spline.rs

//! Spline interpolations
//!
//! # Available splines
//!
//! * Cubic spline
//! * Cubic Hermite spline
//! * B-spline
//!
//! # `Spline<T>` trait
//!
//! ## Methods
//!
//! * `fn eval(&self, t: f64) -> T` : Evaluate the spline at t
//!   * For Cubic Spline, t means x (domain) and `T = f64`
//!   * For B-Spline, t means parameter of curve and `T = (f64, f64)`
//! * `fn eval_vec(&self, v: &[f64]) -> Vec<T>` : Evaluate spline values for an array v
//! * `fn eval_with_cond<F>(&self, t: f64, cond: F) -> T` : Evaluate the spline at t, with a condition
//! * `fn eval_vec_with_cond<F>(&self, v: &[f64], cond: F) -> Vec<T>` : Evaluate spline values for an array v, with a condition
//!
//! # `PolynomialSpline` trait
//!
//! ## Methods
//!
//! * `fn polynomial_at(&self, x: f64) -> &Polynomial` : Get the polynomial at x
//! * `fn number_of_polynomials(&self) -> usize` : Get the number of polynomials
//! * `fn get_ranged_polynomials(&self) -> &Vec<(Range<f64>, Polynomial)>` : Get the polynomials
//!
//! # Low-level interface
//!
//! ## Members
//!
//! * `CubicSpline`: Structure for cubic spline
//!   * `fn from_nodes(node_x: &[f64], node_y: &[f64]) -> Result<Self>` : Create a cubic spline from nodes
//!   * `fn extend_with_nodes(&mut self, node_x: Vec<f64>, node_y: Vec<f64>) -> Result<()>` : Extend the spline with nodes
//! * `CubicHermiteSpline`: Structure for cubic Hermite spline
//!   * `fn from_nodes_with_slopes(node_x: &[f64], node_y: &[f64], m: &[f64]) -> Result<Self>` : Create a Cubic Hermite spline from nodes with slopes
//!   * `fn from_nodes(node_x: &[f64], node_y: &[f64], slope_method: SlopeMethod) -> Result<Self>` : Create a Cubic Hermite spline from nodes with slope estimation methods
//!   * `SlopeMethod`: Enum for slope estimation methods
//!     * `Akima`: Akima's method to estimate slopes ([Akima (1970)](https://dl.acm.org/doi/abs/10.1145/321607.321609))
//!     * `Quadratic`: Using quadratic interpolation to estimate slopes
//! * `BSpline`: Structure for B-Spline
//!   * `fn open(degree: usize, knots: Vec<f64>, control_points: Vec<Vec<f64>>) -> Result<Self>` : Create an open B-Spline
//!   * `fn clamped(degree: usize, knots: Vec<f64>, control_points: Vec<Vec<f64>>) -> Result<Self>`
//!     : Create a clamped B-Spline
//!   * `fn cox_de_boor(t: f64, i: f64)` : Cox-de Boor recursion formula (Here, use iteration
//!   instead of recursion)
//!
//! ## Usage (Cubic Spline Family)
//!
//! ```rust
//! use peroxide::fuga::*;
//!
//! fn main() -> Result<(), Box<dyn Error>> {
//!     let x = seq(0, 10, 1);
//!     let y = x.fmap(|t| t.sin());
//!     
//!     let cs = CubicSpline::from_nodes(&x, &y)?;
//!     let cs_akima = CubicHermiteSpline::from_nodes(&x, &y, Akima)?;
//!     let cs_quad = CubicHermiteSpline::from_nodes(&x, &y, Quadratic)?;
//!
//!     cs.polynomial_at(0f64).print();
//!     cs_akima.polynomial_at(0f64).print();
//!     cs_quad.polynomial_at(0f64).print();
//!     // -0.1523x^3 + 0.9937x
//!     // 0.1259x^3 - 0.5127x^2 + 1.2283x
//!     // -0.0000x^3 - 0.3868x^2 + 1.2283x
//!
//!     let new_x = seq(4, 6, 0.1);
//!     let new_y = new_x.fmap(|t| t.sin());
//!
//!     let y_cs = cs.eval_vec(&new_x);
//!     let y_akima = cs_akima.eval_vec(&new_x);
//!     let y_quad = cs_quad.eval_vec(&new_x);
//!
//!     let mut df = DataFrame::new(vec![]);
//!     df.push("x", Series::new(new_x));
//!     df.push("y", Series::new(new_y));
//!     df.push("y_cs", Series::new(y_cs));
//!     df.push("y_akima", Series::new(y_akima));
//!     df.push("y_quad", Series::new(y_quad));
//!
//!     df.print();
//!     //          x       y    y_cs y_akima  y_quad
//!     //  r[0]    5 -0.9589 -0.9589 -0.9589 -0.9589
//!     //  r[1]  5.2 -0.8835 -0.8826 -0.8583 -0.8836
//!     //  r[2]  5.4 -0.7728 -0.7706 -0.7360 -0.7629
//!     //  r[3]  5.6 -0.6313 -0.6288 -0.5960 -0.6120
//!     //  r[4]  5.8 -0.4646 -0.4631 -0.4424 -0.4459
//!     //  r[5]    6 -0.2794 -0.2794 -0.2794 -0.2794
//!
//!     Ok(())
//! }
//! ```
//!
//! ## Usage (B-Spline)
//!
//! ```rust
//! use peroxide::fuga::*;
//!
//! fn main() -> Result<(), Box<dyn Error>> {
//!     let knots = vec![0f64, 1f64, 2f64, 3f64];
//!     let degree = 3;
//!     let control_points = vec![
//!         vec![0f64, 2f64],
//!         vec![0.2, -1f64],
//!         vec![0.4, 1f64],
//!         vec![0.6, -1f64],
//!         vec![0.8, 1f64],
//!         vec![1f64, 2f64],
//!     ];
//!
//!  
//!     let spline = BSpline::clamped(degree, knots, control_points.clone())?;
//!     let t = linspace(0f64, 3f64, 200);
//!     let (x, y): (Vec<f64>, Vec<f64>) = spline.eval_vec(&t).into_iter().unzip();
//!
//!     # #[cfg(feature = "plot")]
//!     # {
//!         let control_x = control_points.iter().map(|v| v[0]).collect::<Vec<f64>>();
//!         let control_y = control_points.iter().map(|v| v[1]).collect::<Vec<f64>>();
//!
//!         let mut plt = Plot2D::new();
//!         plt
//!             .insert_pair((x.clone(), y.clone()))
//!             .insert_pair((control_x.clone(), control_y.clone()))
//!             .set_plot_type(vec![(1, PlotType::Scatter)])
//!             .set_color(vec![(0, "darkblue"), (1, "red")])
//!             .set_xlabel(r"$x$")
//!             .set_ylabel(r"$y$")
//!             .set_style(PlotStyle::Nature)
//!             .set_dpi(600)
