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Введение ................................................................................................................. 1
ЧАСТЬ I. ЛЕКЦИИ ................................................................................................... 3
Лекция № 1. Теория погрешностей .................................................................... 5
1.1.1. Источники и классификация погрешностей ......................................................... 5
1.1.2. Абсолютная и относительная погрешности. Формы записи данных ................. 7
1.1.3. Вычислительная погрешность ............................................................................... 9
1.1.4. Понятие погрешности машинных вычислений .................................................. 10
Лекция № 2. Решение уравнений с одной переменной ................................ 15
1.2.1. Постановка задачи ................................................................................................ 15
1.2.2. Отделение корней ................................................................................................. 16
1.2.3. Метод половинного деления................................................................................ 17
1.2.4. Метод простой итерации...................................................................................... 20
1.2.5. Оценка погрешности метода простой итерации ................................................ 23
1.2.6. Преобразование уравнения к итерационному виду ........................................... 24
1.2.7. Решение уравнений методом простой итерации в пакете Mathcad .................. 24
Лекция № 3. Методы решения систем
линейных алгебраических уравнений ............................................................. 27
1.3.1. Общие сведения и основные определения ......................................................... 27
1.3.2. Метод Гаусса и его реализация в пакете Mathcad.............................................. 28
1.3.3. Вычисление определителей ................................................................................. 32
1.3.4. Решение систем линейных уравнений методом простой итерации ................. 34
1.3.5. Метод Зейделя....................................................................................................... 40
IV
Оглавление
Лекция № 4. Методы решения систем нелинейных уравнений.................. 44
1.4.1. Векторная запись нелинейных систем. Метод простых итераций ................... 44
1.4.2. Метод Ньютона решения систем нелинейных уравнений ................................ 48
1.4.3. Решение нелинейных систем методами спуска ................................................. 52
1.4.4. Модифицированный метод Ньютона .................................................................. 67
Лекция № 5. Интерполирование функций....................................................... 73
1.5.1. Постановка задачи ................................................................................................ 73
1.5.2. Интерполяционный полином Лагранжа ............................................................. 76
1.5.3. Интерполяционный полином Ньютона для равноотстоящих узлов ................. 78
1.5.4. Погрешность интерполяции ................................................................................ 82
1.5.5. Сплайн-интерполяция .......................................................................................... 83
Лекция № 6. Численное дифференцирование и интегрирование ............. 90
1.6.1. Дифференцирование функций, заданных аналитически ................................... 90
1.6.2. Особенности задачи численного дифференцирования функций,
заданных таблично ......................................................................................................... 93
1.6.3. Интегрирование функций, заданных аналитически
(формула прямоугольников, формула трапеций, формула Симпсона) ...................... 94
1.6.4. Погрешность численного интегрирования ......................................................... 98
1.6.5. Вычисление интегралов методом Монте-Карло .............................................. 101
Лекция № 7. Методы обработки экспериментальных данных ................. 104
1.7.1. Метод наименьших квадратов ........................................................................... 104
1.7.2. Нахождение приближающей функции в виде линейной функции
и квадратичного трехчлена .......................................................................................... 108
1.7.3. Нахождение приближающей функции в виде элементарных функций ......... 112
1.7.4. Аппроксимация линейной комбинацией функций .......................................... 115
1.7.5. Аппроксимация функцией произвольного вида............................................... 117
Лекция № 8. Преобразование Фурье ............................................................. 120
1.8.1. Разложение периодических функций в ряд Фурье .......................................... 120
1.8.2. Эффект Гиббса .................................................................................................... 123
1.8.3. Спектральный анализ дискретных функций конечной длительности............ 128
1.8.4. Быстрое преобразование Фурье ........................................................................ 129
Лекция № 9. Численные методы решения
обыкновенных дифференциальных уравнений ......................................... 134
1.9.1. Постановка задачи .............................................................................................. 134
1.9.2. Метод Пикара...................................................................................................... 137
Оглавление
1.9.3.
1.9.4.
V
Метод Эйлера...................................................................................................... 139
Метод Рунге—Кутты.......................................................................................... 144
Лекция № 10. Численные методы решения
дифференциальных уравнений в частных производных ........................ 150
1.10.1.
1.10.2.
1.10.3.
1.10.4.
1.10.5.
1.10.6.
Примеры уравнений ......................................................................................... 150
Типы уравнений ................................................................................................ 151
Численные методы решения эллиптических уравнений ............................... 153
Явные разностные схемы ................................................................................. 158
Неявная разностная схема для уравнения параболического типа ................ 163
Решение уравнений методом Монте-Карло ................................................... 168
Лекция № 11. Численные методы решения интегральных уравнений ... 174
1.11.1.
1.11.2.
1.11.3.
Общие сведения об интегральных уравнениях .............................................. 174
Квадратурный метод решения интегральных уравнений Фредгольма ........ 180
Квадратурный метод решения интегральных уравнений Вольтерры .......... 184
ЧАСТЬ II. ЛАБОРАТОРНЫЕ РАБОТЫ ............................................................. 193
Лабораторная работа № 1. Теория приближенных вычислений ............. 195
2.1.1.
2.1.2.
2.1.3.
2.1.4.
2.1.5.
2.1.6.
2.1.7.
2.1.8.
Абсолютная и относительная погрешности ..................................................... 195
Погрешность округленного числа..................................................................... 198
Погрешности арифметических действий.......................................................... 200
Погрешности элементарных функций .............................................................. 202
Способ границ ..................................................................................................... 205
Обратная задача теории погрешностей ............................................................ 210
Вопросы по теме ................................................................................................. 212
Задания к лабораторной работе № 1 ................................................................. 213
Лабораторная работа № 2. Численные методы решения
скалярных уравнений ....................................................................................... 220
2.2.1.
