faer_evd/

hessenberg_cplx_evd.rs

// adapted from <T>LAPACK implementation
//
// https://github.com/tlapack/tlapack
// https://github.com/tlapack/tlapack/blob/master/include/tlapack/lapack/lahqr.hpp

use crate::hessenberg::{make_hessenberg_in_place, make_hessenberg_in_place_req};
use core::slice;
use dyn_stack::{PodStack, SizeOverflow, StackReq};
use faer_core::{
    householder::{
        apply_block_householder_sequence_on_the_right_in_place_req,
        apply_block_householder_sequence_on_the_right_in_place_with_conj,
        apply_block_householder_sequence_transpose_on_the_left_in_place_req,
        apply_block_householder_sequence_transpose_on_the_left_in_place_with_conj,
        make_householder_in_place_v2,
    },
    mul::matmul,
    temp_mat_req, unzipped,
    zip::Diag,
    zipped, ComplexField, Conj, Entity, MatMut, MatRef, Parallelism, RealField, SimdCtx,
};
use faer_entity::*;
use reborrow::*;

fn max<T: PartialOrd>(a: T, b: T) -> T {
    if a > b {
        a
    } else {
        b
    }
}

fn min<T: PartialOrd>(a: T, b: T) -> T {
    if a < b {
        a
    } else {
        b
    }
}

fn abs1<E: ComplexField>(a: E) -> E::Real {
    a.faer_real().faer_abs().faer_add(a.faer_imag().faer_abs())
}

fn lahqr_shiftcolumn<E: ComplexField>(h: MatRef<'_, E>, mut v: MatMut<'_, E>, s1: E, s2: E) {
    let n = h.nrows();

    if n == 2 {
        let s = abs1(h.read(0, 0).faer_sub(s2)).faer_add(abs1(h.read(1, 0)));
        if s == E::Real::faer_zero() {
            v.write(0, 0, E::faer_zero());
            v.write(1, 0, E::faer_zero());
        } else {
            let s_inv = s.faer_inv();
            let h10s = h.read(1, 0).faer_scale_real(s_inv);
            v.write(
                0,
                0,
                (h10s.faer_mul(h.read(0, 1))).faer_add(
                    h.read(0, 0)
                        .faer_sub(s1)
                        .faer_mul((h.read(0, 0).faer_sub(s2)).faer_scale_real(s_inv)),
                ),
            );
            v.write(
                1,
                0,
                h10s.faer_mul(
                    h.read(0, 0)
                        .faer_add(h.read(1, 1))
                        .faer_sub(s1)
                        .faer_sub(s2),
                ),
            );
        }
    } else {
        let s = abs1(h.read(0, 0).faer_sub(s2))
            .faer_add(abs1(h.read(1, 0)))
            .faer_add(abs1(h.read(2, 0)));
        if s == E::Real::faer_zero() {
            v.write(0, 0, E::faer_zero());
            v.write(1, 0, E::faer_zero());
            v.write(2, 0, E::faer_zero());
        } else {
            let s_inv = s.faer_inv();
            let h10s = h.read(1, 0).faer_scale_real(s_inv);
            let h20s = h.read(2, 0).faer_scale_real(s_inv);
            v.write(
                0,
                0,
                ((h.read(0, 0).faer_sub(s1))
                    .faer_mul((h.read(0, 0).faer_sub(s2)).faer_scale_real(s_inv)))
                .faer_add(h.read(0, 1).faer_mul(h10s))
                .faer_add(h.read(0, 2).faer_mul(h20s)),
            );
            v.write(
                1,
                0,
                (h10s.faer_mul(
                    h.read(0, 0)
                        .faer_add(h.read(1, 1).faer_sub(s1).faer_sub(s2)),
                ))
                .faer_add(h.read(1, 2).faer_mul(h20s)),
            );
            v.write(
                2,
                0,
                (h20s.faer_mul(
                    h.read(0, 0)
                        .faer_add(h.read(2, 2).faer_sub(s1).faer_sub(s2)),
                ))
                .faer_add(h10s.faer_mul(h.read(2, 1))),
            );
        }
    }
}

// ret: (eig1_re eig1_im) (eig2_re eig2_im)
fn lahqr_eig22<E: ComplexField>(mut a00: E, mut a01: E, mut a10: E, mut a11: E) -> (E, E) {
    let zero = E::Real::faer_zero();
    let half = E::Real::faer_from_f64(0.5);

    let s = abs1(a00)
        .faer_add(abs1(a01))
        .faer_add(abs1(a10))
        .faer_add(abs1(a11));
    if s == zero {
        return (E::faer_zero(), E::faer_zero());
    }

    let s_inv = s.faer_inv();

    a00 = a00.faer_scale_real(s_inv);
    a01 = a01.faer_scale_real(s_inv);
    a10 = a10.faer_scale_real(s_inv);
    a11 = a11.faer_scale_real(s_inv);

    let tr = (a00.faer_add(a11)).faer_scale_power_of_two(half);
    let det = ((a00.faer_sub(tr)).faer_mul(a00.faer_sub(tr))).faer_add(a01.faer_mul(a10));

    let rtdisc = det.faer_sqrt();
    (
        (tr.faer_add(rtdisc)).faer_scale_real(s),
        (tr.faer_sub(rtdisc)).faer_scale_real(s),
    )
}

fn rotg<E: ComplexField>(a: E, b: E, epsilon: E::Real, zero_threshold: E::Real) -> (E::Real, E, E) {
    let safmin = zero_threshold;
    let safmax = zero_threshold.faer_inv();
    let rtmin = zero_threshold.faer_div(epsilon).faer_sqrt();
    let rtmax = rtmin.faer_inv();

    // quick return
    if b == E::faer_zero() {
        return (E::Real::faer_one(), E::faer_zero(), E::faer_one());
    }

    let (c, s, r);
    if a == E::faer_zero() {
        c = E::Real::faer_zero();
        let g1 = max(b.faer_real().faer_abs(), b.faer_imag().faer_abs());
        if g1 > rtmin && g1 < rtmax {
            // Use unscaled algorithm
            let g2 = b
                .faer_real()
                .faer_abs2()
                .faer_add(b.faer_imag().faer_abs2());
            let d = g2.faer_sqrt();
            s = b.faer_conj().faer_scale_real(d.faer_inv());
            r = E::faer_from_real(d);
        } else {
            // Use scaled algorithm
            let u = min(safmax, max(safmin, g1));
            let uu = u.faer_inv();
            let gs = b.faer_scale_real(uu);
            let g2 = gs
                .faer_real()
                .faer_abs2()
                .faer_add(gs.faer_imag().faer_abs2());
            let d = g2.faer_sqrt();
            s = gs.faer_conj().faer_scale_real(d.faer_inv());
            r = E::faer_from_real(d.faer_mul(u));
        }
    } else {
        let f1 = max(E::faer_real(a).faer_abs(), E::faer_imag(a).faer_abs());
        let g1 = max(E::faer_real(b).faer_abs(), E::faer_imag(b).faer_abs());
        if f1 > rtmin && f1 < rtmax && g1 > rtmin && g1 < rtmax {
            // Use unscaled algorithm
            let f2 = a
                .faer_real()
                .faer_abs2()
                .faer_add(a.faer_imag().faer_abs2());
            let g2 = b
                .faer_real()
                .faer_abs2()
                .faer_add(b.faer_imag().faer_abs2());
            let h2 = f2.faer_add(g2);
            let d = if f2 > rtmin && h2 < rtmax {
                f2.faer_mul(h2).faer_sqrt()
            } else {
                f2.faer_sqrt().faer_mul(h2.faer_sqrt())
            };
            let p = d.faer_inv();
            c = f2.faer_scale_real(p);
            s = b.faer_conj().faer_mul(a.faer_scale_real(p));

            r = a.faer_scale_real(h2.faer_mul(p));
        } else {
            // Use scaled algorithm
            let u = min(safmax, max(safmin, max(f1, g1)));
            let uu = u.faer_inv();
            let gs = b.faer_scale_real(uu);
            let g2 = gs
                .faer_real()
                .faer_abs2()
                .faer_add(gs.faer_imag().faer_abs2());
            let (f2, h2, w);
            let fs;
            if f1.faer_scale_real(uu) < rtmin {
                // a is not well-scaled when scaled by g1.
                let v = min(safmax, max(safmin, f1));
                let vv = v.faer_inv();
                w = v.faer_mul(uu);
                fs = a.faer_scale_real(vv);
                f2 = fs
                    .faer_real()
                    .faer_abs2()
                    .faer_add(fs.faer_imag().faer_abs2());
                h2 = (f2.faer_mul(w).faer_mul(w)).faer_add(g2);
            } else {
                // Otherwise use the same scaling for a and b.
                w = E::Real::faer_one();
                fs = a.faer_scale_real(uu);
                f2 = fs
                    .faer_real()
                    .faer_abs2()
                    .faer_add(fs.faer_imag().faer_abs2());
                h2 = f2.faer_add(g2);
            }
            let d = if f2 > rtmin && h2 < rtmax {
                f2.faer_mul(h2).faer_sqrt()
            } else {
                f2.faer_sqrt().faer_mul(h2.faer_sqrt())
            };
            let p = d.faer_inv();
            c = (f2.faer_mul(p)).faer_mul(w);
            s = gs.faer_conj().faer_mul(fs.faer_scale_real(p));
            r = (fs.faer_scale_real(h2.faer_scale_real(p))).faer_scale_real(u);
        }
    }

    (c, s, r)
}

pub fn rot<E: ComplexField>(x: MatMut<'_, E>, y: MatMut<'_, E>, c: E::Real, s: E) {
    let n = x.nrows();
    if n == 0 || (c == E::Real::faer_one() && s == E::faer_zero()) {
        return;
    }

    zipped!(x, y).for_each(|unzipped!(mut x, mut y)| {
        let mut x_ = x.read();
        let mut y_ = y.read();

        (x_, y_) = (
            (x_.faer_scale_real(c)).faer_add(y_.faer_mul(s)),
            (y_.faer_scale_real(c)).faer_sub(x_.faer_mul(s.faer_conj())),
        );

        x.write(x_);
        y.write(y_);
    });
}

pub struct Rot<'a, E: ComplexField> {
    ai: MatMut<'a, E>,
    aj: MatMut<'a, E>,
    c: E::Real,
    s: E,
}