//!             .set_path("example_data/b_spline_test.png")
//!             .savefig()?;
//!     # }
//!
//!     let mut df = DataFrame::new(vec![]);
//!     df.push("t", Series::new(t));
//!     df.push("x", Series::new(x));
//!     df.push("y", Series::new(y));
//!     df.print();
//!
//!     Ok(())
//! }
//! ```
//!
//! - Result for above code is:
//!   ![b_spline_test](https://github.com/Axect/Peroxide/blob/master/example_data/b_spline_test.png?raw=true)
//!
//! # High-level interface
//!
//! ## Functions
//!
//! * `fn cubic_spline(node_x: &[f64], node_y: &[f64]) -> CubicSpline` : Create a cubic spline from nodes
//! * `fn cubic_hermite_spline(node_x: &[f64], node_y: &[f64], m: &[f64]) -> CubicHermiteSpline` : Create a cubic Hermite spline from nodes with slopes
//!
//! ## Usage
//!
//! ```rust
//! use peroxide::fuga::*;
//!
//! fn main() -> Result<(), Box<dyn Error>> {
//!     let x = seq(0, 10, 1);
//!     let y = x.fmap(|t| t.sin());
//!     
//!     let cs = cubic_spline(&x, &y)?;
//!     let cs_akima = cubic_hermite_spline(&x, &y, Akima)?;
//!     let cs_quad = cubic_hermite_spline(&x, &y, Quadratic)?;
//!
//!     cs.polynomial_at(0f64).print();
//!     cs_akima.polynomial_at(0f64).print();
//!     cs_quad.polynomial_at(0f64).print();
//!     // -0.1523x^3 + 0.9937x
//!     // 0.1259x^3 - 0.5127x^2 + 1.2283x
//!     // -0.0000x^3 - 0.3868x^2 + 1.2283x
//!
//!     let new_x = seq(4, 6, 0.1);
//!     let new_y = new_x.fmap(|t| t.sin());
//!
//!     let y_cs = cs.eval_vec(&new_x);
//!     let y_akima = cs_akima.eval_vec(&new_x);
//!     let y_quad = cs_quad.eval_vec(&new_x);
//!
//!     let mut df = DataFrame::new(vec![]);
//!     df.push("x", Series::new(new_x));
//!     df.push("y", Series::new(new_y));
//!     df.push("y_cs", Series::new(y_cs));
//!     df.push("y_akima", Series::new(y_akima));
//!     df.push("y_quad", Series::new(y_quad));
//!
//!     df.print();
//!     //          x       y    y_cs y_akima  y_quad
//!     //  r[0]    5 -0.9589 -0.9589 -0.9589 -0.9589
//!     //  r[1]  5.2 -0.8835 -0.8826 -0.8583 -0.8836
//!     //  r[2]  5.4 -0.7728 -0.7706 -0.7360 -0.7629
//!     //  r[3]  5.6 -0.6313 -0.6288 -0.5960 -0.6120
//!     //  r[4]  5.8 -0.4646 -0.4631 -0.4424 -0.4459
//!     //  r[5]    6 -0.2794 -0.2794 -0.2794 -0.2794
//!
//!     Ok(())
//! }
//! ```
//!
//! # Calculus with polynomial splines
//!
//! ## Usage
//!
//! ```rust
//! use peroxide::fuga::*;
//! use std::f64::consts::PI;
//!
//! fn main() -> Result<(), Box<dyn Error>> {
//!     let x = seq(0, 10, 1);
//!     let y = x.fmap(|t| t.sin());
//!     
//!     let cs = cubic_spline(&x, &y)?;
//!     let cs_akima = cubic_hermite_spline(&x, &y, Akima)?;
//!     let cs_quad = cubic_hermite_spline(&x, &y, Quadratic)?;
//!
//!     println!("============ Polynomial at x=0 ============");
//!
//!     cs.polynomial_at(0f64).print();
//!     cs_akima.polynomial_at(0f64).print();
//!     cs_quad.polynomial_at(0f64).print();
//!
//!     println!("============ Derivative at x=0 ============");
//!
//!     cs.derivative().polynomial_at(0f64).print();
//!     cs_akima.derivative().polynomial_at(0f64).print();
//!     cs_quad.derivative().polynomial_at(0f64).print();
//!
//!     println!("============ Integral at x=0 ============");
//!
//!     cs.integral().polynomial_at(0f64).print();
//!     cs_akima.integral().polynomial_at(0f64).print();
//!     cs_quad.integral().polynomial_at(0f64).print();
//!
//!     println!("============ Integrate from x=0 to x=pi ============");
//!
//!     cs.integrate((0f64, PI)).print();
//!     cs_akima.integrate((0f64, PI)).print();
//!     cs_quad.integrate((0f64, PI)).print();
//!
//!     // ============ Polynomial at x=0 ============
//!     // -0.1523x^3 + 0.9937x
//!     // 0.1259x^3 - 0.5127x^2 + 1.2283x
//!     // -0.0000x^3 - 0.3868x^2 + 1.2283x
//!     // ============ Derivative at x=0 ============
//!     // -0.4568x^2 + 0.9937
//!     // 0.3776x^2 - 1.0254x + 1.2283
//!     // -0.0000x^2 - 0.7736x + 1.2283
//!     // ============ Integral at x=0 ============
//!     // -0.0381x^4 + 0.4969x^2
//!     // 0.0315x^4 - 0.1709x^3 + 0.6141x^2
//!     // -0.0000x^4 - 0.1289x^3 + 0.6141x^2
//!     // ============ Integrate from x=0 to x=pi ============
//!     // 1.9961861265456702
//!     // 2.0049920614062775
//!     // 2.004327391790717
//!
//!     Ok(())
//! }
//! ```
//!
//! # B-Spline utils
//!
//! - `UnitCubicBasis`: Single cubic B-Spline basis function
//! - `CubicBSplineBases`: Uniform Cubic B-Spline basis functions
//!
//! # References
//!
//! - Gary D. Knott, *Interpolating Splines*, Birkhäuser Boston, MA, (2000).
/// - [Wikipedia - Irwin-Hall distribution](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution#Special_cases)
use self::SplineError::{NotEnoughNodes, NotEqualNodes, NotEqualSlopes, RedundantNodeX};
#[allow(unused_imports)]
use crate::structure::matrix::*;
#[allow(unused_imports)]
use crate::structure::polynomial::*;
#[allow(unused_imports)]
use crate::structure::vector::*;
use crate::traits::matrix::LinearAlgebra;
#[allow(unused_imports)]
use crate::util::non_macro::*;
use crate::util::useful::zip_range;
use anyhow::{bail, Result};
use peroxide_num::PowOps;
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};
use std::cmp::{max, min};
use std::convert::From;
use std::ops::{Index, Range};

