数值计算——线性方程组的几个例题解决

1、



计算计算向量b的1范数、2范数、无穷范数。

计算A矩阵的1范数、2范数、无穷范数、条件数、F-范数。

clc,clear

b = [2
8
10];
A = [2 4 -2
4 9 -3
-2 -1 7];

fprintf('%s\n','向量b的1范数')
fanshub1 = abs(b(1)) + abs(b(2)) + abs(b(3))
fprintf('%s\n','向量b的2范数')
fanshub2 = sqrt((b(1)).^2 + (b(2)).^2 + (b(3)).^2)
fprintf('%s\n','向量b的无穷范数')
fanshub3 = max(abs(b))

fprintf('%s\n','矩阵A的1范数')
a = zeros(1,3);
sum = 0;
for i = 1:3
for j = 1:3
sum = sum + abs(A(j,i));
end
a(i) = sum;
sum = 0;
end
fanshua1 = max(a)

fprintf('%s\n','矩阵A的2范数')
B = A'*A;
[v d] = eig(B);
fanshua2 = sqrt(max(max(d)))

fprintf('%s\n','矩阵A的无穷范数')
for i = 1:3
for j = 1:3
sum = sum + abs(A(i,j));
end
a(i) = sum;
sum = 0;
end
fanshua3 = max(a)

fprintf('%s\n',' 1范数下矩阵A的条件数')
A_ = inv(A);
for i = 1:3
for j = 1:3
sum = sum + abs(A_(j,i));
end
a(i) = sum;
sum = 0;
end
conda1 = fanshua1*max(a)

fprintf('%s\n',' 2范数下矩阵A的2条件数')
B = A_'*A_;
[v d] = eig(B);
conda2 = sqrt(max(max(d)))*fanshua2

fprintf('%s\n','无穷范数下矩阵A的条件数')
for i = 1:3
for j = 1:3
sum = sum + abs(A_(i,j));
end
a(i) = sum;
sum = 0;
end
conda3 = max(a)*fanshua3

fprintf('%s\n','矩阵A的F范数')
for i = 1:3
for j = 1:3
sum = sum + (A(i,j)).^2;
end
end
fanshuaf = sqrt(sum)
sum = 0;

运行结果如下所示









2、楚列斯基分解

（1）、写出Cholesky分解函数。

用MATLAB代码实现如下所示

Cholesky.m
function [ B ] = Cholesky(A )
%UNTITLED2 此处显示有关此函数的摘要
%   楚列斯基分解
B = A;
n = length(A);
for k = 1:n
B(k,k) = sqrt(B(k,k));
for i = k+1:n
B(i,k) = B(i,k)/B(k,k);
end
for j = k+1:n
for i = k+1:n
B(i,j) = B(i,j)-B(i,k)*B(j,k);
end
end
end

for i = 1:n-1
for j = i+1:n
B(i,j) = 0;
end
end

end

（2）、



解方程组Hx=b（取n=2，5，10）

用MATLAB编写代码如下

Fiteration.m
function [ x ] = Fiteration( L,b )
%UNTITLED8 此处显示有关此函数的摘要
% 此处显示详细说明
[n1,n]=size(L);
x = zeros(n,1);
if n1==n
j = 1;
while j<n+1 && L(j,j)~=0
x(j,1) = b(j)/L(j,j);
for i = j+1:n
b(i) = b(i)-L(i,j)*x(j,1);
end
j=j+1;
end
end

end

Biteration.m
function [ x ] = Biteration( U,b )
%UNTITLED9 此处显示有关此函数的摘要
%   此处显示详细说明
[n1,n] = size(U);
x = zeros(n,1);
if n1==n
j = n;
while j>0 && U(j,j)~=0
x(j,1) = b(j)/U(j,j);
for i = 1:j-1
b(i) = b(i)-U(i,j)*x(j);
end
j = j-1;
end

end

end

test221.m
clc,clear
fprintf('%s\n','n = 2时')
A = [1 1/2
1/2 1/3];
L = Cholesky(A)
x1 = [1
1];
b = A*x1
front = Fiteration( L,b );
x = Biteration(L',front)

fprintf('%s\n','n = 5时')
A = zeros(5,5);
for i = 1:5
for j = 1:5
A(i,j) = 1/(i+j-1);
end
end
L = Cholesky(A)
x1 = ones(5,1);
b = A*x1
front = Fiteration( L,b );
x = Biteration(L',front)

fprintf('%s\n','n = 10时')
A = zeros(10,10);
for i = 1:10
for j = 1:10
A(i,j) = 1/(i+j-1);
end
end
L = Cholesky(A)
x1 = ones(10,1);
b = A*x1
front = Fiteration( L,b );
x = Biteration(L',front)

运行结果如下所示















（3）、计算H矩阵的1范数、2范数、无穷范数、条件数（取n=2，5，10）

用MATLAB代码实现如下

f222.m
function [ ] = f222(n )
%UNTITLED5 此处显示有关此函数的摘要
% 此处显示详细说明

A = zeros(n,n);
for i = 1:n
for j = 1:n
A(i,j) = 1/(i+j-1);
end
end

fprintf('%s\n','矩阵的1范数为')
a = zeros(1,n);
sum = 0;
for i = 1:n
for j = 1:n
sum = sum + abs(A(j,i));
end
a(i) = sum;
sum = 0;
end
fanshua1 = max(a)

fprintf('%s\n','矩阵A的2范数')
B = A'*A;
[v d] = eig(B);
fanshua2 = sqrt(max(max(d)))

fprintf('%s\n','矩阵A的无穷范数')
for i = 1:n
for j = 1:n
sum = sum + abs(A(i,j));
end
a(i) = sum;
sum = 0;
end
fanshua3 = max(a)

fprintf('%s\n','1范数下矩阵A的条件数')
A_ = inv(A);
for i = 1:n
for j = 1:n
sum = sum + abs(A_(j,i));
end
a(i) = sum;
sum = 0;
end
conda1 = fanshua1*max(a)

fprintf('%s\n','2范数下矩阵A的2条件数')
B = A_'*A_;
[v d] = eig(B);
conda2 = sqrt(max(max(d)))*fanshua2

fprintf('%s\n','无穷范数下矩阵A的条件数')
for i = 1:n
for j = 1:n
sum = sum + abs(A_(i,j));
end
a(i) = sum;
sum = 0;
end
conda3 = max(a)*fanshua3

end
test222.m
clc,clear
fprintf('%s\n','n = 2时')
f222(2)
fprintf('%s\n','n = 5时')
f222(5)
fprintf('%s\n','n = 10时')
f222(10)