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

    #[inline(always)]
    fn with_simd<S: pulp::Simd>(self, simd: S) -> Self::Output {
        let Self { ai, aj, c, s } = self;
        debug_assert!(ai.nrows() == aj.nrows());
        debug_assert!(ai.ncols() == 1);
        debug_assert!(aj.ncols() == 1);

        let n = ai.nrows();
        let ai = ai.as_ptr_mut();
        let aj = aj.as_ptr_mut();

        let ai = unsafe {
            E::faer_map(
                ai,
                #[inline(always)]
                |ptr| slice::from_raw_parts_mut(ptr, n),
            )
        };
        let aj = unsafe {
            E::faer_map(
                aj,
                #[inline(always)]
                |ptr| slice::from_raw_parts_mut(ptr, n),
            )
        };

        let (ai_head, ai_tail) = faer_core::simd::slice_as_mut_simd::<E, S>(ai);
        let (aj_head, aj_tail) = faer_core::simd::slice_as_mut_simd::<E, S>(aj);

        let c = E::Real::faer_simd_splat(simd, c);
        let s = E::faer_simd_splat(simd, s);

        for (ai, aj) in E::faer_into_iter(ai_head).zip(E::faer_into_iter(aj_head)) {
            let mut ai_ = E::faer_deref(E::faer_rb(E::faer_as_ref(&ai)));
            let mut aj_ = E::faer_deref(E::faer_rb(E::faer_as_ref(&aj)));

            let tmp = E::faer_simd_conj_mul_adde(
                simd,
                s,
                into_copy::<E, _>(E::faer_copy(&aj_)),
                E::faer_simd_scale_real(simd, c, into_copy::<E, _>(E::faer_copy(&ai_))),
            );

            aj_ = from_copy::<E, _>(E::faer_simd_mul_adde(
                simd,
                E::faer_simd_neg(simd, s),
                into_copy::<E, _>(ai_),
                E::faer_simd_scale_real(simd, c, into_copy::<E, _>(aj_)),
            ));
            ai_ = from_copy::<E, _>(tmp);

            E::faer_map(
                E::faer_zip(ai, ai_),
                #[inline(always)]
                |(ai, ai_): (&mut _, _)| *ai = ai_,
            );
            E::faer_map(
                E::faer_zip(aj, aj_),
                #[inline(always)]
                |(aj, aj_): (&mut _, _)| *aj = aj_,
            );
        }

        let mut ai_ = E::faer_partial_load(simd, E::faer_rb(E::faer_as_ref(&ai_tail)));
        let mut aj_ = E::faer_partial_load(simd, E::faer_rb(E::faer_as_ref(&aj_tail)));

        let tmp = E::faer_simd_conj_mul_adde(simd, s, aj_, E::faer_simd_scale_real(simd, c, ai_));

        aj_ = E::faer_simd_mul_adde(
            simd,
            E::faer_simd_neg(simd, s),
            ai_,
            E::faer_simd_scale_real(simd, c, aj_),
        );
        ai_ = tmp;

        E::faer_partial_store(simd, ai_tail, ai_);
        E::faer_partial_store(simd, aj_tail, aj_);
    }
}

pub fn lahqr<E: ComplexField>(
    want_t: bool,
    a: MatMut<'_, E>,
    z: Option<MatMut<'_, E>>,
    w: MatMut<'_, E>,
    ilo: usize,
    ihi: usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
) -> isize {
    assert!(a.nrows() == a.ncols());
    assert!(ilo <= ihi);

    let n = a.nrows();
    let nh = ihi - ilo;

    assert!(w.nrows() == n);
    assert!(w.ncols() == 1);

    if let Some(z) = z.rb() {
        assert!(z.nrows() == n);
        assert!(z.ncols() == n);
    }

    let mut a = a;
    let mut z = z;
    let mut w = w;

    let zero = E::Real::faer_zero();
    let eps = epsilon;
    let small_num = zero_threshold.faer_div(eps);
    let non_convergence_limit = 10;
    let dat1 = E::Real::faer_from_f64(0.75);
    let dat2 = E::Real::faer_from_f64(-0.4375);

    if nh == 0 {
        return 0;
    }

    if nh == 1 {
        w.write(ilo, 0, a.read(ilo, ilo));
    }

    // itmax is the total number of QR iterations allowed.
    // For most matrices, 3 shifts per eigenvalue is enough, so
    // we set itmax to 30 times nh as a safe limit.
    let itmax = 30 * Ord::max(10, nh);

    // k_defl counts the number of iterations since a deflation
    let mut k_defl = 0usize;

    // istop is the end of the active subblock.
    // As more and more eigenvalues converge, it eventually
    // becomes ilo+1 and the loop ends.
    let mut istop = ihi;

    // istart is the start of the active subblock. Either
    // istart = ilo, or H(istart, istart-1) = 0. This means
    // that we can treat this subblock separately.
    let mut istart = ilo;

    let arch = E::Simd::default();
    for iter in 0..itmax + 1 {
        if iter == itmax {
            return istop as isize;
        }

        if istart + 1 >= istop {
            if istart + 1 == istop {
                w.write(istart, 0, a.read(istart, istart));
            }
            // All eigenvalues have been found, exit and return 0.
            break;
        }

        // Determine range to apply rotations
        let istart_m;
        let istop_m;
        if !want_t {
            istart_m = istart;
            istop_m = istop;
        } else {
            istart_m = 0;
            istop_m = n;
        }

        // Check if active subblock has split
        for i in (istart + 1..istop).rev() {
            if abs1(a.read(i, i - 1)) < small_num {
                // A(i,i-1) is negligible, take i as new istart.
                a.write(i, i - 1, E::faer_zero());
                istart = i;
                break;
            }

            let mut tst = abs1(a.read(i - 1, i - 1)).faer_add(abs1(a.read(i, i)));
            if tst == zero {
                if i >= ilo + 2 {
                    tst = tst.faer_add(a.read(i - 1, i - 2).faer_abs());
                }
                if i + 1 < ihi {
                    tst = tst.faer_add(a.read(i + 1, i).faer_abs());
                }
            }

            if abs1(a.read(i, i - 1)) <= eps.faer_mul(tst) {
                //
                // The elementwise deflation test has passed
                // The following performs second deflation test due
                // to Ahues & Tisseur (LAWN 122, 1997). It has better
                // mathematical foundation and improves accuracy in some
                // examples.
                //
                // The test is |A(i,i-1)|*|A(i-1,i)| <=
                // eps*|A(i,i)|*|A(i-1,i-1)| The multiplications might overflow
                // so we do some scaling first.
                //

                let ab = max(abs1(a.read(i, i - 1)), abs1(a.read(i - 1, i)));
                let ba = min(abs1(a.read(i, i - 1)), abs1(a.read(i - 1, i)));
                let aa = max(
                    abs1(a.read(i, i)),
                    abs1(a.read(i, i).faer_sub(a.read(i - 1, i - 1))),
                );
                let bb = min(
                    abs1(a.read(i, i)),
                    abs1(a.read(i, i).faer_sub(a.read(i - 1, i - 1))),
                );
                let s = aa.faer_add(ab);
                if ba.faer_mul(ab.faer_div(s))
                    <= max(small_num, eps.faer_mul(bb.faer_mul(aa.faer_div(s))))
                {
                    // A(i,i-1) is negligible, take i as new istart.
                    a.write(i, i - 1, E::faer_zero());
                    istart = i;
                    break;
                }
            }
        }

        if istart + 1 >= istop {
            k_defl = 0;
            w.write(istart, 0, a.read(istart, istart));
            istop = istart;
            istart = ilo;
            continue;
        }

        // Determine shift
        let (a00, a01, a10, a11);
        k_defl += 1;

        if k_defl % non_convergence_limit == 0 {
            // Exceptional shift
            let mut s = a.read(istop - 1, istop - 2).faer_abs();
            if istop > ilo + 2 {
                s = s.faer_add(a.read(istop - 2, istop - 3).faer_abs());
            };
            a00 = E::faer_from_real(dat1.faer_mul(s)).faer_add(a.read(istop - 1, istop - 1));
            a01 = E::faer_from_real(dat2.faer_mul(s));
            a10 = E::faer_from_real(s);
            a11 = a00;
        } else {
            // Wilkinson shift
            a00 = a.read(istop - 2, istop - 2);
            a10 = a.read(istop - 1, istop - 2);
            a01 = a.read(istop - 2, istop - 1);
            a11 = a.read(istop - 1, istop - 1);
        }

        let (mut s1, s2) = lahqr_eig22(a00, a01, a10, a11);

        if abs1(s1.faer_sub(a.read(istop - 1, istop - 1)))
            > abs1(s2.faer_sub(a.read(istop - 1, istop - 1)))
        {
            s1 = s2;
        }

        // We have already checked whether the subblock has split.
        // If it has split, we can introduce any shift at the top of the new
        // subblock. Now that we know the specific shift, we can also check
        // whether we can introduce that shift somewhere else in the subblock.
        let mut istart2 = istart;
        if istart + 2 < istop {
            for i in (istart + 1..istop - 1).rev() {
                let h00 = a.read(i, i).faer_sub(s1);
                let h10 = a.read(i + 1, i);

                let (_cs, sn, _r) = rotg(h00, h10, eps, zero_threshold);
                if abs1(sn.faer_conj().faer_mul(a.read(i, i - 1)))
                    <= eps.faer_mul(abs1(a.read(i, i - 1)).faer_add(abs1(a.read(i, i + 1))))
                {
                    istart2 = i;
                    break;
                }
            }
        }

        for i in istart2..istop - 1 {
            let (cs, sn, r);
            if i == istart2 {
                let h00 = a.read(i, i).faer_sub(s1);
                let h10 = a.read(i + 1, i);
                (cs, sn, _) = rotg(h00, h10, eps, zero_threshold);
                if i > istart {
                    a.write(i, i - 1, a.read(i, i - 1).faer_scale_real(cs));
                }
            } else {
                (cs, sn, r) = rotg(a.read(i, i - 1), a.read(i + 1, i - 1), eps, zero_threshold);
                a.write(i, i - 1, r);
                a.write(i + 1, i - 1, E::faer_zero());
            }

            // Apply G from the left to A
            let ai = unsafe {
                a.rb()
                    .row(i)
                    .subcols(i, istop_m - i)
                    .const_cast()
                    .as_2d_mut()
            };
            let aip1 = unsafe {
                a.rb()
                    .row(i + 1)
                    .subcols(i, istop_m - i)
                    .const_cast()
                    .as_2d_mut()
            };

            zipped!(ai, aip1).for_each(|unzipped!(mut ai, mut aip1)| {
                let ai_ = ai.read();
                let aip1_ = aip1.read();
                let tmp = (ai_.faer_scale_real(cs)).faer_add(aip1_.faer_mul(sn));
                aip1.write(
                    (ai_.faer_mul(sn.faer_conj().faer_neg())).faer_add(aip1_.faer_scale_real(cs)),
                );
                ai.write(tmp);
            });

