Чисельне диференціювання табличних функцій у довільно розташованих вузлах інтерполяції
dc.citation.epage | 41 | |
dc.citation.issue | 1 | |
dc.citation.journalTitle | Український журнал інформаційних технологій | |
dc.citation.spage | 25 | |
dc.citation.volume | 5 | |
dc.contributor.affiliation | Національний університет “Львівська політехніка” | |
dc.contributor.affiliation | Lviv Polytechnic National University | |
dc.contributor.author | Грицюк, Юрій Іванович | |
dc.contributor.author | Тушницький, Р. Б. | |
dc.contributor.author | Hrytsiuk, Yu. I. | |
dc.contributor.author | Tushnytskyy, R. B. | |
dc.coverage.placename | Львів | |
dc.coverage.placename | Lviv | |
dc.date.accessioned | 2024-04-01T07:54:32Z | |
dc.date.available | 2024-04-01T07:54:32Z | |
dc.date.created | 2023-02-28 | |
dc.date.issued | 2023-02-28 | |
dc.description.abstract | Розроблено методику чисельного диференціювання таблично-заданих функцій з використанням многочлена Тейлора n-го степеня, яка дає можливість обчислювати похідні k-го порядку (k £ n) у будь-яких точках між довільно розташованими вузлами інтерполяції від однієї, двох і багатьох незалежних змінних. Проаналізовано останні дослідження та публікації, що дало змогу встановити складність задачі обчислення похідних від функції за значеннями незалежних змінних на деякому інтервалі значень таблично-заданої функції. Наведено постановку задачі чисельного диференціювання таблично-заданих функцій з використанням многочлена Тейлора n-го степеня від однієї, двох і багатьох незалежних змінних. Встановлено, що будь-яку таблично-задану функцію спочатку потрібно згладити деякою функцією, аналітичний вираз якої є глобальним (локальним) інтерполяційним многочленом або многочленом, який отримано за МНК із деякою похибкою. Під похідною від такої таблично-заданої функції розуміють похідну від її інтерполянти. Розроблено метод чисельного диференціювання таблично-заданих функцій, сутність якого зводиться до добутку вектора-рядка Тейлора n-го степеня на матрицю k-го порядку його диференціювання (k £ n) і на вектор-стовпець коефіцієнтів відповідної інтерполянти. Наведено деякі постановки задач чисельного диференціювання таблично-заданих функцій з використанням многочлена Тейлора n-го степеня, відповідні алгоритми їх розв’язання та конкретні приклади реалізації. Встановлено, що для обчислення похідної k-го порядку від таблично-заданої функції за прийнятим значенням незалежної змінної потрібно виконати такі дії: за даними таблиці сформувати матричне рівняння, розв’язати його та отримати значення коефіцієнтів інтерполянти; підставити у відповідний матричний вираз коефіцієнти інтерполянти та значення незалежної змінної та виконати дії множення матриць, вказані у виразі. Здійснено перевірку правильності виконання розрахунків із використанням відповідних центральних різницевих формул. Встановлено, що обчислені похідні k-го порядку з використанням формул центральних скінченних різниць практично збігаються зі значеннями, отриманими за допомогою інтерполяційного многочлена Тейлора n-го степеня, тобто значення похідних обчислено правильно. | |
dc.description.abstract | A methodology has been developed for numerically differentiating table-given functions using a Taylor polynomial of degree n, which enables the computation of k-th order derivatives (k £ n) at any point between arbitrarily located interpolation nodes in one, two, or multiple independent variables. Recent research and publications have been analysed, allowing for the assessment of the task complexity of computing derivatives of a function based on the values of independent variables within a certain interval of a table-given function. The formulation of the problem of numerical differentiation of periodic table-given functions using the Taylor polynomial of the nth order from one, two, and multiple independent variables is described. It is established that any tabulated function should be initially smoothed by some function whose analytical expression is a global (local) interpolating polynomial or a polynomial obtained by least squares approximation with some error. The derivative of such a table-given function is understood as the derivative of its interpolant. A method of numerical differentiation of table-given functions is developed, the essence of which is reduced to the product of the Taylor row vector of the n-th degree by the matrix of the k-th order of its differentiation (k £ n) and on the column vector of the coefficients of the corresponding interpolant. Some problem formulations of numerical differentiation of table-given functions using Taylor polynomials of degree n, corresponding solution algorithms, and specific implementation examples are provided. It has been established that to compute the k-th order derivative of a table-given function at a given value of the independent variable, the following steps need to be performed: based on the given table data, form a matrix equation, solve it to obtain the coefficients of the interpolant; substitute into the corresponding matrix expression the obtained interpolant coefficients and the independent variable value, and perform the matrix multiplication operations specified in the expression. The verification of the accuracy of the calculations using the appropriate central difference formulas was made. It was established that the calculated derivatives of the k-th order using the formulas of central finite differences practically coincide with the values obtained using the Taylor polynomial interpolation of the n-th order, that is, the values of the derivatives are calculated correctly. | |
dc.format.extent | 25-41 | |
dc.format.pages | 17 | |
dc.identifier.citation | Грицюк Ю. І. Чисельне диференціювання табличних функцій у довільно розташованих вузлах інтерполяції / Ю. І. Грицюк, Р. Б. Тушницький // Український журнал інформаційних технологій. — Львів : Видавництво Львівської політехніки, 2023. — Том 5. — № 1. — С. 25–41. | |
dc.identifier.citationen | Hrytsiuk Yu. I. Numerical differentiation of table-given functions at arbitrarily located interpolation nodes / Yu. I. Hrytsiuk, R. B. Tushnytskyy // Ukrainian Journal of Information Technology. — Lviv : Lviv Politechnic Publishing House, 2023. — Vol 5. — No 1. — P. 25–41. | |
dc.identifier.doi | doi.org/10.23939/ujit2023.01.025 | |
dc.identifier.issn | 2707-1898 | |
dc.identifier.uri | https://ena.lpnu.ua/handle/ntb/61566 | |
dc.language.iso | uk | |
dc.publisher | Видавництво Львівської політехніки | |
dc.publisher | Lviv Politechnic Publishing House | |
dc.relation.ispartof | Український журнал інформаційних технологій, 1 (5), 2023 | |
dc.relation.ispartof | Ukrainian Journal of Information Technology, 1 (5), 2023 | |
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dc.rights.holder | © Національний університет “Львівська політехніка”, 2023 | |
dc.subject | зашумлені дані | |
dc.subject | згладжування функцій | |
dc.subject | інтерполяція табличних фунакції | |
dc.subject | формули центральних скінченних різниць | |
dc.subject | обчислення похідних | |
dc.subject | taylor polynomial | |
dc.subject | smoothing functions | |
dc.subject | interpolation of tabular functions | |
dc.subject | formulas of central finite differences | |
dc.subject | calculation of derivatives | |
dc.title | Чисельне диференціювання табличних функцій у довільно розташованих вузлах інтерполяції | |
dc.title.alternative | Numerical differentiation of table-given functions at arbitrarily located interpolation nodes | |
dc.type | Article |
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