faer 0.20.2

// Algorithm ported from Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
// Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

use super::jacobi::{jacobi_svd, Skip};
use crate::{
    assert,
    linalg::{jacobi::JacobiRotation, temp_mat_req, temp_mat_uninit, temp_mat_zeroed},
    unzipped,
    utils::{simd::SimdFor, thread::join_raw},
    zipped_rw, ColMut, ColRef, MatMut, Parallelism,
};
use coe::Coerce;
use core::{iter::zip, mem::swap};
use dyn_stack::{PodStack, SizeOverflow, StackReq};
use faer_entity::*;
use reborrow::*;

#[allow(dead_code)]
fn bidiag_to_mat<E: RealField>(diag: &[E], subdiag: &[E]) -> crate::Mat<E> {
    let mut mat = crate::Mat::<E>::zeros(diag.len() + 1, diag.len());

    for (i, d) in diag.iter().enumerate() {
        mat.write(i, i, *d);
    }
    for (i, d) in subdiag.iter().enumerate() {
        mat.write(i + 1, i, *d);
    }

    mat
}

fn compute_svd_of_m<E: RealField>(
    mut um: Option<MatMut<'_, E>>,
    mut vm: Option<MatMut<'_, E>>,
    diag: &mut [E],
    col0: &[E],
    outer_perm: &[usize],
    epsilon: E,
    _consider_zero_threshold: E,
    stack: &mut PodStack,
) {
    let n = diag.len();

    diag[0] = E::faer_zero();
    let mut actual_n = n;
    while actual_n > 1 && diag[actual_n - 1] == E::faer_zero() {
        actual_n -= 1;
        assert!(col0[actual_n] == E::faer_zero());
    }

    let (perm, stack) = stack.collect(
        col0.iter()
            .take(actual_n)
            .map(|x| x.faer_abs())
            .enumerate()
            .filter(|(_, x)| *x != E::faer_zero())
            .map(|(i, _)| i),
    );
    let perm = &*perm;
    let (col0_perm, stack) = stack.collect(perm.iter().map(|&p| col0[p]));
    let (diag_perm, stack) = stack.collect(perm.iter().map(|&p| diag[p]));

    let (mut shifts, stack) = temp_mat_uninit::<E>(n, 1, stack);
    let shifts = shifts.as_mut();
    let (mut mus, stack) = temp_mat_uninit::<E>(n, 1, stack);
    let mus = mus.as_mut();
    let (mut singular_vals, stack) = temp_mat_uninit::<E>(n, 1, stack);
    let singular_vals = singular_vals.as_mut();
    let (mut zhat, stack) = temp_mat_uninit::<E>(n, 1, stack);
    let zhat = zhat.as_mut();

    let mut shifts = shifts.col_mut(0);
    let mut mus = mus.col_mut(0);
    let mut s = singular_vals.col_mut(0);
    let mut zhat = zhat.col_mut(0);

    compute_singular_values(
        shifts.rb_mut(),
        mus.rb_mut(),
        s.rb_mut(),
        diag,
        diag_perm,
        col0,
        col0_perm,
        epsilon,
    );
    perturb_col0(
        zhat.rb_mut(),
        col0,
        diag,
        perm,
        s.rb(),
        shifts.rb(),
        mus.rb(),
    );

    let (col_perm, stack) = stack.make_with(actual_n, |i| i);
    let (col_perm_inv, _) = stack.make_with(actual_n, |i| i);

    for i in 0..actual_n - 1 {
        if s.read(i) > s.read(i + 1) {
            let si = s.read(i);
            let sj = s.read(i + 1);
            s.write(i, sj);
            s.write(i + 1, si);

            col_perm.swap(i, i + 1);
        }
    }
    for (i, p) in col_perm.iter().copied().enumerate() {
        col_perm_inv[p] = i;
    }

    compute_singular_vectors(
        um.rb_mut(),
        vm.rb_mut(),
        zhat.rb(),
        diag,
        perm,
        outer_perm,
        col_perm_inv,
        actual_n,
        shifts.rb(),
        mus.rb(),
    );

    for (idx, diag) in diag[..actual_n].iter_mut().enumerate() {
        *diag = s.read(actual_n - idx - 1);
    }

    for (idx, diag) in diag[actual_n..n].iter_mut().enumerate() {
        *diag = s.read(actual_n + idx);
    }
}

#[inline(never)]
fn compute_singular_vectors<E: RealField>(
    mut um: Option<MatMut<E>>,
    mut vm: Option<MatMut<E>>,
    zhat: ColRef<E>,
    diag: &[E],
    perm: &[usize],
    outer_perm: &[usize],
    col_perm_inv: &[usize],
    actual_n: usize,
    shifts: ColRef<E>,
    mus: ColRef<E>,
) {
    let n = diag.len();

    for k in 0..n {
        let actual_k = if k >= actual_n {
            k
        } else {
            actual_n - col_perm_inv[k] - 1
        };
        let mut u = um.rb_mut().map(|u| u.col_mut(actual_k));
        let mut v = vm.rb_mut().map(|v| v.col_mut(actual_k));

        if zhat.read(k) == E::faer_zero() {
            if let Some(mut u) = u.rb_mut() {
                u.write(outer_perm[k], E::faer_one());
            }
            if let Some(mut v) = v.rb_mut() {
                v.write(outer_perm[k], E::faer_one());
            }
            continue;
        }

        let mu = mus.read(k);
        let shift = shifts.read(k);

        assert_eq!(zhat.row_stride(), 1);

        if let Some(mut u) = u.rb_mut() {
            assert_eq!(u.row_stride(), 1);
            for &i in perm {
                u.write(
                    outer_perm[i],
                    zhat.read(i)
                        .faer_div(diag[i].faer_sub(shift).faer_sub(mu))
                        .faer_div(diag[i].faer_add(shift.faer_add(mu))),
                );
            }
            u.write(n, E::faer_zero());
            let norm_inv = u.norm_l2().faer_inv();
            zipped_rw!(u.rb_mut())
                .for_each(|unzipped!(mut x)| x.write(x.read().faer_mul(norm_inv)));
        }

        if let Some(mut v) = v {
            assert_eq!(v.row_stride(), 1);
            for &i in &perm[1..] {
                v.write(
                    outer_perm[i],
                    diag[i]
                        .faer_mul(zhat.read(i))
                        .faer_div(diag[i].faer_sub(shift).faer_sub(mu))
                        .faer_div(diag[i].faer_add(shift.faer_add(mu))),
                );
            }
            v.write(outer_perm[0], E::faer_one().faer_neg());
            let norm_inv = v.norm_l2().faer_inv();
            zipped_rw!(v.rb_mut())
                .for_each(|unzipped!(mut x)| x.write(x.read().faer_mul(norm_inv)));
        }
    }
    if let Some(mut um) = um {
        um.write(n, n, E::faer_one());
    }
}

fn perturb_col0<E: RealField>(
    mut zhat: ColMut<E>,
    col0: &[E],
    diag: &[E],
    perm: &[usize],
    s: ColRef<E>,
    shifts: ColRef<E>,
    mus: ColRef<E>,
) {
    let n = diag.len();
    let m = perm.len();
    if m == 0 {
        zipped_rw!(zhat).for_each(|unzipped!(mut x)| x.write(E::faer_zero()));
        return;
    }

    let last_idx = perm[m - 1];
    for k in 0..n {
        if col0[k] == E::faer_zero() {
            zhat.write(k, E::faer_zero());
            continue;
        }

        let dk = diag[k];
        let mut prod = (s.read(last_idx).faer_add(dk)).faer_mul(
            mus.read(last_idx)
                .faer_add(shifts.read(last_idx).faer_sub(dk)),
        );

        for l in 0..m {
            let i = perm[l];
            if i == k {
                continue;
            }
            if i >= k && l == 0 {
                prod = E::faer_zero();
                break;
            }
            let j = if i < k {
                i
            } else if l > 0 {
                perm[l - 1]
            } else {
                i
            };

            let term = ((s.read(j).faer_add(dk)).faer_div(diag[i].faer_add(dk))).faer_mul(
                (mus.read(j).faer_add(shifts.read(j).faer_sub(dk))).faer_div(diag[i].faer_sub(dk)),
            );
            prod = prod.faer_mul(term);
        }

        let tmp = prod.faer_sqrt();
        if col0[k] > E::faer_zero() {
            zhat.write(k, tmp);
        } else {
            zhat.write(k, tmp.faer_neg());
        }
    }
}

fn compute_singular_values<E: RealField>(
    shifts: ColMut<E>,
    mus: ColMut<E>,
    s: ColMut<E>,
    diag: &[E],
    diag_perm: &[E],
    col0: &[E],
    col0_perm: &[E],
    epsilon: E,
) {
    if E::IS_F64 {
        struct ImplF64<'a> {
            shifts: ColMut<'a, f64>,
            mus: ColMut<'a, f64>,
            s: ColMut<'a, f64>,
            diag: &'a [f64],
            diag_perm: &'a [f64],
            col0: &'a [f64],
            col0_perm: &'a [f64],
            epsilon: f64,
        }
        impl pulp::WithSimd for ImplF64<'_> {
            type Output = ();

            #[inline(always)]
            fn with_simd<S: pulp::Simd>(self, simd: S) -> Self::Output {
                let Self {
                    shifts,
                    mus,
                    s,
                    diag,
                    diag_perm,
                    col0,
                    col0_perm,
                    epsilon,
                } = self;
                compute_singular_values_generic::<f64>(
                    simd, shifts, mus, s, diag, diag_perm, col0, col0_perm, epsilon,
                )
            }
        }

        <f64 as ComplexField>::Simd::default().dispatch(ImplF64 {
            shifts: shifts.coerce(),
            mus: mus.coerce(),
            s: s.coerce(),
            diag: diag.coerce(),
            diag_perm: diag_perm.coerce(),
            col0: col0.coerce(),
            col0_perm: col0_perm.coerce(),
            epsilon: coe::coerce_static(epsilon),
        });
    } else if E::IS_F32 {
        struct ImplF32<'a> {
            shifts: ColMut<'a, f32>,
            mus: ColMut<'a, f32>,
            s: ColMut<'a, f32>,
            diag: &'a [f32],
            diag_perm: &'a [f32],
            col0: &'a [f32],
            col0_perm: &'a [f32],
            epsilon: f32,
        }
        impl pulp::WithSimd for ImplF32<'_> {
            type Output = ();

            #[inline(always)]
            fn with_simd<S: pulp::Simd>(self, simd: S) -> Self::Output {
                let Self {
                    shifts,
                    mus,
                    s,
                    diag,
                    diag_perm,
                    col0,
                    col0_perm,
                    epsilon,
                } = self;
                compute_singular_values_generic::<f32>(
                    simd, shifts, mus, s, diag, diag_perm, col0, col0_perm, epsilon,
                )
            }
        }

        <f64 as ComplexField>::Simd::default().dispatch(ImplF32 {
            shifts: shifts.coerce(),
            mus: mus.coerce(),
            s: s.coerce(),
            diag: diag.coerce(),
            diag_perm: diag_perm.coerce(),
            col0: col0.coerce(),
            col0_perm: col0_perm.coerce(),
            epsilon: coe::coerce_static(epsilon),
        });
    } else {
        compute_singular_values_generic(
            pulp::Scalar::new(),
            shifts,
            mus,
            s,
            diag,
            diag_perm,
            col0,
            col0_perm,
            epsilon,
        );
    }
}

#[inline(always)]
fn compute_singular_values_generic<E: RealField>(
    simd: impl pulp::Simd,
    mut shifts: ColMut<E>,
    mut mus: ColMut<E>,
    mut s: ColMut<E>,
    diag: &[E],
    diag_perm: &[E],
    col0: &[E],
    col0_perm: &[E],
    epsilon: E,
) {
    simd.vectorize(
        #[inline(always)]
        || {
            let n = diag.len();
            let mut actual_n = n;
            while actual_n > 1 && col0[actual_n - 1] == E::faer_zero() {
                actual_n -= 1;
            }
            let actual_n = actual_n;

            let two = E::faer_one().faer_add(E::faer_one());
            let eight = two
                .faer_scale_power_of_two(two)
                .faer_scale_power_of_two(two);
            let one_half = two.faer_inv();

            'kth_iter: for k in 0..n {
                s.write(k, E::faer_zero());
                shifts.write(k, E::faer_zero());
                mus.write(k, E::faer_zero());

                if col0[k] == E::faer_zero() || actual_n == 1 {
                    s.write(k, if k == 0 { col0[0] } else { diag[k] });
                    shifts.write(k, s.read(k));
                    mus.write(k, E::faer_zero());
                    continue 'kth_iter;
                }

                let last_k = k == actual_n - 1;
                let left = diag[k];
                let right = if last_k {
                    let mut norm2 = E::faer_zero();
                    for &x in col0 {
                        norm2 = norm2.faer_add(x.faer_mul(x));
                    }
                    diag[actual_n - 1].faer_add(norm2.faer_sqrt())
                } else {
                    let mut l = k + 1;
                    while col0[l] == E::faer_zero() {
                        l += 1;
                    }
                    diag[l]
                };

                let mid = left.faer_add(right.faer_sub(left).faer_scale_power_of_two(one_half));
                let [mut f_mid, f_max, f_mid_left_shift, f_mid_right_shift] = secular_eq_multi_fast(
                    [
                        mid,
                        if last_k {
                            right.faer_sub(left)
                        } else {
                            (right.faer_sub(left)).faer_scale_power_of_two(one_half)
                        },
                        one_half.faer_mul(right.faer_sub(left)),
                        one_half.faer_mul(right.faer_sub(left)).faer_neg(),
                    ],
                    col0_perm,
                    diag_perm,
                    [E::faer_zero(), left, left, right],
                );

                let (mut shift, mu) = if last_k || f_mid > E::faer_zero() {
                    (left, one_half.faer_mul(right.faer_sub(left)))
                } else {
                    (right, one_half.faer_mul(right.faer_sub(left)).faer_neg())
                };

                if f_mid_left_shift <= E::faer_zero() && f_mid_right_shift >= E::faer_zero() {
                    s.write(k, shift.faer_add(mu));
                    shifts.write(k, shift);
                    mus.write(k, mu);
                    continue 'kth_iter;
                }

                if !last_k {
                    if shift == left {
                        if f_mid_left_shift < E::faer_zero() {
                            shift = right;
                            f_mid = f_mid_right_shift;
                        }
                    } else if f_mid_right_shift > E::faer_zero() {
                        shift = left;
                        f_mid = f_mid_left_shift;
                    }
                }

                enum SecantError {
                    OutOfBounds,
                    PrecisionLimitReached,
                }

                let secant = {
                    #[inline(always)]
                    |mut mu_cur: E, mut mu_prev: E, mut f_cur: E, mut f_prev: E| {
                        if f_prev.faer_abs() < f_cur.faer_abs() {
                            swap(&mut f_prev, &mut f_cur);
                            swap(&mut mu_prev, &mut mu_cur);
                        }

                        let mut left_candidate = None;
                        let mut right_candidate = None;

                        let mut use_bisection = false;
                        let same_sign = f_prev.faer_mul(f_cur) > E::faer_zero();
                        if !same_sign {
                            let (min, max) = if mu_cur < mu_prev {
                                (mu_cur, mu_prev)
                            } else {
                                (mu_prev, mu_cur)
                            };
                            left_candidate = Some(min);
                            right_candidate = Some(max);
                        }

                        let mut err = SecantError::PrecisionLimitReached;

                        while f_cur != E::faer_zero()
                            && ((mu_cur.faer_sub(mu_prev)).faer_abs()
                                > eight.faer_mul(epsilon).faer_mul(
                                    if mu_cur.faer_abs() > mu_prev.faer_abs() {
                                        mu_cur.faer_abs()
                                    } else {
                                        mu_prev.faer_abs()
                                    },
                                ))
                            && ((f_cur.faer_sub(f_prev)).faer_abs() > epsilon)
                            && !use_bisection
                        {
                            // rational interpolation: fit a function of the form a / mu + b through
                            // the two previous iterates and use its
                            // zero to compute the next iterate
                            let a = (f_cur.faer_sub(f_prev))
                                .faer_mul(mu_prev.faer_mul(mu_cur))
                                .faer_div(mu_prev.faer_sub(mu_cur));
                            let b = f_cur.faer_sub(a.faer_div(mu_cur));
                            let mu_zero = a.faer_div(b).faer_neg();
                            let f_zero = secular_eq(mu_zero, col0_perm, diag_perm, shift);

                            if f_zero < E::faer_zero() {
                                left_candidate = Some(mu_zero);
                            } else {
                                right_candidate = Some(mu_zero);
                            }

                            mu_prev = mu_cur;
                            f_prev = f_cur;
                            mu_cur = mu_zero;
                            f_cur = f_zero;

