The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11


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1 The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd GussSiedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic University of Gz
2 Introduction Certin mtrices hve prticulr structures tht cn be eploited to develop efficient solution schemes A bnded mtri is squre mtri tht hs ll elements equl to zero, with the eception of bnd centered on the min digonl These mtrices typiclly occur in solution of differentil equtions The dimensions of bnded system cn be quntified by two prmeters: the bnd width BW nd hlfbndwidth HBW These two vlues re relted by BW=HBW+
3 Bnded mtri
4 4 Tridigonl Systems A tridigonl system hs bndwidth of : r r r r f e g f e g f e g f An efficient LU decomposition method, clled Thoms lgorithm, cn be used to solve such n eqution The lgorithm consists of three steps: decomposition, forwrd nd bck substitution, nd hs ll the dvntges of LU decomposition
5 Fig
6 Cholesky Decomposition This method is suitble for only symmetric systems where: nd A A ij ji A L * L T l l l l l l * l l l l l l T l [ L ] l l l l l 6
7 Cholesky Decomposition i l l ki ij kj j i ki ki l l ii ij l kj j ki k kk kk l ii kj j k l kk kk l kj j l for i,,, k l for i,,, k l l
8 Pseudocode for Cholesky s LU Decomposition lgorithm (cont d)
9 GussSiedel Itertive or pproimte methods provide n lterntive to the elimintion methods The Guss Seidel method is the most commonly used itertive method The system [A]{X}={B} is reshped by solving the first eqution for, the second eqution for, nd the third for, nd n th eqution for n We will limit ourselves to set of equtions
10 GussSiedel b b b b b b Now we cn strt the solution process by choosing guesses for the s A simple wy to obtin initil guesses is to ssume tht they re zero These zeros cn be substituted into eqution to clculte new =b /
11 GussSiedel New is substituted to clculte nd The procedure is repeted until the convergence criterion is stisfied: new old i i i, % s new i
12 Jcobi itertion Method An lterntive pproch, clled Jcobi itertion, utilizes somewht different technique This technique includes computing set of new s on the bsis of set of old s Thus, s the new vlues re generted, they re not immeditely used but re retined for the net itertion
13 GussSiedel The GussSeidel method The Jcobi itertion method
14 Convergence Criterion for GussSeidel Method The gusssiedel method is similr to the technique of fiedpoint itertion The GussSeidel method hs two fundmentl problems s ny itertive method: It is sometimes nonconvergent, nd If it converges, converges very slowly Sufficient conditions for convergence of two liner equtions, u(,y) nd v(,y) re: u u y v v y
15 Convergence Criterion for GussSeidel Method (cont d) Similrly, in cse of two simultneous equtions, the GussSeidel lgorithm cn be epressed s: b u (, ) b v (, ) u u v v
16 Convergence Criterion for GussSeidel Method (cont d) Substitution into convergence criterion of two liner equtions yield:, In other words, the bsolute vlues of the slopes must be less thn unity for convergence: Tht is, the digonl element must be greter thn the offdigonl element for ech row For n equtions n ii i j ji, j
17 GussSiedel Method Emple Guess,, = zero for the first guess Iter, (%), (%), (%)
18 Improvement of Convergence Using Reltion new new old i i i Where is weighting fctor tht is ssigned vlue between [, ] If = the method is unmodified If is between nd (under reltion) this is employed to mke non convergent system to converge If is between nd (over reltion) this is employed to ccelerte the convergence
19 GussSiedel Method Emple Rerrnge so tht the equtions re digonlly dominnt
20 GussSiedel Method Emple itertion unknown vlue mimum 5 % % 769 % % % % % 4% % % % 9%
21 GussSiedel Method Emple The sme computtion cn be developed with reltion where = new new old First itertion: i i i 8 Reltion yields: 8 8 () Reltion yields: (7) () Reltion yields: () (5) () 4 () ( 4857) () 77749
22 GussSiedel Method Emple Iter unknown vlue reltion mimum 5 % 7 88 % % % % % % 85% % % % 57% % % % 98%
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