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Метод простої ітерації Розглянемо метод простої ітерації. Нехай задано систему лінійних рівнянь

[pic] (1)

[pic] (2)

Нехай діагональні елементи [pic] (i=1, 2, …,n) матриці А відмінні від нуля. Тоді, розв’язавши перше рівняння системи (1) відносно[pic], а друге відносно [pic] і т.д., дістанемо систему

[pic] (3)

де [pic], [pic], (4)

Ввівши до розгляду матриці [pic], [pic], (5)

Систему (3) запишемо у вигляді [pic]. (6)

Систему (6) називають системою нормального виду.

Розв’яжемо її методом послідовних наближень. За початкове наближення візьмемо, наприклад, стовпець вільних членів, тобто [pic].

Тоді послідовно знаходимо

[pic] (k = 0, 1, 2, …) (7)

або в розгорнутому вигляді

[pic], [pic], i = 1,2,…n; k = 0, 1, 2… (8)

Якщо послідовність наближень [pic], [pic], …,[pic], … має границю [pic], то ця границя і буде розв’язком системи (6).

Справді, перейшовши до границі, коли [pic], у рівності (7), маємо

[pic][pic][pic][pic],

або [pic].

Таким чином, вектор [pic] є розв’язком системи (6).

Метод послідовних наближень, який визначається формулами (7) або (8), називається методом простої ітерації або просто методом ітерації.

Праву частину рівності (6) можна розглядати як задання деякого оператора [pic], який відображає вектор [pic] n-вимірного векторного простору у вектор y цього самого простору. При цьому

[pic], i = 1,2,…n.

Тому знаходження розв’язку системи (6) зводиться до відшукання нерухомої точки відображення [pic], яке визначається формулою

[pic]. (9)

Якщо відображення [pic] є стискуючим, то розв’язок рівняння [pic], тобто рівняння (6), можна знайти методом послідовних наближень за формулами (7), починаючи з довільного вектора [pic].

Теорема 1. Якщо матриця [pic] системи рівнянь (6) задовольняє одну з умов [pic], [pic] або [pic], то система (6) має єдиний розв’язок [pic], який можна дістати як границю послідовності [pic], побудованою за формулою (7), починаючи з початкового довільного вектора [pic].

Метод простої ітерації слід закінчити, якщо стане справедливою нерівність [pic], (10)

Теорема 2. Якщо елементи матриці А задовольняють одну з умов

[pic], [pic], i = 1,2,…n;

або [pic], то система рівнянь (1) має єдиний розв’язок [pic], який можна дістати як границю послідовності [pic], побудованої за формулою [pic], i = 1,2,…n; k = 0, 1, 2…, починаючи з довільного початкового наближення [pic].

Метод ітерації легко реалізовується на ЕОМ. Алгоритм розв’язування системи виду (2) передбачає:

Обчислення величини [pic].

Перевірку умови [pic]. Якщо ця умова не виконується, то процес обчислень закінчується і видається повідомлення про те, що метод застосувати не можна.

Обчислення допустимої похибки[pic].

Вибір початкового наближення [pic] ([pic] = 1, 2, …, [pic]).

Обчислення наступного наближення [pic] ([pic] = 1, 2, …, [pic]) через попередні [pic] ([pic] = 1, 2, …, [pic]).

Перевірку умови [pic]. Якщо ця умова виконується то процес ітерацій завершується, інакше переходять до виконання п.5.

Завдання

Розв’язати систему лінійних рівнянь методом ітерації з точністю до 0,001.

№ 1 [pic] № 2 [pic] № 3 [pic]№4 № 4 [pic] № 5 [pic] № 6 [pic] № 7 [pic] № 8 [pic] № 9 [pic] № 10 [pic] № 11 [pic] № 12 [pic] № 13 [pic] № 14 [pic] № 15 [pic] № 16 [pic]

№ 17 [pic] № 18 [pic] № 19 [pic] № 20 [pic] № 21 [pic] № 22 [pic] № 23 [pic] № 24 [pic] № 25 [pic] № 26 [pic] № 27 [pic] № 28 [pic] № 29 [pic] № 30 [pic] № 31 [pic] № 32 [pic]

№ 33 [pic] № 34 [pic]

№ 35 [pic] № 36 [pic]

Метод Зейделя

Метод Зейделя – деяка модифікація методу простої ітерації. У методі простої ітерації при обчисленні компонентів [pic] вектора-стовпця [pic] на[pic]-му кроці використовується значення [pic] вектора-стовпця [pic], обчисленого в попередньому кроці. Метод Зейделя відрізняється від методу простої ітерації тільки тим, що при обчисленні [pic]-го наближення компоненти [pic] враховуються значення [pic], обчислені на цьому ж кроці.