                            if shift == left
                                && (mu_cur < E::faer_zero() || mu_cur > (right.faer_sub(left)))
                            {
                                err = SecantError::OutOfBounds;
                                use_bisection = true;
                            }
                            if shift == right
                                && (mu_cur < (right.faer_sub(left)).faer_neg()
                                    || mu_cur > E::faer_zero())
                            {
                                err = SecantError::OutOfBounds;
                                use_bisection = true;
                            }
                            if f_cur.faer_abs() > f_prev.faer_abs() {
                                // find mu such that a / mu + b = -k * f_zero
                                // a / mu = -f_zero - b
                                // mu = -a / (f_zero + b)
                                let mut k = E::faer_one();
                                for _ in 0..4 {
                                    let mu_opposite =
                                        a.faer_neg().faer_div(k.faer_mul(f_zero).faer_add(b));
                                    let f_opposite =
                                        secular_eq(mu_opposite, col0_perm, diag_perm, shift);
                                    if f_zero < E::faer_zero() && f_opposite >= E::faer_zero() {
                                        // this will be our right candidate
                                        right_candidate = Some(mu_opposite);
                                        break;
                                    } else if f_zero > E::faer_zero()
                                        && f_opposite <= E::faer_zero()
                                    {
                                        // this will be our left candidate
                                        left_candidate = Some(mu_opposite);
                                        break;
                                    }
                                    k = k.faer_scale_power_of_two(two);
                                }
                                use_bisection = true;
                            }
                        }
                        (use_bisection, mu_cur, left_candidate, right_candidate, err)
                    }
                };

                let (mut left_shifted, mut f_left, mut right_shifted, mut f_right) =
                    if shift == left {
                        (
                            E::faer_zero(),
                            E::faer_zero().faer_inv().faer_neg(),
                            if last_k {
                                right.faer_sub(left)
                            } else {
                                (right.faer_sub(left)).faer_mul(one_half)
                            },
                            if last_k { f_max } else { f_mid },
                        )
                    } else {
                        (
                            (right.faer_sub(left))
                                .faer_neg()
                                .faer_scale_power_of_two(one_half),
                            f_mid,
                            E::faer_zero(),
                            E::faer_zero().faer_inv(),
                        )
                    };

                assert!(
                    PartialOrd::partial_cmp(&f_left, &E::faer_zero())
                        != Some(core::cmp::Ordering::Greater)
                );
                assert!(
                    PartialOrd::partial_cmp(&f_right, &E::faer_zero())
                        != Some(core::cmp::Ordering::Less)
                );

                let mut iteration_count = 0;
                let mut f_prev = f_mid;
                // try to find non zero starting bounds

                let half0 = one_half;
                let half1 = half0.faer_scale_power_of_two(half0);
                let half2 = half1.faer_scale_power_of_two(half1);
                let half3 = half2.faer_scale_power_of_two(half2);
                let half4 = half3.faer_scale_power_of_two(half3);
                let half5 = half4.faer_scale_power_of_two(half4);
                let half6 = half5.faer_scale_power_of_two(half5);
                let half7 = half6.faer_scale_power_of_two(half6);

                let mu_values = if shift == left {
                    [
                        right_shifted.faer_scale_power_of_two(half7),
                        right_shifted.faer_scale_power_of_two(half6),
                        right_shifted.faer_scale_power_of_two(half5),
                        right_shifted.faer_scale_power_of_two(half4),
                        right_shifted.faer_scale_power_of_two(half3),
                        right_shifted.faer_scale_power_of_two(half2),
                        right_shifted.faer_scale_power_of_two(half1),
                        right_shifted.faer_scale_power_of_two(half0),
                    ]
                } else {
                    [
                        left_shifted.faer_scale_power_of_two(half7),
                        left_shifted.faer_scale_power_of_two(half6),
                        left_shifted.faer_scale_power_of_two(half5),
                        left_shifted.faer_scale_power_of_two(half4),
                        left_shifted.faer_scale_power_of_two(half3),
                        left_shifted.faer_scale_power_of_two(half2),
                        left_shifted.faer_scale_power_of_two(half1),
                        left_shifted.faer_scale_power_of_two(half0),
                    ]
                };
                let f_values =
                    secular_eq_multi_fast(mu_values, col0_perm, diag_perm, [(); 8].map(|_| shift));

                if shift == left {
                    let mut i = 0;
                    for (idx, (mu, f)) in zip(mu_values, f_values).enumerate() {
                        if f < E::faer_zero() {
                            left_shifted = mu;
                            f_left = f;
                            i = idx + 1;
                        }
                    }
                    if i < f_values.len() {
                        right_shifted = mu_values[i];
                        f_right = f_values[i];
                    }
                } else {
                    let mut i = 0;
                    for (idx, (mu, f)) in zip(mu_values, f_values).enumerate() {
                        if f > E::faer_zero() {
                            right_shifted = mu;
                            f_right = f;
                            i = idx + 1;
                        }
                    }
                    if i < f_values.len() {
                        left_shifted = mu_values[i];
                        f_left = f_values[i];
                    }
                }

                // try bisection just to get a good guess for secant
                while right_shifted.faer_sub(left_shifted)
                    > two.faer_mul(epsilon).faer_mul(
                        if left_shifted.faer_abs() > right_shifted.faer_abs() {
                            left_shifted.faer_abs()
                        } else {
                            right_shifted.faer_abs()
                        },
                    )
                {
                    let mid_shifted_arithmetic =
                        (left_shifted.faer_add(right_shifted)).faer_scale_power_of_two(one_half);
                    let mut mid_shifted_geometric = left_shifted
                        .faer_abs()
                        .faer_sqrt()
                        .faer_mul(right_shifted.faer_abs().faer_sqrt());
                    if left_shifted < E::faer_zero() {
                        mid_shifted_geometric = mid_shifted_geometric.faer_neg();
                    }
                    let mid_shifted = if mid_shifted_geometric == E::faer_zero() {
                        mid_shifted_arithmetic
                    } else {
                        mid_shifted_geometric
                    };
                    let f_mid = secular_eq(mid_shifted, col0_perm, diag_perm, shift);

                    if f_mid == E::faer_zero() {
                        s.write(k, shift.faer_add(mid_shifted));
                        shifts.write(k, shift);
                        mus.write(k, mid_shifted);
                        continue 'kth_iter;
                    } else if f_mid > E::faer_zero() {
                        right_shifted = mid_shifted;
                        f_prev = f_right;
                        f_right = f_mid;
                    } else {
                        left_shifted = mid_shifted;
                        f_prev = f_left;
                        f_left = f_mid;
                    }

                    if iteration_count == 4 {
                        break;
                    }

                    iteration_count += 1;
                }

                // try secant with the guess from bisection
                let args = if left_shifted == E::faer_zero() {
                    (
                        right_shifted.faer_add(right_shifted),
                        right_shifted,
                        f_prev,
                        f_right,
                    )
                } else if right_shifted == E::faer_zero() {
                    (
                        left_shifted.faer_add(left_shifted),
                        left_shifted,
                        f_prev,
                        f_left,
                    )
                } else {
                    (left_shifted, right_shifted, f_left, f_right)
                };

                let (use_bisection, mut mu_cur, left_candidate, right_candidate, _err) =
                    secant(args.0, args.1, args.2, args.3);

                match (left_candidate, right_candidate) {
                    (Some(left), Some(right)) if left < right => {
                        if left > left_shifted {
                            left_shifted = left;
                        }
                        if right < right_shifted {
                            right_shifted = right;
                        }
                    }
                    _ => (),
                }

                // secant failed, use bisection again
                if use_bisection {
                    while (right_shifted.faer_sub(left_shifted))
                        > two.faer_mul(epsilon).faer_mul(
                            if left_shifted.faer_abs() > right_shifted.faer_abs() {
                                left_shifted.faer_abs()
                            } else {
                                right_shifted.faer_abs()
                            },
                        )
                    {
                        let mid_shifted = (left_shifted.faer_add(right_shifted))
                            .faer_scale_power_of_two(one_half);
                        let f_mid = secular_eq(mid_shifted, col0_perm, diag_perm, shift);

                        if f_mid == E::faer_zero() {
                            break;
                        } else if f_mid > E::faer_zero() {
                            right_shifted = mid_shifted;
                        } else {
                            left_shifted = mid_shifted;
                        }
                    }

                    mu_cur = (left_shifted.faer_add(right_shifted)).faer_mul(one_half);
                }

                s.write(k, shift.faer_add(mu_cur));
                shifts.write(k, shift);
                mus.write(k, mu_cur);
            }
        },
    );
}

#[inline(always)]
fn secular_eq_multi_fast<const N: usize, E: RealField>(
    mu: [E; N],
    col0_perm: &[E],
    diag_perm: &[E],
    shift: [E; N],
) -> [E; N] {
    let mut res0 = [(); N].map(|_| E::faer_one());
    for (c0, d0) in col0_perm.iter().cloned().zip(diag_perm.iter().cloned()) {
        for ((res0, mu), shift) in res0
            .iter_mut()
            .zip(mu.iter().cloned())
            .zip(shift.iter().cloned())
        {
            *res0 = (*res0).faer_add((c0.faer_mul(c0)).faer_div(
                (d0.faer_sub(shift).faer_sub(mu)).faer_mul(d0.faer_add(shift).faer_add(mu)),
            ));
        }
    }
    res0
}

#[inline(always)]
fn secular_eq<E: RealField>(mu: E, col0_perm: &[E], diag_perm: &[E], shift: E) -> E {
    let mut res0 = E::faer_one();
    let mut res1 = E::faer_zero();
    let mut res2 = E::faer_zero();
    let mut res3 = E::faer_zero();
    let mut res4 = E::faer_zero();
    let mut res5 = E::faer_zero();
    let mut res6 = E::faer_zero();
    let mut res7 = E::faer_zero();

    let (col0_head, col0_perm) = pulp::as_arrays::<8, _>(col0_perm);
    let (diag_head, diag_perm) = pulp::as_arrays::<8, _>(diag_perm);
    for ([c0, c1, c2, c3, c4, c5, c6, c7], [d0, d1, d2, d3, d4, d5, d6, d7]) in
        col0_head.iter().zip(diag_head)
    {
        res0 = res0.faer_add(
            (c0.faer_div(d0.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c0.faer_div(d0.faer_add(shift).faer_add(mu))),
        );
        res1 = res1.faer_add(
            (c1.faer_div(d1.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c1.faer_div(d1.faer_add(shift).faer_add(mu))),
        );
        res2 = res2.faer_add(
            (c2.faer_div(d2.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c2.faer_div(d2.faer_add(shift).faer_add(mu))),
        );
        res3 = res3.faer_add(
            (c3.faer_div(d3.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c3.faer_div(d3.faer_add(shift).faer_add(mu))),
        );
        res4 = res4.faer_add(
            (c4.faer_div(d4.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c4.faer_div(d4.faer_add(shift).faer_add(mu))),
        );
        res5 = res5.faer_add(
            (c5.faer_div(d5.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c5.faer_div(d5.faer_add(shift).faer_add(mu))),
        );
        res6 = res6.faer_add(
            (c6.faer_div(d6.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c6.faer_div(d6.faer_add(shift).faer_add(mu))),
        );
        res7 = res7.faer_add(
            (c7.faer_div(d7.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c7.faer_div(d7.faer_add(shift).faer_add(mu))),
        );
    }

    let (col0_head, col0_perm) = pulp::as_arrays::<4, _>(col0_perm);
    let (diag_head, diag_perm) = pulp::as_arrays::<4, _>(diag_perm);
    for ([c0, c1, c2, c3], [d0, d1, d2, d3]) in col0_head.iter().zip(diag_head) {
        res0 = res0.faer_add(
            (c0.faer_div(d0.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c0.faer_div(d0.faer_add(shift).faer_add(mu))),
        );
        res1 = res1.faer_add(
            (c1.faer_div(d1.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c1.faer_div(d1.faer_add(shift).faer_add(mu))),
        );
        res2 = res2.faer_add(
            (c2.faer_div(d2.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c2.faer_div(d2.faer_add(shift).faer_add(mu))),
        );
        res3 = res3.faer_add(
            (c3.faer_div(d3.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c3.faer_div(d3.faer_add(shift).faer_add(mu))),
        );
    }

    let (col0_head, col0_perm) = pulp::as_arrays::<2, _>(col0_perm);
    let (diag_head, diag_perm) = pulp::as_arrays::<2, _>(diag_perm);
    for ([c0, c1], [d0, d1]) in col0_head.iter().zip(diag_head) {
        res0 = res0.faer_add(
            (c0.faer_div(d0.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c0.faer_div(d0.faer_add(shift).faer_add(mu))),
        );
        res1 = res1.faer_add(
            (c1.faer_div(d1.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c1.faer_div(d1.faer_add(shift).faer_add(mu))),
        );
    }

    for (c0, d0) in col0_perm.iter().zip(diag_perm) {
        res0 = res0.faer_add(
            (c0.faer_div(d0.faer_sub(shift).faer_sub(mu)))
                .faer_mul(c0.faer_div(d0.faer_add(shift).faer_add(mu))),
        );
    }

    ((res0.faer_add(res1)).faer_add(res2.faer_add(res3)))
        .faer_add((res4.faer_add(res5)).faer_add(res6.faer_add(res7)))
}

fn deflate<E: RealField>(
    diag: &mut [E],
    col0: &mut [E],
    jacobi_coeffs: &mut [JacobiRotation<E>],
    jacobi_indices: &mut [usize],
    transpositions: &mut [usize],
    perm: &mut [usize],
    k: usize,
    epsilon: E,
    consider_zero_threshold: E,
    stack: &mut PodStack,
) -> (usize, usize) {
    let n = diag.len();
    let mut jacobi_0i = 0;
    let mut jacobi_ij = 0;

    let mut max_diag = E::faer_zero();
    let mut max_col0 = E::faer_zero();
    for d in diag[1..].iter() {
        max_diag = if d.faer_abs() > max_diag {
            d.faer_abs()
        } else {
            max_diag
        };
    }
    for d in col0.iter() {
        max_col0 = if d.faer_abs() > max_col0 {
            d.faer_abs()
        } else {
            max_col0
        };
    }

    let epsilon_strict = epsilon.faer_mul(max_diag);
    let epsilon_strict = if epsilon_strict > consider_zero_threshold {
        &epsilon_strict
    } else {
        &consider_zero_threshold
    };

    let two = E::faer_one().faer_add(E::faer_one());
    let eight = two
        .faer_scale_power_of_two(two)
        .faer_scale_power_of_two(two);
    let epsilon_coarse = eight.faer_mul(epsilon).faer_mul(if max_diag > max_col0 {
        max_diag
    } else {
        max_col0
    });

    // condition 4.1
    if diag[0] < epsilon_coarse {
        diag[0] = epsilon_coarse;
        col0[0] = epsilon_coarse;
    }

    // condition 4.2
    for x in &mut col0[1..] {
        if x.faer_abs() < *epsilon_strict {
            *x = E::faer_zero();
        }
    }

    // condition 4.3
    for i in 1..n {
        if diag[i] < epsilon_coarse {
            if let Some(rot) = deflation43(diag, col0, i) {
                jacobi_coeffs[jacobi_0i] = rot;
                jacobi_indices[jacobi_0i] = i;
                jacobi_0i += 1;
            }
        }
    }

    let mut total_deflation = true;
    for c in col0[1..].iter() {
        if PartialOrd::partial_cmp(&c.faer_abs(), &consider_zero_threshold)
            != Some(core::cmp::Ordering::Less)
        {
            total_deflation = false;
            break;
        }
    }

    perm[0] = 0;
    let mut p = 1;

    for (d, i) in diag[1..].iter().zip(1..n) {
        if d.faer_abs() < consider_zero_threshold {
            perm[p] = i;
            p += 1;
        }
    }

    let mut i = 1;
    let mut j = k + 1;

    for p in &mut perm[p..] {
        if i >= k + 1 {
            *p = j;
            j += 1;
        } else if j >= n {
            *p = i;
            i += 1;
        } else if diag[i] < diag[j] {
            *p = j;
            j += 1;
        } else {
            *p = i;
            i += 1;
        }
    }

    if total_deflation {
        for i in 1..n {
            let pi = perm[i];
            if diag[pi].faer_abs() < consider_zero_threshold || diag[pi] > diag[0] {
                perm[i - 1] = perm[i];
            } else {
                perm[i - 1] = 0;
                break;
            }
        }
    }

    let (real_ind, stack) = stack.make_with(n, |i| i);
    let (real_col, _) = stack.make_with(n, |i| i);

    for i in (if total_deflation { 0 } else { 1 })..n {
        let pi = perm[n - (if total_deflation { i + 1 } else { i })];
        let j = real_col[pi];

        diag.swap(i, j);

        if i != 0 && j != 0 {
            col0.swap(i, j);
        }

        transpositions[i] = j;

        let real_i = real_ind[i];
        real_col[real_i] = j;
        real_col[pi] = i;
        real_ind[j] = real_i;
        real_ind[i] = pi;
    }
    col0[0] = diag[0];
    for (i, p) in perm.iter_mut().enumerate() {
        *p = i;
    }
    for (i, j) in transpositions.iter().copied().enumerate() {
        perm.swap(i, j);
    }

    // condition 4.4
    let mut i = n - 1;
    while i > 0
        && (diag[i].faer_abs() < consider_zero_threshold
            || col0[i].faer_abs() < consider_zero_threshold)
    {
        i -= 1;
    }
    while i > 1 {
        if diag[i].faer_sub(diag[i - 1]) < *epsilon_strict {
            if let Some(rot) = deflation44(diag, col0, i - 1, i) {
                jacobi_coeffs[jacobi_0i + jacobi_ij] = rot;
                jacobi_indices[jacobi_0i + jacobi_ij] = i;
                jacobi_ij += 1;
            }
        }
        i -= 1;
    }