2.2.2.
2.2.3.
2.2.4.
2.2.5.
Метод хорд .......................................................................................................... 220
Метод касательных ............................................................................................. 223
Метод простой итерации.................................................................................... 226
Вопросы по теме ................................................................................................. 230
Задания к лабораторной работе № 2 ................................................................. 231
Лабораторная работа № 3. Численные методы решения
систем линейных уравнений ........................................................................... 234
2.3.1.
2.3.2.
Метод Гаусса—Жордана.................................................................................... 235
Метод простой итерации.................................................................................... 236
Оглавление
VI
2.3.3.
2.3.4.
2.3.5.
Метод Зейделя..................................................................................................... 241
Вопросы по теме ................................................................................................. 243
Задание к лабораторной работе № 3 ................................................................. 244
Лабораторная работа № 4. Численные методы решения
систем нелинейных уравнений....................................................................... 248
2.4.1.
2.4.2.
Метод Ньютона................................................................................................... 248
Задание к лабораторной работе № 4 ................................................................. 252
Лабораторная работа № 5. Приближение значения
таблично заданной функции в точке
с помощью интерполяционных многочленов ............................................. 255
2.5.1.
2.5.2.
2.5.3.
2.5.4.
2.5.5.
Интерполяционный полином Лагранжа ........................................................... 256
Интерполяционные полиномы Ньютона .......................................................... 259
Интерполирование сплайнами........................................................................... 262
Вопросы по теме ................................................................................................. 266
Задание к лабораторной работе № 5 ................................................................. 266
Лабораторная работа № 6. Обратное интерполирование ......................... 273
2.6.1.
2.6.2.
2.6.3.
2.6.4.
Обратное интерполирование с использованием формул Ньютона ................ 273
Обратное интерполирование с использованием формулы Лагранжа ............ 277
Вопросы по теме ................................................................................................. 278
Задание к лабораторной работе № 6 ................................................................. 278
Лабораторная работа № 7. Дискретный вариант
среднеквадратичных приближений. Метод наименьших квадратов ...... 282
2.7.1.
2.7.2.
2.7.3.
2.7.4.
2.7.5.
2.7.6.
2.7.7.
2.7.8.
Линейная функция .............................................................................................. 282
Квадратичная функция ....................................................................................... 285
Степенная функция............................................................................................. 287
Показательная функция...................................................................................... 288
Логарифмическая функция ................................................................................ 290
Гиперболическая функция ................................................................................. 291
Вопросы по теме ................................................................................................. 293
Задание для лабораторной работы № 7 ............................................................ 293
Лабораторная работа № 8. Численное дифференцирование................... 298
2.8.1. Дифференцирование с помощью интерполяционной
формулы Лагранжа ....................................................................................................... 298
2.8.2. Вопросы по теме ................................................................................................. 302
2.8.3. Задание к лабораторной работе № 8 ................................................................. 303
Оглавление
VII
Лабораторная работа № 9. Численное интегрирование ............................ 304
2.9.1.
2.9.2.
2.9.3.
2.9.4.
2.9.5.
2.9.6.
Метод прямоугольников .................................................................................... 305
Метод Симпсона ................................................................................................. 306
Метод трапеций .................................................................................................. 307
Метод Монте-Карло ........................................................................................... 308
Вопросы по теме ................................................................................................. 309
Задание к лабораторной работе № 9 ................................................................. 310
Лабораторная работа № 10. Решение обыкновенных
дифференциальных уравнений ..................................................................... 314
2.10.1.
2.10.2.
2.10.3.
2.10.4.
2.10.5.
2.10.6.
Метод Пикара.................................................................................................... 314
Метод Эйлера и его модификации .................................................................. 317
Метод Рунге—Кутты........................................................................................ 320
Метод Адамса ................................................................................................... 322
Вопросы по теме ............................................................................................... 324
Задание к лабораторной работе № 10 ............................................................. 325
Лабораторная работа № 11. Численное решение
дифференциальных уравнений в частных производных ........................ 329
2.11.1.
2.11.2.
2.11.3.
2.11.4.
2.11.5.
Решение задачи Дирихле для уравнения Лапласа методом сеток ................ 329
Решение уравнения теплопроводности методом сеток ................................. 335
Решение уравнения колебания струны методом сеток.................................. 338
Вопросы по теме ............................................................................................... 341
Задание к лабораторной работе № 11 ............................................................. 342
Лабораторная работа № 12. Численное решение
интегральных уравнений ................................................................................. 352
2.12.1.
Задание к лабораторной работе № 12 ............................................................. 352
ПРИЛОЖЕНИЯ .................................................................................................... 355
Приложение 1. Основные приемы работы с пакетом Mathcad ................ 357
П1.1. Основы работы с Mathcad .................................................................................. 357
П1.2. Справочная информация в Mathcad .................................................................. 367
П1.3. Основы программирования................................................................................ 375
П1.4. Графические возможности Mathcad .................................................................. 380
Приложение 2. Аналитические методы решения
дифференциальных уравнений в частных производных ........................ 394
П2.1. Точные методы решения дифференциальных
уравнений в частных производных ............................................................................. 394
VIII
Оглавление
П2.2. Приближенные методы решения дифференциальных уравнений
в частных производных ................................................................................................ 428
П2.3. Метод обратной задачи рассеяния .................................................................... 439
Приложение 3. Описание компакт-диска....................................................... 445
Литература .......................................................................................................... 447
Предметный указатель ..................................................................................... 449
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