            // Apply G**H from the right to A
            let nrows_ = Ord::min(i + 3, istop) - istart_m;
            let ai = unsafe {
                a.rb()
                    .col(i)
                    .subrows(istart_m, nrows_)
                    .const_cast()
                    .as_2d_mut()
            };
            let aip1 = unsafe {
                a.rb()
                    .col(i + 1)
                    .subrows(istart_m, nrows_)
                    .const_cast()
                    .as_2d_mut()
            };

            if a.row_stride() == 1 {
                arch.dispatch(Rot {
                    ai,
                    aj: aip1,
                    c: cs,
                    s: sn,
                });
            } else {
                zipped!(ai, aip1).for_each(|unzipped!(mut ai, mut aip1)| {
                    let ai_ = ai.read();
                    let aip1_ = aip1.read();
                    let tmp = (ai_.faer_scale_real(cs)).faer_add(aip1_.faer_mul(sn.faer_conj()));
                    aip1.write((ai_.faer_mul(sn.faer_neg())).faer_add(aip1_.faer_scale_real(cs)));
                    ai.write(tmp);
                });
            }
            if let Some(z) = z.rb_mut() {
                // Apply G**H to Z from the right
                let zi = unsafe { z.rb().col(i).const_cast().as_2d_mut() };
                let zip1 = unsafe { z.rb().col(i + 1).const_cast().as_2d_mut() };

                if z.row_stride() == 1 {
                    arch.dispatch(Rot {
                        ai: zi,
                        aj: zip1,
                        c: cs,
                        s: sn,
                    });
                } else {
                    zipped!(zi, zip1).for_each(|unzipped!(mut zi, mut zip1)| {
                        let zi_ = zi.read();
                        let zip1_ = zip1.read();
                        let tmp =
                            (zi_.faer_scale_real(cs)).faer_add(zip1_.faer_mul(sn.faer_conj()));
                        zip1.write(
                            (zi_.faer_mul(sn.faer_neg())).faer_add(zip1_.faer_scale_real(cs)),
                        );
                        zi.write(tmp);
                    });
                }
            }
        }
    }
    0
}

#[derive(Default, Clone, Copy, Debug)]
#[non_exhaustive]
pub struct EvdParams {
    /// Function that returns the number of shifts to use for a given matrix size
    pub recommended_shift_count:
        Option<fn(matrix_dimension: usize, active_block_dimension: usize) -> usize>,
    /// Function that returns the deflation window to use for a given matrix size
    pub recommended_deflation_window:
        Option<fn(matrix_dimension: usize, active_block_dimension: usize) -> usize>,
    /// Threshold to switch between blocked and unblocked code
    pub blocking_threshold: Option<usize>,
    /// Threshold of percent of aggressive-early-deflation window that must converge to skip a
    /// sweep
    pub nibble_threshold: Option<usize>,
}

pub fn default_recommended_shift_count(dim: usize, _active_block_dim: usize) -> usize {
    let n = dim;
    if n < 30 {
        2
    } else if n < 60 {
        4
    } else if n < 150 {
        12
    } else if n < 590 {
        32
    } else if n < 3000 {
        64
    } else if n < 6000 {
        128
    } else {
        256
    }
}

pub fn default_recommended_deflation_window(dim: usize, _active_block_dim: usize) -> usize {
    let n = dim;
    if n < 30 {
        2
    } else if n < 60 {
        4
    } else if n < 150 {
        10
    } else if n < 590 {
        #[cfg(feature = "std")]
        {
            (n as f64 / (n as f64).log2()) as usize
        }
        #[cfg(not(feature = "std"))]
        {
            libm::log2(n as f64 / (n as f64)) as usize
        }
    } else if n < 3000 {
        96
    } else if n < 6000 {
        192
    } else {
        384
    }
}

pub fn default_blocking_threshold() -> usize {
    75
}

pub fn default_nibble_threshold() -> usize {
    50
}

fn aggressive_early_deflation<E: ComplexField>(
    want_t: bool,
    mut a: MatMut<'_, E>,
    mut z: Option<MatMut<'_, E>>,
    mut s: MatMut<'_, E>,
    ilo: usize,
    ihi: usize,
    nw: usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
    parallelism: Parallelism,
    mut stack: PodStack<'_>,
    params: EvdParams,
) -> (usize, usize) {
    let n = a.nrows();

    // Because we will use the lower triangular part of A as workspace,
    // We have a maximum window size
    let nw_max = (n - 3) / 3;
    let eps = epsilon;
    let small_num = zero_threshold
        .faer_div(eps)
        .faer_mul(E::Real::faer_from_f64(n as f64));

    // Size of the deflation window
    let jw = Ord::min(Ord::min(nw, ihi - ilo), nw_max);
    // First row index in the deflation window
    let kwtop = ihi - jw;

    // s is the value just outside the window. It determines the spike
    // together with the orthogonal schur factors.
    let mut s_spike = if kwtop == ilo {
        E::faer_zero()
    } else {
        a.read(kwtop, kwtop - 1)
    };

    if kwtop + 1 == ihi {
        // 1x1 deflation window, not much to do
        s.write(kwtop, 0, a.read(kwtop, kwtop));
        let mut ns = 1;
        let mut nd = 0;
        if abs1(s_spike) <= max(small_num, eps.faer_mul(abs1(a.read(kwtop, kwtop)))) {
            ns = 0;
            nd = 1;
            if kwtop > ilo {
                a.write(kwtop, kwtop - 1, E::faer_zero());
            }
        }
        return (ns, nd);
    }

    // Define workspace matrices
    // We use the lower triangular part of A as workspace
    // TW and WH overlap, but WH is only used after we no longer need
    // TW so it is ok.
    let mut v = unsafe { a.rb().submatrix(n - jw, 0, jw, jw).const_cast() };
    let mut tw = unsafe { a.rb().submatrix(n - jw, jw, jw, jw).const_cast() };
    let mut wh = unsafe {
        a.rb()
            .submatrix(n - jw, jw, jw, n - 2 * jw - 3)
            .const_cast()
    };
    let mut wv = unsafe { a.rb().submatrix(jw + 3, 0, n - 2 * jw - 3, jw).const_cast() };
    let mut a = unsafe { a.rb().const_cast() };

    // Convert the window to spike-triangular form. i.e. calculate the
    // Schur form of the deflation window.
    // If the QR algorithm fails to convergence, it can still be
    // partially in Schur form. In that case we continue on a smaller
    // window (note the use of infqr later in the code).
    let a_window = a.rb().submatrix(kwtop, kwtop, ihi - kwtop, ihi - kwtop);
    let mut s_window = unsafe { s.rb().subrows(kwtop, ihi - kwtop).const_cast() };
    zipped!(tw.rb_mut())
        .for_each_triangular_lower(Diag::Include, |unzipped!(mut x)| x.write(E::faer_zero()));
    for j in 0..jw {
        for i in 0..Ord::min(j + 2, jw) {
            tw.write(i, j, a_window.read(i, j));
        }
    }
    v.fill_zero();
    v.rb_mut()
        .diagonal_mut()
        .column_vector_mut()
        .fill(E::faer_one());

    let infqr = if jw
        < params
            .blocking_threshold
            .unwrap_or(default_blocking_threshold())
    {
        lahqr(
            true,
            tw.rb_mut(),
            Some(v.rb_mut()),
            s_window.rb_mut(),
            0,
            jw,
            epsilon,
            zero_threshold,
        )
    } else {
        let infqr = multishift_qr(
            true,
            tw.rb_mut(),
            Some(v.rb_mut()),
            s_window.rb_mut(),
            0,
            jw,
            epsilon,
            zero_threshold,
            parallelism,
            stack.rb_mut(),
            params,
        )
        .0;
        for j in 0..jw {
            for i in j + 2..jw {
                tw.write(i, j, E::faer_zero());
            }
        }
        infqr
    };
    let infqr = infqr as usize;

    // Deflation detection loop
    // one eigenvalue block at a time, we will check if it is deflatable
    // by checking the bottom spike element. If it is not deflatable,
    // we move the block up. This moves other blocks down to check.
    let mut ns = jw;
    let nd;
    let mut ilst = infqr;
    while ilst < ns {
        // 1x1 eigenvalue block
        #[allow(clippy::disallowed_names)]
        let mut foo = abs1(tw.read(ns - 1, ns - 1));
        if foo == E::Real::faer_zero() {
            foo = abs1(s_spike);
        }
        if abs1(s_spike).faer_mul(abs1(v.read(0, ns - 1))) <= max(small_num, eps.faer_mul(foo)) {
            // Eigenvalue is deflatable
            ns -= 1;
        } else {
            // Eigenvalue is not deflatable.
            // Move it up out of the way.
            let ifst = ns - 1;
            schur_move(
                tw.rb_mut(),
                Some(v.rb_mut()),
                ifst,
                &mut ilst,
                epsilon,
                zero_threshold,
            );
            ilst += 1;
        }
    }

    if ns == 0 {
        s_spike = E::faer_zero();
    }

    if ns == jw {
        // Agressive early deflation didn't deflate any eigenvalues
        // We don't need to apply the update to the rest of the matrix
        nd = jw - ns;
        ns -= infqr;
        return (ns, nd);
    }

    // sorting diagonal blocks of T improves accuracy for graded matrices.
    // Bubble sort deals well with exchange failures.
    let mut sorted = false;
    // Window to be checked (other eigenvalue are sorted)
    let mut sorting_window_size = jw;
    while !sorted {
        sorted = true;

        // Index of last eigenvalue that was swapped
        let mut ilst = 0;

        // Index of the first block
        let mut i1 = ns;

        while i1 + 1 < sorting_window_size {
            // Check if there is a next block
            if i1 + 1 == jw {
                ilst -= 1;
                break;
            }

            // Index of the second block
            let i2 = i1 + 1;

            // Size of the second block

            let ev1 = abs1(tw.read(i1, i1));
            let ev2 = abs1(tw.read(i2, i2));

            if ev1 > ev2 {
                i1 = i2;
            } else {
                sorted = false;
                let ierr = schur_swap(tw.rb_mut(), Some(v.rb_mut()), i1, epsilon, zero_threshold);
                if ierr == 0 {
                    i1 += 1;
                } else {
                    i1 = i2;
                }
                ilst = i1;
            }
        }
        sorting_window_size = ilst;
    }

    // Recalculate the eigenvalues
    let mut i = 0;
    while i < jw {
        s.write(kwtop + i, 0, tw.read(i, i));
        i += 1;
    }