    (jacobi_0i, jacobi_ij)
}

fn deflation43<E: RealField>(
    diag: &mut [E],
    col0: &mut [E],
    i: usize,
) -> Option<JacobiRotation<E>> {
    let p = col0[0];
    let q = col0[i];
    if p == E::faer_zero() && q == E::faer_zero() {
        diag[i] = E::faer_zero();
        return None;
    }
    let rot = JacobiRotation::make_givens(p, q);

    let r = rot.c * p - rot.s * q;

    col0[0] = r;
    diag[0] = r;
    col0[i] = E::faer_zero();
    diag[i] = E::faer_zero();

    Some(rot)
}

fn deflation44<E: RealField>(
    diag: &mut [E],
    col0: &mut [E],
    i: usize,
    j: usize,
) -> Option<JacobiRotation<E>> {
    let p = col0[i];
    let q = col0[j];
    if p == E::faer_zero() && q == E::faer_zero() {
        diag[i] = diag[j];
        return None;
    }
    let rot = JacobiRotation::make_givens(p, q);
    let r = rot.c * p - rot.s * q;

    col0[i] = r;
    col0[j] = E::faer_zero();
    diag[i] = diag[j];

    Some(rot)
}

fn bidiag_svd_qr_algorithm_impl<E: RealField>(
    diag: &mut [E],
    subdiag: &mut [E],
    mut u: Option<MatMut<'_, E>>,
    mut v: Option<MatMut<'_, E>>,
    epsilon: E,
    consider_zero_threshold: E,
) {
    let n = diag.len();
    let max_iter = 30usize.saturating_mul(n).saturating_mul(n);

    let epsilon = epsilon.faer_scale_real(E::faer_from_f64(128.0));

    if let Some(mut u) = u.rb_mut() {
        u.fill_zero();
        u.diagonal_mut().column_vector_mut().fill(E::faer_one());
    }
    if let Some(mut v) = v.rb_mut() {
        v.fill_zero();
        v.diagonal_mut().column_vector_mut().fill(E::faer_one());
    }

    u = u.map(|u| u.submatrix_mut(0, 0, n, n));
    v = v.map(|v| v.submatrix_mut(0, 0, n, n));

    let mut max_val = E::faer_zero();

    for x in &*diag {
        let val = x.faer_abs();
        if val > max_val {
            max_val = val;
        }
    }
    for x in &*subdiag {
        let val = x.faer_abs();
        if val > max_val {
            max_val = val;
        }
    }

    if max_val == E::faer_zero() {
        return;
    }

    for x in &mut *diag {
        *x = (*x).faer_div(max_val);
    }
    for x in &mut *subdiag {
        *x = (*x).faer_div(max_val);
    }

    struct Impl<'a, E: Entity> {
        epsilon: E,
        consider_zero_threshold: E,
        max_iter: usize,
        diag: &'a mut [E],
        subdiag: &'a mut [E],
        u: Option<MatMut<'a, E>>,
        v: Option<MatMut<'a, E>>,
    }

    impl<E: RealField> pulp::WithSimd for Impl<'_, E> {
        type Output = ();

        #[inline(always)]
        fn with_simd<S: pulp::Simd>(self, simd: S) -> Self::Output {
            let Self {
                epsilon,
                consider_zero_threshold,
                max_iter,
                diag,
                subdiag,
                mut u,
                mut v,
            } = self;
            let n = diag.len();
            let arch = E::Simd::default();
            let epsilon2 = E::faer_mul(epsilon, epsilon);

            for iter in 0..max_iter {
                let _ = iter;
                for i in 0..n - 1 {
                    if subdiag[i].faer_abs2()
                        <= epsilon2
                            .faer_mul(E::faer_mul(diag[i].faer_abs(), diag[i + 1].faer_abs()))
                            + consider_zero_threshold
                    {
                        subdiag[i] = E::faer_zero();
                    }
                }
                for i in 0..n {
                    if diag[i].faer_abs() <= epsilon {
                        diag[i] = E::faer_zero();
                    }
                }

                let mut end = n;
                while end > 1 && subdiag[end - 2].faer_abs2() <= consider_zero_threshold {
                    end -= 1;
                }

                if end == 1 {
                    break;
                }

                let mut start = end - 1;
                while start > 0 && subdiag[start - 1] != E::faer_zero() {
                    start -= 1;
                }

                let mut found_zero_diag = false;
                for i in start..end - 1 {
                    if diag[i] == E::faer_zero() {
                        found_zero_diag = true;
                        let mut val = subdiag[i];
                        subdiag[i] = E::faer_zero();
                        for j in i + 1..end {
                            let rot = JacobiRotation::make_givens(diag[j], val);
                            diag[j] = rot
                                .c
                                .faer_mul(diag[j])
                                .faer_sub(rot.s.faer_mul(val))
                                .faer_abs();

                            if j < end - 1 {
                                (val, subdiag[j]) = (
                                    rot.s.faer_mul(subdiag[j]).faer_neg(),
                                    rot.c.faer_mul(subdiag[j]),
                                );
                            }

                            if let Some(v) = v.rb_mut() {
                                unsafe {
                                    rot.apply_on_the_right_in_place_arch(
                                        arch,
                                        v.rb().col(i).const_cast(),
                                        v.rb().col(j).const_cast(),
                                    );
                                }
                            }
                        }
                    }
                }

                if found_zero_diag {
                    continue;
                }

                let t00 = if end - start == 2 {
                    diag[end - 2].faer_abs2()
                } else {
                    diag[end - 2]
                        .faer_abs2()
                        .faer_add(subdiag[end - 3].faer_abs2())
                };
                let t11 = diag[end - 1]
                    .faer_abs2()
                    .faer_add(subdiag[end - 2].faer_abs2());
                let t01 = diag[end - 2].faer_mul(subdiag[end - 2]);

                let mu;
                if false {
                    let delta = E::faer_sub(
                        t00.faer_add(t11).faer_abs2(),
                        t00.faer_mul(t11)
                            .faer_sub(t01.faer_abs2())
                            .faer_scale_power_of_two(E::faer_from_f64(4.0)),
                    );

                    mu = if delta > E::faer_zero() {
                        let lambda0 = t00
                            .faer_add(t11)
                            .faer_add(delta.faer_sqrt())
                            .faer_scale_power_of_two(E::faer_from_f64(0.5));
                        let lambda1 = t00
                            .faer_add(t11)
                            .faer_sub(delta.faer_sqrt())
                            .faer_scale_power_of_two(E::faer_from_f64(0.5));

                        if lambda0.faer_sub(t11).faer_abs() < lambda1.faer_sub(t11).faer_abs() {
                            lambda0
                        } else {
                            lambda1
                        }
                    } else {
                        t11
                    };
                } else {
                    let t01_2 = t01.faer_abs2();
                    if t01_2 > consider_zero_threshold {
                        let d = (t00.faer_sub(t11)).faer_mul(E::faer_from_f64(0.5));
                        let mut delta = d.faer_abs2().faer_add(t01_2).faer_sqrt();
                        if d < E::faer_zero() {
                            delta = delta.faer_neg();
                        }

                        mu = t11.faer_sub(t01_2.faer_div(d.faer_add(delta)));
                    } else {
                        mu = t11
                    }
                }

                let mut y = diag[start].faer_abs2().faer_sub(mu);
                let mut z = diag[start].faer_mul(subdiag[start]);

                let simde = SimdFor::<E, S>::new(simd);
                let u_offset = simde.align_offset_ptr(
                    u.rb()
                        .map(|mat| mat.as_ptr())
                        .unwrap_or(E::faer_map(E::UNIT, |()| core::ptr::null())),
                    diag.len(),
                );
                let v_offset = simde.align_offset_ptr(
                    v.rb()
                        .map(|mat| mat.as_ptr())
                        .unwrap_or(E::faer_map(E::UNIT, |()| core::ptr::null())),
                    diag.len(),
                );

                for k in start..end - 1 {
                    let rot = JacobiRotation::make_givens(y, z);
                    if k > start {
                        subdiag[k - 1] = rot.c.faer_mul(y).faer_sub(rot.s.faer_mul(z)).faer_abs();
                    }

                    let mut diag_k = diag[k];

                    (diag_k, subdiag[k]) = (
                        simde.scalar_mul_add_e(
                            rot.c,
                            diag_k,
                            simde.scalar_mul(rot.s.faer_neg(), subdiag[k]),
                        ),
                        simde.scalar_mul_add_e(rot.s, diag_k, simde.scalar_mul(rot.c, subdiag[k])),
                    );

                    y = diag_k;
                    (z, diag[k + 1]) = (
                        simde.scalar_mul(rot.s.faer_neg(), diag[k + 1]),
                        simde.scalar_mul(rot.c, diag[k + 1]),
                    );

                    if let Some(u) = u.rb_mut() {
                        unsafe {
                            rot.apply_on_the_right_in_place_with_simd_and_offset(
                                simd,
                                u_offset,
                                u.rb().col(k).const_cast(),
                                u.rb().col(k + 1).const_cast(),
                            );
                        }
                    }

                    let rot = JacobiRotation::make_givens(y, z);

                    diag_k = rot.c.faer_mul(y).faer_sub(rot.s.faer_mul(z)).faer_abs();
                    diag[k] = diag_k;
                    (subdiag[k], diag[k + 1]) = (
                        simde.scalar_mul_add_e(
                            rot.c,
                            subdiag[k],
                            simde.scalar_mul(rot.s.faer_neg(), diag[k + 1]),
                        ),
                        simde.scalar_mul_add_e(
                            rot.s,
                            subdiag[k],
                            simde.scalar_mul(rot.c, diag[k + 1]),
                        ),
                    );

                    if k < end - 2 {
                        y = subdiag[k];
                        (z, subdiag[k + 1]) = (
                            simde.scalar_mul(rot.s.faer_neg(), subdiag[k + 1]),
                            simde.scalar_mul(rot.c, subdiag[k + 1]),
                        );
                    }

                    if let Some(v) = v.rb_mut() {
                        unsafe {
                            rot.apply_on_the_right_in_place_with_simd_and_offset(
                                simd,
                                v_offset,
                                v.rb().col(k).const_cast(),
                                v.rb().col(k + 1).const_cast(),
                            );
                        }
                    }
                }
            }
        }
    }

    E::Simd::default().dispatch(Impl {
        epsilon,
        consider_zero_threshold,
        max_iter,
        diag,
        subdiag,
        u: u.rb_mut(),
        v: v.rb_mut(),
    });

    for (j, d) in diag.iter_mut().enumerate() {
        if *d < E::faer_zero() {
            *d = d.faer_neg();
            if let Some(mut v) = v.rb_mut() {
                for i in 0..n {
                    v.write(i, j, v.read(i, j).faer_neg());
                }
            }
        }
    }

    for k in 0..n {
        let mut max = E::faer_zero();
        let mut max_idx = k;
        for kk in k..n {
            if diag[kk] > max {
                max = diag[kk];
                max_idx = kk;
            }
        }

        if k != max_idx {
            diag.swap(k, max_idx);
            if let Some(u) = u.rb_mut() {
                crate::perm::swap_cols_idx(u, k, max_idx);
            }
            if let Some(v) = v.rb_mut() {
                crate::perm::swap_cols_idx(v, k, max_idx);
            }
        }
    }

    for x in &mut *diag {
        *x = (*x).faer_mul(max_val);
    }
}

/// svd of bidiagonal lower matrix of shape (n + 1, n), with the last row being all zeros
pub fn compute_bidiag_real_svd<E: RealField>(
    diag: &mut [E],
    subdiag: &mut [E],
    mut u: Option<MatMut<'_, E>>,
    v: Option<MatMut<'_, E>>,
    jacobi_fallback_threshold: usize,
    bidiag_qr_fallback_threshold: usize,
    epsilon: E,
    consider_zero_threshold: E,
    parallelism: Parallelism,
    stack: &mut PodStack,
) {
    let n = diag.len();

    if n <= jacobi_fallback_threshold {
        let (mut s, _) = temp_mat_zeroed::<E>(n, n, stack);
        let mut s = s.as_mut();

        for i in 0..n {
            s.write(i, i, diag[i]);
            if i + 1 < n {
                s.write(i + 1, i, subdiag[i]);
            }
        }

        jacobi_svd(
            s.rb_mut(),
            u.rb_mut().map(|u| u.submatrix_mut(0, 0, n, n)),
            v,
            Skip::None,
            epsilon,
            consider_zero_threshold,
        );

        for (i, diag) in diag.iter_mut().enumerate() {
            *diag = s.read(i, i);
        }
        if let Some(mut u) = u {
            zipped_rw!(u.rb_mut().row_mut(n)).for_each(|unzipped!(mut x)| x.write(E::faer_zero()));
            zipped_rw!(u.rb_mut().col_mut(n)).for_each(|unzipped!(mut x)| x.write(E::faer_zero()));
            u.write(n, n, E::faer_one());
        }
    } else if n <= bidiag_qr_fallback_threshold {
        bidiag_svd_qr_algorithm_impl(diag, subdiag, u, v, epsilon, consider_zero_threshold);
    } else {
        match u {
            Some(u) => bidiag_svd_impl(
                diag,
                subdiag,
                u,
                v,
                true,
                jacobi_fallback_threshold,
                epsilon,
                consider_zero_threshold,
                parallelism,
                stack,
            ),
            None => {
                let (mut u, stack) = temp_mat_uninit::<E>(2, n + 1, stack);
                let u = u.as_mut();
                bidiag_svd_impl(
                    diag,
                    subdiag,
                    u,
                    v,
                    false,
                    jacobi_fallback_threshold,
                    epsilon,
                    consider_zero_threshold,
                    parallelism,
                    stack,
                );
            }
        }
    }
}

/// svd of bidiagonal lower matrix
fn bidiag_svd_impl<E: RealField>(
    diag: &mut [E],
    subdiag: &mut [E],
    mut u: MatMut<'_, E>,
    mut v: Option<MatMut<'_, E>>,
    fill_u: bool,
    jacobi_fallback_threshold: usize,
    epsilon: E,
    consider_zero_threshold: E,
    parallelism: Parallelism,
    mut stack: &mut PodStack,
) {
    let n = diag.len();

    let mut max_val = E::faer_zero();

    for x in &*diag {
        let val = x.faer_abs();
        if val > max_val {
            max_val = val;
        }
    }
    for x in &*subdiag {
        let val = x.faer_abs();
        if val > max_val {
            max_val = val;
        }
    }

    if max_val == E::faer_zero() {
        u.fill_zero();
        if u.nrows() == n + 1 {
            u.diagonal_mut().column_vector_mut().fill(E::faer_one());
        } else {
            u.write(0, 0, E::faer_one());
            u.write(1, n, E::faer_one());
        }
        if let Some(mut v) = v {
            v.fill_zero();
            v.diagonal_mut().column_vector_mut().fill(E::faer_one());
        };
        return;
    }

    for x in &mut *diag {
        *x = (*x).faer_div(max_val);
    }
    for x in &mut *subdiag {
        *x = (*x).faer_div(max_val);
    }

    assert!(subdiag.len() == n);
    assert!(n > jacobi_fallback_threshold);

    let k = n / 2;
    let rem = n - 1 - k;

    let (d1, alpha_d2) = diag.split_at_mut(k);
    let (sub_d1, beta_sub_d2) = subdiag.split_at_mut(k);
    let (alpha, d2) = alpha_d2.split_first_mut().unwrap();
    let (beta, sub_d2) = beta_sub_d2.split_first_mut().unwrap();
    let alpha = *alpha;
    let beta = *beta;

    let compact_u = (u.nrows() != n + 1) as usize;

    if k <= jacobi_fallback_threshold || rem <= jacobi_fallback_threshold {
        let (mut u1_alloc, stack) =
            temp_mat_uninit::<E>(k + 1, compact_u * (k + 1), stack.rb_mut());
        let mut u1_alloc = u1_alloc.as_mut();
        let (mut u2_alloc, stack) = temp_mat_uninit::<E>(rem + 1, compact_u * (rem + 1), stack);
        let mut u2_alloc = u2_alloc.as_mut();

        let (_u0, mut u1, mut u2) = if compact_u == 0 {
            let (u1, u2) = u.rb_mut().split_at_row_mut(k + 1);
            let (u0, u1) = u1.split_at_col_mut(1);
            (
                u0,
                u1.submatrix_mut(0, 0, k + 1, k + 1),
                u2.submatrix_mut(0, k, rem + 1, rem + 1),
            )
        } else {
            (
                u.rb_mut().col_mut(0).as_2d_mut(),
                u1_alloc.rb_mut(),
                u2_alloc.rb_mut(),
            )
        };

        let (mut v1, mut v2) = match v.rb_mut() {
            Some(v) => {
                let (v1, v2) = v.split_at_row_mut(k);
                (
                    Some(v1.submatrix_mut(0, 1, k, k + 1)),
                    Some(v2.submatrix_mut(1, k, rem, rem + 1)),
                )
            }
            None => (None, None),
        };

        let (mut matrix1, stack) = temp_mat_zeroed::<E>(k + 1, k + 1, stack);
        let (mut matrix2, _) = temp_mat_zeroed::<E>(rem + 1, rem + 1, stack);
        let mut matrix1 = matrix1.as_mut();
        let mut matrix2 = matrix2.as_mut();

