smartcore 0.3.0

//! # Eigen Decomposition
//!
//! Eigendecomposition is one of the most useful matrix factorization methods in machine learning that decomposes a matrix into eigenvectors and eigenvalues.
//! This decomposition plays an important role in the the [Principal Component Analysis (PCA)](../../decomposition/pca/index.html).
//!
//! Eigendecomposition decomposes a square matrix into a set of eigenvectors and eigenvalues.
//!
//! \\[A = Q \Lambda Q^{-1}\\]
//!
//! where \\(Q\\) is a matrix comprised of the eigenvectors, \\(\Lambda\\) is a diagonal matrix comprised of the eigenvalues along the diagonal,
//! and \\(Q{-1}\\) is the inverse of the matrix comprised of the eigenvectors.
//!
//! Example:
//! ```
//! use smartcore::linalg::basic::matrix::DenseMatrix;
//! use smartcore::linalg::traits::evd::*;
//!
//! let A = DenseMatrix::from_2d_array(&[
//!                  &[0.9000, 0.4000, 0.7000],
//!                  &[0.4000, 0.5000, 0.3000],
//!                  &[0.7000, 0.3000, 0.8000],
//!         ]);
//!
//! let evd = A.evd(true).unwrap();
//! let eigenvectors: DenseMatrix<f64> = evd.V;
//! let eigenvalues: Vec<f64> = evd.d;
//! ```
//!
//! ## References:
//! * ["Numerical Recipes: The Art of Scientific Computing",  Press W.H., Teukolsky S.A., Vetterling W.T, Flannery B.P, 3rd ed., Section 11 Eigensystems](http://numerical.recipes/)
//! * ["Introduction to Linear Algebra", Gilbert Strang, 5rd ed., ch. 6 Eigenvalues and Eigenvectors](https://math.mit.edu/~gs/linearalgebra/)
//!
//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
#![allow(non_snake_case)]

use crate::error::Failed;
use crate::linalg::basic::arrays::Array2;
use crate::numbers::basenum::Number;
use crate::numbers::realnum::RealNumber;
use num::complex::Complex;
use std::fmt::Debug;

#[derive(Debug, Clone)]
/// Results of eigen decomposition
pub struct EVD<T: Number + RealNumber, M: Array2<T>> {
    /// Real part of eigenvalues.
    pub d: Vec<T>,
    /// Imaginary part of eigenvalues.
    pub e: Vec<T>,
    /// Eigenvectors
    pub V: M,
}

/// Trait that implements EVD decomposition routine for any matrix.
pub trait EVDDecomposable<T: Number + RealNumber>: Array2<T> {
    /// Compute the eigen decomposition of a square matrix.
    /// * `symmetric` - whether the matrix is symmetric
    fn evd(&self, symmetric: bool) -> Result<EVD<T, Self>, Failed> {
        self.clone().evd_mut(symmetric)
    }

    /// Compute the eigen decomposition of a square matrix. The input matrix
    /// will be used for factorization.
    /// * `symmetric` - whether the matrix is symmetric
    fn evd_mut(mut self, symmetric: bool) -> Result<EVD<T, Self>, Failed> {
        let (nrows, ncols) = self.shape();
        if ncols != nrows {
            panic!("Matrix is not square: {} x {}", nrows, ncols);
        }

        let n = nrows;
        let mut d = vec![T::zero(); n];
        let mut e = vec![T::zero(); n];

        let mut V;
        if symmetric {
            V = self;
            // Tridiagonalize.
            tred2(&mut V, &mut d, &mut e);
            // Diagonalize.
            tql2(&mut V, &mut d, &mut e);
        } else {
            let scale = balance(&mut self);

            let perm = elmhes(&mut self);

            V = Self::eye(n);

            eltran(&self, &mut V, &perm);

            hqr2(&mut self, &mut V, &mut d, &mut e);
            balbak(&mut V, &scale);
            sort(&mut d, &mut e, &mut V);
        }

