部分选主元的Doolittle分解

用matlab编写函数，输入为线性方程组系数矩阵A，常数项b，输出为排列阵P,单位下三角矩阵L,上三角矩阵U，方程组的解x

具体算法参照施吉林的《计算机数值方法》第三版

function [P,L,U,x] = my_doolittle(A,b)

m=length(A);
P = eye(m);
L = eye(m);

%当r=1时，单独求解
a = [A b];
[num,idx] = max(A(:,1));
a = exchange_row(a,1,idx);  %为了换行方便，调用一个自己构造的换行函数
P = exchange_row(P,1,idx);
a(2:m,1) = a(2:m,1)/a(1,1);

%当r=2:m-1时，依照算法逐层运算
for j=2:m-1
    for i=j:m
        a(i,j) = a(i,j)-a(i,1:j-1)*a(1:j-1,j);
    end
    
    n_max = a(j,j);
    for i=j+1:m
        if a(i,j)>n_max
            idx = i;
            n_max = a(i,j);
        end
    end
    
    a = exchange_row(a,j,idx);
    P = exchange_row(P,j,idx);
    
    a(j+1:m,j) = a(j+1:m,j)/a(j,j);
    
    for col = j+1:m+1
        a(j,col) = a(j,col)-a(j,1:j-1)*a(1:j-1,col);
    end
   
end

%单独求解增广矩阵最后一行倒数两个元素
a(m,m) = a(m,m)-a(m,1:m-1)*a(1:m-1,m);
a(m,m+1) = a(m,m+1)-a(m,1:m-1)*a(1:m-1,m+1);

for i=2:m
    for j=1:i-1
        L(i,j) = a(i,j);
    end
end

U = a(:,1:end-1);
for i=2:m
    for j=1:i-1
        U(i,j) = 0;
    end
end

y = a(:,end);

x(m,1) = y(m,1)/U(m,m);

%利用Ux=y的算法求解x
for i=m-1:-1:1
    x(i,1)=(y(i,1)-U(i,i+1:m)*x(i+1:m,1))/U(i,i);
end

end

%输入矩阵a,要交换的行数i和j,对a矩阵进行行交换
function a = exchange_row(a,i,j)
    temp = a(i,:);
    a(i,:) = a(j,:);
    a(j,:) = temp;
end

在命令行进行测试
测试一：


测试二


测试三