工程数学线性方程组直接解法数值试验.docx
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工程数学线性方程组直接解法数值试验.docx
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工程数学线性方程组直接解法数值试验
functionx=gauss1(a,b)
n=length(b);
fork=1:
n-1
fori=k+1:
n
m(i,k)=a(i,k)/a(k,k);
a(i,k)=0;
b(i)=b(i)-m(i,k)*b(k);
forj=k+1:
n
a(i,j)=a(i,j)-a(k,j)*m(i,k);
end
end
end
x(n)=b(n)/a(n,n);
fori=n-1:
-1:
1
s=0;
forj=i+1:
n
s=s+a(i,j)*x(j);
end
x(i)=(b(i)-s)/a(i,i);
end
x=x';
a=[234;352;4330];b=[6,5,32]'
a(1,2)
a=[123;456;788];b=[61523];
x=gauss1(a,b)
x=
1
1
1
x=
NaN
NaN
NaN
x=
-13
8
2
a=[2-11;-1-23;131]
b=[456]’
x=gauss1(a,b)
x=gauss1(a,b)
x=
1.11111111111111
0.777777777777778
2.55555555555556
a=[12-33;-1831;111]
b=[15-156]’
x=gauss1(A,b))
m=
0
1.5
2
a=
234
00.5-4
0-322
b=
6
-4
20
m=
00
1.50
2-6
a=
234
00.5-4
00-2
b=
6
-4
-4
x=
-13
8
2
functionx=gauss4(a,b)
a=[a,b];
n=length(b);
fork=1:
n-1
s=a(k,k);
p=k;
fori=k+1:
n
ifabs(s) s=a(i,k); p=i; end end ifp~=k forj=k: n+1 t=a(k,j); a(k,j)=a(p,j); a(p,j)=t; end end a,p fori=k+1: n m(i,k)=a(i,k)/a(k,k); a(i,k)=0; forj=k+1: n+1 a(i,j)=a(i,j)-a(k,j)*m(i,k); end end end a x(n)=a(n,n+1)/a(n,n); fori=n-1: -1: 1 s=0; forj=i+1: n s=s+a(i,j)*x(j); end x(i)=(a(i,n+1)-s)/a(i,i); end x=x'; a=[234;352;4330];b=[6,5,32]' x=gauss4(a,b) a=[234;352;4330];b=[6,5,32]' x=gauss4(a,b) b= 6 5 32 a= 433032 3525 2346 p= 3 a= 433032 02.75-20.5-19 01.5-11-10 p= 2 a= 433032 02.75-20.5-19 000.181********81820.363636363636363 x= -13 8 2 x= -13 8 2 functionL=sanjiao3(a) [n,n]=size(a); a(1,1)=sqrt(a(1,1)); fori=2: n a(i,1)=a(i,1)/a(1,1); end fork=2: n-1 a(k,k)=a(k,k)-dot(a(k,1: k-1),a(k,1: k-1)); a(k,k)=sqrt(a(k,k)); fori=k+1: n a(i,k)=(a(i,k)-dot(a(i,1: k-1),a(k,1: k-1)))/a(k,k); end end a(n,n)=a(n,n)-dot(a(n,1: n-1),a(n,1: n-1)); a(n,n)=sqrt(a(n,n)); L=tril(a); a=[9189-27;18450-45;901269;-27-459135] a= 9189-27 18450-45 901269 -27-459135 >>L=sanjiao3(a) L= 3000 6300 3-690 -9363 >>L*L' ans= 9189-27 18450-45 901269 -27-459135 functionx=sanduijiao(a,b,c,f) n=length(a); q (1)=a (1); b=[0,b]; fori=2: n p(i)=b(i)/q(i-1); q(i)=a(i)-p(i)*c(i-1); end fori=2: n f(i)=f(i)-p(i)*f(i-1); end x(n)=f(n)/q(n); fori=n-1: -1: 1 x(i)=(f(i)-c(i)*x(i+1))/q(i); end a=[22222]; >>b=[1111]; >>c=[-1-1-1-1]; f=[12223];x=sanduijiao(a,b,c,f) x= 11111 function[L,U]=sanjiao1(a) [n,n]=size(a); fori=2: n a(i,1)=a(i,1)/a(1,1); end fork=2: n forj=k: n a(k,j)=a(k,j)-dot(a(k,1: k-1),a(1: k-1,j)); end fori=k+1: n a(i,k)=(a(i,k)-dot(a(i,1: k-1),a(1: k-1,k)))/a(k,k); end end U=triu(a); L=tril(a,-1)+eye(n); a=[9189-27;18450-45;901269;-27-459135] function[L,U,P]=sanjiao2(a) [n,n]=size(a); I=[1: n]; fork=1: n forj=k: n a(j,k)=a(j,k)-sum1(a(j,: ),a(: k),1,k-1); s(j)=a(j,k); end m=s(k); t=k; forj=k+1: n ifabs(m) m=s(j); t=j; end end p=I(k);I(k)=I(t);I(t)=p; forj=1: n p=a(k,j);a(k,j)=a(t,j);a(t,j)=p; end forj=k+1: n a(j,k)=a(j,k)/a(k,k); end forj=k+1: n a(k,j)=a(k,j)-sum1(a(k,: ),a(: j),1,k-1); end end L=tril(a,-1)+eye(n); U=triu(a); fori=1: n P(i,I(i))=1; end function[L,U]=sanjiao1(a) [n,n]=size(a); fori=2: n a(i,1)=a(i,1)/a(1,1); end fork=2: n forj=k: n a(k,j)=a(k,j)-dot(a(k,1: k-1),a(1: k-1,j)); end fori=k+1: n a(i,k)=(a(i,k)-dot(a(i,1: k-1),a(1: k-1,k)))/a(k,k); end end U=triu(a); L=tril(a,-1)+eye(n);
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