        Ok(EVD { V, d, e })
    }
}

fn tred2<T: Number + RealNumber, M: Array2<T>>(V: &mut M, d: &mut [T], e: &mut [T]) {
    let (n, _) = V.shape();
    for (i, d_i) in d.iter_mut().enumerate().take(n) {
        *d_i = *V.get((n - 1, i));
    }

    for i in (1..n).rev() {
        let mut scale = T::zero();
        let mut h = T::zero();
        for d_k in d.iter().take(i) {
            scale += d_k.abs();
        }
        if scale == T::zero() {
            e[i] = d[i - 1];
            for (j, d_j) in d.iter_mut().enumerate().take(i) {
                *d_j = *V.get((i - 1, j));
                V.set((i, j), T::zero());
                V.set((j, i), T::zero());
            }
        } else {
            for d_k in d.iter_mut().take(i) {
                *d_k /= scale;
                h += (*d_k) * (*d_k);
            }
            let mut f = d[i - 1];
            let mut g = h.sqrt();
            if f > T::zero() {
                g = -g;
            }
            e[i] = scale * g;
            h -= f * g;
            d[i - 1] = f - g;
            for e_j in e.iter_mut().take(i) {
                *e_j = T::zero();
            }

            for j in 0..i {
                f = d[j];
                V.set((j, i), f);
                g = e[j] + *V.get((j, j)) * f;
                for k in j + 1..=i - 1 {
                    g += *V.get((k, j)) * d[k];
                    e[k] += *V.get((k, j)) * f;
                }
                e[j] = g;
            }
            f = T::zero();
            for j in 0..i {
                e[j] /= h;
                f += e[j] * d[j];
            }
            let hh = f / (h + h);
            for j in 0..i {
                e[j] -= hh * d[j];
            }
            for j in 0..i {
                f = d[j];
                g = e[j];
                for k in j..=i - 1 {
                    V.sub_element_mut((k, j), f * e[k] + g * d[k]);
                }
                d[j] = *V.get((i - 1, j));
                V.set((i, j), T::zero());
            }
        }
        d[i] = h;
    }

    for i in 0..n - 1 {
        V.set((n - 1, i), *V.get((i, i)));
        V.set((i, i), T::one());
        let h = d[i + 1];
        if h != T::zero() {
            for (k, d_k) in d.iter_mut().enumerate().take(i + 1) {
                *d_k = *V.get((k, i + 1)) / h;
            }
            for j in 0..=i {
                let mut g = T::zero();
                for k in 0..=i {
                    g += *V.get((k, i + 1)) * *V.get((k, j));
                }
                for (k, d_k) in d.iter().enumerate().take(i + 1) {
                    V.sub_element_mut((k, j), g * (*d_k));
                }
            }
        }
        for k in 0..=i {
            V.set((k, i + 1), T::zero());
        }
    }
    for (j, d_j) in d.iter_mut().enumerate().take(n) {
        *d_j = *V.get((n - 1, j));
        V.set((n - 1, j), T::zero());
    }
    V.set((n - 1, n - 1), T::one());
    e[0] = T::zero();
}

fn tql2<T: Number + RealNumber, M: Array2<T>>(V: &mut M, d: &mut [T], e: &mut [T]) {
    let (n, _) = V.shape();
    for i in 1..n {
        e[i - 1] = e[i];
    }
    e[n - 1] = T::zero();

    let mut f = T::zero();
    let mut tst1 = T::zero();
    for l in 0..n {
        tst1 = T::max(tst1, d[l].abs() + e[l].abs());

        let mut m = l;

        loop {
            if m < n {
                if e[m].abs() <= tst1 * T::epsilon() {
                    break;
                }
                m += 1;
            } else {
                break;
            }
        }

        if m > l {
            let mut iter = 0;
            loop {
                iter += 1;
                if iter >= 30 {
                    panic!("Too many iterations");
                }

                let mut g = d[l];
                let mut p = (d[l + 1] - g) / (T::two() * e[l]);
                let mut r = p.hypot(T::one());
                if p < T::zero() {
                    r = -r;
                }
                d[l] = e[l] / (p + r);
                d[l + 1] = e[l] * (p + r);
                let dl1 = d[l + 1];
                let mut h = g - d[l];
                for d_i in d.iter_mut().take(n).skip(l + 2) {
                    *d_i -= h;
                }
                f += h;

                p = d[m];
                let mut c = T::one();
                let mut c2 = c;
                let mut c3 = c;
                let el1 = e[l + 1];
                let mut s = T::zero();
                let mut s2 = T::zero();
                for i in (l..m).rev() {
                    c3 = c2;
                    c2 = c;
                    s2 = s;
                    g = c * e[i];
                    h = c * p;
                    r = p.hypot(e[i]);
                    e[i + 1] = s * r;
                    s = e[i] / r;
                    c = p / r;
                    p = c * d[i] - s * g;
                    d[i + 1] = h + s * (c * g + s * d[i]);