    // Reduce A back to Hessenberg form (if neccesary)
    if s_spike != E::faer_zero() {
        // Reflect spike back
        {
            let mut vv = wv.rb_mut().col_mut(0).subrows_mut(0, ns).as_2d_mut();
            for i in 0..ns {
                vv.write(i, 0, v.read(0, i).faer_conj());
            }
            let head = vv.read(0, 0);
            let tail = vv.rb_mut().subrows_mut(1, ns - 1);
            let tail_norm = tail.rb().norm_l2();
            let (tau, beta) = make_householder_in_place_v2(Some(tail), head, tail_norm);
            vv.write(0, 0, E::faer_one());
            let tau = tau.faer_inv();

            {
                let mut tw_slice = tw.rb_mut().submatrix_mut(0, 0, ns, jw);
                let tmp = vv.rb().adjoint() * tw_slice.rb();
                matmul(
                    tw_slice.rb_mut(),
                    vv.rb(),
                    tmp.as_ref(),
                    Some(E::faer_one()),
                    tau.faer_neg(),
                    parallelism,
                );
            }

            {
                let mut tw_slice2 = tw.rb_mut().submatrix_mut(0, 0, jw, ns);
                let tmp = tw_slice2.rb() * vv.rb();
                matmul(
                    tw_slice2.rb_mut(),
                    tmp.as_ref(),
                    vv.rb().adjoint(),
                    Some(E::faer_one()),
                    tau.faer_neg(),
                    parallelism,
                );
            }

            {
                let mut v_slice = v.rb_mut().submatrix_mut(0, 0, jw, ns);
                let tmp = v_slice.rb() * vv.rb();
                matmul(
                    v_slice.rb_mut(),
                    tmp.as_ref(),
                    vv.rb().adjoint(),
                    Some(E::faer_one()),
                    tau.faer_neg(),
                    parallelism,
                );
            }
            vv.write(0, 0, beta);
        }

        // Hessenberg reduction
        {
            let mut householder = wv.rb_mut().col_mut(0).subrows_mut(0, ns - 1);
            make_hessenberg_in_place(
                tw.rb_mut().submatrix_mut(0, 0, ns, ns),
                householder.rb_mut().as_2d_mut(),
                parallelism,
                stack.rb_mut(),
            );
            apply_block_householder_sequence_transpose_on_the_left_in_place_with_conj(
                tw.rb().submatrix(1, 0, ns - 1, ns - 1),
                householder.rb().transpose().as_2d(),
                Conj::Yes,
                unsafe { tw.rb().submatrix(1, ns, ns - 1, jw - ns).const_cast() },
                parallelism,
                stack.rb_mut(),
            );
            apply_block_householder_sequence_on_the_right_in_place_with_conj(
                tw.rb().submatrix(1, 0, ns - 1, ns - 1),
                householder.rb().transpose().as_2d(),
                Conj::No,
                v.rb_mut().submatrix_mut(0, 1, jw, ns - 1),
                parallelism,
                stack.rb_mut(),
            );
        }
    }

    // Copy the deflation window back into place
    if kwtop > 0 {
        a.write(kwtop, kwtop - 1, s_spike.faer_mul(v.read(0, 0).faer_conj()));
    }
    for j in 0..jw {
        for i in 0..Ord::min(j + 2, jw) {
            a.write(kwtop + i, kwtop + j, tw.read(i, j));
        }
    }

    // Store number of deflated eigenvalues
    nd = jw - ns;
    ns -= infqr;

    //
    // Update rest of the matrix using matrix matrix multiplication
    //
    let (istart_m, istop_m);
    if want_t {
        istart_m = 0;
        istop_m = n;
    } else {
        istart_m = ilo;
        istop_m = ihi;
    }

    // Horizontal multiply
    if ihi < istop_m {
        let mut i = ihi;
        while i < istop_m {
            let iblock = Ord::min(istop_m - i, wh.ncols());
            let mut a_slice = a.rb_mut().submatrix_mut(kwtop, i, ihi - kwtop, iblock);
            let mut wh_slice = wh
                .rb_mut()
                .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
            matmul(
                wh_slice.rb_mut(),
                v.rb().adjoint(),
                a_slice.rb(),
                None,
                E::faer_one(),
                parallelism,
            );
            a_slice.copy_from(wh_slice.rb());
            i += iblock;
        }
    }

    // Vertical multiply
    if istart_m < kwtop {
        let mut i = istart_m;
        while i < kwtop {
            let iblock = Ord::min(kwtop - i, wv.nrows());
            let mut a_slice = a.rb_mut().submatrix_mut(i, kwtop, iblock, ihi - kwtop);
            let mut wv_slice = wv
                .rb_mut()
                .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
            matmul(
                wv_slice.rb_mut(),
                a_slice.rb(),
                v.rb(),
                None,
                E::faer_one(),
                parallelism,
            );
            a_slice.copy_from(wv_slice.rb());
            i += iblock;
        }
    }
    // Update Z (also a vertical multiplication)
    if let Some(mut z) = z.rb_mut() {
        let mut i = 0;
        while i < n {
            let iblock = Ord::min(n - i, wv.nrows());
            let mut z_slice = z.rb_mut().submatrix_mut(i, kwtop, iblock, ihi - kwtop);
            let mut wv_slice = wv
                .rb_mut()
                .submatrix_mut(0, 0, z_slice.nrows(), z_slice.ncols());
            matmul(
                wv_slice.rb_mut(),
                z_slice.rb(),
                v.rb(),
                None,
                E::faer_one(),
                parallelism,
            );
            z_slice.copy_from(wv_slice.rb());
            i += iblock;
        }
    }

    (ns, nd)
}

fn schur_move<E: ComplexField>(
    mut a: MatMut<E>,
    mut q: Option<MatMut<E>>,
    ifst: usize,
    ilst: &mut usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
) -> isize {
    let n = a.nrows();

    // Quick return
    if n == 0 {
        return 0;
    }

    let mut here = ifst;
    if ifst < *ilst {
        while here != *ilst {
            // Size of the next eigenvalue block
            let ierr = schur_swap(a.rb_mut(), q.rb_mut(), here, epsilon, zero_threshold);
            if ierr != 0 {
                // The swap failed, return with error
                *ilst = here;
                return 1;
            }
            here += 1;
        }
    } else {
        while here != *ilst {
            // Size of the next eigenvalue block
            let ierr = schur_swap(a.rb_mut(), q.rb_mut(), here - 1, epsilon, zero_threshold);
            if ierr != 0 {
                // The swap failed, return with error
                *ilst = here;
                return 1;
            }
            here -= 1;
        }
    }

    0
}

fn schur_swap<E: ComplexField>(
    mut a: MatMut<E>,
    q: Option<MatMut<E>>,
    j0: usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
) -> isize {
    let n = a.nrows();

    let j1 = j0 + 1;
    let j2 = j0 + 2;

    //
    // In the complex case, there can only be 1x1 blocks to swap
    //
    let t00 = a.read(j0, j0);
    let t11 = a.read(j1, j1);
    //
    // Determine the transformation to perform the interchange
    //
    let (cs, sn, _) = rotg(a.read(j0, j1), t11.faer_sub(t00), epsilon, zero_threshold);

    a.write(j1, j1, t00);
    a.write(j0, j0, t11);

    // Apply transformation from the left
    if j2 < n {
        let row1 = unsafe {
            a.rb()
                .row(j0)
                .subcols(j2, n - j2)
                .transpose()
                .const_cast()
                .as_2d_mut()
        };
        let row2 = unsafe {
            a.rb()
                .row(j1)
                .subcols(j2, n - j2)
                .transpose()
                .const_cast()
                .as_2d_mut()
        };
        rot(row1.transpose_mut(), row2.transpose_mut(), cs, sn);
    }
    // Apply transformation from the right
    if j0 > 0 {
        let col1 = unsafe { a.rb().col(j0).subrows(0, j0).const_cast().as_2d_mut() };
        let col2 = unsafe { a.rb().col(j1).subrows(0, j0).const_cast().as_2d_mut() };

        rot(col1, col2, cs, sn.faer_conj());
    }
    if let Some(q) = q {
        let col1 = unsafe { q.rb().col(j0).const_cast().as_2d_mut() };
        let col2 = unsafe { q.rb().col(j1).const_cast().as_2d_mut() };
        rot(col1, col2, cs, sn.faer_conj());
    }

    0
}

pub fn multishift_qr_req<E: Entity>(
    n: usize,
    nh: usize,
    want_t: bool,
    want_z: bool,
    parallelism: Parallelism,
    params: EvdParams,
) -> Result<StackReq, SizeOverflow> {
    let nsr = (params
        .recommended_shift_count
        .unwrap_or(default_recommended_shift_count))(n, nh);

    let _ = want_t;
    let _ = want_z;

    if n <= 3 {
        return Ok(StackReq::empty());
    }

    let nw_max = (n - 3) / 3;

    StackReq::try_any_of([
        make_hessenberg_in_place_req::<E>(nw_max, 1, parallelism)?,
        apply_block_householder_sequence_transpose_on_the_left_in_place_req::<E>(
            nw_max, nw_max, nw_max,
        )?,
        apply_block_householder_sequence_on_the_right_in_place_req::<E>(nw_max, nw_max, nw_max)?,
        temp_mat_req::<E>(3, nsr)?,
    ])
}

/// returns err code, number of aggressive early deflations, number of qr sweeps
pub fn multishift_qr<E: ComplexField>(
    want_t: bool,
    a: MatMut<'_, E>,
    z: Option<MatMut<'_, E>>,
    w: MatMut<'_, E>,
    ilo: usize,
    ihi: usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
    parallelism: Parallelism,
    stack: PodStack<'_>,
    params: EvdParams,
) -> (isize, usize, usize) {
    assert!(a.nrows() == a.ncols());
    assert!(ilo <= ihi);

    let n = a.nrows();
    let nh = ihi - ilo;

    assert!(w.nrows() == n);
    assert!(w.ncols() == 1);

    if let Some(z) = z.rb() {
        assert!(z.nrows() == n);
        assert!(z.ncols() == n);
    }

    let mut a = a;
    let mut z = z;
    let mut w = w;
    let mut stack = stack;

    let non_convergence_limit_window = 5;
    let non_convergence_limit_shift = 6;
    let dat1 = E::Real::faer_from_f64(0.75);
    let dat2 = E::Real::faer_from_f64(-0.4375);

    // This routine uses the space below the subdiagonal as workspace
    // For small matrices, this is not enough
    // if n < nmin, the matrix will be passed to lahqr
    let nmin = Ord::max(
        15,
        params
            .blocking_threshold
            .unwrap_or(default_blocking_threshold()),
    );
    let nibble = params
        .nibble_threshold
        .unwrap_or(default_nibble_threshold());