        for j in 0..k {
            matrix1.write(j, j, d1[j]);
            matrix1.write(j + 1, j, sub_d1[j]);
        }
        for j in 0..rem {
            matrix2.write(j, j + 1, d2[j]);
            matrix2.write(j + 1, j + 1, sub_d2[j]);
        }

        jacobi_svd(
            matrix1.rb_mut(),
            Some(u1.rb_mut()),
            v1.rb_mut(),
            Skip::Last,
            epsilon,
            consider_zero_threshold,
        );
        for j in 0..matrix1.ncols() {
            for i in 0..matrix1.nrows() {
                if i != j {
                    matrix1.write(i, j, E::faer_zero());
                }
            }
        }
        jacobi_svd(
            matrix2.rb_mut(),
            Some(u2.rb_mut()),
            v2.rb_mut(),
            Skip::First,
            epsilon,
            consider_zero_threshold,
        );
        for j in 0..matrix2.ncols() {
            for i in 0..matrix1.nrows() {
                if i != j {
                    matrix1.write(i, j, E::faer_zero());
                }
            }
        }

        if cfg!(debug_assertions) {
            if let Some(v1) = v1 {
                zipped_rw!(v1.col_mut(k)).for_each(|unzipped!(mut x)| x.write(E::faer_nan()));
            }
            if let Some(v2) = v2 {
                zipped_rw!(v2.col_mut(0)).for_each(|unzipped!(mut x)| x.write(E::faer_nan()));
            }
        }

        for j in 0..k {
            diag[j + 1] = matrix1.read(j, j);
        }
        for j in 0..rem {
            diag[j + k + 1] = matrix2.read(j + 1, j + 1);
        }

        if compact_u == 1 {
            // need to copy the first and last rows
            //
            // NOTE: we handle the rotation of (Q1, q1) here, so no need to handle it later when
            // compact_u == 1
            for (row, row1, row2) in [(0, 0, 0), (1, k, rem)] {
                zipped_rw!(
                    u.rb_mut().row_mut(row).subcols_mut(1, k),
                    u1_alloc.rb().row(row1).subcols(0, k),
                )
                .for_each(|unzipped!(mut dst, src)| dst.write(src.read()));
                u.write(row, 0, u1_alloc.read(row1, k));

                zipped_rw!(
                    u.rb_mut().row_mut(row).subcols_mut(k + 1, rem),
                    u2_alloc.rb().row(row2).subcols(1, rem),
                )
                .for_each(|unzipped!(mut dst, src)| dst.write(src.read()));
                u.write(row, n, u2_alloc.read(row2, 0));
            }
        } else {
            let (_, u2) = if compact_u == 0 {
                let (u1, u2) = u.rb_mut().split_at_row_mut(k + 1);
                (u1, u2)
            } else {
                let (u1, u2) = u.rb_mut().split_at_row_mut(1);
                (u1, u2)
            };

            let (left, right) = u2.split_at_col_mut(k + 1);
            let left = left.col_mut(k);
            let right = right.col_mut(rem);
            zipped_rw!(right, left).for_each(|unzipped!(mut right, mut left)| {
                right.write(left.read());

                if cfg!(debug_assertions) {
                    left.write(E::faer_nan());
                }
            });
        }
    } else {
        let (mut u1, mut u2) = if compact_u == 0 {
            let (u1, u2) = u.rb_mut().split_at_row_mut(k + 1);
            (
                u1.submatrix_mut(0, 1, k + 1, k + 1),
                u2.submatrix_mut(0, k + 1, rem + 1, rem + 1),
            )
        } else {
            // NOTE: need to handle rotation of Q1, q1
            let (u1, u2) = u.rb_mut().split_at_col_mut(k + 1);
            (u1, u2)
        };

        let (mut v1, mut v2) = match v.rb_mut() {
            Some(v) => {
                let (v1, v2) = v.split_at_row_mut(k);
                (
                    Some(v1.submatrix_mut(0, 1, k, k)),
                    Some(v2.submatrix_mut(1, k + 1, rem, rem)),
                )
            }
            None => (None, None),
        };

        let stack_bytes = stack.len_bytes();
        let (mem1, stack2) = stack.rb_mut().make_raw::<u8>(stack_bytes / 2);
        let stack1 = PodStack::new(mem1);

        join_raw(
            |parallelism| {
                bidiag_svd_impl(
                    d1,
                    sub_d1,
                    u1.rb_mut(),
                    v1.rb_mut(),
                    true,
                    jacobi_fallback_threshold,
                    epsilon,
                    consider_zero_threshold,
                    parallelism,
                    stack1,
                );
            },
            |parallelism| {
                bidiag_svd_impl(
                    d2,
                    sub_d2,
                    u2.rb_mut(),
                    v2.rb_mut(),
                    true,
                    jacobi_fallback_threshold,
                    epsilon,
                    consider_zero_threshold,
                    parallelism,
                    stack2,
                );
            },
            parallelism,
        );

        if compact_u == 1 {
            // handle rotation of Q1, q1
            for i in (0..k).rev() {
                crate::perm::swap_cols_idx(u1.rb_mut(), i, i + 1);
            }
        }

        for i in (0..k).rev() {
            diag[i + 1] = diag[i];
        }
    }

    if let Some(mut v) = v.rb_mut() {
        v.write(k, 0, E::faer_one());
    };

    let lambda = if compact_u == 0 {
        u.read(k, k + 1)
    } else {
        // we already rotated u
        u.read(1, 0)
    };
    let phi = if compact_u == 0 {
        u.read(k + 1, n)
    } else {
        u.read(0, n)
    };

    let al = alpha.faer_mul(lambda);
    let bp = beta.faer_mul(phi);

    let r0 = ((al.faer_mul(al)).faer_add(bp.faer_mul(bp))).faer_sqrt();
    let (c0, s0) = if r0 == E::faer_zero() {
        (E::faer_one(), E::faer_zero())
    } else {
        (al.faer_div(r0), bp.faer_div(r0))
    };

    let col0 = subdiag;
    diag[0] = r0;
    col0[0] = r0;

    if compact_u == 0 {
        let (u1, u2) = if compact_u == 0 {
            let (u1, u2) = u.rb_mut().split_at_row_mut(k + 1);
            (u1, u2)
        } else {
            let (u1, u2) = u.rb_mut().split_at_row_mut(1);
            (u1, u2)
        };

        let (mut u0_top, u1) = u1.split_at_col_mut(1);
        let (u1, mut un_top) = u1.split_at_col_mut(n - 1);
        let (mut u0_bot, u2) = u2.split_at_col_mut(1);
        let (u2, mut un_bot) = u2.split_at_col_mut(n - 1);

        for j in 0..k {
            col0[j + 1] = alpha.faer_mul(u1.read(k, j));
        }
        for j in 0..rem {
            col0[j + 1 + k] = beta.faer_mul(u2.read(0, j + k));
        }

        zipped_rw!(
            u0_top.rb_mut().col_mut(0),
            un_top.rb_mut().col_mut(0),
            u1.col_mut(k),
        )
        .for_each(|unzipped!(mut dst0, mut dstn, mut src)| {
            let src_ = src.read();
            dst0.write(c0.faer_mul(src_));
            dstn.write(s0.faer_neg().faer_mul(src_));
            if cfg!(debug_assertions) {
                src.write(E::faer_nan());
            }
        });

        zipped_rw!(u0_bot.rb_mut().col_mut(0), un_bot.rb_mut().col_mut(0),).for_each(
            |unzipped!(mut dst0, mut dstn)| {
                let src_ = dstn.read();
                dst0.write(s0.faer_mul(src_));
                dstn.write(c0.faer_mul(src_));
            },
        );
    } else {
        for j in 0..k {
            col0[j + 1] = alpha.faer_mul(u.read(1, j + 1));
            u.write(1, j + 1, E::faer_zero());
        }
        for j in 0..rem {
            col0[j + 1 + k] = beta.faer_mul(u.read(0, j + k + 1));
            u.write(0, j + k + 1, E::faer_zero());
        }

        let q10 = u.read(0, 0);
        let q21 = u.read(1, n);

        u.write(0, 0, c0.faer_mul(q10));
        u.write(0, n, s0.faer_neg().faer_mul(q10));
        u.write(1, 0, s0.faer_mul(q21));
        u.write(1, n, c0.faer_mul(q21));
    }

    let (perm, stack) = stack.rb_mut().make_with(n, |_| 0usize);
    let (jacobi_coeffs, stack) = stack.make_with(n, |_| JacobiRotation {
        c: E::faer_zero(),
        s: E::faer_zero(),
    });
    let (jacobi_indices, mut stack) = stack.make_with(n, |_| 0);

    let (jacobi_0i, jacobi_ij) = {
        let (transpositions, stack) = stack.rb_mut().make_with(n, |_| 0usize);
        deflate(
            diag,
            col0,
            jacobi_coeffs,
            jacobi_indices,
            transpositions,
            perm,
            k,
            epsilon,
            consider_zero_threshold,
            stack,
        )
    };

    let allocate_vm = v.is_some() as usize;
    let allocate_um = fill_u as usize;
    let (mut um, stack) = temp_mat_zeroed::<E>(n + 1, allocate_um * (n + 1), stack);
    let (mut vm, mut stack) = temp_mat_zeroed::<E>(n, allocate_vm * n, stack);
    let mut um = um.as_mut();
    let mut vm = vm.as_mut();

    compute_svd_of_m(
        fill_u.then_some(um.rb_mut()),
        v.is_some().then_some(vm.rb_mut()),
        diag,
        col0,
        perm,
        epsilon,
        consider_zero_threshold,
        stack.rb_mut(),
    );

    for (rot, &i) in jacobi_coeffs[jacobi_0i..][..jacobi_ij]
        .iter()
        .zip(&jacobi_indices[jacobi_0i..][..jacobi_ij])
        .rev()
    {
        let (i, j) = (i - 1, i);
        let actual_i = perm[i];
        let actual_j = perm[j];

        if fill_u {
            let (row_i, row_j) = um.rb_mut().two_rows_mut(actual_i, actual_j);
            rot.apply_on_the_left_in_place(row_i, row_j);
        }

        if v.is_some() {
            let (row_i, row_j) = vm.rb_mut().two_rows_mut(actual_i, actual_j);
            rot.apply_on_the_left_in_place(row_i, row_j);
        }
    }

    if fill_u {
        for (rot, &i) in jacobi_coeffs[..jacobi_0i]
            .iter()
            .zip(&jacobi_indices[..jacobi_0i])
            .rev()
        {
            let (row_0, row_i) = um.rb_mut().two_rows_mut(0, i);
            rot.apply_on_the_left_in_place(row_0, row_i);
        }
    }

    let _v_is_none = v.is_none();

    let mut update_v = |parallelism: Parallelism<'_>, stack: &mut PodStack| {
        let (mut combined_v, _) = temp_mat_uninit::<E>(n, allocate_vm * n, stack);
        let mut combined_v = combined_v.as_mut();
        let v_rhs = vm.rb();

        if let Some(mut v) = v.rb_mut() {
            let mut combined_v = combined_v.rb_mut();
            let (mut combined_v1, combined_v2) = combined_v.rb_mut().split_at_row_mut(k);
            let mut combined_v2 = combined_v2.submatrix_mut(1, 0, rem, n);

            let v_lhs = v.rb();
            let v_lhs1 = v_lhs.submatrix(0, 1, k, k);
            let v_lhs2 = v_lhs.submatrix(k + 1, k + 1, rem, rem);
            let (v_rhs1, v_rhs2) = v_rhs.split_at_row(1).1.split_at_row(k);

            join_raw(
                |parallelism| {
                    crate::linalg::matmul::matmul(
                        combined_v1.rb_mut(),
                        v_lhs1,
                        v_rhs1,
                        None,
                        E::faer_one(),
                        parallelism,
                    )
                },
                |parallelism| {
                    crate::linalg::matmul::matmul(
                        combined_v2.rb_mut(),
                        v_lhs2,
                        v_rhs2,
                        None,
                        E::faer_one(),
                        parallelism,
                    )
                },
                parallelism,
            );

            crate::linalg::matmul::matmul(
                combined_v.rb_mut().submatrix_mut(k, 0, 1, n),
                v_lhs.submatrix(k, 0, 1, 1),
                v_rhs.submatrix(0, 0, 1, n),
                None,
                E::faer_one(),
                parallelism,
            );

            zipped_rw!(v.rb_mut(), combined_v.rb())
                .for_each(|unzipped!(mut dst, src)| dst.write(src.read()));
        }
    };

    let mut update_u = |parallelism: Parallelism<'_>, stack: &mut PodStack| {
        let (mut combined_u, _) = temp_mat_uninit::<E>(n + 1, allocate_um * (n + 1), stack);
        let mut combined_u = combined_u.as_mut();

        if fill_u {
            let (mut combined_u1, mut combined_u2) = combined_u.rb_mut().split_at_row_mut(k + 1);
            let u_lhs = u.rb();
            let u_rhs = um.rb();
            let (u_lhs1, u_lhs2) = (
                u_lhs.submatrix(0, 0, k + 1, k + 1),
                u_lhs.submatrix(k + 1, k + 1, rem + 1, rem + 1),
            );
            let (u_rhs1, u_rhs2) = u_rhs.split_at_row(k + 1);

            join_raw(
                |parallelism| {
                    // matrix matrix
                    crate::linalg::matmul::matmul(
                        combined_u1.rb_mut(),
                        u_lhs1,
                        u_rhs1,
                        None,
                        E::faer_one(),
                        parallelism,
                    );
                    // rank 1 update
                    crate::linalg::matmul::matmul(
                        combined_u1.rb_mut(),
                        u_lhs.col(n).subrows(0, k + 1).as_2d(),
                        u_rhs2.row(rem).as_2d(),
                        Some(E::faer_one()),
                        E::faer_one(),
                        parallelism,
                    );
                },
                |parallelism| {
                    // matrix matrix
                    crate::linalg::matmul::matmul(
                        combined_u2.rb_mut(),
                        u_lhs2,
                        u_rhs2,
                        None,
                        E::faer_one(),
                        parallelism,
                    );
                    // rank 1 update
                    crate::linalg::matmul::matmul(
                        combined_u2.rb_mut(),
                        u_lhs.col(0).subrows(k + 1, rem + 1).as_2d(),
                        u_rhs1.row(0).as_2d(),
                        Some(E::faer_one()),
                        E::faer_one(),
                        parallelism,
                    );
                },
                parallelism,
            );

            zipped_rw!(u.rb_mut(), combined_u.rb())
                .for_each(|unzipped!(mut dst, src)| dst.write(src.read()));
        }
    };

    if compact_u == 1 {
        update_v(parallelism, stack.rb_mut());
        if fill_u {
            let (mut combined_u, _) = temp_mat_uninit::<E>(2, n + 1, stack);
            let mut combined_u = combined_u.as_mut();
            crate::linalg::matmul::matmul(
                combined_u.rb_mut(),
                u.rb(),
                um.rb(),
                None,
                E::faer_one(),
                parallelism,
            );
            zipped_rw!(u.rb_mut(), combined_u.rb())
                .for_each(|unzipped!(mut dst, src)| dst.write(src.read()));
        }
    } else {
        match parallelism {
            #[cfg(feature = "rayon")]
            Parallelism::Rayon(_) if !_v_is_none => {
                let req_v = crate::linalg::temp_mat_req::<E>(n, n).unwrap();
                let (mem_v, stack_u) =
                    stack.make_aligned_raw::<u8>(req_v.size_bytes(), req_v.align_bytes());
                let stack_v = PodStack::new(mem_v);
                crate::utils::thread::join_raw(
                    move |parallelism| update_v(parallelism, stack_v),
                    move |parallelism| update_u(parallelism, stack_u),
                    parallelism,
                );
            }
            _ => {
                update_v(parallelism, stack.rb_mut());
                update_u(parallelism, stack);
            }
        }
    }

    for x in &mut *diag {
        *x = (*x).faer_mul(max_val);
    }
}

pub fn bidiag_real_svd_req<E: Entity>(
    n: usize,
    jacobi_fallback_threshold: usize,
    compute_u: bool,
    compute_v: bool,
    parallelism: Parallelism,
) -> Result<StackReq, SizeOverflow> {
    if n <= jacobi_fallback_threshold {
        temp_mat_req::<E>(n, n)
    } else {
        let _ = parallelism;
        let perm = StackReq::try_new::<usize>(n)?;
        let jacobi_coeffs = StackReq::try_new::<JacobiRotation<E>>(n)?;
        let jacobi_indices = perm;
        let transpositions = perm;
        let real_ind = perm;
        let real_col = perm;

        let um = temp_mat_req::<E>(n + 1, n + 1)?;
        let vm = temp_mat_req::<E>(n, if compute_v { n } else { 0 })?;

        let combined_u = temp_mat_req::<E>(if compute_u { n + 1 } else { 2 }, n + 1)?;
        let combined_v = vm;

        let prologue = StackReq::try_all_of([perm, jacobi_coeffs, jacobi_indices])?;

        StackReq::try_all_of([
            prologue,
            um,
            vm,
            combined_u,
            combined_v,
            transpositions,
            real_ind,
            real_col,
        ])
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::{assert, Mat};
    use assert_approx_eq::assert_approx_eq;

    macro_rules! make_stack {
        ($req: expr) => {
            ::dyn_stack::PodStack::new(&mut ::dyn_stack::GlobalPodBuffer::new($req.unwrap()))
        };
    }