pub trait Spline<T> {
    fn eval(&self, t: f64) -> T;
    fn eval_vec(&self, v: &[f64]) -> Vec<T> {
        v.iter().map(|&t| self.eval(t)).collect()
    }
    fn eval_with_cond<F: Fn(T) -> T>(&self, t: f64, cond: F) -> T {
        cond(self.eval(t))
    }
    fn eval_vec_with_cond<F: Fn(T) -> T + Copy>(&self, t: &[f64], cond: F) -> Vec<T> {
        t.iter().map(|&x| self.eval_with_cond(x, cond)).collect()
    }
}

/// Trait for spline interpolation
///
/// # Available Splines
///
/// - `CubicSpline`
/// - `CubicHermiteSpline`
impl<P: PolynomialSpline> Spline<f64> for P {
    fn eval(&self, x: f64) -> f64 {
        self.polynomial_at(x).eval(x)
    }
}

pub trait PolynomialSpline {
    fn polynomial_at<T: Into<f64> + Copy>(&self, x: T) -> &Polynomial {
        let x = x.into();

        let poly = self.get_ranged_polynomials();

        let index = match poly.binary_search_by(|(range, _)| {
            if range.contains(&x) {
                core::cmp::Ordering::Equal
            } else if x < range.start {
                core::cmp::Ordering::Greater
            } else {
                core::cmp::Ordering::Less
            }
        }) {
            Ok(index) => index,
            Err(index) => max(0, min(index, poly.len() - 1)),
        };

        &poly[index].1
    }

    fn number_of_polynomials(&self) -> usize {
        self.get_ranged_polynomials().len()
    }

    fn get_ranged_polynomials(&self) -> &Vec<(Range<f64>, Polynomial)>;
}

// =============================================================================
// High level functions
// =============================================================================
/// Cubic Spline (Natural)
///
/// # Description
///
/// Implement traits of Natural cubic splines, by Arne Morten Kvarving.
///
/// # Type
/// `(&[f64], &[f64]) -> Cubic Spline`
///
/// # Examples
/// ```
/// #[macro_use]
/// extern crate peroxide;
/// use peroxide::fuga::*;
///
/// fn main() -> Result<(), Box<dyn Error>> {
///     let x = c!(0.9, 1.3, 1.9, 2.1);
///     let y = c!(1.3, 1.5, 1.85, 2.1);
///
///     let s = cubic_spline(&x, &y)?;
///
///     let new_x = c!(1, 1.5, 2.0);
///
///     // Generate Cubic polynomial
///     for t in new_x.iter() {
///         s.polynomial_at(*t).print();
///     }
///     // -0.2347x^3 + 0.6338x^2 - 0.0329x + 0.9873
///     // 0.9096x^3 - 3.8292x^2 + 5.7691x - 1.5268
///     // -2.2594x^3 + 14.2342x^2 - 28.5513x + 20.2094
///
///     // Evaluation
///     for t in new_x.iter() {
///         s.eval(*t).print();
///     }
///
///     Ok(())
/// }
/// ```
pub fn cubic_spline(node_x: &[f64], node_y: &[f64]) -> Result<CubicSpline> {
    CubicSpline::from_nodes(node_x, node_y)
}

