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矩阵LU分解的高斯消元法
A=[1,-1,1,-4;5,-4,3,12;2,1,1,11;2,-1,7,-1] L=eye(length(A)) %开始消元过程 for k=1:(length(A)) a=A(k,k) for i=k+1:(length(A)) c=-A(i,k) L(i,k)=-c./a for j=1: (length(A)) A(i,j)=A(i,j)+c.*A(k,j)./ a end end end L U=A A = 1 -1 1 -4 5 -4 3 12 2 1 1 11 2 -1 7 -1 L = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 a = 1 c = -5 L = 1 0 0 0 5 1 0 0 0 0 1 0 0 0 0 1 A = 1 -1 1 -4 0 -4 3 12 2 1 1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 3 12 2 1 1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 -2 12 2 1 1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 -2 32 2 1 1 11 2 -1 7 -1 c = -2 L = 1 0 0 0 5 1 0 0 2 0 1 0 0 0 0 1 A = 1 -1 1 -4 0 1 -2 32 0 1 1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 11 2 -1 7 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 2 -1 7 -1 c = -2 L = 1 0 0 0 5 1 0 0 2 0 1 0 2 0 0 1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 0 -1 7 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 0 1 7 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 0 1 5 -1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 0 1 5 7 a = 1 c = -3 L = 1 0 0 0 5 1 0 0 2 3 1 0 2 0 0 1 A = 1 -1 1 -4 0 1 -2 32 0 3 -1 19 0 1 5 7 A = 1 -1 1 -4 0 1 -2 32 0 0 -1 19 0 1 5 7 A = 1 -1 1 -4 0 1 -2 32 0 0 5 19 0 1 5 7 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 1 5 7 c = -1 L = 1 0 0 0 5 1 0 0 2 3 1 0 2 1 0 1 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 1 5 7 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 5 7 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 7 7 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 7 -25 a = 5 c = -7 L = 1.0000 0 0 0 5.0000 1.0000 0 0 2.0000 3.0000 1.0000 0 2.0000 1.0000 1.4000 1.0000 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 7 -25 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 7 -25 A = 1 -1 1 -4 0 1 -2 32 0 0 5 -77 0 0 0 -25 A = 1.0000 -1.0000 1.0000 -4.0000 0 1.0000 -2.0000 32.0000 0 0 5.0000 -77.0000 0 0 0 82.8000 a = 82.8000 L = 1.0000 0 0 0 5.0000 1.0000 0 0 2.0000 3.0000 1.0000 0 2.0000 1.0000 1.4000 1.0000 U = 1.0000 -1.0000 1.0000 -4.0000 0 1.0000 -2.0000 32.0000 0 0 5.0000 -77.0000 0 0 0 82.8000 >> A=L*U A = 1 -1 1 -4 5 -4 3 12 2 1 1 11 2 -1 7 -1
矩阵LU分解的高斯消元法
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