    // to avoid overflowing the stack
    macro_rules! vec_static {
    ($($x:expr),+ $(,)?) => (
        {
            static ARRAY: &[f64] = &[$($x),+];
            ARRAY.to_vec()
        }
    );
    }

    #[test]
    fn test_svd_n() {
        for n in [9, 16, 32, 64, 128, 256, 512, 1024] {
            let diag = (0..n).map(|_| rand::random::<f64>()).collect::<Vec<_>>();
            let subdiag = (0..n).map(|_| rand::random::<f64>()).collect::<Vec<_>>();
            dbg!(&diag, &subdiag);

            let n = diag.len();
            let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
            let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
            let s = {
                let mut diag = diag.clone();
                let mut subdiag = subdiag.clone();
                compute_bidiag_real_svd(
                    &mut diag,
                    &mut subdiag,
                    Some(u.as_mut()),
                    Some(v.as_mut()),
                    5,
                    0,
                    f64::EPSILON,
                    f64::MIN_POSITIVE,
                    Parallelism::None,
                    make_stack!(bidiag_real_svd_req::<f64>(
                        n,
                        5,
                        true,
                        true,
                        Parallelism::None
                    )),
                );
                Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
            };

            let reconstructed = &u * &s * v.transpose();
            for j in 0..n {
                for i in 0..n + 1 {
                    let target = if i == j {
                        diag[j]
                    } else if i == j + 1 {
                        subdiag[j]
                    } else {
                        0.0
                    };

                    assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
                }
            }
        }
    }

    #[test]
    fn test_svd_4() {
        let diag = vec_static![1.0, 2.0, 3.0, 4.0];
        let subdiag = vec_static![1.0, 1.0, 1.0];