                    for k in 0..n {
                        h = *V.get((k, i + 1));
                        V.set((k, i + 1), s * *V.get((k, i)) + c * h);
                        V.set((k, i), c * *V.get((k, i)) - s * h);
                    }
                }
                p = -s * s2 * c3 * el1 * e[l] / dl1;
                e[l] = s * p;
                d[l] = c * p;

                if e[l].abs() <= tst1 * T::epsilon() {
                    break;
                }
            }
        }
        d[l] += f;
        e[l] = T::zero();
    }

    for i in 0..n - 1 {
        let mut k = i;
        let mut p = d[i];
        for (j, d_j) in d.iter().enumerate().take(n).skip(i + 1) {
            if *d_j > p {
                k = j;
                p = *d_j;
            }
        }
        if k != i {
            d[k] = d[i];
            d[i] = p;
            for j in 0..n {
                p = *V.get((j, i));
                V.set((j, i), *V.get((j, k)));
                V.set((j, k), p);
            }
        }
    }
}

fn balance<T: Number + RealNumber, M: Array2<T>>(A: &mut M) -> Vec<T> {
    let radix = T::two();
    let sqrdx = radix * radix;

    let (n, _) = A.shape();

    let mut scale = vec![T::one(); n];

    let t = T::from(0.95).unwrap();

    let mut done = false;
    while !done {
        done = true;
        for (i, scale_i) in scale.iter_mut().enumerate().take(n) {
            let mut r = T::zero();
            let mut c = T::zero();
            for j in 0..n {
                if j != i {
                    c += A.get((j, i)).abs();
                    r += A.get((i, j)).abs();
                }
            }
            if c != T::zero() && r != T::zero() {
                let mut g = r / radix;
                let mut f = T::one();
                let s = c + r;
                while c < g {
                    f *= radix;
                    c *= sqrdx;
                }
                g = r * radix;
                while c > g {
                    f /= radix;
                    c /= sqrdx;
                }
                if (c + r) / f < t * s {
                    done = false;
                    g = T::one() / f;
                    *scale_i *= f;
                    for j in 0..n {
                        A.mul_element_mut((i, j), g);
                    }
                    for j in 0..n {
                        A.mul_element_mut((j, i), f);
                    }
                }
            }
        }
    }

    scale
}

fn elmhes<T: Number + RealNumber, M: Array2<T>>(A: &mut M) -> Vec<usize> {
    let (n, _) = A.shape();
    let mut perm = vec![0; n];

    for (m, perm_m) in perm.iter_mut().enumerate().take(n - 1).skip(1) {
        let mut x = T::zero();
        let mut i = m;
        for j in m..n {
            if A.get((j, m - 1)).abs() > x.abs() {
                x = *A.get((j, m - 1));
                i = j;
            }
        }
        *perm_m = i;
        if i != m {
            for j in (m - 1)..n {
                A.swap((i, j), (m, j));
            }
            for j in 0..n {
                A.swap((j, i), (j, m));
            }
        }
        if x != T::zero() {
            for i in (m + 1)..n {
                let mut y = *A.get((i, m - 1));
                if y != T::zero() {
                    y /= x;
                    A.set((i, m - 1), y);
                    for j in m..n {
                        A.sub_element_mut((i, j), y * *A.get((m, j)));
                    }
                    for j in 0..n {
                        A.add_element_mut((j, m), y * *A.get((j, i)));
                    }
                }
            }
        }
    }

    perm
}

fn eltran<T: Number + RealNumber, M: Array2<T>>(A: &M, V: &mut M, perm: &[usize]) {
    let (n, _) = A.shape();
    for mp in (1..n - 1).rev() {
        for k in mp + 1..n {
            V.set((k, mp), *A.get((k, mp - 1)));
        }
        let i = perm[mp];
        if i != mp {
            for j in mp..n {
                V.set((mp, j), *V.get((i, j)));
                V.set((i, j), T::zero());
            }
            V.set((i, mp), T::one());
        }
    }
}