    // Recommended number of shifts
    let nsr = (params
        .recommended_shift_count
        .unwrap_or(default_recommended_shift_count))(n, nh);

    // Recommended deflation window size
    let nwr = (params
        .recommended_deflation_window
        .unwrap_or(default_recommended_deflation_window))(n, nh);

    // Tiny matrices must use lahqr
    if n < nmin {
        let err = lahqr(want_t, a, z, w, ilo, ihi, epsilon, zero_threshold);
        return (err, 0, 0);
    }
    if nh == 0 {
        return (0, 0, 0);
    }

    let nw_max = (n - 3) / 3;

    // itmax is the total number of QR iterations allowed.
    // For most matrices, 3 shifts per eigenvalue is enough, so
    // we set itmax to 30 times nh as a safe limit.
    let itmax = 30 * Ord::max(10, nh);

    // k_defl counts the number of iterations since a deflation
    let mut k_defl = 0;

    // istop is the end of the active subblock.
    // As more and more eigenvalues converge, it eventually
    // becomes ilo+1 and the loop ends.
    let mut istop = ihi;

    let mut info = 0;
    let mut nw = 0;

    let mut count_aed = 0;
    let mut count_sweep = 0;

    for iter in 0..itmax + 1 {
        if iter == itmax {
            // The QR algorithm failed to converge, return with error.
            info = istop as isize;
            break;
        }

        if ilo + 1 >= istop {
            if ilo + 1 == istop {
                w.write(ilo, 0, a.read(ilo, ilo));
            }
            // All eigenvalues have been found, exit and return 0.
            break;
        }

        // istart is the start of the active subblock. Either
        // istart = ilo, or H(istart, istart-1) = 0. This means
        // that we can treat this subblock separately.
        let mut istart = ilo;

        // Find active block
        for i in (ilo + 1..istop).rev() {
            if a.read(i, i - 1) == E::faer_zero() {
                istart = i;
                break;
            }
        }

        //
        // Agressive early deflation
        //
        let nh = istop - istart;
        let nwupbd = Ord::min(nh, nw_max);
        if k_defl < non_convergence_limit_window {
            nw = Ord::min(nwupbd, nwr);
        } else {
            // There have been no deflations in many iterations
            // Try to vary the deflation window size.
            nw = Ord::min(nwupbd, 2 * nw);
        }
        if nh <= 4 {
            // n >= nmin, so there is always enough space for a 4x4 window
            nw = nh;
        }
        if nw < nw_max {
            if nw + 1 >= nh {
                nw = nh
            };
            let kwtop = istop - nw;
            if (kwtop > istart + 2)
                && (abs1(a.read(kwtop, kwtop - 1)) > abs1(a.read(kwtop - 1, kwtop - 2)))
            {
                nw += 1;
            }
        }

        let (ls, ld) = aggressive_early_deflation(
            want_t,
            a.rb_mut(),
            z.rb_mut(),
            w.rb_mut(),
            istart,
            istop,
            nw,
            epsilon,
            zero_threshold,
            parallelism,
            stack.rb_mut(),
            params,
        );
        count_aed += 1;

        istop -= ld;

        if ld > 0 {
            k_defl = 0;
        }

        // Skip an expensive QR sweep if there is a (partly heuristic)
        // reason to expect that many eigenvalues will deflate without it.
        // Here, the QR sweep is skipped if many eigenvalues have just been
        // deflated or if the remaining active block is small.
        if ld > 0 && (100 * ld > nwr * nibble || (istop - istart) <= Ord::min(nmin, nw_max)) {
            continue;
        }

        k_defl += 1;
        let mut ns = Ord::min(nh - 1, Ord::min(ls, nsr));
        let mut i_shifts = istop - ls;

        if k_defl % non_convergence_limit_shift == 0 {
            ns = nsr;
            // TODO: compare with original fortran impl
            // unsure about this part
            for i in (i_shifts + 1..istop - 1).step_by(2) {
                let ss = abs1(a.read(i, i - 1)).faer_add(abs1(a.read(i - 1, i - 2)));
                let aa = E::faer_from_real(dat1.faer_mul(ss)).faer_add(a.read(i, i));
                let bb = E::faer_from_real(ss);
                let cc = E::faer_from_real(dat2.faer_mul(ss));
                let dd = aa;
                let (s1, s2) = lahqr_eig22(aa, bb, cc, dd);
                w.write(i, 0, s1);
                w.write(i + 1, 0, s2);
            }
        } else {
            if ls <= nsr / 2 {
                // Got nsr/2 or fewer shifts? Then use multi/double shift qr to
                // get more
                let mut temp = a.rb_mut().submatrix_mut(n - nsr, 0, nsr, nsr);
                let mut shifts = w.rb_mut().subrows_mut(istop - nsr, nsr);
                let ierr = lahqr(
                    false,
                    temp.rb_mut(),
                    None,
                    shifts.rb_mut(),
                    0,
                    nsr,
                    epsilon,
                    zero_threshold,
                ) as usize;

                ns = nsr - ierr;

                if ns < 2 {
                    // In case of a rare QR failure, use eigenvalues
                    // of the trailing 2x2 submatrix
                    let aa = a.read(istop - 2, istop - 2);
                    let bb = a.read(istop - 2, istop - 1);
                    let cc = a.read(istop - 1, istop - 2);
                    let dd = a.read(istop - 1, istop - 1);
                    let (s1, s2) = lahqr_eig22(aa, bb, cc, dd);
                    w.write(istop - 2, 0, s1);
                    w.write(istop - 1, 0, s2);
                    ns = 2;
                }

                i_shifts = istop - ns;
            }

            // Sort the shifts (helps a little)
            // Bubble sort keeps complex conjugate pairs together
            let mut sorted = false;
            let mut k = istop;
            while !sorted && k > i_shifts {
                sorted = true;
                for i in i_shifts..k - 1 {
                    if abs1(w.read(i, 0)) < abs1(w.read(i + 1, 0)) {
                        sorted = false;
                        let wi = w.read(i, 0);
                        let wip1 = w.read(i + 1, 0);
                        w.write(i, 0, wip1);
                        w.write(i + 1, 0, wi);
                    }
                }
                k -= 1;
            }

            // Shuffle shifts into pairs of real shifts
            // and pairs of complex conjugate shifts
            // assuming complex conjugate shifts are
            // already adjacent to one another. (Yes,
            // they are.)
            for i in (i_shifts + 2..istop).rev().step_by(2) {
                if w.read(i, 0).faer_imag() != w.read(i - 1, 0).faer_imag().faer_neg() {
                    let tmp = w.read(i, 0);
                    w.write(i, 0, w.read(i - 1, 0));
                    w.write(i - 1, 0, w.read(i - 2, 0));
                    w.write(i - 2, 0, tmp);
                }
            }

            // Since we shuffled the shifts, we will only drop
            // Real shifts
            if ns % 2 == 1 {
                ns -= 1;
            }
            i_shifts = istop - ns;
        }

        let mut shifts = w.rb_mut().subrows_mut(i_shifts, ns);
        multishift_qr_sweep(
            want_t,
            a.rb_mut(),
            z.rb_mut(),
            shifts.rb_mut(),
            istart,
            istop,
            epsilon,
            zero_threshold,
            parallelism,
            stack.rb_mut(),
        );
        count_sweep += 1;
    }

    (info, count_aed, count_sweep)
}

fn move_bulge<E: ComplexField>(
    mut h: MatMut<'_, E>,
    mut v: MatMut<'_, E>,
    s1: E,
    s2: E,
    epsilon: E::Real,
) {
    // Perform delayed update of row below the bulge
    // Assumes the first two elements of the row are zero
    let v0 = v.read(0, 0).faer_real();
    let v1 = v.read(1, 0);
    let v2 = v.read(2, 0);
    let refsum = v2.faer_scale_real(v0).faer_mul(h.read(3, 2));

    h.write(3, 0, refsum.faer_neg());
    h.write(3, 1, refsum.faer_neg().faer_mul(v1.faer_conj()));
    h.write(3, 2, h.read(3, 2).faer_sub(refsum.faer_mul(v2.faer_conj())));

    // Generate reflector to move bulge down
    v.write(0, 0, h.read(1, 0));
    v.write(1, 0, h.read(2, 0));
    v.write(2, 0, h.read(3, 0));

    let head = v.read(0, 0);
    let tail = v.rb_mut().subrows_mut(1, 2);
    let tail_norm = tail.rb().norm_l2();
    let (tau, beta) = make_householder_in_place_v2(Some(tail), head, tail_norm);
    v.write(0, 0, tau.faer_inv());

    // Check for bulge collapse
    if h.read(3, 0) != E::faer_zero()
        || h.read(3, 1) != E::faer_zero()
        || h.read(3, 2) != E::faer_zero()
    {
        // The bulge hasn't collapsed, typical case
        h.write(1, 0, beta);
        h.write(2, 0, E::faer_zero());
        h.write(3, 0, E::faer_zero());
    } else {
        // The bulge has collapsed, attempt to reintroduce using
        // 2-small-subdiagonals trick
        let mut vt_storage = E::faer_map(E::faer_zero().faer_into_units(), |zero_unit| {
            [zero_unit, zero_unit, zero_unit]
        });
        let vt_ptr = E::faer_map(E::faer_as_mut(&mut vt_storage), |array| array.as_mut_ptr());
        let mut vt = unsafe { faer_core::mat::from_raw_parts_mut::<'_, E>(vt_ptr, 3, 1, 1, 3) };

        let h2 = h.rb().submatrix(1, 1, 3, 3);
        lahqr_shiftcolumn(h2, vt.rb_mut(), s1, s2);

        let head = vt.read(0, 0);
        let tail = vt.rb_mut().subrows_mut(1, 2);
        let tail_norm = tail.rb().norm_l2();
        let (tau, _) = make_householder_in_place_v2(Some(tail), head, tail_norm);
        vt.write(0, 0, tau.faer_inv());
        let vt0 = vt.read(0, 0);
        let vt1 = vt.read(1, 0);
        let vt2 = vt.read(2, 0);

        let refsum = (vt0.faer_conj().faer_mul(h.read(1, 0)))
            .faer_add(vt1.faer_conj().faer_mul(h.read(2, 0)));

