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% assignment3.m

function [x] = assignment3(A,b)
%  inputs:
%         A:系数矩阵，维度为n*n
%         b:载荷矩阵，维度为n*1
%  outputs:
%         x:未知量矩阵，维度n*1

[A_row,A_col] = size(A);
x=zeros(A_row,1);
x1_change=[]; % x1_change，x2_change记录x的位置变换
x2_change=[];

for i1=1:A_col-1 % 找出第i1行系数并对后面行进行消元
	max_val=0;
	for i2=i1:A_row % 找出右下方矩阵中最大的系数
		for i3=i1:A_col
			if max_val < abs(A(i2,i3))
				max_val = abs(A(i2,i3));  
				max_i = i2;  
				max_j = i3;
			end
		end
	end
    
	if max_i ~= i1 % 行交换矩阵
		for i2 = i1:A_col
			temp = A(max_i,i2);
			A(max_i,i2) = A(i1,i2);
			A(i1,i2) = temp;
		end
		temp = b(i1);
		b(i1) = b(max_i);
		b(max_i) = temp; 
	end
    
	if max_j ~= i1 % 列交换矩阵
		for i2 = 1:A_col
			temp = A(i2,max_j);
			A(i2,max_j) = A(i2,i1);
			A(i2,i1) = temp;
		end
		x1_change = [x1_change max_j]; % 记录x的交换
		x2_change = [x2_change i1];
	end
    
	if A(i1,i1) ~= 0 % 消元
		for i2 = i1+1:A_row
			L_i1i2=A(i2,i1)/A(i1,i1);
			A(i2,:)=-A(i1,:).*L_i1i2+A(i2,:);
			b(i2)=-b(i1)*L_i1i2+b(i2);
		end
	end
end

for i1 = A_row:-1:1 %回代求解
	x(i1)=b(i1);
	for i2 = A_col:-1:i1+1
		x(i1)=x(i1)-A(i1,i2)*x(i2);
	end
	x(i1)=x(i1)/A(i1,i1);
end

for i1=length(x1_change):-1:1 %调转x为原有顺序
	temp=x(x1_change(i1));
	x(x1_change(i1))=x(x2_change(i1));
	x(x2_change(i1))=temp;
end
end

% 命令行输入
% assignment3([2 3 4; 5 6 7; 8 9 11], [2 3 4])