NumericalComputing-Chapter3

# NumericalComputing-Chapter3 - Numerical Computing chapter...

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Numerical Computing chapter 3- Iterative methods for Linear Systems Dr. Nguyen V.M. Man Email: Faculty of Computer Science and Engineering HCMUT Vietnam April 1, 2010

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The structure of the chapter ——————————————————————————— I Introduction I Conceptual Foundations I Matrix-Splitting I Iterative methods for Linear Systems
Introduction ——————————————————————————— Basic observation : the numerical solutions of diﬃcult nonlinear problems are nearly always traced back to those of linear systems. Secondly, many applications lead to large band matrices in which the bands are sparse; the iterative schemes oﬀer greater eﬃciency. The idea behind iterative methods for solving a system A u = b ... (1) is: generate a sequence { u ( m ) } m of approximate solution vectors ensuring that they converges to u , the solution of (1).

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Conceptual Foundations ——————————————————————————— Suppose an iterative scheme for A u = b produces a sequence { u ( m ) } of iterates. The aim : determine conditions under which the errors e ( m ) := u ( m ) - u converges to 0 when m goes to . An approach is to associate with the iterative scheme a matrix G that relates errors at successive iterations in the following formula: e ( m + 1) = G e ( m ) = G 2 e ( m - 1) ··· = G m +1 e (0)
Conceptual Foundations ——————————————————————————— I A matrix G C n × n is convergent if lim m →∞ G m = 0 I If we could ﬁnd a matrix G to be convergent, then we are done, since u ( m ) u e ( m ) 0 G m 0 (of course, the initial error e (0) := u (0) - u depending on an initial guess u (0) which must be nonzero) What techniques can be employed?

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Matrix-Splitting Techniques ——————————————————————————— Motivation : one good tactic for solving a linear system A u = b involves splitting A as A = M + ( A - M ) . Put N = M - A then A = M - N , and A u = b ( M - N ) u = b M u = N u + b If the system M z = f is presumably easy to solve, then we write u = M - 1 N u + M - 1 b ⇐⇒ u = M
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NumericalComputing-Chapter3 - Numerical Computing chapter...

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