        let n = diag.len();
        let mut u = Mat::from_fn(n, n, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            bidiag_svd_qr_algorithm_impl(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                f64::EPSILON,
                f64::MIN_POSITIVE,
            );
            Mat::from_fn(n, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_64() {
        let diag = vec_static![
            0.5488135039273248,
            0.6027633760716439,
            0.4236547993389047,
            0.4375872112626925,
            0.9636627605010293,
            0.7917250380826646,
            0.5680445610939323,
            0.07103605819788694,
            0.02021839744032572,
            0.7781567509498505,
            0.978618342232764,
            0.46147936225293185,
            0.11827442586893322,
            0.1433532874090464,
            0.5218483217500717,
            0.26455561210462697,
            0.45615033221654855,
            0.018789800436355142,
            0.6120957227224214,
            0.9437480785146242,
            0.359507900573786,
            0.6976311959272649,
            0.6667667154456677,
            0.2103825610738409,
            0.31542835092418386,
            0.5701967704178796,
            0.9883738380592262,
            0.2088767560948347,
            0.6531083254653984,
            0.4663107728563063,
            0.15896958364551972,
            0.6563295894652734,
            0.1965823616800535,
            0.8209932298479351,
            0.8379449074988039,
            0.9764594650133958,
            0.9767610881903371,
            0.7392635793983017,
            0.2828069625764096,
            0.29614019752214493,
            0.317983179393976,
            0.06414749634878436,
            0.5666014542065752,
            0.5232480534666997,
            0.5759464955561793,
            0.31856895245132366,
            0.13179786240439217,
            0.2894060929472011,
            0.5865129348100832,
            0.8289400292173631,
            0.6778165367962301,
            0.7351940221225949,
            0.24875314351995803,
            0.592041931271839,
            0.2230816326406183,
            0.44712537861762736,
            0.6994792753175043,
            0.8137978197024772,
            0.8811031971111616,
            0.8817353618548528,
            0.7252542798196405,
            0.9560836347232239,
            0.4238550485581797,
            0.019193198309333526,
        ];
        let subdiag = vec_static![
            0.7151893663724195,
            0.5448831829968969,
            0.6458941130666561,
            0.8917730007820798,
            0.3834415188257777,
            0.5288949197529045,
            0.925596638292661,
            0.08712929970154071,
            0.832619845547938,
            0.8700121482468192,
            0.7991585642167236,
            0.7805291762864555,
            0.6399210213275238,
            0.9446689170495839,
            0.4146619399905236,
            0.7742336894342167,
            0.5684339488686485,
            0.6176354970758771,
            0.6169339968747569,
            0.6818202991034834,
            0.43703195379934145,
            0.06022547162926983,
            0.6706378696181594,
            0.1289262976548533,
            0.3637107709426226,
            0.43860151346232035,
            0.10204481074802807,
            0.16130951788499626,
            0.2532916025397821,
            0.24442559200160274,
            0.11037514116430513,
            0.1381829513486138,
            0.3687251706609641,
            0.09710127579306127,
            0.09609840789396307,
            0.4686512016477016,
            0.604845519745046,
            0.039187792254320675,
            0.1201965612131689,
            0.11872771895424405,
            0.41426299451466997,
            0.6924721193700198,
            0.2653894909394454,
            0.09394051075844168,
            0.9292961975762141,
            0.6674103799636817,
            0.7163272041185655,
            0.18319136200711683,
            0.020107546187493552,
            0.004695476192547066,
            0.27000797319216485,
            0.9621885451174382,
            0.5761573344178369,
            0.5722519057908734,
            0.952749011516985,
            0.8464086724711278,
            0.29743695085513366,
            0.39650574084698464,
            0.5812728726358587,
            0.6925315900777659,
            0.5013243819267023,
            0.6439901992296374,
            0.6063932141279244,
            0.30157481667454933,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                15,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    15,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_128() {
        let diag = vec_static![
            0.21604803719303378,
            0.3911685871373043,
            0.4717353802588816,
            0.5258511967180588,
            0.3495587375007383,
            0.29956620660149913,
            0.9617586737752963,
            0.1358382160501187,
            0.7927594027427639,
            0.7002703649721469,
            0.5011867621846828,
            0.5508360458872776,
            0.6671001529243077,
            0.6182640702775855,
            0.537113258218727,
            0.6494319004775305,
            0.22394544080467793,
            0.48963764985534675,
            0.4960750561790864,
            0.6762779313777806,
            0.4942507487962028,
            0.30598289084328734,
            0.7477750830615565,
            0.4134175601075717,
            0.16210479706508774,
            0.8554869501826141,
            0.9633922725373281,
            0.5178447186808554,
            0.4808128823427542,
            0.21235530384938095,
            0.34390969950363515,
            0.5222397627933266,
            0.8078540262388403,
            0.3084527067162488,
            0.8510243010197533,
            0.7492658574080864,
            0.2971760706318315,
            0.5821109217348188,
            0.9355688927263782,
            0.6568884170143395,
            0.7143623902994362,
            0.8745547764908594,
            0.3166725157072694,
            0.06280104609776738,
            0.9988219571244557,
            0.3034566490500038,
            0.6043605519679998,
            0.5327046414132618,
            0.8160784550544813,
            0.33220426954591,
            0.3160884461616036,
            0.08177180318496124,
            0.5859174456552851,
            0.225028943377522,
            0.6862486995947498,
            0.3697307197174694,
            0.7873879339970076,
            0.21908989285489933,
            0.5410943047067103,
            0.6302243946164361,
            0.17747192668740164,
            0.6281781273604742,
            0.5854835895783808,
            0.6512696242357562,
            0.559113383282545,
            0.7596325146050337,
            0.09312745837133651,
            0.501703867727036,
            0.949275885265856,
            0.620974047588036,
            0.3874150582755552,
            0.7083379430913563,
            0.75288059477905,
            0.1270527976228708,
            0.23126586686009443,
            0.12024520920717885,
            0.5679798123160427,
            0.17978590193238875,
            0.6968486822029739,
            0.7157516389948776,
            0.863508815070862,
            0.15864367506453403,
            0.11417600568460162,
            0.9651132813595767,
            0.10920826282790252,
            0.28700997153205676,
            0.7054115856120382,
            0.3490250121285702,
            0.29128696537701393,
            0.9304161241740285,
            0.2268455711369768,
            0.7658439715371834,
            0.06071820836196018,
            0.027168754664580574,
            0.4433866338712438,
            0.8175541779071445,
            0.1195115235309906,
            0.5543104624561522,
            0.3831276253977298,
            0.4944969243226346,
            0.5069163526882893,
            0.2519761931522101,
            0.3802289988930322,
            0.12792877754948118,
            0.964418293878996,
            0.5028833771104924,
            0.7140891912929843,
            0.929920514299548,
            0.9622470160446642,
            0.9165762824392009,
            0.957409262046926,
            0.046890401426795014,
            0.9559558333706967,
            0.10165169845100896,
            0.4030729711821963,
            0.7457966905965205,
            0.45506350389528505,
            0.22855385350793034,
            0.5774367409801651,
            0.3395031602763888,
            0.8750661230154188,
            0.5436696130661226,
            0.14750222902451415,
            0.4702601766248026,
            0.380398914581512,
            0.9870908933390458,
            0.46972043263478913,
            0.7629347676594994,
        ];
        let subdiag = vec_static![
            0.5656024571426115,
            0.45845513013153916,
            0.8464475246274293,
            0.5477997098985157,
            0.8505749005789477,
            0.036655381821360744,
            0.2164923701172936,
            0.992162216073592,
            0.2442305366823595,
            0.5417621610202344,
            0.4608606260638025,
            0.41227354070140787,
            0.9159226592102876,
            0.41719392867697913,
            0.30450224568165174,
            0.913124938919393,
            0.8998452924705163,
            0.8615757311218621,
            0.2152284123127688,
            0.4290329466601026,
            0.5684735244446122,
            0.5679483074313831,
            0.34457343811812624,
            0.5415296298206568,
            0.8356244784272918,
            0.8166459785498866,
            0.5080772280859633,
            0.8956149463267449,
            0.04644596806209644,
            0.3433783039423306,
            0.6431291583487241,
            0.4639223708461163,
            0.8505923626098933,
            0.04989146706988801,
            0.7831842489006767,
            0.27932627070175853,
            0.4802138742827945,
            0.5760948972572593,
            0.5545322682351106,
            0.21873418294084268,
            0.431998289203515,
            0.28960427805813527,
            0.20805283252719742,
            0.4598338664038314,
            0.03434413796223135,
            0.8739945690639832,
            0.5815729275918338,
            0.19359623734135956,
            0.7141431528881316,
            0.5483053379185088,
            0.3859904909506836,
            0.32779561877996766,
            0.7473245156367854,
            0.7527840401770349,
            0.5332180778014928,
            0.18845765841689788,
            0.8863719798459543,
            0.04292531786719711,
            0.7455031922487241,
            0.46581152947672433,
            0.2528295777598255,
            0.3175198547141883,
            0.45895268958636226,
            0.5910400449400208,
            0.5907671590866751,
            0.7200711220634236,
            0.06754760808821003,
            0.4622760636221317,
            0.6725272498965258,
            0.842299904498038,
            0.675399181783893,
            0.8815937503757029,
            0.4030870020741989,
            0.7417783045865814,
            0.27829985877736574,
            0.16223373545195352,
            0.047472863984402536,
            0.9762183713220033,
            0.850015796049705,
            0.9883602721062648,
            0.7183825826458256,
            0.6776325302074241,
            0.5606755750903751,
            0.4901171002893656,
            0.726732063504903,
            0.5776852831317164,
            0.7123622484466453,
            0.755897963752805,
            0.5446524557651221,
            0.3337687893180128,
            0.3075795178050936,
            0.8001257236535856,
            0.18895675106770227,
            0.3579844168353461,
            0.3527072586559228,
            0.0817878204567436,
            0.9969511917959174,
            0.6404417022308447,
            0.28418361844714657,
            0.6511969684943811,
            0.2866537500197578,
            0.33561205627307966,
            0.2534861786545628,
            0.9960188356208826,
            0.10107370019257966,
            0.6295541630401397,
            0.5638169140807354,
            0.7619672332144566,
            0.19599633482758116,
            0.004262801281641138,
            0.5637409510062904,
            0.15132931034408448,
            0.6357856412777408,
            0.3570943637285525,
            0.8986379725558856,
            0.6123637576833882,
            0.21915528413252194,
            0.35614983876651884,
            0.9830902701738703,
            0.2656035472280712,
            0.8894933827255144,
            0.5619386899890909,
            0.019131462959508294,
            0.36198104450872315,
            0.8387140376274692,
            0.2967905230261907,
            0.009493153610368066,
            0.4451542437520205,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                40,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    40,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_1024_0() {
        let diag = vec_static![
            0.8845251118613418,
            0.34455256888844044,
            0.210674711024898,
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            0.7797586491845876,
            0.09633675865715385,
            0.15677008169215234,
            0.23394539969965789,
            0.7323421797485367,
            0.2695584209878511,
            0.8923611906456285,
            0.009384104149009809,
            0.20087989891307956,
            0.37650593207094696,
            0.9675672321088212,
            0.2460276680731922,
            0.8829436148568359,
            0.9011339590969748,
            0.9893348013640839,
            0.34402326744018086,
            0.2918830938824284,
            0.032443773270217524,
            0.20236048346275737,
            0.10644854768113488,
            0.39314962067289483,
            0.16844774292617437,
            0.0193640642109717,
            0.32131633915309843,
            0.9031289996826208,
            0.7996900072509187,
            0.9544912609903554,
            0.21238118940953343,
            0.5588780213505607,
            0.359423748193081,
            0.14306783265825895,
            0.03646714126807182,
            0.8108290011843635,
            0.5024978136603396,
            0.30684702940974795,
            0.08078866745769053,
            0.584846202681692,
            0.9731481619668385,
            0.015849082319753238,
            0.47318193572395295,
            0.45751317022352533,
            0.1347836174790057,
            0.07869729277442472,
            0.3730223469708691,
            0.8108036315360886,
            0.6191676048612971,
            0.3026033801925956,
            0.12952357665540415,
            0.9899082523324236,
            0.7427677782218289,
            0.5197759491989283,
            0.9967841038515952,
            0.5035569325914083,
            0.3834211729816368,
            0.3309587035330964,
            0.05340317428058894,
            0.6463414755064464,
            0.5887364268384743,
            0.1394604722694749,
            0.457951772941251,
            0.45288455141492356,
            0.08565262580204513,
            0.6084418050718142,
            0.8203820932095913,
            0.6461410253957783,
            0.2280002938711736,
            0.30887824900482863,
            0.9628221732310465,
            0.44209913122842126,
            0.35857995161780576,
            0.07182970907223107,
            0.171047907339957,
            0.24878787204659314,
            0.06756423474769113,
            0.8732787479160486,
            0.11342075749834801,
            0.9967087053746094,
            0.29132559183022533,
            0.7537365927002493,
            0.29658308252612886,
            0.8269346653596071,
            0.8714825210553282,
            0.07360104666475487,
            0.33337023074576533,
            0.9799778581582539,
            0.08375046754795701,
            0.8660403417189309,
            0.28286840561868565,
            0.8252295280561422,
            0.21889588864658394,
            0.003964186364546207,
            0.027000390024012777,
            0.4606990121656196,
            0.3297646362154407,
            0.8929719909234096,
            0.18414480606970096,
            0.5377843931338276,
            0.506843731844477,
            0.3338619626598003,
            0.300351599526833,
            0.1808568608406541,
            0.41477945200762745,
            0.5759119192571923,
            0.11284300062398211,
            0.749403315530038,
            0.3859482139006477,
            0.6051165122948982,
            0.971891394749644,
            0.6605478455153012,
            0.5861700067837108,
            0.07225848311996741,
            0.26352858091537257,
            0.8742611615172913,
            0.8420666125054983,
            0.30095829631010684,
            0.9035078353304578,
            0.6468305240161736,
            0.40631323088985216,
            0.26893323723422446,
            0.14348472814734614,
            0.8446063514723486,
            0.8841258110212956,
            0.259918014410066,
            0.6701006927979326,
            0.9864420189950698,
            0.7476246814792673,
            0.9505952494844244,
            0.3799973827243003,
            0.5547180102231944,
            0.7105616395309934,
            0.5383483239781777,
            0.31000439759468,
            0.7059163158915425,
            0.6856815369544714,
            0.39866345825125704,
            0.042513276582041515,
            0.337803158592229,
            0.39465691436662753,
            0.40306375626322977,
            0.8235182077717283,
            0.6090460629372503,
            0.6764030144841834,
            0.9635539428291812,
            0.7042014067502246,
            0.5127830179290652,
            0.9131677796050657,
            0.5311464655158742,
            0.8535060094374574,
            0.6649277719671861,
            0.8158423948021637,
            0.3003102858044917,
            0.9543186871563114,
            0.6693534602340823,
            0.9280256323955216,
            0.07378036482014472,
            0.7734809173963871,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                40,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    40,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_1024_1() {
        let diag = vec_static![
            0.4841629363684429,
            0.09500211557292393,
            0.09054687338157585,
            0.9349389484044085,
            0.2263540442491473,
            0.9883453587159688,
            0.27611412994054574,
            0.6934210307229077,
            0.7122722731446822,
            0.5764931807281632,
            0.7220922855179277,
            0.7690715441721571,
            0.18151810699724935,
            0.9703738941665442,
            0.46756918888654386,
            0.13829262641884732,
            0.26785351679939384,
            0.6826209585171479,
            0.7450857828877535,
            0.9612828629996442,
            0.8597056636877132,
            0.3147189885973717,
            0.9873882525578956,
            0.6390222666607266,
            0.6720524846258185,
            0.5710278204457248,
            0.012892307681411919,
            0.7804378611928872,
            0.2207373153565294,
            0.6251662057647014,
            0.13965841961139402,
            0.2611627557457328,
            0.21717079350115864,
            0.43746631711094563,
            0.3551960866238878,
            0.6076465429515558,
            0.10918455630762303,
            0.45967703949396843,
            0.7809114058971589,
            0.9954329677023613,
            0.6578987749953844,
            0.4524030893215839,
            0.8561365938081589,
            0.2702994410284004,
            0.5910189554474038,
            0.9215907321467688,
            0.7637726604998021,
            0.9996866856048456,
            0.4508665111686113,
            0.8440519119195404,
            0.5636981910772803,
            0.19691331749849916,
            0.14415353894000804,
            0.5491145376147477,
            0.5355242003504422,
            0.3569416212159129,
            0.7716747444655886,
            0.6481040816879275,
            0.0861928588345302,
            0.5644989365621479,
            0.4595649555376554,
            0.9924614965479639,
            0.34457880027775034,
            0.2709464498548626,
            0.6175995628458563,
            0.06410776075791469,
            0.8668177074089747,
            0.5133288724007797,
            0.26197543298622306,
            0.636716245285106,
            0.07784169215536019,
            0.23103012381763322,
            0.820507398231268,
            0.023267897611993882,
            0.9922903501796004,
            0.28413038646415656,
            0.4902041468658134,
            0.5353728340280559,
            0.39068757308715574,
            0.24110407030206493,
            0.7982266250190029,
            0.7498583088545846,
            0.8054303600922323,
            0.727152156075704,
            0.4433406657279716,
            0.11813500917759767,
            0.19322091867414604,
            0.6073386848510687,
            0.1892620575417534,
            0.9276884465180404,
            0.07416066041374547,
            0.11055221860974573,
            0.02742841195102952,
            0.9046475741926477,
            0.33798444110426407,
            0.25146251279015375,
            0.32543843780189374,
            0.23502095326757122,
            0.8151869673056854,
            0.06550480813538595,
            0.27213001583974805,
            0.2645917156255321,
            0.5965380863001157,
            0.6850808677017469,
            0.5485690599538193,
            0.7758605807485486,
            0.2819101125296871,
            0.21744462405804543,
            0.3210524068951949,
            0.8219967960710458,
            0.7168902509705798,
            0.20198849159812726,
            0.886790290413851,
            0.8405592175465555,
            0.6418755287944,
            0.2717157722359368,
            0.8785193775058516,
            0.2883033828479482,
            0.10560195836536401,
            0.9457172912478532,
            0.24413132566350226,
            0.4798067689848272,
            0.9758153512537662,
            0.9461639347527152,
            0.5015824600544672,
            0.4743546024442775,
            0.9439300463479419,
            0.8210197093977561,
            0.9756341250452755,
            0.3034443196917138,
            0.800696521458751,
            0.2469174591716501,
            0.24844908962357637,
            0.6729189361923051,
            0.34976823478261343,
            0.4548480895382889,
            0.06654283936888206,
            0.0538242709114648,
            0.6604648475989987,
            0.3761972351630566,
            0.17973288886807337,
            0.7785638677716654,
            0.3078309007281408,
            0.34383089894111674,
            0.9740115657118916,
            0.4142242073362077,
            0.46788525339663745,
            0.5102644222807242,
            0.6821466796495953,
            0.9234699208966476,
            0.25792002693706884,
            0.4473866152881345,
            0.5316752542174584,
            0.6753097405426745,
            0.9875155323493832,
            0.4846734070202009,
            0.25690620257834906,
            0.17444353295771786,
            0.1459026770332964,
            0.03789979000177324,
            0.009118098382703943,
            0.3369152724463358,
            0.6194036398104508,
            0.8802810559789717,
            0.5818695975885244,
            0.26902985408430724,
            0.8102535632453027,
            0.14367013617397928,
            0.5154528225856342,
            0.5525644118916562,
            0.11068694771654464,
            0.06648831260453159,
            0.30229383062322257,
            0.11043335057841508,
            0.7879174452882572,
            0.6847400055584101,
            0.7093025409825697,
            0.5672680847728112,
            0.9220621299078362,
            0.2308678094100194,
            0.11095431934483901,
            0.5076880016394446,
            0.18043287756259263,
            0.3633969552240899,
            0.0074109593813797,
            0.09540490121094358,
            0.5571906415926369,
            0.43964175899243474,
            0.6761714080491062,
            0.4050321473554952,
            0.4602021774929316,
            0.07357681703454355,
            0.5647139088824532,
            0.6632368351901867,
            0.038928065259385525,
            0.44302015465469524,
            0.30679321194639764,
            0.7896327936646736,
            0.36808308055633976,
            0.9630604740173925,
            0.5525981053006727,
            0.4390151783304377,
            0.5986597671093334,
            0.7053735351729479,
            0.3744694351975014,
            0.06532632305974129,
            0.7215457382122921,
            0.008636006836127419,
            0.6280218545206359,
            0.33553959659277455,
            0.25002369784269374,
            0.2222248299935472,
            0.4568576034259546,
            0.4735434309364681,
            0.7560202412217341,
            0.2878333331705232,
            0.8562148011742544,
            0.9300668918750148,
            0.5038181128961342,
            0.8752815415361238,
            0.33283717808771585,
            0.0533181654462932,
            0.6436829043555116,
            0.6420080671552673,
            0.35854475824887166,
            0.27660688607425366,
            0.09673817671242757,
            0.7732513019446627,
            0.8411109662991648,
            0.749581543720349,
            0.9699796229928836,
            0.3521026232752017,
            0.7805969271105201,
            0.488079532732169,
            0.3153834803216925,
            0.07515270460745327,
            0.9732385585699602,
            0.4379249859226594,
            0.47941135538293433,
            0.13531367195050192,
            0.7146259058201841,
            0.4191544792989341,
            0.9918711123891871,
            0.30163052510779575,
            0.15398133758394816,
            0.8062176573881823,
            0.48038219879486577,
            0.3693644458963321,
            0.2627309145392913,
            0.7075668012609716,
            0.4565937211789599,
            0.5928922749230827,
            0.3384720695832677,
            0.7510771063489731,
            0.09656778467148652,
            0.6111109196994062,
            0.4147274155510545,
            0.0825902347020141,
            0.23342446109854698,
            0.8755256481911295,
            0.3410479507631834,
            0.49101534840303174,
            0.9696314869369961,
            0.510646800941389,
            0.21155499454128557,
            0.04077883012959882,
            0.38616102006555564,
            0.27156802107462563,
            0.06772016667809999,
            0.08411279644339542,
            0.8105121747384336,
            0.8226062374641675,
            0.44436768427765405,
            0.8725783369785883,
            0.779280645376425,
            0.3493455256878125,
            0.8904257905232991,
            0.17885649868819264,
            0.37091811243722606,
            0.9379307789143213,
            0.5731629435054184,
            0.24143821812058197,
            0.4049562680058758,
            0.42299735446198383,
            0.2527001487162306,
            0.9054521922102531,
            0.8277991529134413,
            0.8896621223418875,
            0.2839601147382753,
            0.46967983028760474,
            0.6504612353037362,
            0.34359028291120974,
            0.49029076200099064,
            0.14605930621786978,
            0.7971485298165085,
            0.5670859996593979,
            0.5839996800579184,
            0.6921622958466791,
            0.013697871099110137,
            0.5716412855888595,
            0.33906402404202585,
            0.22346066558462552,
            0.3951530817090858,
            0.3137771706646575,
            0.17589811380719156,
            0.6129863037725723,
            0.5529667540622067,
            0.12509740175859485,
            0.08164145825063607,
            0.6066900494595772,
            0.841451822666421,
            0.674802798921404,
            0.6185795283245251,
            0.9582067789561727,
            0.2569207105730952,
            0.6883723276402567,
            0.9437634272458615,
            0.6907567856518468,
            0.30993500261298346,
            0.35730588644284866,
            0.03729512159713966,
            0.6212835719526415,
            0.9352121927645822,
            0.003243518888762509,
            0.5186143509666988,
            0.6353445184976869,
            0.3617368415649064,
            0.0490224440259972,
            0.050915012421679506,
            0.2242457283292124,
            0.8147096772905736,
            0.6409998571108203,
            0.4429200379026843,
            0.9765405935181827,
            0.947474779514954,
            0.18982873752274343,
            0.9001090977751718,
            0.24092805773510084,
            0.32038670586129525,
            0.7243688544237985,
            0.16277724068208277,
            0.7593379678662593,
            0.03968818460330925,
            0.7244607120881216,
            0.5165074643832736,
            0.5509416401806246,
            0.27586276721428116,
            0.21355188867529862,
            0.6848900102063389,
            0.6036452778970051,
            0.24640329117263937,
            0.3724137096019503,
            0.6206648555568799,
            0.7958800469169095,
            0.4900216823225789,
            0.7667675482271813,
            0.7289915357163391,
            0.1869296356076564,
            0.4524808232640244,
            0.7127991694765413,
            0.8523560180647201,
            0.08452399270145938,
            0.08783929331698737,
            0.9187616736013637,
            0.8344792646462194,
            0.9271740035408323,
            0.6370044327004926,
            0.873942787396045,
            0.9326891534142052,
            0.583677916072454,
            0.5922397691297683,
            0.034166387868455494,
            0.49377929786225716,
            0.06968856674405388,
            0.3615198138197385,
            0.6147672290102518,
            0.7483870374797253,
            0.5383849201838203,
            0.20836389375701803,
            0.051848999025362885,
            0.7320297267131506,
            0.924153708258932,
            0.24940540066017358,
            0.7254900302438296,
            0.8398185409002303,
            0.777945701086813,
            0.6028385015259239,
            0.8381119231050299,
            0.27366756479468357,
            0.46285049762980857,
            0.007202203008645491,
            0.20677467863560528,
            0.28019168675955963,
            0.6564077343354984,
            0.8408712038064403,
            0.5349118002964522,
            0.43356705996070866,
            0.5525629375884271,
            0.8029332651737107,
            0.3789630801802527,
            0.17394534806832818,
            0.988718962514845,
            0.008255988120142943,
            0.015895226625594283,
            0.6206393567272559,
            0.9209847878321579,
            0.39110624455115695,
            0.7437450370797415,
            0.8941157743879106,
            0.011860557459840382,
            0.4065207233329732,
            0.23282699080782854,
            0.4136910892382927,
            0.7206841713816652,
            0.9464858269922742,
            0.9756111958533134,
            0.8903027302325448,
            0.7803498010740468,
            0.8488613163761867,
            0.07892513274645052,
            0.30522172054483054,
            0.4993261339902775,
            0.1307441671901306,
            0.6724515744593036,
            0.4119707684405064,
            0.38835171094008847,
            0.2895036626226579,
            0.718862164114217,
            0.7422150014781085,
            0.577025210006751,
            0.8438049461114684,
            0.04479570356686591,
            0.006571236569574479,
            0.17682956223658808,
            0.967370326521429,
            0.6761115997966827,
            0.896496576282121,
            0.42414571451691774,
            0.2643999567897386,
            0.7674469660502053,
            0.13691540715717643,
            0.6323850606911723,
            0.722449896798871,
            0.12800291638316297,
            0.5857947189453917,
            0.4163336920317575,
            0.7884813514972842,
            0.1508727072569006,
            0.19160043939953175,
            0.23301374790352558,
            0.9359396775007895,
            0.9774923318040876,
            0.36863638896238904,
            0.6610912680758803,
            0.46057923250460875,
            0.9590829904391315,
            0.1991152018103779,
            0.28851906877515865,
            0.6000178123986674,
            0.1718467848396532,
            0.7441859910583363,
            0.4622654532527547,
            0.5555247226894746,
            0.7677773119779349,
            0.6189662457058476,
            0.6988753748655162,
            0.4750032009040738,
            0.5857617813696513,
            0.3775093165711094,
            0.44160821335373923,
            0.12090759035787657,
            0.6841147475130123,
            0.03774648075440323,
            0.8954128692100448,
            0.5935413509246164,
            0.06702705653530572,
            0.28700587543321343,
            0.0697394877285672,
            0.9125516672429073,
            0.6719964366102898,
            0.9959123901824811,
            0.504950180087362,
            0.9477042666997653,
            0.09952647180930152,
            0.23616454829702027,
            0.8089820248976782,
            0.5723582906192601,
            0.45900223107602056,
            0.7190603249230394,
            0.7312224218219942,
            0.2440096836115907,
            0.3214679621992278,
            0.6436465041307681,
            0.8041091255879695,
            0.9295477549605089,
            0.22298283586216816,
            0.9521137131163172,
            0.9113445716477977,
            0.9091525207587764,
            0.98119554906578,
            0.9623632967357395,
            0.4176562224573084,
            0.43224238227876044,
            0.8594532789544238,
            0.16938535196509052,
            0.5192280607612347,
            0.9258356358139087,
            0.9614365095053088,
            0.9978293940686767,
            0.7113526084695919,
            0.12023427192990421,
            0.6130024834770222,
        ];
        let subdiag = vec_static![
            0.7089936750607991,
            0.4483258132447713,
            0.9981541517505396,
            0.4357762689203716,
            0.08785321714493932,
            0.5830593471239359,
            0.8920829832035405,
            0.41879264651643233,
            0.6111872756826352,
            0.8355133394363216,
            0.9793257049723074,
            0.3077792382592349,
            0.026969652678317857,
            0.4182365345593675,
            0.7165887422928039,
            0.32051113319939784,
            0.3040695673093029,
