8月19日 题目四解答说明

function linear_interpolation(x, x_points, y_points)
    # 找到x所在的区间
    for i in 1:length(x_points)-1
        if x >= x_points[i] && x <= x_points[i+1]
            # 线性插值公式：y = y1 + (y2-y1)*(x-x1)/(x2-x1)
            x1, x2 = x_points[i], x_points[i+1]
            y1, y2 = y_points[i], y_points[i+1]
            return y1 + (y2 - y1) * (x - x1) / (x2 - x1)
        end
    end
    return 0.0  # 如果超出范围
end

function digit_sum(n)
    return sum(digits(n))
end

function satisfies_conditions(n)
    return n % 7 == 0 && digit_sum(n) > 20
end

# 使用Syslab (Julia) 解决数学问题

println("=== 8月19日 题目四解答 ===")
println()

# ========================================
# 题目1：对sin函数在[0,2π]内进行线性一维插值，并绘图
# ========================================

println("1. sin函数线性插值问题")
println("=" ^ 50)

# 定义原始sin函数
function original_sin(x)
    return sin(x)
end

# 线性插值函数
function linear_interpolation(x, x_points, y_points)
    # 找到x所在的区间
    for i in 1:length(x_points)-1
        if x >= x_points[i] && x <= x_points[i+1]
            # 线性插值公式：y = y1 + (y2-y1)*(x-x1)/(x2-x1)
            x1, x2 = x_points[i], x_points[i+1]
            y1, y2 = y_points[i], y_points[i+1]
            return y1 + (y2 - y1) * (x - x1) / (x2 - x1)
        end
    end
    return 0.0  # 如果超出范围
end

# 设置插值点
n_points = 10  # 插值点数量
x_interp = range(0, 2π, n_points)  # 均匀分布的插值点
y_interp = [sin(x) for x in x_interp]  # 对应的sin值

println("插值点数量: ", n_points)
println("插值点 x: ", collect(x_interp))
println("插值点 y: ", [round(y, digits=3) for y in y_interp])
println()

# 生成绘图用的密集点
x_plot = range(0, 2π, 1000)
y_original = [original_sin(x) for x in x_plot]
y_interpolated = [linear_interpolation(x, x_interp, y_interp) for x in x_plot]

# 计算插值误差
max_error = maximum(abs.(y_original - y_interpolated))
mean_error = sum(abs.(y_original - y_interpolated)) / length(y_original)

println("插值误差分析:")
println("最大误差: ", round(max_error, digits=6))
println("平均误差: ", round(mean_error, digits=6))
println()

# 绘图：使用简单的文本图形显示
println("绘图：文本形式的sin函数与线性插值对比")
println("=" ^ 60)

# 创建简单的文本图形
function draw_text_plot(x_vals, y_true, y_lin, x_nodes, y_nodes)
    # 选择一些关键点来显示
    step = length(x_vals) ÷ 20  # 显示20个点
    display_points = 1:step:length(x_vals)
    
    println("x值\t\t原始sin(x)\t插值结果\t误差")
    println("-" ^ 60)
    
    for i in display_points
        x_val = x_vals[i]
        orig_val = y_true[i]
        interp_val = y_lin[i]
        error_val = abs(orig_val - interp_val)
        
        # 检查是否是插值节点
        is_node = false
        for j in 1:length(x_nodes)
            if abs(x_val - x_nodes[j]) < 0.01
                is_node = true
                break
            end
        end
        
        node_mark = is_node ? " *" : "  "
        
        println(round(x_val, digits=3), node_mark, "\t", 
                round(orig_val, digits=3), "\t\t", 
                round(interp_val, digits=3), "\t\t", 
                round(error_val, digits=6))
    end
end

# 显示文本图形
draw_text_plot(collect(x_plot), y_original, y_interpolated, collect(x_interp), y_interp)

println()
println("图例说明：")
println("* 标记的点为插值节点（误差为0）")
println("其他点为插值结果（有误差）")
println()
println()

# 显示插值效果对比
println("插值效果对比 (部分点):")
println("x值\t\t原始sin(x)\t插值结果\t误差")
println("-" ^ 60)
for i in 1:10
    x_val = x_plot[i*100]
    orig_val = y_original[i*100]
    interp_val = y_interpolated[i*100]
    error_val = abs(orig_val - interp_val)
    println(round(x_val, digits=3), "\t\t", round(orig_val, digits=3), "\t\t", 
            round(interp_val, digits=3), "\t\t", round(error_val, digits=6))
end
println()

# ========================================
# 题目2：找出最小的正整数，满足以下条件：
# (1) 该数能被7整除
# (2) 该数的各位数字之和大于20
# ========================================

println("2. 最小正整数问题")
println("=" ^ 50)

# 计算数字各位之和的函数
function digit_sum(n)
    return sum(digits(n))
end

# 检查是否满足条件的函数
function satisfies_conditions(n)
    return n % 7 == 0 && digit_sum(n) > 20
end

println("寻找满足条件的最小正整数:")
println("条件1: 能被7整除")
println("条件2: 各位数字之和 > 20")
println()

# 搜索满足条件的最小正整数
function find_smallest_number()
    candidate = 7
    count = 0
    max_search = 10000  # 设置搜索上限

    println("搜索过程:")
    println("候选数\t\t各位数字之和\t是否满足条件")
    println("-" ^ 50)

    while candidate <= max_search
        count += 1
        d_sum = digit_sum(candidate)
        satisfies = satisfies_conditions(candidate)
        
        if count <= 10 || satisfies  # 显示前10个和满足条件的
            status = satisfies ? "✓" : "✗"
            println(candidate, "\t\t", d_sum, "\t\t", status)
        end
        
        if satisfies
            break
        end
        
        candidate += 7
    end

    println()

    if candidate <= max_search
        println("找到满足条件的最小正整数: ", candidate)
        println("验证:")
        println("- 能被7整除: ", candidate, " ÷ 7 = ", candidate ÷ 7, " ✓")
        println("- 各位数字之和: ", digit_sum(candidate), " > 20 ✓")
        return candidate
    else
        println("在搜索范围内未找到满足条件的数")
        return nothing
    end
end

# 执行搜索
result = find_smallest_number()

println()
println("=== 解答完成 ===")