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• W2183639498 abstract "ion of iterative methods In our procedures we have considered Ax = b and x = Bx + c as equivalent linear systems of equations, where B is the iteration matrix and xk+1 = Bxk + c. Some advantages of iterative methods over direct methods are: (i) Iterative methods are faster (depends on B, accuracy is required) (ii) Iterative methods require less memory (sparsity of A can be preserved). By a spars matrix we mean a matrix with most zero elements, i.e. a matrix that has very few non-zero elements. Questions: For a qualitative study of an iteration procedure the following questions are of vital interest; (Q1) For a given A, what is a good choice for B? (Q2) When does x → x? (Q3) What is the rate of convergence? The error at step k is ek = x (k)−x and that of step (k+1) is ek+1 = x(k+1)−x. Then we have ek+1 = x (k+1)−x = (Bx(k)+c)−(Bx+c) = B(x(k)−x) = Bek. Iterating, we have ek = Bek−1 = B ·Bek−2 = Bek−3 = . . . = Bek−k = B e0. 4.2. ITERATIVE METHODS 105 Thus we have shown that ek = B e0. Let now L =   0 . . . . . . 0 a21 0 . . . 0 . . . . . . . . . . . . an1 . . . an,n−1 0   , U =   0 a12 . . . a1n . . . . . . . . . . . . . . . . . . 0 an−1,n 0 . . . . . . 0  " @default.
• W2183639498 created "2016-06-24" @default.
• W2183639498 creator A5040028686 @default.
• W2183639498 date "2009-01-01" @default.
• W2183639498 modified "2023-09-26" @default.
• W2183639498 title "An Introduction to the Finite Element Method (FEM) for Differential Equations" @default.
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