fn hqr2<T: Number + RealNumber, M: Array2<T>>(A: &mut M, V: &mut M, d: &mut [T], e: &mut [T]) {
    let (n, _) = A.shape();
    let mut z = T::zero();
    let mut s = T::zero();
    let mut r = T::zero();
    let mut q = T::zero();
    let mut p = T::zero();
    let mut anorm = T::zero();

    for i in 0..n {
        for j in i32::max(i as i32 - 1, 0)..n as i32 {
            anorm += A.get((i, j as usize)).abs();
        }
    }

    let mut nn = n - 1;
    let mut t = T::zero();
    'outer: loop {
        let mut its = 0;
        loop {
            let mut l = nn;
            while l > 0 {
                s = A.get((l - 1, l - 1)).abs() + A.get((l, l)).abs();
                if s == T::zero() {
                    s = anorm;
                }
                if A.get((l, l - 1)).abs() <= T::epsilon() * s {
                    A.set((l, l - 1), T::zero());
                    break;
                }
                l -= 1;
            }
            let mut x = *A.get((nn, nn));
            if l == nn {
                d[nn] = x + t;
                A.set((nn, nn), x + t);
                if nn == 0 {
                    break 'outer;
                } else {
                    nn -= 1;
                }
            } else {
                let mut y = *A.get((nn - 1, nn - 1));
                let mut w = *A.get((nn, nn - 1)) * *A.get((nn - 1, nn));
                if l == nn - 1 {
                    p = T::half() * (y - x);
                    q = p * p + w;
                    z = q.abs().sqrt();
                    x += t;
                    A.set((nn, nn), x);
                    A.set((nn - 1, nn - 1), y + t);
                    if q >= T::zero() {
                        z = p + <T as RealNumber>::copysign(z, p);
                        d[nn - 1] = x + z;
                        d[nn] = x + z;
                        if z != T::zero() {
                            d[nn] = x - w / z;
                        }
                        x = *A.get((nn, nn - 1));
                        s = x.abs() + z.abs();
                        p = x / s;
                        q = z / s;
                        r = (p * p + q * q).sqrt();
                        p /= r;
                        q /= r;
                        for j in nn - 1..n {
                            z = *A.get((nn - 1, j));
                            A.set((nn - 1, j), q * z + p * *A.get((nn, j)));
                            A.set((nn, j), q * *A.get((nn, j)) - p * z);
                        }
                        for i in 0..=nn {
                            z = *A.get((i, nn - 1));
                            A.set((i, nn - 1), q * z + p * *A.get((i, nn)));
                            A.set((i, nn), q * *A.get((i, nn)) - p * z);
                        }
                        for i in 0..n {
                            z = *V.get((i, nn - 1));
                            V.set((i, nn - 1), q * z + p * *V.get((i, nn)));
                            V.set((i, nn), q * *V.get((i, nn)) - p * z);
                        }
                    } else {
                        d[nn] = x + p;
                        e[nn] = -z;
                        d[nn - 1] = d[nn];
                        e[nn - 1] = -e[nn];
                    }