            0.12374943078010614,
            0.5760995454778959,
            0.8349949587005299,
            0.8659898386028743,
            0.7380424842517467,
            0.3393928531765691,
            0.7692042361831338,
            0.7796769522976207,
            0.03394625254827366,
            0.8571181864186348,
            0.6332478925233004,
            0.5184070284563695,
            0.019874111122098692,
            0.34934923133474194,
            0.8996435373419481,
            0.9380698449825945,
            0.42729146961334785,
            0.8503760643862777,
            0.02100272123108793,
            0.860005740282607,
            0.5326670023777708,
            0.5375140475812259,
            0.47919176360919014,
            0.7828043406116267,
            0.3794436036617802,
            0.9565916535824568,
            0.3980599768323815,
            0.8370299250814085,
            0.0849340864800805,
            0.7982582487709378,
            0.8809610530914053,
            0.36691575235983476,
            0.33283272535856,
            0.20354784446582086,
            0.5905114553771318,
            0.9047458258061869,
            0.0741901700696852,
            0.7621042915415703,
            0.9271099058676796,
            0.9611052970248738,
            0.21963155934129597,
            0.1558133881330961,
            0.19737060099724124,
            0.6639174149441336,
            0.8533931201297167,
            0.020945116530869057,
            0.6309209706848019,
            0.662729381326571,
            0.5035887768043217,
            0.1860476975995281,
            0.7977891924110831,
            0.6120596386351329,
            0.4508756288936743,
            0.4280982611244475,
            0.6565556625529626,
            0.05492439555509321,
            0.8468248520937989,
            0.20170352295274463,
            0.7393170393911367,
            0.48708726141266956,
            0.5460887373535718,
            0.656782452876117,
            0.568726037595071,
            0.026929796216858648,
            0.4245161336525607,
            0.9785776113872318,
            0.27746635862377156,
            0.14235497780854722,
            0.5841314393096457,
            0.9132559628467195,
            0.7254633296207957,
            0.3994675683815655,
            0.6542374938945021,
            0.1902512272011253,
            0.3277110299931433,
            0.33880621714171744,
            0.7594202382313123,
            0.0741024563031083,
            0.4449907034648557,
            0.17929774764796536,
            0.12210400948657041,
            0.6787006921534013,
            0.7281232868795051,
            0.7792629004817995,
            0.08064014445414724,
            0.9724962652917525,
            0.5641512037393203,
            0.9530186413776301,
            0.7883920219250815,
            0.450092735561583,
            0.5860790804560074,
            0.6807750490420452,
            0.987717614690177,
            0.21946639995166828,
            0.3756875030946536,
            0.0017060266273737357,
            0.3528908661861736,
            0.23522785730653006,
            0.6636060441904992,
            0.9407607887743468,
            0.7781934785793816,
            0.053397574393358016,
            0.5877392899593834,
            0.36009459733262517,
            0.3946933470154008,
            0.003646320901408595,
            0.597476378931147,
            0.43062018218454423,
            0.754668884154315,
            0.7355323365994135,
            0.5992070407324596,
            0.80570947364751,
            0.8795556637529897,
            0.8470422441081501,
            0.8946910951244957,
            0.6124018879965861,
            0.7088007839992834,
            0.9492294886442566,
            0.2251524815149637,
            0.4393504360847734,
            0.20876330443014623,
            0.6237404705834967,
            0.568434233274606,
            0.9168918462734527,
            0.3169042147373512,
            0.6276703450566922,
            0.08898882215107173,
            0.7321272744066755,
            0.8346756063313672,
            0.9565539131638344,
            0.003786651390763507,
            0.9521878269851143,
            0.8212172699917235,
            0.6238752560966074,
            0.09062018280410378,
            0.5532464578105598,
            0.4732513968021569,
            0.08469215363970584,
            0.6052403882500162,
            0.9941187028387413,
            0.7991286021147076,
            0.41618648415719917,
            0.22088579734088232,
            0.35506367821590523,
            0.3195577484429003,
            0.4312013001520172,
            0.34701617555070774,
            0.02232731969316082,
            0.6455567189539291,
            0.03371765698821416,
            0.9994490916392196,
            0.38346264435008326,
            0.3900943382330475,
            0.41645227148290087,
            0.4072897445006035,
            0.49980001378406624,
            0.9624024912309374,
            0.4243441122717946,
            0.10097127806713302,
            0.27318128155213284,
            0.01980432947685362,
            0.3919061611532205,
            0.5400372964620943,
            0.2837105264404538,
            0.5103853253037687,
            0.22502206457461582,
            0.42980394628690555,
            0.4255706600799779,
            0.05882680912038962,
            0.045980087565794414,
            0.4593492445739672,
            0.16748295392635637,
            0.5815760659541308,
            0.11804087865064283,
            0.9808592457277086,
            0.08447454175042679,
            0.14528681629095253,
            0.7966020134503727,
            0.39988958241786354,
            0.3966452292589152,
            0.7632471794259268,
            0.8364344797266556,
            0.05406706032400832,
            0.6196158413543642,
            0.8134083748522049,
            0.44315471096621983,
            0.7381281011866444,
            0.18178582271615507,
            0.39598457602595705,
            0.800755518965327,
            0.0996500182280885,
            0.6928794391981185,
            0.9271894035862747,
            0.420985543811215,
            0.8299212709536242,
            0.43361990762578595,
            0.8474588422074529,
            0.5384961461705045,
            0.04579216102837591,
            0.8144388324215048,
            0.7503331242965274,
            0.7443007483027947,
            0.8591172111741237,
            0.3129572042807881,
            0.43629364584579977,
            0.040775882175336786,
            0.5265987324893766,
            0.3278197764720532,
            0.9638233406714964,
            0.593089005448218,
            0.8531017811247185,
            0.3833716681860213,
            0.734770506472232,
            0.3961906892927213,
            0.7378952710775825,
            0.37054254273069753,
            0.25018676528013395,
            0.9155778839394266,
            0.5775373957855319,
            0.9299799078638226,
            0.1559513162598678,
            0.019987346908604264,
            0.35431474983114275,
            0.07348835806127052,
            0.05331421841012918,
            0.37541187674860843,
            0.3661350181166707,
            0.46649040220621407,
            0.5391607833956678,
            0.21034606925423682,
            0.058248534604802504,
            0.25723227398593673,
            0.3993796060279372,
            0.6796992856613469,
            0.08247673723983995,
            0.023451831694683567,
            0.13695099159078417,
            0.7384727200594983,
            0.6820765144990079,
            0.18829448527021186,
            0.41752496350142543,
            0.44795608890648975,
            0.05387150818596265,
            0.09342223756050061,
            0.36871457265674346,
            0.38200253190773514,
            0.9780514611196112,
            0.8456093037811834,
            0.1013726609210367,
            0.46219518711839835,
            0.6917545713401585,
            0.038618164075749584,
            0.40274635983558005,
            0.7018794912964974,
            0.6478854695078218,
            0.7477266217538235,
            0.6155100513086282,
            0.7365073270103238,
            0.12209169396041908,
            0.3190988421650174,
            0.8152293593221053,
            0.903108385570188,
            0.7992173115598106,
            0.7778936768007642,
            0.5419106038480274,
            0.41508926093445864,
            0.6245464965280143,
            0.21031078381218093,
            0.9157838433489752,
            0.2068179324203827,
            0.6403528464071976,
            0.22552934946201353,
            0.3498260607478918,
            0.5922724980687619,
            0.38056497907580034,
            0.016171926437021922,
            0.4563335453363586,
            0.8915640222527538,
            0.9009568027990429,
            0.39953462973284104,
            0.31682222212460576,
            0.8549860892263802,
            0.3874141470807463,
            0.7162930394888214,
            0.17862862530412082,
            0.7807735411730311,
            0.6005852379109198,
            0.5788955529636066,
            0.1998735921588788,
            0.0746258058981547,
            0.6755307862214068,
            0.959465854838282,
            0.7843160832503643,
            0.25173786054792646,
            0.8678783883274283,
            0.3354412471856949,
            0.298729267655819,
            0.7240785327926429,
            0.16587799922976687,
            0.7738582127496083,
            0.055824188508107664,
            0.8474459279899753,
            0.20327555814239462,
            0.6358785119454993,
            0.9923670488531944,
            0.3927013509323347,
            0.24821353641176047,
            0.7192887012700124,
            0.3391411570681473,
            0.8611950034309545,
            0.7322169437654343,
            0.6618650095669462,
            0.30558339768650067,
            0.7952130699383725,
            0.747194229022527,
            0.7877859162499721,
            0.2628035609220535,
            0.8908939970244619,
            0.1670668384252767,
            0.3197013127719599,
            0.4346388067636745,
            0.016611764360168424,
            0.3155599914778735,
            0.9711458254001295,
            0.5669388313504983,
            0.8225768349717703,
            0.10813802446203624,
            0.5150844011353757,
            0.5546695430534467,
            0.05753382687024333,
            0.3564421503451065,
            0.38259748941509086,
            0.4517495518340209,
            0.8539300200628299,
            0.6928383607839574,
            0.662790903000381,
            0.3728097457050529,
            0.07341040804273502,
            0.7135621444049983,
            0.31494535845890126,
            0.8463389381085897,
            0.3552370271076457,
            0.5334047271865571,
            0.44829450788280034,
            0.7392049863313708,
            0.9406772837478246,
            0.8205141626131973,
            0.4750865936497728,
            0.9879160960776046,
            0.4536776307377438,
            0.498044342363688,
            0.5982072579860978,
            0.47692296384000243,
            0.482346372060111,
            0.3520092892408174,
            0.6252355284088753,
            0.7908458992815688,
            0.4951939932647895,
            0.36334123906868776,
            0.6086985051092341,
            0.767302370706033,
            0.9117783574459476,
            0.4891381026342524,
            0.1379600694717562,
            0.8358452329194815,
            0.7878489151861279,
            0.0585353932698931,
            0.4137802559208035,
            0.4964892266394356,
            0.6340343086640661,
            0.8123945627818064,
            0.4276739901862191,
            0.46116608952790905,
            0.6117911541905424,
            0.06256859422580863,
            0.2897511039044913,
            0.9575747514837571,
            0.6260440704739371,
            0.28954243876188457,
            0.5730799397713243,
            0.5402210629491576,
            0.6899624495178694,
            0.3952333785839468,
            0.419967752739726,
            0.8817800062416639,
            0.9552369145214008,
            0.9404590446314967,
            0.8209434622238633,
            0.32398948251105153,
            0.23313715271019653,
            0.635024474315615,
            0.6206876961727517,
            0.09676694992655821,
            0.8247916367437644,
            0.6633543963118202,
            0.33219769457683146,
            0.6525608003946799,
            0.010264844795985328,
            0.20339365066885373,
            0.5293482755601088,
            0.6850695045131971,
            0.1837872683403884,
            0.19661655300335612,
            0.3215943227053656,
            0.4237431770336323,
            0.7069257266285722,
            0.00014216843975400906,
            0.12789839251588386,
            0.9870745527308604,
            0.3358551482188855,
            0.02649377917959217,
            0.6037087692892408,
            0.14468946436750985,
            0.8455686901890028,
            0.5489341054502427,
            0.6416105810992518,
            0.8716835798812409,
            0.9855863643288766,
            0.7621222238453107,
            0.10971708819658998,
            0.5992404260859467,
            0.0960184757175312,
            0.7478831288740773,
            0.21281437228159317,
            0.039989908116329964,
            0.6666889228105682,
            0.7082334278192032,
            0.4965891745189013,
            0.8448474491099335,
            0.51899221270943,
            0.053533044882776104,
            0.1832037916939313,
            0.7086558266051965,
            0.09232277641153386,
            0.6600839540684303,
            0.013429106260064105,
            0.755355951658492,
            0.1521493574747187,
            0.7359029926157673,
            0.31386400087456223,
            0.2786935794976002,
            0.8574892490208118,
            0.7048444522850063,
            0.11099956938971611,
            0.41377696534215747,
            0.4972042095004061,
            0.8123024006102493,
            0.019842136393627086,
            0.9110135852923356,
            0.7333099706166278,
            0.9354587752844313,
            0.9746672511177458,
            0.49312307327819005,
            0.009712103267392691,
            0.4179387564431378,
            0.4264449180491885,
            0.3178185085304487,
            0.4568848031303122,
            0.6746225129344577,
            0.612490689125883,
            0.43797541886896396,
            0.8206451670821743,
            0.5755039811379289,
            0.6969733354065955,
            0.5409265436547918,
            0.4319598200039658,
            0.2595330341771661,
            0.7496365412646532,
            0.44554171051435454,
            0.933953734036019,
            0.6394466211797196,
            0.6253318254844847,
            0.622598526858425,
            0.42819693534628167,
            0.49608801391604496,
            0.004078043936252662,
            0.31546465727721773,
            0.5620275115036629,
            0.24981547755110622,
            0.46638455050467276,
            0.8203192948951312,
            0.9531460405175395,
            0.08767861482470207,
            0.10182665848106964,
            0.41722139013286297,
            0.9050081587484748,
            0.7997471863258789,
            0.07654727069374845,
            0.015678303100936986,
            0.7531058353281517,
            0.11733320889337284,
            0.6678888964137968,
            0.25222433839055036,
            0.39593237313221374,
            0.2639734864670895,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                40,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    40,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_1024_2() {
        let diag = vec_static![
            0.009879042533147642,
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            0.8134685055868323,
            0.36202799069146974,
            0.9757682306946027,
            0.6473127267758995,
            0.2986449225504132,
            0.8881611353540096,
            0.19239744013030524,
            0.40733335986526575,
            0.5321704089004125,
            0.09898594195868682,
            0.9507441811197719,
            0.844880344564211,
            0.03731918415138358,
            0.3385622493858168,
            0.8504023048799197,
            0.6989467669923123,
            0.2549675349670987,
            0.48572089933797746,
            0.07288131474554993,
            0.09191220411594636,
            0.31033556092635084,
            0.30395736119114725,
            0.3846459319653658,
            0.392841391931931,
            0.24653075820003234,
            0.8544516886192999,
            0.8245326833288107,
            0.9958164191489396,
            0.19854830880772956,
            0.7766367108118657,
            0.09761322283985896,
            0.9417275796434791,
            0.7850343271348909,
            0.28609148479516555,
            0.5125689627715785,
            0.9258578542694729,
            0.275764740840005,
            0.6016164455627839,
            0.3462210019447792,
            0.3233452195422518,
            0.8409645559476775,
            0.7524922981717692,
            0.6339260467857786,
            0.9314935276749724,
            0.972677031160904,
            0.31299813790851694,
            0.8760082255657017,
            0.761333890179974,
            0.45938005370147283,
            0.520200318837958,
            0.6076441106575183,
            0.8724900021090676,
            0.8611036210839981,
            0.6552473684667536,
            0.6252833327380347,
            0.5185353691425693,
            0.9040227518479278,
            0.27970653915447574,
            0.9845390078155203,
            0.346673916502631,
            0.6623590611948782,
            0.006379473313236073,
            0.3734655087767689,
            0.7744014191561744,
            0.16961823466111714,
            0.3659953549414826,
            0.11118971782813591,
            0.759433495713853,
            0.1487394051434685,
            0.46865406255457365,
            0.8750582025359577,
            0.5073586413621533,
            0.41813721709898055,
            0.8391732370379675,
            0.5352984693712416,
            0.06008548992839924,
            0.350790921866179,
            0.21069799207917228,
            0.7949364947712025,
            0.39135641034895996,
            0.08709451091563836,
            0.8814193716323061,
            0.27314551566775214,
            0.9830695721780692,
            0.503725832798613,
            0.6084501117465266,
            0.47272365964120555,
            0.3892550561813739,
            0.09819293907928683,
            0.2384714035720421,
            0.08522948546776465,
            0.047686853879890134,
            0.5882039696597616,
            0.8672446036284184,
            0.27450605832508945,
            0.8613595906290894,
            0.8745126749842754,
            0.9851395552590442,
            0.4978119091583112,
            0.8993476815825009,
            0.7448498189517259,
            0.8369711602636397,
            0.8224174091515873,
            0.6822207719042813,
            0.1488219227059281,
            0.5778325840739624,
            0.4795873368483069,
            0.7090204945767113,
            0.15922992305806227,
            0.27438393598215605,
            0.2793210940950517,
            0.8884162176404946,
            0.6608015270385535,
            0.5282031642912364,
            0.9994618773829554,
            0.13146227355187012,
            0.14029156682207644,
            0.600131650519978,
            0.6390150442010555,
            0.31119136217286747,
            0.4016278174225866,
            0.07902734097141195,
            0.18629172890496581,
            0.8417267599628195,
            0.0009516721844141651,
            0.8728767092552249,
            0.5705298847912903,
            0.2329828050038638,
            0.1633535986935607,
            0.5443617553931892,
            0.9586315815409794,
            0.023573402817029132,
            0.630137650018288,
            0.3893839335340684,
            0.6032743106801581,
            0.6895868748541909,
            0.25749567302946685,
            0.9422588879367039,
            0.45962380488382426,
            0.13146747514986146,
            0.26518064000063035,
            0.7877056840494667,
            0.3289307264763762,
            0.6165600739898811,
            0.38624729302603034,
            0.8218929084592703,
            0.7273333375682015,
            0.6951739643262745,
            0.2620961872208437,
            0.867643751647387,
            0.5043461487128705,
            0.6162961579478152,
            0.9288355286850705,
            0.9298555904716704,
            0.019198104824075357,
            0.9225521021579641,
            0.0888584112444325,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                40,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    40,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_1024_3() {
        let diag = vec_static![
            0.060466268434667514,
            0.24070433133590274,
            0.05184276387772846,
            0.9898738802383618,
            0.5867224416658019,
            0.8920795530759005,
            0.8213260193280432,
            0.1274585126399569,
            0.5137105305003337,
            0.8929508711344447,
            0.34644482421179423,
            0.15662711364303783,
            0.13204459060563245,
            0.2452181097021333,
            0.5529961084728586,
            0.6059885561225518,
            0.335661034135773,
            0.8178650650229435,
            0.6928232210699408,
            0.6993333722077023,
            0.3614916242067844,
            0.7887570130392116,
            0.14888470133705733,
            0.15853694539661844,
            0.5334912050093811,
            0.3612993548593889,
            0.6903211472739849,
            0.64703468631002,
            0.829497094858506,
            0.2522343254330478,
            0.9505881136261928,
            0.7340919699859023,
            0.4445327799200357,
            0.276725070716636,
            0.8813837074592618,
            0.8641241736265327,
            0.37601439289318583,
            0.7406106431867137,
            0.8187897453440819,
            0.44679682056462267,
            0.6866279118940135,
            0.6322556970967156,
            0.5177506730663544,
            0.9643854839179787,
            0.6465100827097693,
            0.339197465636223,
            0.7513292599815653,
            0.5602603062963017,
            0.4553307839097991,
            0.22844009431280932,
            0.6235416261588994,
            0.2262804849429565,
            0.9136387602980167,
            0.9915699182157715,
            0.7259256237506346,
            0.3929852244527421,
            0.2155371379405563,
            0.8086877453574028,
            0.8647281751947866,
            0.606292073442245,
            0.30240091664534696,
            0.9225948840817058,
            0.6810625155003264,
            0.05179968431711934,
            0.595659576422282,
            0.7564014939381208,
            0.13794493869812707,
            0.36285564872228715,
            0.32342613546406007,
            0.79124383797685,
            0.03550783493526544,
            0.7487151700009632,
            0.6765223339310117,
            0.5937152610233598,
            0.07403103674795586,
            0.7606782796859048,
            0.18578166030690013,
            0.7013672177585895,
            0.2033016281334249,
            0.36779130656982795,
            0.6624345349315481,
            0.17276643968296868,
            0.7326612329736877,
            0.39683700964818025,
            0.9249585764714129,
            0.9058620912528339,
            0.5273041580770942,
            0.25056807692037486,
            0.48806466395096515,
            0.06552243597979834,
            0.978315141677103,
            0.502782492170536,
            0.5975218887295574,
            0.8954320387435921,
            0.18424721934266852,
            0.7682726736917449,
            0.43241327339139235,
            0.9531509198549333,
            0.9046304333439057,
            0.209761648896709,
            0.8410391805702537,
            0.10657531625818184,
            0.05568939898497982,
            0.676138899758238,
            0.532991981445195,
            0.6550486984863335,
            0.5855669735264619,
            0.7507383900993289,
            0.6547493864972989,
            0.39773180941485786,
            0.21977443508356898,
            0.20152541150538528,
            0.23513555154348653,
            0.6260455323467065,
            0.5686434999835956,
            0.3645862855059864,
            0.5982150616652503,
            0.681662158287625,
            0.5023884533718146,
            0.08596469818006636,
            0.865840915557725,
            0.5096733454968728,
            0.7376731774487919,
            0.23500068740937674,
            0.7760973488757089,
            0.30075646064672434,
            0.9565535059308128,
            0.144637054217305,
            0.766299241844306,
            0.9182047949601909,
            0.8981438083995072,
            0.10243727574939943,
            0.6444340540568536,
            0.39948153880812554,
            0.22358680321222313,
            0.5940835708596859,
            0.3193867223358725,
            0.8098588935943911,
            0.010859229221052202,
            0.41213053744762584,
            0.5781544958693299,
            0.2559845296086064,
            0.7173914473935831,
            0.3618426918318993,
            0.6545691051178975,
            0.16389232580120328,
            0.029853796333192628,
            0.1112042501011069,
            0.4131986127015722,
            0.0675709858058694,
            0.47697288079451994,
            0.7579635090222712,
            0.939295939451755,
            0.06104837646390948,
            0.6746647049785921,
            0.41976327511961575,
            0.9033609580534747,
            0.2726505261765354,
            0.6810347096753436,
            0.5761914873814167,
            0.5605593648243876,
            0.2040106732498883,
            0.06559590462849507,
            0.19867228693975358,
            0.3795569308165738,
            0.3576230887086853,
            0.6646129207450863,
            0.19159996290089576,
            0.26289150671048334,
            0.22984353830414195,
            0.10998864141249054,
            0.14726051758768155,
            0.09891300931363878,
            0.20699396396845948,
            0.4396333193710559,
            0.66049328467152,
            0.7484210613887542,
            0.6214537395445267,
            0.19715195461724988,
            0.26926297891853945,
            0.013177250804681129,
            0.5401145872074694,
            0.8876912160030009,
            0.3958473022669018,
            0.8525317391364969,
            0.9646251671547991,
            0.7639431892415399,
            0.6176857025453736,
            0.2440159000876102,
            0.4338943641942402,
            0.6325069093330767,
            0.42747587311352664,
            0.611567551714215,
            0.6500639751807277,
            0.837380450954655,
            0.2998521440987161,
            0.17958444155475328,
            0.3889472877193627,
            0.6898239835481556,
            0.8370090796660907,
            0.6064193851083068,
            0.4274634932199487,
            0.6087718098881021,
            0.508256409689673,
            0.339079489379435,
            0.10509571352293445,
            0.005572687883438454,
            0.021981179575465415,
            0.29212956768147014,
            0.7867865404724902,
            0.1807384308077843,
            0.8665190424853524,
            0.6184357021120035,
            0.7536704913156237,
            0.7271598615669912,
            0.49475392709260113,
            0.728471317065983,
            0.3269837398505393,
            0.4704197531972808,
            0.5738584097539486,
            0.758731532875281,
            0.7260845747787328,
            0.6016048024552125,
            0.8010556210240337,
            0.004072879110130434,
            0.8437986241962506,
            0.5146240323523598,
            0.13904073839194253,
            0.6089836301203322,
            0.897844926432131,
            0.49057508349361,
            0.29005256571070226,
            0.15772152541113738,
            0.4738990343662315,
            0.5021527238202769,
            0.6937852389489244,
            0.009814520346564715,
            0.15675068544014148,
            0.44385575330248883,
            0.4485313346936677,
            0.4217206515389108,
            0.7577444148847231,
            0.7258868622262982,
            0.6193382669120356,
            0.039861150798117695,
            0.4349519716620509,
            0.5532646897637169,
            0.28283121796858735,
            0.0023393446168564758,
            0.54027399425744,
            0.5054603742880899,
            0.8414459593178738,
            0.8888400513837226,
            0.6477615598506755,
            0.5463852474548928,
            0.22949832244208968,
            0.31690470916831925,
            0.33430272710595077,
            0.0018161848198701147,
            0.49139883689044017,
            0.6797887885789651,
            0.5861639317229459,
            0.035282919686503766,
            0.10547619370542094,
            0.7271339305759082,
            0.5016067603875377,
            0.8157682787043935,
            0.4560885642814806,
            0.6929899653460532,
            0.5490310236928294,
            0.4530007155322374,
            0.44049548891776324,
            0.3073456560671243,
            0.7919866041386854,
            0.9399916206326354,
            0.7527419916944253,
            0.10112484431007052,
            0.2243815401473951,
            0.07039176683656823,
            0.6169715161346244,
            0.08123319473394963,
            0.33386307361579226,
            0.7833580512725923,
            0.2505341520737143,
            0.30782230756455664,
            0.7479846627003448,
            0.2317759693352216,
            0.933348786188565,
            0.2750733182757722,
            0.7252317550631081,
            0.06447834497830485,
            0.032088850870325425,
            0.8873228725504719,
            0.15311031466542857,
            0.2687565257110237,
            0.004116203327798051,
            0.9334773509255564,
            0.4411582352055272,
            0.915699640538774,
            0.6540205822944993,
            0.22129013965221733,
            0.9197389724854845,
            0.7413607462811231,
            0.33557913718762133,
            0.48682708653142126,
            0.23261276643023798,
            0.7298006313346186,
            0.05533463818311113,
            0.5593987881884823,
            0.014043027025292698,
            0.8466474369742201,
            0.476037950820811,
            0.8444410961569767,
            0.16171101653838926,
            0.42377692467080386,
            0.9780260328429565,
            0.2702756266464895,
            0.6443988832387432,
            0.8993272243844188,
            0.7327024724314705,
            0.6054206077823142,
            0.588745513467837,
            0.6099464369729185,
            0.5242293066667245,
            0.6367910856473106,
            0.95096174945863,
            0.7939574092983276,
            0.3852863174746787,
            0.20255106392741273,
            0.4245303071023879,
            0.9967055343008029,
            0.4458664645461715,
            0.9257682958149203,
            0.3415930338870309,
            0.19630239658931292,
            0.5492196374119545,
            0.15687901061103193,
            0.6952090496329576,
            0.3956802166028377,
            0.5576427063755885,
            0.2040497200375847,
            0.5924369085502799,
            0.31777150634999896,
            0.6188717500402732,
            0.10233206975978115,
            0.6843201135492385,
            0.97041224730265,
            0.2995654214923589,
            0.6445544485180777,
            0.8301737642729924,
            0.1087683656777727,
            0.20019953992353745,
            0.22780836215874622,
            0.1276194439987589,
            0.38899213955877954,
            0.5064763107596547,
            0.6461906101897172,
            0.29746881883163756,
            0.12374457000837158,
            0.8069582603483658,
            0.25061357630791214,
            0.271756743228987,
            0.5239738641031194,
            0.6553247838521854,
            0.683817735836212,
            0.06802304767960465,
            0.7418517643983699,
            0.07774073352161648,
            0.9620799864233502,
            0.7358697033778088,
            0.6321490186410873,
            0.6177412024364181,
            0.868224449638578,
            0.7861974420960143,
            0.9434823199966517,
            0.6471369630040992,
            0.4048133530399751,
            0.21870779533881557,
            0.04034528679441152,
            0.40085799212995166,
            0.7373559016434784,
            0.8017355475036948,
            0.8967226537967593,
            0.7176831089070658,
            0.2613162507363975,
            0.3295930554915326,
            0.9184791699454984,
            0.640401400967848,
            0.8506077837886161,
            0.7664496800476533,
            0.1840288042109116,
            0.19215292024269792,
            0.9621850056142566,
            0.7270739851985532,
            0.4000222354425286,
            0.7398694017088451,
            0.8102704435115223,
            0.04869313905506856,
            0.9793000884909765,
            0.1741974674289617,
            0.3761252028210257,
            0.5186546213017674,
            0.6544630055785722,
            0.035486878857104,
            0.8302105873193061,
            0.027163249430116165,
            0.22238221247779077,
            0.6629828858480685,
            0.28167067761823206,
            0.15507531405462183,
            0.5299310060229058,
            0.17615517303353367,
            0.9160813969780187,
            0.21545598837949875,
            0.12327465612788735,
            0.22341207675958197,
            0.7350951031975246,
            0.20212244417713343,
            0.10806891511769379,
            0.07983934140633298,
            0.5784829838569209,
            0.378211476865968,
            0.1600334499763354,
            0.005458695509232947,
            0.04659586838146379,
            0.16941413051975984,
            0.8176494023491518,
            0.8193983869596879,
            0.8108230208574312,
            0.6743040619295617,
            0.9291104455200995,
            0.4551862464547173,
            0.5081503416196177,
            0.5863629766904948,
            0.879725318736957,
            0.44780357058335174,
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        let subdiag = vec_static![
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            0.7972941049881628,
            0.4354805939118921,
            0.6289399489190155,
        ];

