154 lines
4.1 KiB
Rust
154 lines
4.1 KiB
Rust
//! 插值函数。
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//!
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//! 重构自 TLUSTY `yint.f` 和 `lagran.f`
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/// 使用 3 点进行二次插值。
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///
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/// 给定 3 个 x 值和 3 个 y 值的数组，插值求 `xl0` 处的 y 值。
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///
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/// # 参数
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///
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/// * `xl` - 3 个 x 坐标的数组
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/// * `yl` - 3 个 y 坐标的数组（f(x) 值）
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/// * `xl0` - 要插值到的 x 值
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///
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/// # 返回值
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///
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/// `xl0` 处的插值 y 值。
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///
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/// # Panics
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///
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/// 如果输入数组不正好有 3 个元素则 panic。
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///
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/// # 示例
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///
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/// ```
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/// use tlusty_rust::math::yint;
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///
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/// let xl = [0.0, 1.0, 2.0];
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/// let yl = [0.0, 1.0, 4.0]; // f(x) = x^2
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/// let result = yint(&xl, &yl, 0.5);
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/// assert!((result - 0.25).abs() < 1e-10);
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/// ```
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pub fn yint(xl: &[f64], yl: &[f64], xl0: f64) -> f64 {
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assert!(xl.len() == 3 && yl.len() == 3, "yint 需要大小为 3 的数组");
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// Fortran 使用 1 索引数组: XL(1), XL(2), XL(3)
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let x1 = xl[0];
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let x2 = xl[1];
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let x3 = xl[2];
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let y1 = yl[0];
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let y2 = yl[1];
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let y3 = yl[2];
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// A0 = (x2-x1)*(x3-x2)*(x3-x1)
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let a0 = (x2 - x1) * (x3 - x2) * (x3 - x1);
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// A1 = (xl0-x2)*(xl0-x3)*(x3-x2)
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let a1 = (xl0 - x2) * (xl0 - x3) * (x3 - x2);
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// A2 = (xl0-x1)*(x3-xl0)*(x3-x1)
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let a2 = (xl0 - x1) * (x3 - xl0) * (x3 - x1);
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// A3 = (xl0-x1)*(xl0-x2)*(x2-x1)
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let a3 = (xl0 - x1) * (xl0 - x2) * (x2 - x1);
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(y1 * a1 + y2 * a2 + y3 * a3) / a0
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}
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/// 三点 Lagrange 插值。
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///
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/// # 参数
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///
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/// * `x0`, `x1`, `x2` - 三个点的 x 坐标
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/// * `y0`, `y1`, `y2` - 三个点的 y 坐标
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/// * `x` - 要插值到的 x 值
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///
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/// # 返回值
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///
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/// `x` 处的插值 y 值。
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///
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/// # 示例
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///
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/// ```
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/// use tlusty_rust::math::lagran;
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///
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/// // 使用点 (0,0), (1,1), (2,4) 插值 f(x) = x^2
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/// let result = lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 0.5);
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/// assert!((result - 0.25).abs() < 1e-10);
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/// ```
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pub fn lagran(x0: f64, x1: f64, x2: f64, y0: f64, y1: f64, y2: f64, x: f64) -> f64 {
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// Lagrange 基多项式
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let xl0 = (x - x1) * (x - x2) / (x0 - x1) / (x0 - x2);
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let xl1 = (x - x0) * (x - x2) / (x1 - x0) / (x1 - x2);
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let xl2 = (x - x0) * (x - x1) / (x2 - x0) / (x2 - x1);
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y0 * xl0 + y1 * xl1 + y2 * xl2
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use approx::assert_relative_eq;
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#[test]
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fn test_yint_quadratic() {
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// 测试 f(x) = x^2
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let xl = [0.0, 1.0, 2.0];
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let yl = [0.0, 1.0, 4.0];
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// 中点
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let result = yint(&xl, &yl, 0.5);
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assert_relative_eq!(result, 0.25, epsilon = 1e-10);
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// 另一点
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let result = yint(&xl, &yl, 1.5);
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assert_relative_eq!(result, 2.25, epsilon = 1e-10);
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// 已知点（应返回精确值）
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let result = yint(&xl, &yl, 1.0);
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assert_relative_eq!(result, 1.0, epsilon = 1e-10);
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}
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#[test]
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fn test_yint_linear() {
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// 测试 f(x) = 2x + 1
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let xl = [0.0, 1.0, 2.0];
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let yl = [1.0, 3.0, 5.0];
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let result = yint(&xl, &yl, 0.5);
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assert_relative_eq!(result, 2.0, epsilon = 1e-10);
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}
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#[test]
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fn test_lagran_quadratic() {
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// 测试 f(x) = x^2
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let result = lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 0.5);
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assert_relative_eq!(result, 0.25, epsilon = 1e-10);
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let result = lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 1.5);
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assert_relative_eq!(result, 2.25, epsilon = 1e-10);
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}
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#[test]
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fn test_lagran_at_known_points() {
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// 在已知点应返回精确值
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assert_relative_eq!(lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 0.0), 0.0, epsilon = 1e-10);
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assert_relative_eq!(lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 1.0), 1.0, epsilon = 1e-10);
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assert_relative_eq!(lagran(0.0, 1.0, 2.0, 0.0, 1.0, 4.0, 2.0), 4.0, epsilon = 1e-10);
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}
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#[test]
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fn test_yint_lagran_equivalence() {
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// yint 和 lagran 对相同数据应给出相同结果
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let xl = [1.0, 2.0, 3.0];
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let yl = [1.0, 8.0, 27.0]; // f(x) = x^3
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let x = 2.5;
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let yint_result = yint(&xl, &yl, x);
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let lagran_result = lagran(xl[0], xl[1], xl[2], yl[0], yl[1], yl[2], x);
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assert_relative_eq!(yint_result, lagran_result, epsilon = 1e-10);
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}
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}
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