Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 94

-2.5000
-2.0000
-1.5000
-1.0000
-0.5000
0.0000
0.5000
1.0000
1.5000
2.0000
2.5000
3.0000
3.5000
4.0000
4.5000
5.0000
numIter =
259
omega =
1.7055

The convergence is very slow, because the coefficient matrix lacks diagonal
dominance—substituting the elements of A in Eq. (2.30) produces an equalityrather
than the desiredinequality. If we weretochangeeach diagonalterm of the coefficient
matrix from2to 4, A wouldbe diagonallydominant and the solutionwould converge
in only22iterations.

EXAMPLE 2.18
Solve Example 2.17 with the conjugate gradient method, also using n = 20.

Solution For the given A , the components of the vector Av are

( Av ) 1 = 2 v 1 − v 2 + v n

( Av ) i =− v i − 1 + 2 v i − v i + 1 , i = 2 , 3 ,..., n − 1

( Av ) n =− v n − 1 + 2 v n + v 1

which areevaluatedbythe following function:

function Av = fex2 18(v)

% Computes the product A*v in Example 2.18

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