        if abs1(h.read(2, 0).faer_sub(refsum.faer_mul(vt1))).faer_add(abs1(refsum.faer_mul(vt2)))
            > epsilon.faer_mul(
                abs1(h.read(0, 0))
                    .faer_add(abs1(h.read(1, 1)))
                    .faer_add(abs1(h.read(2, 2))),
            )
        {
            // Starting a new bulge here would create non-negligible fill. Use
            // the old one.
            h.write(1, 0, beta);
            h.write(2, 0, E::faer_zero());
            h.write(3, 0, E::faer_zero());
        } else {
            // Fill-in is negligible, use the new reflector.
            h.write(1, 0, h.read(1, 0).faer_sub(refsum));
            h.write(2, 0, E::faer_zero());
            h.write(3, 0, E::faer_zero());
            v.write(0, 0, vt.read(0, 0));
            v.write(1, 0, vt.read(1, 0));
            v.write(2, 0, vt.read(2, 0));
        }
    }
}

fn multishift_qr_sweep<E: ComplexField>(
    want_t: bool,
    a: MatMut<E>,
    mut z: Option<MatMut<E>>,
    s: MatMut<E>,
    ilo: usize,
    ihi: usize,
    epsilon: E::Real,
    zero_threshold: E::Real,
    parallelism: Parallelism,
    stack: PodStack<'_>,
) {
    let n = a.nrows();

    let eps = epsilon;
    let small_num = zero_threshold
        .faer_div(eps)
        .faer_mul(E::Real::faer_from_f64(n as f64));
    assert!(n >= 12);

    let (mut v, _stack) = faer_core::temp_mat_zeroed::<E>(3, s.nrows() / 2, stack);
    let mut v = v.as_mut();

    let n_block_max = (n - 3) / 3;
    let n_shifts_max = Ord::min(ihi - ilo - 1, Ord::max(2, 3 * (n_block_max / 4)));

    let mut n_shifts = Ord::min(s.nrows(), n_shifts_max);
    if n_shifts % 2 == 1 {
        n_shifts -= 1;
    }
    let n_bulges = n_shifts / 2;

    let n_block_desired = Ord::min(2 * n_shifts, n_block_max);

    // Define workspace matrices
    // We use the lower triangular part of A as workspace

    // U stores the orthogonal transformations
    let mut u = unsafe {
        a.rb()
            .submatrix(n - n_block_desired, 0, n_block_desired, n_block_desired)
            .const_cast()
    };
    // Workspace for horizontal multiplications
    let mut wh = unsafe {
        a.rb()
            .submatrix(
                n - n_block_desired,
                n_block_desired,
                n_block_desired,
                n - 2 * n_block_desired - 3,
            )
            .const_cast()
    };
    // Workspace for vertical multiplications
    let mut wv = unsafe {
        a.rb()
            .submatrix(
                n_block_desired + 3,
                0,
                n - 2 * n_block_desired - 3,
                n_block_desired,
            )
            .const_cast()
    };
    let mut a = unsafe { a.rb().const_cast() };

    // i_pos_block points to the start of the block of bulges
    let mut i_pos_block;

    //
    // The following code block introduces the bulges into the matrix
    //
    {
        // Near-the-diagonal bulge introduction
        // The calculations are initially limited to the window:
        // A(ilo:ilo+n_block,ilo:ilo+n_block) The rest is updated later via
        // level 3 BLAS

        let n_block = Ord::min(n_block_desired, ihi - ilo);
        let mut istart_m = ilo;
        let mut istop_m = ilo + n_block;
        let mut u2 = u.rb_mut().submatrix_mut(0, 0, n_block, n_block);
        u2.fill_zero();
        u2.rb_mut()
            .diagonal_mut()
            .column_vector_mut()
            .fill(E::faer_one());

        for i_pos_last in ilo..ilo + n_block - 2 {
            // The number of bulges that are in the pencil
            let n_active_bulges = Ord::min(n_bulges, ((i_pos_last - ilo) / 2) + 1);

            for i_bulge in 0..n_active_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let mut v = v.rb_mut().col_mut(i_bulge).as_2d_mut();
                if i_pos == ilo {
                    // Introduce bulge
                    let h = a.rb().submatrix(ilo, ilo, 3, 3);

                    let s1 = s.read(s.nrows() - 1 - 2 * i_bulge, 0);
                    let s2 = s.read(s.nrows() - 1 - 2 * i_bulge - 1, 0);
                    lahqr_shiftcolumn(h, v.rb_mut(), s1, s2);

                    debug_assert!(v.nrows() == 3);
                    let head = v.read(0, 0);
                    let tail = v.rb_mut().subrows_mut(1, 2);
                    let tail_norm = tail.rb().norm_l2();
                    let (tau, _) = make_householder_in_place_v2(Some(tail), head, tail_norm);
                    v.write(0, 0, tau.faer_inv());
                } else {
                    // Chase bulge down
                    let mut h = a.rb_mut().submatrix_mut(i_pos - 1, i_pos - 1, 4, 4);
                    let s1 = s.read(s.nrows() - 1 - 2 * i_bulge, 0);
                    let s2 = s.read(s.nrows() - 1 - 2 * i_bulge - 1, 0);
                    move_bulge(h.rb_mut(), v.rb_mut(), s1, s2, epsilon);
                }

                // Apply the reflector we just calculated from the right
                // We leave the last row for later (it interferes with the
                // optimally packed bulges)

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                for j in istart_m..i_pos + 3 {
                    let sum = a
                        .read(j, i_pos)
                        .faer_add(v1.faer_mul(a.read(j, i_pos + 1)))
                        .faer_add(v2.faer_mul(a.read(j, i_pos + 2)));
                    a.write(j, i_pos, a.read(j, i_pos).faer_sub(sum.faer_scale_real(v0)));
                    a.write(
                        j,
                        i_pos + 1,
                        a.read(j, i_pos + 1)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1.faer_conj())),
                    );
                    a.write(
                        j,
                        i_pos + 2,
                        a.read(j, i_pos + 2)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2.faer_conj())),
                    );
                }

                // Apply the reflector we just calculated from the left
                // We only update a single column, the rest is updated later
                let sum = a
                    .read(i_pos, i_pos)
                    .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, i_pos)))
                    .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, i_pos)));
                a.write(
                    i_pos,
                    i_pos,
                    a.read(i_pos, i_pos).faer_sub(sum.faer_scale_real(v0)),
                );
                a.write(
                    i_pos + 1,
                    i_pos,
                    a.read(i_pos + 1, i_pos)
                        .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                );
                a.write(
                    i_pos + 2,
                    i_pos,
                    a.read(i_pos + 2, i_pos)
                        .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                );

                // Test for deflation.
                if (i_pos > ilo) && (a.read(i_pos, i_pos - 1) != E::faer_zero()) {
                    let mut tst1 =
                        abs1(a.read(i_pos - 1, i_pos - 1)).faer_add(abs1(a.read(i_pos, i_pos)));
                    if tst1 == E::Real::faer_zero() {
                        if i_pos > ilo + 1 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 2)));
                        }
                        if i_pos > ilo + 2 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 3)));
                        }
                        if i_pos > ilo + 3 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 4)));
                        }
                        if i_pos < ihi - 1 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 1, i_pos)));
                        }
                        if i_pos < ihi - 2 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 2, i_pos)));
                        }
                        if i_pos < ihi - 3 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 3, i_pos)));
                        }
                    }
                    if abs1(a.read(i_pos, i_pos - 1)) < max(small_num, eps.faer_mul(tst1)) {
                        let ab = max(
                            abs1(a.read(i_pos, i_pos - 1)),
                            abs1(a.read(i_pos - 1, i_pos)),
                        );
                        let ba = min(
                            abs1(a.read(i_pos, i_pos - 1)),
                            abs1(a.read(i_pos - 1, i_pos)),
                        );
                        let aa = max(
                            abs1(a.read(i_pos, i_pos)),
                            abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                        );
                        let bb = min(
                            abs1(a.read(i_pos, i_pos)),
                            abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                        );
                        let s = aa.faer_add(ab);
                        if ba.faer_mul(ab.faer_div(s))
                            <= max(small_num, eps.faer_mul(bb.faer_mul(aa.faer_div(s))))
                        {
                            a.write(i_pos, i_pos - 1, E::faer_zero());
                        }
                    }
                }
            }

            // Delayed update from the left
            for i_bulge in 0..n_active_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                for j in i_pos + 1..istop_m {
                    let sum = a
                        .read(i_pos, j)
                        .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, j)))
                        .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, j)));
                    a.write(i_pos, j, a.read(i_pos, j).faer_sub(sum.faer_scale_real(v0)));
                    a.write(
                        i_pos + 1,
                        j,
                        a.read(i_pos + 1, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                    );
                    a.write(
                        i_pos + 2,
                        j,
                        a.read(i_pos + 2, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                    );
                }
            }

            // Accumulate the reflectors into U
            for i_bulge in 0..n_active_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                let i1 = 0;
                let i2 = Ord::min(u2.nrows(), (i_pos_last - ilo) + (i_pos_last - ilo) + 3);

                for j in i1..i2 {
                    let sum = u2
                        .read(j, i_pos - ilo)
                        .faer_add(v1.faer_mul(u2.read(j, i_pos - ilo + 1)))
                        .faer_add(v2.faer_mul(u2.read(j, i_pos - ilo + 2)));

                    u2.write(
                        j,
                        i_pos - ilo,
                        u2.read(j, i_pos - ilo).faer_sub(sum.faer_scale_real(v0)),
                    );
                    u2.write(
                        j,
                        i_pos - ilo + 1,
                        u2.read(j, i_pos - ilo + 1)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1.faer_conj())),
                    );
                    u2.write(
                        j,
                        i_pos - ilo + 2,
                        u2.read(j, i_pos - ilo + 2)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2.faer_conj())),
                    );
                }
            }
        }

        // Update rest of the matrix
        if want_t {
            istart_m = 0;
            istop_m = n;
        } else {
            istart_m = ilo;
            istop_m = ihi;
        }

