Open Library - открытая библиотека учебной информации

Открытая библиотека для школьников и студентов. Лекции, конспекты и учебные материалы по всем научным направлениям.


Образование Chapter 04.06
просмотров - 83


After reading this chapter, you should be able to:

1. Solve a set of simultaneous linear equations using Naïve Gauss Elimination,

  1. Learn the pitfalls of Naïve Gauss Elimination Method,
  2. Understand the effect of round off error on a solving set of linear equation by Naïve Gauss Elimination Method,
  3. Learn how to modify Naïve Gauss Elimination method to Gaussian Elimination with Partial Pivoting Method to avoid pitfalls of the former method,
  4. Find the determinant of a square matrix using Gaussian Elimination,
  5. Understand the relationship between determinant of co-efficient matrix and the solution of simultaneous linear equations.

How are a set of equations solved numerically?

One of the most popular techniques for solving simultaneous linear equations is the Gaussianelimination method. The approach is designed to solve a general set of n equations and n unknowns

. .

. .

. .

Gaussianelimination consists of two steps

1. Forward Elimination of Unknowns: In this step, the unknown is eliminated in each equation starting with the first equation. This way, the equations are “reduced” to one equation and one unknown in each equation.

2. Back Substitution: In this step, starting from the last equation, each of the unknowns is found.

Forward Elimination of Unknowns:

In the first step of forward elimination, the first unknown, x1 is eliminated from all rows below the first row. The first equation is selected as the pivot equation to eliminate x1. So, to eliminate x1 in the second equation, one divides the first equation by a11 (hence called the pivot element) and then multiply it by a21. That is, same as multiplying the first equation by a21/ a11 to give

Now, this equation can be subtracted from the second equation to give



This procedure of eliminating , is now repeated for the third equation to the nth equation to reduce the set of equations as

. . .

. . .

. . .

This is the end of the first step of forward elimination. Now for the second step of forward elimination, we start with the second equation as the pivot equation and as the pivot element. So, to eliminate x2 in the third equation, one divides the second equation by (the pivot element) and then multiply it by . That is, same as multiplying the second equation by and subtracting from the third equation. This makes the coefficient of x2 zero in the third equation. The same procedure is now repeated for the fourth equation till the nth equation to give

. .

. .

. .

The next steps of forward elimination are conducted by using the third equation as a pivot equation and so on. That is, there will be a total of (n-1) steps of forward elimination. At the end of (n-1) steps of forward elimination, we get a set of equations that look like

. .

. .

. .

Back Substitution:

Now the equations are solved starting from the last equation as it has only one unknown.

Then the second last equation, that is the (n-1)th equation, has two unknowns - xn and xn-1, but xn is already known. This reduces the (n-1)th equation also to one unknown. Back substitution hence can be represented for all equations by the formula

for i = n – 1, n – 2,…,


Читайте также

  • - Using the words from task I try to make up the game described in the chapter.

    TASKS CAMP LAURENCE CHAPTER 12 Translate the following passage into English. Compare it with the original. Dwell upon the results of the experiment. - Я поняла, Марми! - воскликнула Мег. - Ведь ты нарочно ушла из дома, чтобы посмотреть, как мы управимся без тебя и Ханны. - Ну да! Я хотела, чтобы вы поняли, насколько уют в нашем... [читать подробенее]

  • - Reread the description of the girls in the chapter and present each of them

    Explain the words in bold type. Find in the text and translate the following passages: 1. “We shouldn't enjoy ourselves half so much as we do now.” - So Meg went down, wearing an injured look, and wasn't at all agreeable at breakfast time. 2. Jo gave her sister an encouraging pat on the shoulder… - Other friends told the Marches that they had lost all chance of being remembered in the rich old lady's will, but the unworldly Marches only said… 3. The old lady wouldn't speak to... [читать подробенее]

  • - Pick up some more examples from Chapters 17 and 18 to prove that critical situations change us all out of recognition and help to distinguish between real and false values.

    Tasks Find in the text and translate into Russian the following words and expressions: to do one’s best to devote oneself to… to feel very anxious to keep house to miss smb. irksome to keep the watch despondently as the winter night began to wane. Recall the situations from the story suggested by the following sentences: Meg stayed at home, lest she should infect the Kings. … Beth was very patient, and bore her pain uncomplainingly as long as she could... [читать подробенее]

  • - Chapter III

    The Financier, by Theodore Dreiser It was in his thirteenth year that young Cowperwood entered into his first business venture. Walking along Front Street one day, a street of importing and wholesale establishments, he saw an auctioneer’s flag hanging out before a wholesale grocery and from the interior came the auctioneer’s voice: “What am I bid for this exceptional lot of Java coffee, twenty-two bags all told, which is now selling in the market for seven dollars and thirty-two... [читать подробенее]


    І Incombination of sentences into larger units we may observe two different types of grammatical relationship based upon relative position and interaction of sentences. These are co-ordination and subordination. This classification remains the prevalent scheme of the structural classification of sentences in the grammars of all types in various languages. A very important syntactic concept developed along with this classification is the concept of syndeton and asyndeton. Sentences... [читать подробенее]

  • - Chapter VIII THE ADVERB

    Adverbs make up a rather complicated group of words varying widely in form and distribution. Considered in their morphemic structure, adverbs may be classified in eight groups. 1—2. The two largest groups are those formed from derived and base adjectives by adding the suffix -ly, e. g.: hopefully, physically, strangely, falsely, etc. 3. The third group consists of those that are formed by means of the derivational prefix -a (phonemically [э]) to nouns, adjectives or verbs. Of about... [читать подробенее]

  • - Chapter VII THE PRONOUN

    PERSONAL PRONOUNS In the category of person English makes distinction between three classes of personal pronouns denoting respectively the person (s) speaking (first person); the person (s) spoken to (second person) another person(s) or thing(s) — third person. Person distinctions are naturally closely related to the category of number. There is no formal distinction of persons in plural, e. g.: we speak, you speak, they speak. There is no distinction of number in the 1st and 2nd persons... [читать подробенее]

  • - Chapter V THE VERB

    Revision Material 1. Comment on analytical and inflected form of comparison. 2. Give comments on the classification of adjectives in terms of meaning. 3. Comment on the distinction between base adjectives and derived adjectives. 4. Illustrate the statement that relative adjectives can develop qualitative meanings. Give examples of such metaphoric extension. 5. Comment on the noticeable change going on in present-day in the formation of the comparative and superlative of dissyllabic... [читать подробенее]

  • - Chapter IV THE ADJECTIVE

    An adjectiveis a word which expresses the attributes of substances (good, young, easy, soft, loud, hard, wooden, flaxen). As a class of lexical words adjectives are identified by their ability to fill the position between noun-determiner and noun and the position after a copula-verb and a qualifier. Considered in meaning, adjectives fall into two large groups: a) qualitative adjectives, b) relative adjectives. Qualitative adjectivesdenote qualities of size, shape, colour, etc. which an... [читать подробенее]

  • - Chapter HI THE NOUN

    The lexico-grammatical meaning of the noun is denoting "substance" In Modern English the marked formal characteristics of nouns are as follows: nouns have inflections for number and case, they may be associated with the definite or indefinite article. There is no grammatical gender in Modern English 1. The noun does not possess any special gender forms, neither does the accompanying adjective, pronoun or article indicate any gender agreement with the head noun. Unlike many... [читать подробенее]