                    if nn <= 1 {
                        break 'outer;
                    } else {
                        nn -= 2;
                    }
                } else {
                    if its == 30 {
                        panic!("Too many iterations in hqr");
                    }
                    if its == 10 || its == 20 {
                        t += x;
                        for i in 0..nn + 1 {
                            A.sub_element_mut((i, i), x);
                        }
                        s = A.get((nn, nn - 1)).abs() + A.get((nn - 1, nn - 2)).abs();
                        y = T::from_f64(0.75).unwrap() * s;
                        x = T::from_f64(0.75).unwrap() * s;
                        w = T::from_f64(-0.4375).unwrap() * s * s;
                    }
                    its += 1;
                    let mut m = nn - 2;
                    while m >= l {
                        z = *A.get((m, m));
                        r = x - z;
                        s = y - z;
                        p = (r * s - w) / *A.get((m + 1, m)) + *A.get((m, m + 1));
                        q = *A.get((m + 1, m + 1)) - z - r - s;
                        r = *A.get((m + 2, m + 1));
                        s = p.abs() + q.abs() + r.abs();
                        p /= s;
                        q /= s;
                        r /= s;
                        if m == l {
                            break;
                        }
                        let u = A.get((m, m - 1)).abs() * (q.abs() + r.abs());
                        let v = p.abs()
                            * (A.get((m - 1, m - 1)).abs() + z.abs() + A.get((m + 1, m + 1)).abs());
                        if u <= T::epsilon() * v {
                            break;
                        }
                        m -= 1;
                    }
                    for i in m..nn - 1 {
                        A.set((i + 2, i), T::zero());
                        if i != m {
                            A.set((i + 2, i - 1), T::zero());
                        }
                    }
                    for k in m..nn {
                        if k != m {
                            p = *A.get((k, k - 1));
                            q = *A.get((k + 1, k - 1));
                            r = T::zero();
                            if k + 1 != nn {
                                r = *A.get((k + 2, k - 1));
                            }
                            x = p.abs() + q.abs() + r.abs();
                            if x != T::zero() {
                                p /= x;
                                q /= x;
                                r /= x;
                            }
                        }
                        let s = <T as RealNumber>::copysign((p * p + q * q + r * r).sqrt(), p);
                        if s != T::zero() {
                            if k == m {
                                if l != m {
                                    A.set((k, k - 1), -*A.get((k, k - 1)));
                                }
                            } else {
                                A.set((k, k - 1), -s * x);
                            }
                            p += s;
                            x = p / s;
                            y = q / s;
                            z = r / s;
                            q /= p;
                            r /= p;
                            for j in k..n {
                                p = *A.get((k, j)) + q * *A.get((k + 1, j));
                                if k + 1 != nn {
                                    p += r * *A.get((k + 2, j));
                                    A.sub_element_mut((k + 2, j), p * z);
                                }
                                A.sub_element_mut((k + 1, j), p * y);
                                A.sub_element_mut((k, j), p * x);
                            }

                            let mmin = if nn < k + 3 { nn } else { k + 3 };
                            for i in 0..(mmin + 1) {
                                p = x * *A.get((i, k)) + y * *A.get((i, k + 1));
                                if k + 1 != nn {
                                    p += z * *A.get((i, k + 2));
                                    A.sub_element_mut((i, k + 2), p * r);
                                }
                                A.sub_element_mut((i, k + 1), p * q);
                                A.sub_element_mut((i, k), p);
                            }
                            for i in 0..n {
                                p = x * *V.get((i, k)) + y * *V.get((i, k + 1));
                                if k + 1 != nn {
                                    p += z * *V.get((i, k + 2));
                                    V.sub_element_mut((i, k + 2), p * r);
                                }
                                V.sub_element_mut((i, k + 1), p * q);
                                V.sub_element_mut((i, k), p);
                            }
                        }
                    }
                }
            }
            if l + 1 >= nn {
                break;
            }
        }
    }

    if anorm != T::zero() {
        for nn in (0..n).rev() {
            p = d[nn];
            q = e[nn];
            let na = nn.wrapping_sub(1);
            if q == T::zero() {
                let mut m = nn;
                A.set((nn, nn), T::one());
                if nn > 0 {
                    let mut i = nn - 1;
                    loop {
                        let w = *A.get((i, i)) - p;
                        r = T::zero();
                        for j in m..=nn {
                            r += *A.get((i, j)) * *A.get((j, nn));
                        }
                        if e[i] < T::zero() {
                            z = w;
                            s = r;
                        } else {
                            m = i;