pub fn cubic_hermite_spline(
    node_x: &[f64],
    node_y: &[f64],
    slope_method: SlopeMethod,
) -> Result<CubicHermiteSpline> {
    CubicHermiteSpline::from_nodes(node_x, node_y, slope_method)
}

// =============================================================================
// Cubic Spline
// =============================================================================
/// Cubic Spline (Natural)
///
/// # Description
///
/// Implement traits of Natural cubic splines, by Arne Morten Kvarving.
///
/// # Type
/// `(&[f64], &[f64]) -> Cubic Spline`
///
/// # Examples
/// ```
/// #[macro_use]
/// extern crate peroxide;
/// use peroxide::fuga::*;
///
/// fn main() -> Result<(), Box<dyn Error>> {
///     let x = c!(0.9, 1.3, 1.9, 2.1);
///     let y = c!(1.3, 1.5, 1.85, 2.1);
///
///     let s = cubic_spline(&x, &y)?;
///
///     let new_x = c!(1, 1.5, 2.0);
///
///     // Generate Cubic polynomial
///     for t in new_x.iter() {
///         s.polynomial_at(*t).print();
///     }
///     // -0.2347x^3 + 0.6338x^2 - 0.0329x + 0.9873
///     // 0.9096x^3 - 3.8292x^2 + 5.7691x - 1.5268
///     // -2.2594x^3 + 14.2342x^2 - 28.5513x + 20.2094
///
///     // Evaluation
///     for t in new_x.iter() {
///         s.eval(*t).print();
///     }
///
///     Ok(())
/// }
/// ```
#[derive(Debug, Clone, Default)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct CubicSpline {
    polynomials: Vec<(Range<f64>, Polynomial)>,
}

impl PolynomialSpline for CubicSpline {
    fn get_ranged_polynomials(&self) -> &Vec<(Range<f64>, Polynomial)> {
        &self.polynomials
    }
}

#[derive(Debug, Copy, Clone)]
pub enum SplineError {
    NotEnoughNodes,
    NotEqualNodes,
    NotEqualSlopes,
    RedundantNodeX,
}

impl std::fmt::Display for SplineError {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        match self {
            SplineError::NotEnoughNodes => write!(f, "node_x has less than 3 elements"),
            SplineError::NotEqualNodes => write!(f, "node_x and node_y have different lengths"),
            SplineError::NotEqualSlopes => write!(f, "nodes and slopes have different lengths"),
            SplineError::RedundantNodeX => write!(f, "there are redundant nodes in node_x"),
        }
    }
}

impl CubicSpline {
    /// # Examples
    /// ```
    /// #[macro_use]
    /// extern crate peroxide;
    /// use peroxide::fuga::*;
    ///
    /// fn main() -> Result<(), Box<dyn Error>> {
    ///     let x = c!(0.9, 1.3, 1.9, 2.1);
    ///     let y = c!(1.3, 1.5, 1.85, 2.1);
    ///
    ///     let s = CubicSpline::from_nodes(&x, &y)?;
    ///
    ///     for i in 0 .. 4 {
    ///         println!("{}", s.eval(i as f64 / 2.0));
    ///     }
    ///
    ///     Ok(())
    /// }
    /// ```
    pub fn from_nodes(node_x: &[f64], node_y: &[f64]) -> Result<Self> {
        let polynomials = CubicSpline::cubic_spline(node_x, node_y)?;
        Ok(CubicSpline {
            polynomials: zip_range(node_x, &polynomials),
        })
    }

    fn cubic_spline(node_x: &[f64], node_y: &[f64]) -> Result<Vec<Polynomial>> {
        //! Pre calculated variables
        //! node_x: n+1
        //! node_y: n+1
        //! h     : n
        //! b     : n
        //! v     : n
        //! u     : n
        //! z     : n+1
        let n = node_x.len() - 1;
        if n < 2 {
            bail!(NotEnoughNodes);
        }
        if n != node_y.len() - 1 {
            bail!(NotEqualNodes);
        }

        // Pre-calculations
        let mut h = vec![0f64; n];
        let mut b = vec![0f64; n];
        let mut v = vec![0f64; n];
        let mut u = vec![0f64; n];
        for i in 0..n {
            if i == 0 {
                h[i] = node_x[i + 1] - node_x[i];
                b[i] = (node_y[i + 1] - node_y[i]) / h[i];
            } else {
                h[i] = node_x[i + 1] - node_x[i];
                b[i] = (node_y[i + 1] - node_y[i]) / h[i];
                v[i] = 2. * (h[i] + h[i - 1]);
                u[i] = 6. * (b[i] - b[i - 1]);
            }
        }

        // Tri-diagonal matrix
        let mut m = matrix(vec![0f64; (n - 1) * (n - 1)], n - 1, n - 1, Col);
        for i in 0..n - 2 {
            m[(i, i)] = v[i + 1];
            m[(i + 1, i)] = h[i + 1];
            m[(i, i + 1)] = h[i + 1];
        }
        m[(n - 2, n - 2)] = v[n - 1];