Формули для знаходження послідовних наближень мають вигляд

[pic],

.........................................................

[pic]+[pic],

.........................................................

[pic]+[pic], [pic] = 0, 1, 2,…

Зазначимо, що достатні умови збіжності для методу простої ітерації справедливі і для методу Зейделя.

Програма методу Зейделя відрізняється від методу простої ітерації тільки фрагментом обчислення наступних наближень. У програмі методу простої ітерації необхідно одночасно зберігати усі попередні [pic] й наступні [pic] наближення, оскільки найбільшу різницю [pic] можна знайти тільки після закінчення кроку ітерації. Користуючись методом Зейделя, немає потреби зберігати всі знайдені наближення [pic] [pic], оскільки вони відразу використовуються для знаходження [pic]. Тому у програмі методу Зейделя змінну [pic] позначимо змінною [pic], яка зберігає обчислене значення [pic] до того часу, поки воно не присвоїться змінній [pic]

Завдання

Розв’язати систему лінійних рівнянь методом ітерації та методом Зейделя з точністю до 0,001.

№1 [pic] №2 [pic]
№3 [pic] №4 [pic]
№5 [pic] №6 [pic]
№7 [pic] №8 [pic]
№9 [pic] №10 [pic]
№11 [pic] №12 [pic]
№13 [pic] №14 [pic]
№15 [pic] №16 [pic]
№17 [pic] №18 [pic]
№19 [pic] №20 [pic]
№21 [pic] №22 [pic]
№23 [pic] №24 [pic]
№25 [pic] №26 [pic]
№27 [pic] №28 [pic]
№29 [pic] №30 [pic]
№31 [pic] №32 [pic]
№33 [pic] №34 [pic]
№35 [pic] №36 [pic]

Зразок виконання завдання [pic]

Зведемо систему до вигляду, в якому елементи головної діагоналі перевищували б інші елементи рядків:

[pic]

[pic]

[pic]

Норма[pic] матриці, яка складається з коефіцієнтів при невідомих в правих частинах рівнянь, рівна [pic] отже, процес Зейделя збігається.

Розрахунки розташовуємо в таблиці:

|N |[pic] |[pic] |[pic] |N |[pic] |[pic] |[pic] |
|0 |0,19 |0,97 |-0,14 |5 |0,2467 |1,1138 |-0,2237 |
|1 |0,2207 |1,0703 |-0,1915 |6 |0,2472 |1,1143 |-0,2241 |
|2 |0,2354 |1,0988 |-0,2118 |7 |0,2474 |1,1145 |-0,2243 |
|3 |0,2424 |1,1088 |-0,2196 |8 |0,2475 |1,1145 |-0,2243 |
|4 |0,2454 |1,1124 |-0,2226 | |

Відповідь: [pic][pic]0,248; [pic][pic]1,115; [pic][pic]-0,224.

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...Abdul Jamil., Mdm Goh Wan Chen, Ms Saratha Thevi Ramasamy, Ms Norzaireen Shamsul Kamar Synopsis: This course is designed for students who require the necessary skills for tertiary studies. Some basic grammatical concepts are taught and students are to apply them in their writing. Writing will focus on the development of coherent paragraphs. Reading skills will cover such strategies as scanning, skimming, main ideas, contextual clues and inferences. Learning Outcomes: Upon completion of this subject, student will be able to: 1. write summaries as well as process, comparison-contrast and cause-effect essays 2. apply basic grammatical concepts in writing 3. answer questions based on academic texts 4. give oral presentations Textbook: 1. Daise, D., Norloff, C., and Carne, P., (2011). Q: Skills for Success 4 : Reading and Writing Oxford University Press, UK 2. Paterson, K, and Wedge, R., (2013). Oxford Grammar for EAP. Oxford University Press, UK Recommended References: Cambridge International Dictionary of English (1997), Cambridge University Press, UK Mode of Assessment: [1] Class participation 5% [2] Quiz 1 15% [3] Quiz 2 10% [4] Oral Presentation 10% [5] Mid-Term Examination 20% [6] Final Examination 40% Syllabus – FDENG001 |Week |UNIT |Topics | |Week 1 ...

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