        // Horizontal multiply
        if ilo + n_shifts + 1 < istop_m {
            let mut i = ilo + n_block;
            while i < istop_m {
                let iblock = Ord::min(istop_m - i, wh.ncols());
                let mut a_slice = a.rb_mut().submatrix_mut(ilo, i, n_block, iblock);
                let mut wh_slice =
                    wh.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wh_slice.rb_mut(),
                    u2.rb().adjoint(),
                    a_slice.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wh_slice.rb());
                i += iblock;
            }
        }
        // Vertical multiply
        if istart_m < ilo {
            let mut i = istart_m;
            while i < ilo {
                let iblock = Ord::min(ilo - i, wv.nrows());
                let mut a_slice = a.rb_mut().submatrix_mut(i, ilo, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    a_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }
        // Update Z (also a vertical multiplication)
        if let Some(mut z) = z.rb_mut() {
            let mut i = 0;
            while i < n {
                let iblock = Ord::min(n - i, wv.nrows());
                let mut z_slice = z.rb_mut().submatrix_mut(i, ilo, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, z_slice.nrows(), z_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    z_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                z_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }

        i_pos_block = ilo + n_block - n_shifts;
    }

    //
    // The following code block moves the bulges down untill they are low enough
    // to be removed
    //
    while i_pos_block + n_block_desired < ihi {
        // Number of positions each bulge will be moved down
        let n_pos = Ord::min(n_block_desired - n_shifts, ihi - n_shifts - 1 - i_pos_block);
        // Actual blocksize
        let n_block = n_shifts + n_pos;

        let mut u2 = u.rb_mut().submatrix_mut(0, 0, n_block, n_block);
        u2.fill_zero();
        u2.rb_mut()
            .diagonal_mut()
            .column_vector_mut()
            .fill(E::faer_one());

        // Near-the-diagonal bulge chase
        // The calculations are initially limited to the window:
        // A(i_pos_block-1:i_pos_block+n_block,i_pos_block:i_pos_block+n_block)
        // The rest is updated later via level 3 BLAS
        let mut istart_m = i_pos_block;
        let mut istop_m = i_pos_block + n_block;

        for i_pos_last in i_pos_block + n_shifts - 2..i_pos_block + n_shifts - 2 + n_pos {
            for i_bulge in 0..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let mut v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                // Chase bulge down
                let mut h = a.rb_mut().submatrix_mut(i_pos - 1, i_pos - 1, 4, 4);
                let s1 = s.read(s.nrows() - 1 - 2 * i_bulge, 0);
                let s2 = s.read(s.nrows() - 1 - 2 * i_bulge - 1, 0);
                move_bulge(h.rb_mut(), v.rb_mut(), s1, s2, epsilon);

                // Apply the reflector we just calculated from the right
                // We leave the last row for later (it interferes with the
                // optimally packed bulges)

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                for j in istart_m..i_pos + 3 {
                    let sum = a
                        .read(j, i_pos)
                        .faer_add(v1.faer_mul(a.read(j, i_pos + 1)))
                        .faer_add(v2.faer_mul(a.read(j, i_pos + 2)));
                    a.write(j, i_pos, a.read(j, i_pos).faer_sub(sum.faer_scale_real(v0)));
                    a.write(
                        j,
                        i_pos + 1,
                        a.read(j, i_pos + 1)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1.faer_conj())),
                    );
                    a.write(
                        j,
                        i_pos + 2,
                        a.read(j, i_pos + 2)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2.faer_conj())),
                    );
                }

                // Apply the reflector we just calculated from the left
                // We only update a single column, the rest is updated later
                let sum = a
                    .read(i_pos, i_pos)
                    .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, i_pos)))
                    .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, i_pos)));
                a.write(
                    i_pos,
                    i_pos,
                    a.read(i_pos, i_pos).faer_sub(sum.faer_scale_real(v0)),
                );
                a.write(
                    i_pos + 1,
                    i_pos,
                    a.read(i_pos + 1, i_pos)
                        .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                );
                a.write(
                    i_pos + 2,
                    i_pos,
                    a.read(i_pos + 2, i_pos)
                        .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                );

                // Test for deflation.
                if (i_pos > ilo) && (a.read(i_pos, i_pos - 1) != E::faer_zero()) {
                    let mut tst1 =
                        abs1(a.read(i_pos - 1, i_pos - 1)).faer_add(abs1(a.read(i_pos, i_pos)));
                    if tst1 == E::Real::faer_zero() {
                        if i_pos > ilo + 1 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 2)));
                        }
                        if i_pos > ilo + 2 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 3)));
                        }
                        if i_pos > ilo + 3 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 4)));
                        }
                        if i_pos < ihi - 1 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 1, i_pos)));
                        }
                        if i_pos < ihi - 2 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 2, i_pos)));
                        }
                        if i_pos < ihi - 3 {
                            tst1 = tst1.faer_add(abs1(a.read(i_pos + 3, i_pos)));
                        }
                    }
                    if abs1(a.read(i_pos, i_pos - 1)) < max(small_num, eps.faer_mul(tst1)) {
                        let ab = max(
                            abs1(a.read(i_pos, i_pos - 1)),
                            abs1(a.read(i_pos - 1, i_pos)),
                        );
                        let ba = min(
                            abs1(a.read(i_pos, i_pos - 1)),
                            abs1(a.read(i_pos - 1, i_pos)),
                        );
                        let aa = max(
                            abs1(a.read(i_pos, i_pos)),
                            abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                        );
                        let bb = min(
                            abs1(a.read(i_pos, i_pos)),
                            abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                        );
                        let s = aa.faer_add(ab);
                        if ba.faer_mul(ab.faer_div(s))
                            <= max(small_num, eps.faer_mul(bb.faer_mul(aa.faer_div(s))))
                        {
                            a.write(i_pos, i_pos - 1, E::faer_zero());
                        }
                    }
                }
            }

            // Delayed update from the left
            for i_bulge in 0..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                for j in i_pos + 1..istop_m {
                    let sum = a
                        .read(i_pos, j)
                        .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, j)))
                        .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, j)));
                    a.write(i_pos, j, a.read(i_pos, j).faer_sub(sum.faer_scale_real(v0)));
                    a.write(
                        i_pos + 1,
                        j,
                        a.read(i_pos + 1, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                    );
                    a.write(
                        i_pos + 2,
                        j,
                        a.read(i_pos + 2, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                    );
                }
            }

            // Accumulate the reflectors into U
            for i_bulge in 0..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                let i1 = (i_pos - i_pos_block) - (i_pos_last + 2 - i_pos_block - n_shifts);
                let i2 = Ord::min(
                    u2.nrows(),
                    (i_pos_last - i_pos_block) + (i_pos_last + 2 - i_pos_block - n_shifts) + 3,
                );

                for j in i1..i2 {
                    let sum = u2
                        .read(j, i_pos - i_pos_block)
                        .faer_add(v1.faer_mul(u2.read(j, i_pos - i_pos_block + 1)))
                        .faer_add(v2.faer_mul(u2.read(j, i_pos - i_pos_block + 2)));

                    u2.write(
                        j,
                        i_pos - i_pos_block,
                        u2.read(j, i_pos - i_pos_block)
                            .faer_sub(sum.faer_scale_real(v0)),
                    );
                    u2.write(
                        j,
                        i_pos - i_pos_block + 1,
                        u2.read(j, i_pos - i_pos_block + 1)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1.faer_conj())),
                    );
                    u2.write(
                        j,
                        i_pos - i_pos_block + 2,
                        u2.read(j, i_pos - i_pos_block + 2)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2.faer_conj())),
                    );
                }
            }
        }

        // Update rest of the matrix
        if want_t {
            istart_m = 0;
            istop_m = n;
        } else {
            istart_m = ilo;
            istop_m = ihi;
        }

        // Horizontal multiply
        if i_pos_block + n_block < istop_m {
            let mut i = i_pos_block + n_block;
            while i < istop_m {
                let iblock = Ord::min(istop_m - i, wh.ncols());
                let mut a_slice = a.rb_mut().submatrix_mut(i_pos_block, i, n_block, iblock);
                let mut wh_slice =
                    wh.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wh_slice.rb_mut(),
                    u2.rb().adjoint(),
                    a_slice.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wh_slice.rb());
                i += iblock;
            }
        }

        // Vertical multiply
        if istart_m < i_pos_block {
            let mut i = istart_m;
            while i < i_pos_block {
                let iblock = Ord::min(i_pos_block - i, wv.nrows());
                let mut a_slice = a.rb_mut().submatrix_mut(i, i_pos_block, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    a_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }
        // Update Z (also a vertical multiplication)
        if let Some(mut z) = z.rb_mut() {
            let mut i = 0;
            while i < n {
                let iblock = Ord::min(n - i, wv.nrows());
                let mut z_slice = z.rb_mut().submatrix_mut(i, i_pos_block, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, z_slice.nrows(), z_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    z_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                z_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }

        i_pos_block += n_pos;
    }

    //
    // The following code removes the bulges from the matrix
    //
    {
        let n_block = ihi - i_pos_block;

        let mut u2 = u.rb_mut().submatrix_mut(0, 0, n_block, n_block);
        u2.fill_zero();
        u2.rb_mut()
            .diagonal_mut()
            .column_vector_mut()
            .fill(E::faer_one());

        // Near-the-diagonal bulge chase
        // The calculations are initially limited to the window:
        // A(i_pos_block-1:ihi,i_pos_block:ihi) The rest is updated later via
        // level 3 BLAS
        let mut istart_m = i_pos_block;
        let mut istop_m = ihi;

        for i_pos_last in i_pos_block + n_shifts - 2..ihi + n_shifts - 1 {
            let mut i_bulge_start = if i_pos_last + 3 > ihi {
                (i_pos_last + 3 - ihi) / 2
            } else {
                0
            };

            for i_bulge in i_bulge_start..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                if i_pos == ihi - 2 {
                    // Special case, the bulge is at the bottom, needs a smaller
                    // reflector (order 2)
                    let mut v = v.rb_mut().subrows_mut(0, 2).col_mut(i_bulge).as_2d_mut();
                    let mut h = a
                        .rb_mut()
                        .subrows_mut(i_pos, 2)
                        .col_mut(i_pos - 1)
                        .as_2d_mut();
                    let head = h.read(0, 0);
                    let tail = h.rb_mut().subrows_mut(1, 1);
                    let tail_norm = tail.rb().norm_l2();
                    let (tau, beta) = make_householder_in_place_v2(Some(tail), head, tail_norm);
                    v.write(0, 0, tau.faer_inv());
                    v.write(1, 0, h.read(1, 0));
                    h.write(0, 0, beta);
                    h.write(1, 0, E::faer_zero());