        // Calculate z
        let z_inner = m.inv() * Vec::from(&u[1..]);
        let mut z = vec![0f64];
        z.extend(&z_inner);
        z.push(0f64);

        // Declare empty spline
        let mut s: Vec<Polynomial> = Vec::new();

        // Main process
        for i in 0..n {
            // Memoization
            let t_i = node_x[i];
            let t_i1 = node_x[i + 1];
            let z_i = z[i];
            let z_i1 = z[i + 1];
            let h_i = h[i];
            let y_i = node_y[i];
            let y_i1 = node_y[i + 1];
            let temp1 = poly(vec![1f64, -t_i]);
            let temp2 = poly(vec![1f64, -t_i1]);

            let term1 = temp1.powi(3) * (z_i1 / (6f64 * h_i));
            let term2 = temp2.powi(3) * (-z_i / (6f64 * h_i));
            let term3 = temp1 * (y_i1 / h_i - z_i1 * h_i / 6.);
            let term4 = temp2 * (-y_i / h_i + h_i * z_i / 6.0);

            s.push(term1 + term2 + term3 + term4);
        }
        Ok(s)
    }

    /// Extends the spline with the given nodes.
    ///
    /// # Description
    ///
    /// The method ensures that the transition between each polynomial is smooth and that the spline
    /// interpolation of the new nodes is calculated around `x = 0` in order to avoid that
    /// successive spline extensions with large x values become inaccurate.
    pub fn extend_with_nodes(&mut self, node_x: Vec<f64>, node_y: Vec<f64>) -> Result<()> {
        let mut ext_node_x = Vec::with_capacity(node_x.len() + 1);
        let mut ext_node_y = Vec::with_capacity(node_x.len() + 1);

        let (r, polynomial) = &self.polynomials[self.polynomials.len() - 1];
        ext_node_x.push(0.0f64);
        ext_node_y.push(polynomial.eval(r.end));

        // translating the node towards x = 0 increases accuracy tremendously
        let tx = r.end;
        ext_node_x.extend(node_x.into_iter().map(|x| x - tx));
        ext_node_y.extend(node_y);

        let polynomials = zip_range(
            &ext_node_x,
            &CubicSpline::cubic_spline(&ext_node_x, &ext_node_y)?,
        );

        self.polynomials
            .extend(polynomials.into_iter().map(|(r, p)| {
                (
                    Range {
                        start: r.start + tx,
                        end: r.end + tx,
                    },
                    p.translate_x(tx),
                )
            }));

        Ok(())
    }
}

impl Into<Vec<Polynomial>> for CubicSpline {
    fn into(self) -> Vec<Polynomial> {
        self.polynomials
            .into_iter()
            .map(|(_, polynomial)| polynomial)
            .collect()
    }
}

impl From<Vec<(Range<f64>, Polynomial)>> for CubicSpline {
    fn from(polynomials: Vec<(Range<f64>, Polynomial)>) -> Self {
        CubicSpline { polynomials }
    }
}

impl Into<Vec<(Range<f64>, Polynomial)>> for CubicSpline {
    fn into(self) -> Vec<(Range<f64>, Polynomial)> {
        self.polynomials
    }
}

impl Index<usize> for CubicSpline {
    type Output = (Range<f64>, Polynomial);

    fn index(&self, index: usize) -> &Self::Output {
        &self.polynomials[index]
    }
}

impl Calculus for CubicSpline {
    fn derivative(&self) -> Self {
        let mut polynomials: Vec<(Range<f64>, Polynomial)> = self.clone().into();

        polynomials = polynomials
            .into_iter()
            .map(|(r, poly)| (r, poly.derivative()))
            .collect();

        Self::from(polynomials)
    }

    fn integral(&self) -> Self {
        let mut polynomials: Vec<(Range<f64>, Polynomial)> = self.clone().into();

        polynomials = polynomials
            .into_iter()
            .map(|(r, poly)| (r, poly.integral()))
            .collect();

        Self::from(polynomials)
    }

    fn integrate<T: Into<f64> + Copy>(&self, interval: (T, T)) -> f64 {
        let (a, b) = interval;
        let a = a.into();
        let b = b.into();

        let mut s = 0f64;
        for (r, p) in self.polynomials.iter() {
            if r.start > b {
                break;
            } else if r.end < a {
                continue;
            } else {
                // r.start <= b, r.end >= a
                let x = r.start.max(a);
                let y = r.end.min(b);
                s += p.integrate((x, y));
            }
        }
        s
    }
}

