Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 32

transform the originalequations into equivalent equations (equationsthathave the
same solution)thatcan be solvedmoreeasily. The transformationiscarried out by
applying the three operationslistedbelow. These so-called elementary operations do
not change the solution, but theymay affect the determinant of the coefficient matrix
as indicated in parentheses.

Exchanging twoequations (changes sign of | A | ).

| |

Multiplying an equationbyanonzeroconstant(multiplies A by the same

constant).

Multiplying an equationbyanonzeroconstant and then subtracting itfroman-
other equation (leaves

| A | unchanged).

and thenre-
peatedlyrefine the solutionuntil a certain convergence criterionis reached. Iterative
methods are generally less efficientthan their direct counterparts due to the large
number of iterations required. But theydohavesignificantcomputational advan-
tages if the coefficient matrix is very large and sparselypopulated (most coefficients
arezero).

Iterative, or indirect methods ,start with aguess of the solution x ,

Overview of Direct Methods

Table 2.1 lists three popular directmethods, each of which uses elementary operations
to produce its own finalform of easy-to-solveequations.

Method Initialform Finalform

Gauss elimination Ax = b Ux = c

LU decomposition Ax = b LUx = b

Gauss-Jordan elimination Ax = b Ix = c

Table 2.1

In the abovetable U represents an upper triangular matrix, L is a lower triangular
matrix and I denotes the identitymatrix. A square matrix iscalled triangular if it
containsonly zero elements on oneside of the leading diagonal. Thus a3

× 3 upper

triangular matrix has the form

⎡
⎣

⎤
⎦

U 11 U 12 U 13
0 U 22 U 23
00 U 33

U =