need to take into account the possibility of having accidental voltage spikes produced by transition processes. However, the proof ofvalidity of this approach faced several challenges. One of them was that the properties of sums of exponential functions were insufficiently studied for our purposes. For instance, we did know how many oscillations the sum (2) may have based on the parameters of summands, which is a critical issue for this problem. What we needed was some analogue of Descartes Rule of Signs, which allows us to evaluate the maximum number of possible solutions of a polynomial equation, but applied to sum of exponential functions. Note that the number of oscillations of (2) is related to number of solutions of equation: Z jAj exp(-bjt) = C, where C is a constant. This was one incentive to begin the study of properties of sums of exponential functions. One of the results of the presented article is an analogue of Descartes Rule of Signs which can be applied to such a sum of exponential functions or series. Using this result, we show that (2) presents an adequate description of transition processes in certain electric circuits.

It is interesting to note that the above formulated problem, in a mathematical sense, soon intersected with another important practical issue from a different area, namely computations of rates of return and implied volatility. These parameters are widely used for quantitative valuations in many financial applications (see [1], [2]). Evaluation of implied volatility is based on solution of the so called IRR equation (IRR means "internal rate of return"). The same equation is used for computing interest rate in lending financial applications. It is also the basis for evaluation of capital investments (net present value, Modified IRR). One of the forms of this equation is as follows (see [1]):

N

E(R) = £ Cj (1 + R)T. (3)

j=o

Consider an investment portfolio consisting of a collection of assets, such as stocks. Here, E is the ending market value, which is non-negative; Cj are cash flows originating from the buying or selling of stocks, which can be positive (if cash is added to the investment portfolio), or negative (if cash is withdrawn from the portfolio); Tj is the time period from the origination of the cash flow until the end of the investment period; T0 is the length of the overall period; j = 1, 2, ..., N; R is the rate of return that we want to find (this rate is per one unit of time).

Generally, IRR equation has multiple solutions. As we can see from (3), the time period is a positive real number. In special cases, (3) can be a polynomial equation. Then, the maximum possible number of real solutions is defined by Descartes Rule of Signs. However, if the powers in (3) are real, we do not have methods for evaluating the number of real solutions. The IRR equation is a transcendental one and generally can be solved only numerically. Knowing the maximum number of solutions helps us to find the intervals which contain them. In turn, this will help us to choose a good initial value for our iterative method. Information about the number of solutions is also needed to choose the solution that makes business sense (see [1]). These issues are of practical importance and often discussed by financial industry's regu-

latory and standards bodies, such as the European Bond Commission (see [3]). The article presents an analogue of Descartes Rule of Signs for equation (3) and establishes the relationship between the properties of polynomial, power and sums of exponential functions. This knowledge substantially facilitates analysis and finding solutions of IRR equation. For instance, it is possible to unambiguously define the interval in which the mortgage IRR equation has a unique solution. (Mortgage equation (see [4]) is widely used in financial applications.)

2. INTRODUCING A CORRESPONDING EQUATION FOR A POLYNOMIAL

Beside the practical problems described in the previous section, mathematical modeling of real phenomena is often based on usage of elementary functions and their sums (see [1], [5]—[7]). Below, we present the general study of polynomial, power and exponential functions and show the relationships of their properties. Then, we illustrate the application of obtained results considering some practical problems.

Let us consider a polynomial equation for which we will introduce a corresponding equation, compose

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