// =============================================================================
// Cubic Hermite Spline
// =============================================================================
#[derive(Debug, Clone, Default)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct CubicHermiteSpline {
    polynomials: Vec<(Range<f64>, Polynomial)>,
}

impl PolynomialSpline for CubicHermiteSpline {
    fn get_ranged_polynomials(&self) -> &Vec<(Range<f64>, Polynomial)> {
        &self.polynomials
    }
}

impl CubicHermiteSpline {
    pub fn from_nodes_with_slopes(node_x: &[f64], node_y: &[f64], m: &[f64]) -> Result<Self> {
        let n = node_x.len();
        if n < 3 {
            bail!(NotEnoughNodes);
        }
        if n != node_y.len() {
            bail!(NotEqualNodes);
        }
        if n != m.len() {
            bail!(NotEqualSlopes);
        }

        let mut r = vec![Range::default(); node_x.len() - 1];
        let mut u = vec![Polynomial::default(); node_x.len() - 1];

        for i in 0..node_x.len() - 1 {
            let a_i = node_y[i];
            let b_i = m[i];
            let dx = node_x[i + 1] - node_x[i];
            let dy = node_y[i + 1] - node_y[i];
            let c_i = (3f64 * dy / dx - 2f64 * m[i] - m[i + 1]) / dx;
            let d_i = (m[i] + m[i + 1] - 2f64 * dy / dx) / dx.powi(2);

            let p = Polynomial::new(vec![1f64, -node_x[i]]);

            r[i] = Range {
                start: node_x[i],
                end: node_x[i + 1],
            };
            u[i] = p.powi(3) * d_i + p.powi(2) * c_i + p.clone() * b_i;
            u[i].coef[3] += a_i;
        }

        Ok(CubicHermiteSpline {
            polynomials: r.into_iter().zip(u).collect(),
        })
    }

    pub fn from_nodes(node_x: &[f64], node_y: &[f64], slope_method: SlopeMethod) -> Result<Self> {
        match slope_method {
            SlopeMethod::Akima => CubicHermiteSpline::from_nodes_with_slopes(
                node_x,
                node_y,
                &akima_slopes(node_x, node_y)?,
            ),
            SlopeMethod::Quadratic => CubicHermiteSpline::from_nodes_with_slopes(
                node_x,
                node_y,
                &quadratic_slopes(node_x, node_y)?,
            ),
        }
    }
}

impl Into<Vec<Polynomial>> for CubicHermiteSpline {
    fn into(self) -> Vec<Polynomial> {
        self.polynomials
            .into_iter()
            .map(|(_, polynomial)| polynomial)
            .collect()
    }
}

impl From<Vec<(Range<f64>, Polynomial)>> for CubicHermiteSpline {
    fn from(polynomials: Vec<(Range<f64>, Polynomial)>) -> Self {
        CubicHermiteSpline { polynomials }
    }
}

impl Into<Vec<(Range<f64>, Polynomial)>> for CubicHermiteSpline {
    fn into(self) -> Vec<(Range<f64>, Polynomial)> {
        self.polynomials
    }
}

impl Index<usize> for CubicHermiteSpline {
    type Output = (Range<f64>, Polynomial);

    fn index(&self, index: usize) -> &Self::Output {
        &self.polynomials[index]
    }
}

impl Calculus for CubicHermiteSpline {
    fn derivative(&self) -> Self {
        let mut polynomials: Vec<(Range<f64>, Polynomial)> = self.clone().into();

        polynomials = polynomials
            .into_iter()
            .map(|(r, poly)| (r, poly.derivative()))
            .collect();

        Self::from(polynomials)
    }

    fn integral(&self) -> Self {
        let mut polynomials: Vec<(Range<f64>, Polynomial)> = self.clone().into();

        polynomials = polynomials
            .into_iter()
            .map(|(r, poly)| (r, poly.integral()))
            .collect();

        Self::from(polynomials)
    }

    fn integrate<T: Into<f64> + Copy>(&self, interval: (T, T)) -> f64 {
        let (a, b) = interval;
        let a = a.into();
        let b = b.into();

        let mut s = 0f64;
        for (r, p) in self.polynomials.iter() {
            if r.start > b {
                break;
            } else if r.end < a {
                continue;
            } else {
                // r.start <= b, r.end >= a
                let x = r.start.max(a);
                let y = r.end.min(b);
                s += p.integrate((x, y));
            }
        }
        s
    }
}

// =============================================================================
// Estimate Slopes
// =============================================================================
#[derive(Debug, Copy, Clone, PartialEq, Eq)]
pub enum SlopeMethod {
    Akima,
    Quadratic,
}

fn akima_slopes(x: &[f64], y: &[f64]) -> Result<Vec<f64>> {
    if x.len() < 3 {
        bail!(NotEnoughNodes);
    }

    let mut m = vec![0f64; x.len()];
    let mut s = vec![0f64; x.len() + 3]; // -2, -1, 0, ..., x.len()-1, x.len()

    let l_i = lagrange_polynomial(x[0..3].to_vec(), y[0..3].to_vec());
    let l_f = lagrange_polynomial(x[x.len() - 3..].to_vec(), y[y.len() - 3..].to_vec());

    let x_i = x[0] - (x[1] - x[0]) / 10f64;
    let x_ii = x_i - (x[1] - x[0]) / 10f64;
    let x_f = x[x.len() - 1] + (x[x.len() - 1] - x[x.len() - 2]) / 10f64;
    let x_ff = x_f + (x[x.len() - 1] - x[x.len() - 2]) / 10f64;

    let y_i = l_i.eval(x_i);
    let y_ii = l_i.eval(x_ii);
    let y_f = l_f.eval(x_f);
    let y_ff = l_f.eval(x_ff);

    let new_x = concat(&concat(&[x_ii, x_i], x), &[x_f, x_ff]);
    let new_y = concat(&concat(&[y_ii, y_i], y), &[y_f, y_ff]);

    for i in 0..new_x.len() - 1 {
        let dx = new_x[i + 1] - new_x[i];
        if dx == 0f64 {
            bail!(RedundantNodeX);
        }
        s[i] = (new_y[i + 1] - new_y[i]) / dx;
    }

    for (i, m_i) in m.iter_mut().enumerate() {
        let j = i + 2;
        let ds_f = (s[j + 1] - s[j]).abs();
        let ds_i = (s[j - 1] - s[j - 2]).abs();

