matlab高斯消gf,高斯消去、追赶法 matlab(示例代码)

  • Post author:
  • Post category:其他


1. 分别用Gauss消去法、列主元Gauss消去法、三角分解方法求解方程组

20181120210543339390.png

程序:

(1)Guess消去法:

function x=GaussXQByOrder(A,b)

%Gauss消去法

N = size(A);

n = N(1);

x = zeros(n,1);

for i=1:(n-1)

for j=(i+1):n

if(A(i,i)==0)

disp(‘对角元不能为0‘);

return;

end

m = A(j,i)/A(i,i);

A(j,i:n)=A(j,i:n)-m*A(i,i:n);

b(j)=b(j)-m*b(i);

end

end

x(n)=b(n)/A(n,n);

for i=n-1:-1:1

x(i)=(b(i)-sum(A(i,i+1:n)*x(i+1:n)))/A(i,i);

end

命令行输入:

A=[1 -1 2 1;-1 3 0 -3 ;2 0 9 -6;1 -3 -6 19];

b=[1 3 5 7];

b=b‘;

x=GaussXQByOrder(A,b)

运算结果:

x =

-8.000000000000000

0.333333333333333

3.666666666666667

2.000000000000000

(2)列主元Gauss消去法

程序:

function x=GaussXQLineMain(A,b)

%列主元Gauss消去法

N = size(A);

n = N(1);

x = zeros(n,1);

zz=zeros(1,n);

for i=1:(n-1)

[~,p]=max(abs(A(i:n,i)));

zz=A(i,:);

A(i,:)=A(p+i-1,:);

A(p+i-1,:)=zz;

temp=b(i);

b(i)=b(i+p-1);

b(i+p-1)=temp;

for j=(i+1):n

m = A(j,i)/A(i,i);

A(j,i:n)=A(j,i:n)-m*A(i,i:n);

b(j)=b(j)-m*b(i);

end

end

x(n)=b(n)/A(n,n);

for i=n-1:-1:1

x(i)=(b(i)-sum(A(i,i+1:n)*x(i+1:n)))/A(i,i);

end

命令行:

A=[1 -1 2 1;-1 3 0 -3 ;2 0 9 -6;1 -3 -6 19];

b=[1 3 5 7];

b=b‘;

x=GaussXQLineMain(A,b)

运行结果:

x =

-8.000000000000005

0.333333333333332

3.666666666666668

2.000000000000000

(3)三角分解方法

程序:

function x = LU(A,b)

%三角分解

N = size(A);

n = N(1);

L = eye(n,n);

U = zeros(n,n);

x = zeros(n,1);

y = zeros(n,1);

U(1,1:n) = A(1,1:n);

L(1:n,1) = A(1:n,1)/U(1,1);

for k=2:n

for i=k:n

U(k,i) = A(k,i)-L(k,1:(k-1))*U(1:(k-1),i);

end

for j=(k+1):n

L(j,k) = (A(j,k)-L(j,1:(k-1))*U(1:(k-1),k))/U(k,k);

end

end

y(1)=b(1)/L(1,1);

for i=2:n

y(i)=b(i)-sum(L(i,1:i-1)*y(1:i-1));

end

x(n)=y(n)/U(n,n);

for i=n-1:-1:1

x(i)=(y(i)-sum(U(i,i+1:n)*x(i+1:n)))/U(i,i);

end

命令行:

A=[1 -1 2 1;-1 3 0 -3 ;2 0 9 -6;1 -3 -6 19];

b=[1 3 5 7];

b=b‘;

x=LU(A,b)

运行结果:

x =

-8.000000000000000

0.333333333333333

3.666666666666667

2.000000000000000

20181120210543509318.png

程序:function [times,wucha]=zhuiganfa(a,b,c,f)

%追赶法:x为所求解,times为所有乘除运算次数(即时间),wucha为误差的2-范数。

n=length(f);

x=zeros(n,1);

y=zeros(n,1);

times=0;

alpha=zeros(1,n);

p=zeros(1,n-1);

alpha(1)=b(1);

for i=2:n

p(i-1)=c(i-1)/alpha(i-1);

alpha(i)=b(i)-a(i-1)*p(i-1);

times=times+1;

end

y(1)=f(1)/b(1);

for i=2:n

y(i)=(f(i)-a(i-1)*y(i-1))/alpha(i);

times=times+1;

end

x(n)=y(n);

for i=n-1:-1:1

x(i)=y(i)-p(i)*x(i+1);

times=times+1;

end

A=zeros(n,n);

A=diag(b,0)+diag(a,-1)+diag(c,1);

wucha=norm((A*x-f‘),2);

命令行(n=20):

a=repmat(11,1,19);

b=repmat(-19,1,20);

c=repmat(7,1,19);

f1=repmat(1.1,1,18);

f=[0 f1 1];

[times,wucha]=zhuiganfa(a,b,c,f)

运行结果:

times =

57

wucha =

8.009010697694412e-15

n=50时

命令行:

a=repmat(11,1,49);

b=repmat(-19,1,50);

c=repmat(7,1,49);

f1=repmat(1.1,1,48);

f=[0 f1 1];

[times,wucha]=zhuiganfa(a,b,c,f)

运行结果:

times =

147

wucha =

1.292635294609912e-14

命令行(n=100)

a=repmat(11,1,99);

b=repmat(-19,1,100);

c=repmat(7,1,99);

f1=repmat(1.1,1,98);

f=[0 f1 1];

[times,wucha]=zhuiganfa(a,b,c,f)

结果:

times =

297

wucha =

2.599344850768740e-14

20181120210543739796.png

程序:function [count,wucha] = zhouqisanduijaiozhuiganfa(a,b,c,f)

%x为所求解,count为所有乘除运算次数

n=length(f);

x=zeros(n,1);

y=zeros(n,1);

count=0;

l=zeros(1,n-2);

s=zeros(1,n-1);

u=zeros(1,n);

t=zeros(1,n-1);

u(1)=b(1);t(1)=1;

s(1)=1/u(1);y(1)=f(1);

for i=2:n-1

l(i-1)=a(i-1)/u(i-1);

u(i)=b(i)-l(i-1)*c(i-1);

t(i)=-l(i-1)*t(i-1);

s(i)=-s(i-1)*c(i-1)/u(i);

y(i)=f(i)-l(i-1)*y(i-1);

count=count+4;

end

st=0;

for k=1:n-1

st=st+s(k)*t(k);

count=count+1;

end

sy=0;

for k=1:n-2

sy=sy+s(k)*y(k);

count=count+1;

end

u(n)=b(n)-st-s(n-1)*(c(n-1)+t(n-1));

y(n)=f(n)-sy;

x(n)=y(n)/u(n);

for i=n-1:-1:1

x(i)=(y(i)-c(i)*x(i+1)-t(i)*x(n))/u(i);

count=count+1;

end

A=zeros(n,n);

A=diag(b,0)+diag(a,-1)+diag(c,1);

A(n,1)=1;

A(1,n)=1;

wucha=norm((A*x-f‘),2);

命令行:

n=10;

a=repmat(11,1,n-1);b=repmat(-19,1,n);

c=repmat(7,1,n-1);f1=repmat(1.1,1,n-2);f=[0 f1 1];

[count,wucha]= zhouqisanduijaiozhuiganfa(a,b,c,f)

运行结果:

count =

58

wucha =

4.525439045433075

n=30

count =

198

wucha =

5.951269557941316

n=100

count =

688

wucha =

5.993271932634396