sor迭代(算法分析和数值算例)

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1、SOR迭代基本思想Gauss-Seidel迭代的结果作为中间值，记为。SOR方法是将与上次计算的结果做加权平均作为最后结果。迭代格式为：或者算法：1．2．当时，，结果仍然存储在中。迭代次数3．计算误差（真解已知）4．如果，则已达到精确度要求，否则继续第2步数值结果，用Gauss消去法求的其真解为依次取，数值结果见下表0.251.0625-0.4843750.25751.09630625-0.4902011406250.2751.175625-0.5017031250.5156251.0078125-0.4980468750.5320738593751.00789303757813

2、-0.4982615086048830.5707968751.00143828125-0.499434160156250.5019531251.0009765625-0.4997558593750.5010702413951171.00048645756614-0.4999268919185720.493315839843750.998173633789063-0.5005588346923830.5002441406251.0001220703125-0.4999694824218750.5000931555814281.0000282191662-0.499994926807

3、1460.5001661653076171.00007465254028-0.4999235870821840.5000305175781251.00001525878906-0.4999961853027340.5000044717678541.0000016112524-0.4999997372982940.5000039129178161.00001462435077-0.500003619595320.5000038146972661.00000190734863-0.4999995231628420.500003630404680.999998540537497-0.5

4、00000039392656迭代次数656总结从实验结果可以看出，当取松弛参数为1.03时只需五步就能达到所需精度。附录（M文件）function[t,x]=successiive_over_Rellaxatiion(A,b,x0,w,rx)n=length(A);x=x0;%%x0为迭代初值e=norm(rx-x0,inf);%%rx为真解,e为误差t=0;%%t为迭代次数whilee>5*10^(-6)fori=1:ntemp=0;forj=1:ntemp=temp+A(i,j)*x(j,1);endx(i,1)=x(i,1)+w*(b(i,1)-temp)/A(i,i);e

5、nde=norm(rx-x,inf);t=t+1;xend