        *m_i = if ds_f == 0f64 && ds_i == 0f64 {
            (s[j - 1] + s[j]) / 2f64
        } else {
            (ds_f * s[j - 1] + ds_i * s[j]) / (ds_f + ds_i)
        };
    }
    Ok(m)
}

fn quadratic_slopes(x: &[f64], y: &[f64]) -> Result<Vec<f64>> {
    if x.len() < 3 {
        bail!(NotEnoughNodes);
    }
    let mut m = vec![0f64; x.len()];
    let q_i = lagrange_polynomial(x[0..3].to_vec(), y[0..3].to_vec());
    let q_f = lagrange_polynomial(x[x.len() - 3..].to_vec(), y[y.len() - 3..].to_vec());

    m[0] = q_i.derivative().eval(x[0]);
    m[x.len() - 1] = q_f.derivative().eval(x[x.len() - 1]);

    for i in 1..x.len() - 1 {
        let q = lagrange_polynomial(x[i - 1..i + 2].to_vec(), y[i - 1..i + 2].to_vec());
        m[i] = q.derivative().eval(x[i]);
    }

    Ok(m)
}

// =============================================================================
// B-Spline
// =============================================================================
/// Unit Cubic Basis Function
///
/// # Description
/// Unit cubic basis function from Irwin-Hall distribution (n=4).
/// For general interval, we substitute t = 4 * (x - a) / (b - a).
///
/// # Reference
/// [Wikipedia](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution#Special_cases)
#[derive(Debug, Copy, Clone)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct UnitCubicBasis {
    pub x_min: f64,
    pub x_max: f64,
    pub scale: f64,
}

impl UnitCubicBasis {
    pub fn new(x_min: f64, x_max: f64, scale: f64) -> Self {
        Self {
            x_min,
            x_max,
            scale,
        }
    }

    pub fn eval(&self, x: f64) -> f64 {
        let t = 4f64 * (x - self.x_min) / (self.x_max - self.x_min);

        let result = if (0f64..1f64).contains(&t) {
            t.powi(3) / 6f64
        } else if (1f64..2f64).contains(&t) {
            (-3f64 * t.powi(3) + 12f64 * t.powi(2) - 12f64 * t + 4f64) / 6f64
        } else if (2f64..3f64).contains(&t) {
            (3f64 * t.powi(3) - 24f64 * t.powi(2) + 60f64 * t - 44f64) / 6f64
        } else if (3f64..4f64).contains(&t) {
            (4f64 - t).powi(3) / 6f64
        } else {
            0f64
        };

        self.scale * result
    }

    pub fn eval_vec(&self, x: &[f64]) -> Vec<f64> {
        x.iter().map(|x| self.eval(*x)).collect()
    }
}

/// Uniform Cubic B-Spline basis functions
///
/// # Example
///
/// ```rust
/// use peroxide::fuga::*;
/// use core::ops::Range;
///
/// # #[allow(unused_variables)]
/// fn main() -> anyhow::Result<()> {
///     let cubic_b_spline = CubicBSplineBases::from_interval((0f64, 1f64), 5);
///     let x = linspace(0f64, 1f64, 1000);
///     let y = cubic_b_spline.eval_vec(&x);
///
/// #   #[cfg(feature = "plot")] {
///     let mut plt = Plot2D::new();
///     plt.set_domain(x.clone());
///
///     for basis in &cubic_b_spline.bases {
///         plt.insert_image(basis.eval_vec(&x));
///     }
///
///     plt
///         .insert_image(y)
///         .set_xlabel(r"$x$")
///         .set_ylabel(r"$y$")
///         .set_style(PlotStyle::Nature)
///         .tight_layout()
///         .set_dpi(600)
///         .set_path("example_data/cubic_b_spline.png")
///         .savefig()?;
/// #   }
///     Ok(())
/// }
/// ```
#[derive(Debug, Clone)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct CubicBSplineBases {
    pub ranges: Vec<Range<f64>>,
    pub bases: Vec<UnitCubicBasis>,
}

impl CubicBSplineBases {
    /// Create new Cubic B-Spline basis functions
    pub fn new(ranges: Vec<Range<f64>>, bases: Vec<UnitCubicBasis>) -> Self {
        Self { ranges, bases }
    }

    /// Create new Cubic B-Spline basis functions for `[a, b]`
    pub fn from_interval((a, b): (f64, f64), num_bases: usize) -> Self {
        let nodes = linspace(a, b, num_bases + 4);
        let (ranges, bases) = nodes
            .iter()
            .zip(nodes.iter().skip(4))
            .map(|(a, b)| {
                (
                    Range { start: *a, end: *b },
                    UnitCubicBasis::new(*a, *b, 1f64),
                )
            })
            .unzip();

        Self::new(ranges, bases)
    }

    /// Rescale all basis functions
    ///
    /// # Arguments
    /// - `scale_vec` - scale vector
    pub fn rescale(&mut self, scale_vec: &[f64]) -> Result<()> {
        if scale_vec.len() != self.bases.len() {
            bail!("The number of scales should be equal to the number of basis functions");
        }

        for (basis, scale) in self.bases.iter_mut().zip(scale_vec) {
            basis.scale = *scale;
        }