                            if e[i] == T::zero() {
                                t = w;
                                if t == T::zero() {
                                    t = T::epsilon() * anorm;
                                }
                                A.set((i, nn), -r / t);
                            } else {
                                let x = *A.get((i, i + 1));
                                let y = *A.get((i + 1, i));
                                q = (d[i] - p).powf(T::two()) + e[i].powf(T::two());
                                t = (x * s - z * r) / q;
                                A.set((i, nn), t);
                                if x.abs() > z.abs() {
                                    A.set((i + 1, nn), (-r - w * t) / x);
                                } else {
                                    A.set((i + 1, nn), (-s - y * t) / z);
                                }
                            }
                            t = A.get((i, nn)).abs();
                            if T::epsilon() * t * t > T::one() {
                                for j in i..=nn {
                                    A.div_element_mut((j, nn), t);
                                }
                            }
                        }
                        if i == 0 {
                            break;
                        } else {
                            i -= 1;
                        }
                    }
                }
            } else if q < T::zero() {
                let mut m = na;
                if A.get((nn, na)).abs() > A.get((na, nn)).abs() {
                    A.set((na, na), q / *A.get((nn, na)));
                    A.set((na, nn), -(*A.get((nn, nn)) - p) / *A.get((nn, na)));
                } else {
                    let temp = Complex::new(T::zero(), -*A.get((na, nn)))
                        / Complex::new(*A.get((na, na)) - p, q);
                    A.set((na, na), temp.re);
                    A.set((na, nn), temp.im);
                }
                A.set((nn, na), T::zero());
                A.set((nn, nn), T::one());
                if nn >= 2 {
                    for i in (0..nn - 1).rev() {
                        let w = *A.get((i, i)) - p;
                        let mut ra = T::zero();
                        let mut sa = T::zero();
                        for j in m..=nn {
                            ra += *A.get((i, j)) * *A.get((j, na));
                            sa += *A.get((i, j)) * *A.get((j, nn));
                        }
                        if e[i] < T::zero() {
                            z = w;
                            r = ra;
                            s = sa;
                        } else {
                            m = i;
                            if e[i] == T::zero() {
                                let temp = Complex::new(-ra, -sa) / Complex::new(w, q);
                                A.set((i, na), temp.re);
                                A.set((i, nn), temp.im);
                            } else {
                                let x = *A.get((i, i + 1));
                                let y = *A.get((i + 1, i));
                                let mut vr =
                                    (d[i] - p).powf(T::two()) + (e[i]).powf(T::two()) - q * q;
                                let vi = T::two() * q * (d[i] - p);
                                if vr == T::zero() && vi == T::zero() {
                                    vr = T::epsilon()
                                        * anorm
                                        * (w.abs() + q.abs() + x.abs() + y.abs() + z.abs());
                                }
                                let temp =
                                    Complex::new(x * r - z * ra + q * sa, x * s - z * sa - q * ra)
                                        / Complex::new(vr, vi);
                                A.set((i, na), temp.re);
                                A.set((i, nn), temp.im);
                                if x.abs() > z.abs() + q.abs() {
                                    A.set(
                                        (i + 1, na),
                                        (-ra - w * *A.get((i, na)) + q * *A.get((i, nn))) / x,
                                    );
                                    A.set(
                                        (i + 1, nn),
                                        (-sa - w * *A.get((i, nn)) - q * *A.get((i, na))) / x,
                                    );
                                } else {
                                    let temp = Complex::new(
                                        -r - y * *A.get((i, na)),
                                        -s - y * *A.get((i, nn)),
                                    ) / Complex::new(z, q);
                                    A.set((i + 1, na), temp.re);
                                    A.set((i + 1, nn), temp.im);
                                }
                            }
                        }
                        t = T::max(A.get((i, na)).abs(), A.get((i, nn)).abs());
                        if T::epsilon() * t * t > T::one() {
                            for j in i..=nn {
                                A.div_element_mut((j, na), t);
                                A.div_element_mut((j, nn), t);
                            }
                        }
                    }
                }
            }
        }

        for j in (0..n).rev() {
            for i in 0..n {
                z = T::zero();
                for k in 0..=j {
                    z += *V.get((i, k)) * *A.get((k, j));
                }
                V.set((i, j), z);
            }
        }
    }
}

fn balbak<T: Number + RealNumber, M: Array2<T>>(V: &mut M, scale: &[T]) {
    let (n, _) = V.shape();
    for (i, scale_i) in scale.iter().enumerate().take(n) {
        for j in 0..n {
            V.mul_element_mut((i, j), *scale_i);
        }
    }
}

fn sort<T: Number + RealNumber, M: Array2<T>>(d: &mut [T], e: &mut [T], V: &mut M) {
    let n = d.len();
    let mut temp = vec![T::zero(); n];
    for j in 1..n {
        let real = d[j];
        let img = e[j];
        for (k, temp_k) in temp.iter_mut().enumerate().take(n) {
            *temp_k = *V.get((k, j));
        }
        let mut i = j as i32 - 1;
        while i >= 0 {
            if d[i as usize] >= d[j] {
                break;
            }
            d[i as usize + 1] = d[i as usize];
            e[i as usize + 1] = e[i as usize];
            for k in 0..n {
                V.set((k, i as usize + 1), *V.get((k, i as usize)));
            }
            i -= 1;
        }
        d[i as usize + 1] = real;
        e[i as usize + 1] = img;
        for (k, temp_k) in temp.iter().enumerate().take(n) {
            V.set((k, i as usize + 1), *temp_k);
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::linalg::basic::matrix::DenseMatrix;
    use approx::relative_eq;

