用部分旋转matlab代码实现高斯消元

function [x, y] = gaussianEliminationWithPartialPivoting(A, b)
    % A: 系数矩阵
    % b: 常数向量
    % x: 解向量
    % y: 中间变量，用于存储每一步的消元结果

    n = size(A, 1);
    y = b;

    for k = 1:n-1
        % 部分旋转：选择第k列中绝对值最大的元素所在的行，并与第k行交换
        [~, maxRow] = max(abs(A(k:n, k)));
        maxRow = maxRow + k - 1;
        [A([k, maxRow], :), A([maxRow, k], :)] = deal(A([maxRow, k], :), A([k, maxRow], :));
        [y([k, maxRow]), y([maxRow, k])] = deal(y([maxRow, k]), y([k, maxRow]));

        % 消元过程
        for i = k+1:n
            factor = A(i, k) / A(k, k);
            A(i, k:end) = A(i, k:end) - factor * A(k, k:end);
            y(i) = y(i) - factor * y(k);
        end
    end

    % 回代求解
    x = zeros(n, 1);
    for k = n:-1:1
        x(k) = (y(k) - A(k, k+1:end) * x(k+1:end)) / A(k, k);
    end
end