        let n = diag.len();
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| f64::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| f64::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                40,
                0,
                f64::EPSILON,
                f64::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<f64>(
                    n,
                    40,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                assert_approx_eq!(reconstructed.read(i, j), target, 1e-10);
            }
        }
    }

    #[test]
    fn test_svd_josef() {
        if option_env!("CI") == Some("true") {
            // too big for CI
            return;
        }

        let diag = [
            -4982.29,
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            -53520.258,
            -36040.266,
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            -38354.395,
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            2.3570395,
            3.080712,
            -0.28484115,
            -1.1142883,
            0.0,
        ];

        type T = f32;

        let n = diag.len();
        let max = T::max(
            crate::col::from_slice(&diag).norm_max(),
            crate::col::from_slice(&subdiag).norm_max(),
        );
        let mut u = Mat::from_fn(n + 1, n + 1, |_, _| T::NAN);
        let mut v = Mat::from_fn(n, n, |_, _| T::NAN);
        let s = {
            let mut diag = diag.clone();
            let mut subdiag = subdiag.clone();
            compute_bidiag_real_svd(
                &mut diag,
                &mut subdiag,
                Some(u.as_mut()),
                Some(v.as_mut()),
                4,
                128,
                T::EPSILON,
                T::MIN_POSITIVE,
                Parallelism::None,
                make_stack!(bidiag_real_svd_req::<T>(
                    n,
                    4,
                    true,
                    true,
                    Parallelism::None
                )),
            );
            Mat::from_fn(n + 1, n, |i, j| if i == j { diag[i] } else { 0.0 })
        };

        let reconstructed = &u * &s * v.transpose();
        let mut err = 0.0;

        for j in 0..n {
            for i in 0..n + 1 {
                let target = if i == j {
                    diag[j]
                } else if i == j + 1 {
                    subdiag[j]
                } else {
                    0.0
                };

                err = T::max(err, (reconstructed.read(i, j) - target).abs());
                assert_approx_eq!(reconstructed.read(i, j), target, 1e-2 * max);
            }
        }
    }

    #[test]
    fn test_deflation43() {
        let approx_eq = crate::ApproxEq::<f32>::eps();

        let eps = f32::EPSILON;
        let zero_threshold = f32::MIN_POSITIVE;

        let n = 7;
        let k = 3;
        let diag_orig = &*vec![1.0, 5.0, 3.0, 1e-7, 4.0, 2.0, 2e-7_f32];
        let col_orig = &*vec![1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0_f32];

        let n_jacobi = 2;
        let jacobi_coeffs = &mut *vec![JacobiRotation { c: 0.0, s: 0.0 }; n_jacobi];
        let jacobi_indices = &mut *vec![0usize; n_jacobi];
        let perm = &mut *vec![0usize; n];
        let diag = &mut *(diag_orig.to_vec());
        let col = &mut *(col_orig.to_vec());

        let (jacobi_0i, jacobi_ij) = deflate(
            diag,
            col,
            jacobi_coeffs,
            jacobi_indices,
            &mut *vec![0usize; n],
            perm,
            k,
            eps,
            zero_threshold,
            PodStack::new(bytemuck::cast_slice_mut(&mut *vec![0usize; 2 * n])),
        );
        assert!(all(jacobi_0i == 2, jacobi_ij == 0));

        let perm_inv = &mut *vec![0usize; n];
        for (i, p) in perm.iter().enumerate() {
            perm_inv[*p] = i;
        }

        let P = crate::perm::PermRef::new_checked(perm, perm_inv, n);

        let mut M_orig = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M_orig[(i, i)] = diag_orig[i];
            M_orig[(i, 0)] = col_orig[i];
        }

        let mut M = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M[(i, i)] = diag[i];
            M[(i, 0)] = col[i];
        }

        M = P.inverse() * M * P;

        for (&rot, &i) in core::iter::zip(&*jacobi_coeffs, &*jacobi_indices).rev() {
            let (x, y) = M.two_rows_mut(0, i);
            rot.apply_on_the_left_in_place(x, y);
        }

        for j in 0..n {
            for i in 0..n {
                assert!(M[(i, j)] ~ M_orig[(i, j)]);
            }
        }
    }

    #[test]
    fn test_deflation44() {
        let approx_eq = crate::ApproxEq::<f32>::eps();

        let eps = f32::EPSILON;
        let zero_threshold = f32::MIN_POSITIVE;

        let n = 7;
        let k = 3;
        let diag_orig = &*vec![1.0, 5.0, 3.0, 1.0, 4.0, 2.0, 1.0];
        let col_orig = &*vec![1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0_f32];

        let n_jacobi = 1;
        let jacobi_coeffs = &mut *vec![JacobiRotation { c: 0.0, s: 0.0 }; n_jacobi];
        let jacobi_indices = &mut *vec![0usize; n_jacobi];
        let perm = &mut *vec![0usize; n];
        let diag = &mut *(diag_orig.to_vec());
        let col = &mut *(col_orig.to_vec());

        let (jacobi_0i, jacobi_ij) = deflate(
            diag,
            col,
            jacobi_coeffs,
            jacobi_indices,
            &mut *vec![0usize; n],
            perm,
            k,
            eps,
            zero_threshold,
            PodStack::new(bytemuck::cast_slice_mut(&mut *vec![0usize; 2 * n])),
        );
        assert!(all(jacobi_0i == 0, jacobi_ij == 1));

        let perm_inv = &mut *vec![0usize; n];
        for (i, p) in perm.iter().enumerate() {
            perm_inv[*p] = i;
        }

        let P = crate::perm::PermRef::new_checked(perm, perm_inv, n);

        let mut M_orig = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M_orig[(i, i)] = diag_orig[i];
            M_orig[(i, 0)] = col_orig[i];
        }

        let mut M = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M[(i, i)] = diag[i];
            M[(i, 0)] = col[i];
        }

        M = P.inverse() * M * P;

        for (&rot, &i) in core::iter::zip(&*jacobi_coeffs, &*jacobi_indices).rev() {
            let (i, j) = (i - 1, i);
            let (pi, pj) = (perm[i], perm[j]);

            let (x, y) = M.two_rows_mut(pi, pj);
            rot.apply_on_the_left_in_place(x, y);

            let (x, y) = M.two_cols_mut(pi, pj);
            rot.transpose().apply_on_the_right_in_place(x, y);
        }

        for j in 0..n {
            for i in 0..n {
                assert!(M[(i, j)] ~ M_orig[(i, j)]);
            }
        }
    }

    #[test]
    fn test_both_deflation() {
        let approx_eq = crate::ApproxEq::<f32>::eps();

        let eps = f32::EPSILON;
        let zero_threshold = f32::MIN_POSITIVE;

        let n = 7;
        let k = 3;
        let diag_orig = &*vec![1.0, 5.0, 3.0, 2.0, 4.0, 2.0, 0.0];
        let col_orig = &*vec![1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0_f32];

        let n_jacobi = 2;
        let jacobi_coeffs = &mut *vec![JacobiRotation { c: 0.0, s: 0.0 }; n_jacobi];
        let jacobi_indices = &mut *vec![0usize; n_jacobi];
        let perm = &mut *vec![0usize; n];
        let diag = &mut *(diag_orig.to_vec());
        let col = &mut *(col_orig.to_vec());

        let (jacobi_0i, jacobi_ij) = deflate(
            diag,
            col,
            jacobi_coeffs,
            jacobi_indices,
            &mut *vec![0usize; n],
            perm,
            k,
            eps,
            zero_threshold,
            PodStack::new(bytemuck::cast_slice_mut(&mut *vec![0usize; 2 * n])),
        );
        assert!(all(jacobi_0i == 1, jacobi_ij == 1));

        let perm_inv = &mut *vec![0usize; n];
        for (i, p) in perm.iter().enumerate() {
            perm_inv[*p] = i;
        }

        let P = crate::perm::PermRef::new_checked(perm, perm_inv, n);

        let mut M_orig = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M_orig[(i, i)] = diag_orig[i];
            M_orig[(i, 0)] = col_orig[i];
        }

        let mut M = Mat::<f32>::zeros(n, n);
        for i in 0..n {
            M[(i, i)] = diag[i];
            M[(i, 0)] = col[i];
        }

        M = P.inverse() * M * P;

        for (&rot, &i) in core::iter::zip(&jacobi_coeffs[1..], &jacobi_indices[1..]).rev() {
            let (i, j) = (i - 1, i);
            let (pi, pj) = (perm[i], perm[j]);

            let (x, y) = M.two_rows_mut(pi, pj);
            rot.apply_on_the_left_in_place(x, y);

            let (x, y) = M.two_cols_mut(pi, pj);
            rot.transpose().apply_on_the_right_in_place(x, y);
        }
        for (&rot, &i) in core::iter::zip(&jacobi_coeffs[..1], &jacobi_indices[..1]).rev() {
            let (x, y) = M.two_rows_mut(0, i);
            rot.apply_on_the_left_in_place(x, y);
        }

        for j in 0..n {
            for i in 0..n {
                assert!(M[(i, j)] ~ M_orig[(i, j)]);
            }
        }
    }
}