Solving Systems of Linear Equations

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  • Topic: System of linear equations, Linear algebra, Elementary algebra
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Solving systems of linear equations

7.1 Introduction
Let a system of linear equations of the following form:

a11 x1
a21 x1





a12 x2
a22 x2


ai1x1  ai 2 x2


am1 x1  am2 x2





a1n xn
a2 n x n





   ain xn


   amn xn



b1
b2


bi

 bm

(7.1)

be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m; k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In matrix notation, this system has the form:

Ax  b ,

(7.2)

where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the column vector of the free terms, bT  [b1 , b2 , ... , bm ] , x is the column vector of the unknowns, xT  [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.

It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real numbers satisfying (7.1) is referred to as the solution of the system, and the individual numbers, x1, x2, ..., xn, are roots of the system.

A system of linear equations is:


consistent - if it has at least one solution. At the same time it can be -



determined - if it has exactly one, unique solution,
undetermined - if it has infinitely many solutions;

inconsistent - if it does not have any solution.

The further considerations will be limited to most frequently met in technical problems non-homogeneous systems of equations, i.e., systems that satisfy:

m

b

2
i

 0.

(7.3)

i 1

Conditions for existing of solution of such systems give the following theorem:

Theorem 7.1 (Kronecker-Capelly)
The system of equations (7.1) is solvable if and only if, in notation (7.2):

rank(A) = rank(B) = r,

where B is the extended matrix created by inserting vector b as the n+1 column of matrix A. At the same time:


there exists exactly one, unique solution if r = n;



there are infinitely many solutions dependently on the number of parameters, e = n - r, if r < n;



2

the system is inconsistent if rank(A) < rank(B).

(7.4)

Common rank, r, of matrices A and B is referred to as the system order. Calculation of the solution of a non-homogenous system of linear equations involves:

1. Determining the system order, r, and respective non-zero minor, D, of matrix A.

2. Selecting equations, whose coefficients are present in minor D and rejecting other equations. In the selected equations r unknowns at coefficients creating the minor D are left on the left-hand side of the r equations, and the remaining n - r unknowns are considered as parameters and left on the right-hand side of the equations.

3. Solving the obtained system of r equations with r unknowns. The unknowns are linear functions of n - r parameters. The solution is called the general solution.

4. Assuming arbitrary values for the parameters. The solution obtained is called the particular solution.

The above procedure requires skills of solving n-th (or r-th, if r < n) order systems of linear equations. Some methods, both accurate and approximate, are presented in following sections.

3

7.2 Accurate methods
The term accurate methods (referred to also as direct methods) will be used for methods of solving systems of linear equations, which allow finding accurate solutions in a finite number of elementary arithmetical operations, where this number depends on the algorithms used and the system order.

7.2.1 Cramer formulae
A Cramer system, i.e., system of n equations in n unknowns, is given by (7.2). Provided the matrix, A, of coefficients is non-singular, i.e., det A  0 , there exists an inverse matrix A-1. In order to find the solution of the system, both sides of (7.2) are premultiplied by A-1, what gives:

x  A1b .

(7.5)

The inverse matrix is calculated from the following equation:

A 1 

AD
,
det A

where AD is the transposed matrix of...
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