        Ok(())
    }

    pub fn eval(&self, x: f64) -> f64 {
        self.ranges
            .iter()
            .enumerate()
            .filter(|(_, range)| range.contains(&x))
            .fold(0f64, |acc, (i, _)| {
                let basis = &self.bases[i];
                acc + basis.eval(x)
            })
    }

    pub fn eval_vec(&self, x: &[f64]) -> Vec<f64> {
        x.iter().map(|x| self.eval(*x)).collect()
    }
}

/// B-Spline
///
/// # Description
/// : B-Spline is a generalization of Bezier curve.
///
/// # Caution
/// - Let K = the number of knots, C = the number of control points
/// - For open, K = C + degree + 1 (C = K - degree - 1)
/// - For clamped, K + 2 * degree = C + degree + 1 (C = K + degree - 1)
///
/// # Example
/// ```
/// use peroxide::fuga::*;
/// use anyhow::Result;
///
/// fn main() -> Result<()> {
///     let degree = 3;
///     let knots = vec![0f64, 0.5, 1f64];
///     let control_points = vec![
///         vec![0f64, 0f64],
///         vec![0.3, 0.5],
///         vec![0.5, 0.7],
///         vec![0.7, 0.3],
///         vec![1f64, 1f64],
///     ];
///     let spline = BSpline::clamped(degree, knots, control_points)?;
///     
///     let t = linspace(0f64, 1f64, 100);
///     let (x, y): (Vec<f64>, Vec<f64>) = spline.eval_vec(&t).into_iter().unzip();
///     assert_eq!(x.len(), 100);
///     assert_eq!(y.len(), 100);
///
///     Ok(())
/// }
/// ```
#[derive(Debug, Clone, Default)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct BSpline {
    pub degree: usize,
    pub knots: Vec<f64>,
    pub control_points: Vec<Vec<f64>>,
}

impl BSpline {
    /// Create new open B-Spline
    ///
    /// # Arguments
    /// - `degree` - Degree of B-Spline
    /// - `knots` - Knots (length = K)
    /// - `control_points` - Control points (length = C)
    ///
    /// # Caution
    /// - K = C + degree + 1 (C = K - degree - 1)
    pub fn open(degree: usize, knots: Vec<f64>, control_points: Vec<Vec<f64>>) -> Result<Self> {
        if knots.len() != control_points.len() + degree + 1 {
            bail!("The number of knots ({}) should be equal to the number of control points ({}) + degree ({}) + 1", knots.len(), control_points.len(), degree);
        }

        Ok(Self {
            degree,
            knots,
            control_points,
        })
    }

    /// Create new clamped B-Spline
    ///
    /// # Arguments
    /// - `degree` - Degree of B-Spline
    /// - `knots` - Knots (length = K)
    /// - `control_points` - Control points (length = C)
    ///
    /// # Caution
    /// - K + 2 * degree = C + degree + 1 (C = K + degree - 1)
    pub fn clamped(degree: usize, knots: Vec<f64>, control_points: Vec<Vec<f64>>) -> Result<Self> {
        let mut knots = knots.clone();

        if knots.len() != control_points.len() + 1 - degree {
            bail!("For clamped, the number of knots ({}) should be equal to the number of control points ({}) + 1 - degree ({})", knots.len(), control_points.len(), degree);
        }
        for _ in 0..degree {
            knots.insert(0, knots[0]);
            knots.push(knots[knots.len() - 1]);
        }
        if knots.len() != control_points.len() + degree + 1 {
            bail!("The number of knots ({}) should be equal to the number of control points ({}) + degree ({}) + 1", knots.len(), control_points.len(), degree);
        }

        Ok(Self {
            degree,
            knots,
            control_points,
        })
    }

    /// Obtain basis function via Cox-de Boor algorithm
    #[allow(non_snake_case)]
    pub fn cox_de_boor(&self, t: f64, i: usize) -> f64 {
        let p = self.degree;
        let mut B = vec![vec![0f64; p + 1]; p + 1];

        // Initialize 0th degree basis functions
        for (j, B_j) in B.iter_mut().enumerate() {
            if (self.knots[i + j] <= t && t < self.knots[i + j + 1])
                || (i + j == self.knots.len() - (p + 2) && t == self.knots[i + j + 1])
            {
                B_j[0] = 1f64;
            } else {
                B_j[0] = 0f64;
            }
        }

        // Compute the basis functions for higher degree
        for k in 1..=p {
            for j in 0..=(p - k) {
                let a = if self.knots[i + j + k] == self.knots[i + j] {
                    0f64
                } else {
                    (t - self.knots[i + j]) / (self.knots[i + j + k] - self.knots[i + j])
                };

                let b = if self.knots[i + j + k + 1] == self.knots[i + j + 1] {
                    0f64
                } else {
                    (self.knots[i + j + k + 1] - t)
                        / (self.knots[i + j + k + 1] - self.knots[i + j + 1])
                };

                B[j][k] = a * B[j][k - 1] + b * B[j + 1][k - 1];
            }
        }

        B[0][p]
    }
}

impl Spline<(f64, f64)> for BSpline {
    #[allow(non_snake_case)]
    fn eval(&self, t: f64) -> (f64, f64) {
        let n = self.control_points.len();

        let mut x = 0f64;
        let mut y = 0f64;
        for i in 0..n {
            let B = self.cox_de_boor(t, i);
            x += B * self.control_points[i][0];
            y += B * self.control_points[i][1];
        }

        (x, y)
    }
}