    #[cfg_attr(
        all(target_arch = "wasm32", not(target_os = "wasi")),
        wasm_bindgen_test::wasm_bindgen_test
    )]
    #[test]
    fn decompose_symmetric() {
        let A = DenseMatrix::from_2d_array(&[
            &[0.9000, 0.4000, 0.7000],
            &[0.4000, 0.5000, 0.3000],
            &[0.7000, 0.3000, 0.8000],
        ]);

        let eigen_values: Vec<f64> = vec![1.7498382, 0.3165784, 0.1335834];

        let eigen_vectors = DenseMatrix::from_2d_array(&[
            &[0.6881997, -0.07121225, 0.7220180],
            &[0.3700456, 0.89044952, -0.2648886],
            &[0.6240573, -0.44947578, -0.6391588],
        ]);

        let evd = A.evd(true).unwrap();

        assert!(relative_eq!(
            eigen_vectors.abs(),
            evd.V.abs(),
            epsilon = 1e-4
        ));
        for i in 0..eigen_values.len() {
            assert!((eigen_values[i] - evd.d[i]).abs() < 1e-4);
        }
        for i in 0..eigen_values.len() {
            assert!((0f64 - evd.e[i]).abs() < std::f64::EPSILON);
        }
    }
    #[cfg_attr(
        all(target_arch = "wasm32", not(target_os = "wasi")),
        wasm_bindgen_test::wasm_bindgen_test
    )]
    #[test]
    fn decompose_asymmetric() {
        let A = DenseMatrix::from_2d_array(&[
            &[0.9000, 0.4000, 0.7000],
            &[0.4000, 0.5000, 0.3000],
            &[0.8000, 0.3000, 0.8000],
        ]);

        let eigen_values: Vec<f64> = vec![1.79171122, 0.31908143, 0.08920735];

        let eigen_vectors = DenseMatrix::from_2d_array(&[
            &[0.7178958, 0.05322098, 0.6812010],
            &[0.3837711, -0.84702111, -0.1494582],
            &[0.6952105, 0.43984484, -0.7036135],
        ]);

        let evd = A.evd(false).unwrap();

        assert!(relative_eq!(
            eigen_vectors.abs(),
            evd.V.abs(),
            epsilon = 1e-4
        ));
        for i in 0..eigen_values.len() {
            assert!((eigen_values[i] - evd.d[i]).abs() < 1e-4);
        }
        for i in 0..eigen_values.len() {
            assert!((0f64 - evd.e[i]).abs() < std::f64::EPSILON);
        }
    }
    #[cfg_attr(
        all(target_arch = "wasm32", not(target_os = "wasi")),
        wasm_bindgen_test::wasm_bindgen_test
    )]
    #[test]
    fn decompose_complex() {
        let A = DenseMatrix::from_2d_array(&[
            &[3.0, -2.0, 1.0, 1.0],
            &[4.0, -1.0, 1.0, 1.0],
            &[1.0, 1.0, 3.0, -2.0],
            &[1.0, 1.0, 4.0, -1.0],
        ]);

        let eigen_values_d: Vec<f64> = vec![0.0, 2.0, 2.0, 0.0];
        let eigen_values_e: Vec<f64> = vec![2.2361, 0.9999, -0.9999, -2.2361];

        let eigen_vectors = DenseMatrix::from_2d_array(&[
            &[-0.9159, -0.1378, 0.3816, -0.0806],
            &[-0.6707, 0.1059, 0.901, 0.6289],
            &[0.9159, -0.1378, 0.3816, 0.0806],
            &[0.6707, 0.1059, 0.901, -0.6289],
        ]);

        let evd = A.evd(false).unwrap();

        assert!(relative_eq!(
            eigen_vectors.abs(),
            evd.V.abs(),
            epsilon = 1e-4
        ));
        for i in 0..eigen_values_d.len() {
            assert!((eigen_values_d[i] - evd.d[i]).abs() < 1e-4);
        }
        for i in 0..eigen_values_e.len() {
            assert!((eigen_values_e[i] - evd.e[i]).abs() < 1e-4);
        }
    }
}