06_Jacobi_Gauss_Seidel.ppt - Linear Systems u2013...

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Linear Systems – Iterative methods1. Jacobi Method2. Gauss-Siedel Method1
Iterative MethodsIterative methods can be expressed in the general form: x(k)=F(x(k-1))where ss.t. F(s)=sis called a Fixed PointHopefully: x(k)s (solution of my problem)Will it converge? How rapidly?2
Iterative MethodsStationary: x(k+1)=Gx(k)+cwhere Gand cdo not depend on iteration count (k)Non Stationary:x(k+1)=x(k)+akp(k)where computation involves information that change at each iteration
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Iterative – StationaryJacobiIn the i-thequation solve for the value of xiwhile assuming the other entries of xremain fixed:In matrix terms the method becomes:where D, -L and -U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of MiiijjijiiNjijijmxmbxbxm1iiijkjijikimxmbx)1()(bDxULDxkk111)(4
Iterative – StationaryGauss-SeidelLike Jacobi, but now assume that previously computed results are used as soon as they are available:In matrix terms the method becomes:where D, -L and -U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of MiiijjijiiNjijijmxmbxbxm1iiijkjijijkjijikimxmxmbx)1()()()(11)(bUxLDxkk
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Successive Overrelaxation (SOR)Devised by extrapolation applied to Gauss-Seidel in the form of weighted average:In matrix terms the method becomes:where D, -L and -U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of Mwis chosen to increase convergenceiiijkjijijkjijikimxmxmbx)1()()(bwLDwxDwwUwLDxkk111)()())1(()1()()()1(kikikixwxwx6
7Jacobi iterationnnnnnnnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111002010nxxxx)(101021211111nnxaxabax11111ijnijkjijkjijiiikixaxabax)(10110220111nnnnnnnnnxaxaxabax)(1020323012122212nnxaxaxabax
8Gauss-Seidel (GS) iterationnnnnnnnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111002010nxxxx111111ijnijkjijkjijiiikixaxabax)(101021211111nnxaxabax)(11111221111nnnnnnnnnxaxaxabax)(1020323112122212nnxaxaxabaxUse the latestupdate
Gauss-Seidel MethodAn iterativemethod.Basic Procedure:-Algebraically solve each linear equation for xi -Assume an initial guess solution array-Solve for each xiand repeat-Use absolute relative approximate error after each iteration to check if error is within a pre-specified tolerance.9
Gauss-Seidel MethodWhy?The Gauss-Seidel Method allows the user to control round-off error.Elimination methods such as Gaussian Elimination and LU

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