                    let t0 = v.read(0, 0).faer_conj();
                    let v1 = v.read(1, 0);
                    let t1 = t0.faer_mul(v1);
                    // Apply the reflector we just calculated from the right
                    for j in istart_m..i_pos + 2 {
                        let sum = a.read(j, i_pos).faer_add(v1.faer_mul(a.read(j, i_pos + 1)));
                        a.write(
                            j,
                            i_pos,
                            a.read(j, i_pos).faer_sub(sum.faer_mul(t0.faer_conj())),
                        );
                        a.write(
                            j,
                            i_pos + 1,
                            a.read(j, i_pos + 1).faer_sub(sum.faer_mul(t1.faer_conj())),
                        );
                    }
                    // Apply the reflector we just calculated from the left
                    for j in i_pos..istop_m {
                        let sum = a
                            .read(i_pos, j)
                            .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, j)));
                        a.write(i_pos, j, a.read(i_pos, j).faer_sub(sum.faer_mul(t0)));
                        a.write(
                            i_pos + 1,
                            j,
                            a.read(i_pos + 1, j).faer_sub(sum.faer_mul(t1)),
                        );
                    }
                    // Accumulate the reflector into U
                    // The loop bounds should be changed to reflect the fact
                    // that U2 starts off as diagonal
                    for j in 0..u2.nrows() {
                        let sum = u2
                            .read(j, i_pos - i_pos_block)
                            .faer_add(v1.faer_mul(u2.read(j, i_pos - i_pos_block + 1)));
                        u2.write(
                            j,
                            i_pos - i_pos_block,
                            u2.read(j, i_pos - i_pos_block)
                                .faer_sub(sum.faer_mul(t0.faer_conj())),
                        );
                        u2.write(
                            j,
                            i_pos - i_pos_block + 1,
                            u2.read(j, i_pos - i_pos_block + 1)
                                .faer_sub(sum.faer_mul(t1.faer_conj())),
                        );
                    }
                } else {
                    let mut v = v.rb_mut().col_mut(i_bulge).as_2d_mut();
                    let mut h = a.rb_mut().submatrix_mut(i_pos - 1, i_pos - 1, 4, 4);
                    let s1 = s.read(s.nrows() - 1 - 2 * i_bulge, 0);
                    let s2 = s.read(s.nrows() - 1 - 2 * i_bulge - 1, 0);
                    move_bulge(h.rb_mut(), v.rb_mut(), s1, s2, epsilon);

                    {
                        let t0 = v.read(0, 0).faer_conj();
                        let v1 = v.read(1, 0);
                        let t1 = t0.faer_mul(v1);
                        let v2 = v.read(2, 0);
                        let t2 = t0.faer_mul(v2);
                        // Apply the reflector we just calculated from the right
                        // (but leave the last row for later)
                        for j in istart_m..i_pos + 3 {
                            let sum = a
                                .read(j, i_pos)
                                .faer_add(v1.faer_mul(a.read(j, i_pos + 1)))
                                .faer_add(v2.faer_mul(a.read(j, i_pos + 2)));
                            a.write(
                                j,
                                i_pos,
                                a.read(j, i_pos).faer_sub(sum.faer_mul(t0.faer_conj())),
                            );
                            a.write(
                                j,
                                i_pos + 1,
                                a.read(j, i_pos + 1).faer_sub(sum.faer_mul(t1.faer_conj())),
                            );
                            a.write(
                                j,
                                i_pos + 2,
                                a.read(j, i_pos + 2).faer_sub(sum.faer_mul(t2.faer_conj())),
                            );
                        }
                    }

                    let v0 = v.read(0, 0).faer_real();
                    let v1 = v.read(1, 0);
                    let v2 = v.read(2, 0);
                    // Apply the reflector we just calculated from the left
                    // We only update a single column, the rest is updated later
                    let sum = a
                        .read(i_pos, i_pos)
                        .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, i_pos)))
                        .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, i_pos)));
                    a.write(
                        i_pos,
                        i_pos,
                        a.read(i_pos, i_pos).faer_sub(sum.faer_scale_real(v0)),
                    );
                    a.write(
                        i_pos + 1,
                        i_pos,
                        a.read(i_pos + 1, i_pos)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                    );
                    a.write(
                        i_pos + 2,
                        i_pos,
                        a.read(i_pos + 2, i_pos)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                    );

                    // Test for deflation.
                    if (i_pos > ilo) && (a.read(i_pos, i_pos - 1) != E::faer_zero()) {
                        let mut tst1 =
                            abs1(a.read(i_pos - 1, i_pos - 1)).faer_add(abs1(a.read(i_pos, i_pos)));
                        if tst1 == E::Real::faer_zero() {
                            if i_pos > ilo + 1 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 2)));
                            }
                            if i_pos > ilo + 2 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 3)));
                            }
                            if i_pos > ilo + 3 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos - 1, i_pos - 4)));
                            }
                            if i_pos < ihi - 1 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos + 1, i_pos)));
                            }
                            if i_pos < ihi - 2 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos + 2, i_pos)));
                            }
                            if i_pos < ihi - 3 {
                                tst1 = tst1.faer_add(abs1(a.read(i_pos + 3, i_pos)));
                            }
                        }
                        if abs1(a.read(i_pos, i_pos - 1)) < max(small_num, eps.faer_mul(tst1)) {
                            let ab = max(
                                abs1(a.read(i_pos, i_pos - 1)),
                                abs1(a.read(i_pos - 1, i_pos)),
                            );
                            let ba = min(
                                abs1(a.read(i_pos, i_pos - 1)),
                                abs1(a.read(i_pos - 1, i_pos)),
                            );
                            let aa = max(
                                abs1(a.read(i_pos, i_pos)),
                                abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                            );
                            let bb = min(
                                abs1(a.read(i_pos, i_pos)),
                                abs1(a.read(i_pos, i_pos).faer_sub(a.read(i_pos - 1, i_pos - 1))),
                            );
                            let s = aa.faer_add(ab);
                            if ba.faer_mul(ab.faer_div(s))
                                <= max(small_num, eps.faer_mul(bb.faer_mul(aa.faer_div(s))))
                            {
                                a.write(i_pos, i_pos - 1, E::faer_zero());
                            }
                        }
                    }
                }
            }

            i_bulge_start = if i_pos_last + 4 > ihi {
                (i_pos_last + 4 - ihi) / 2
            } else {
                0
            };

            // Delayed update from the left
            for i_bulge in i_bulge_start..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                for j in i_pos + 1..istop_m {
                    let sum = a
                        .read(i_pos, j)
                        .faer_add(v1.faer_conj().faer_mul(a.read(i_pos + 1, j)))
                        .faer_add(v2.faer_conj().faer_mul(a.read(i_pos + 2, j)));
                    a.write(i_pos, j, a.read(i_pos, j).faer_sub(sum.faer_scale_real(v0)));
                    a.write(
                        i_pos + 1,
                        j,
                        a.read(i_pos + 1, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1)),
                    );
                    a.write(
                        i_pos + 2,
                        j,
                        a.read(i_pos + 2, j)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2)),
                    );
                }
            }

            // Accumulate the reflectors into U
            for i_bulge in i_bulge_start..n_bulges {
                let i_pos = i_pos_last - 2 * i_bulge;
                let v = v.rb_mut().col_mut(i_bulge).as_2d_mut();

                let v0 = v.read(0, 0).faer_real();
                let v1 = v.read(1, 0);
                let v2 = v.read(2, 0);

                let i1 = (i_pos - i_pos_block) - (i_pos_last + 2 - i_pos_block - n_shifts);
                let i2 = Ord::min(
                    u2.nrows(),
                    (i_pos_last - i_pos_block) + (i_pos_last + 2 - i_pos_block - n_shifts) + 3,
                );

                for j in i1..i2 {
                    let sum = u2
                        .read(j, i_pos - i_pos_block)
                        .faer_add(v1.faer_mul(u2.read(j, i_pos - i_pos_block + 1)))
                        .faer_add(v2.faer_mul(u2.read(j, i_pos - i_pos_block + 2)));

                    u2.write(
                        j,
                        i_pos - i_pos_block,
                        u2.read(j, i_pos - i_pos_block)
                            .faer_sub(sum.faer_scale_real(v0)),
                    );
                    u2.write(
                        j,
                        i_pos - i_pos_block + 1,
                        u2.read(j, i_pos - i_pos_block + 1)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v1.faer_conj())),
                    );
                    u2.write(
                        j,
                        i_pos - i_pos_block + 2,
                        u2.read(j, i_pos - i_pos_block + 2)
                            .faer_sub(sum.faer_scale_real(v0).faer_mul(v2.faer_conj())),
                    );
                }
            }
        }

        // Update rest of the matrix
        if want_t {
            istart_m = 0;
            istop_m = n;
        } else {
            istart_m = ilo;
            istop_m = ihi;
        }

        debug_assert!(i_pos_block + n_block == ihi);

        // Horizontal multiply
        if ihi < istop_m {
            let mut i = ihi;
            while i < istop_m {
                let iblock = Ord::min(istop_m - i, wh.ncols());
                let mut a_slice = a.rb_mut().submatrix_mut(i_pos_block, i, n_block, iblock);
                let mut wh_slice =
                    wh.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wh_slice.rb_mut(),
                    u2.rb().adjoint(),
                    a_slice.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wh_slice.rb());
                i += iblock;
            }
        }

        // Vertical multiply
        if istart_m < i_pos_block {
            let mut i = istart_m;
            while i < i_pos_block {
                let iblock = Ord::min(i_pos_block - i, wv.nrows());
                let mut a_slice = a.rb_mut().submatrix_mut(i, i_pos_block, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, a_slice.nrows(), a_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    a_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                a_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }
        // Update Z (also a vertical multiplication)
        if let Some(mut z) = z.rb_mut() {
            let mut i = 0;
            while i < n {
                let iblock = Ord::min(n - i, wv.nrows());
                let mut z_slice = z.rb_mut().submatrix_mut(i, i_pos_block, iblock, n_block);
                let mut wv_slice =
                    wv.rb_mut()
                        .submatrix_mut(0, 0, z_slice.nrows(), z_slice.ncols());
                matmul(
                    wv_slice.rb_mut(),
                    z_slice.rb(),
                    u2.rb(),
                    None,
                    E::faer_one(),
                    parallelism,
                );
                z_slice.copy_from(wv_slice.rb());
                i += iblock;
            }
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use assert_approx_eq::assert_approx_eq;
    use faer_core::{assert, c64, mat, Mat};

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

    #[test]
    fn test_3() {
        let n = 3;
        let h = mat![
            [
                c64::new(0.997386, 0.677592),
                c64::new(0.646064, 0.936948),
                c64::new(0.090948, 0.674011),
            ],
            [
                c64::new(0.212396, 0.976794),
                c64::new(0.460270, 0.926436),
                c64::new(0.494441, 0.888187),
            ],
            [
                c64::new(0.000000, 0.000000),
                c64::new(0.616652, 0.840012),
                c64::new(0.768245, 0.349193),
            ],
        ];

        let mut q = Mat::from_fn(n, n, |i, j| {
            if i == j {
                c64::faer_one()
            } else {
                c64::faer_zero()
            }
        });
        let mut w = Mat::zeros(n, 1);
        let mut t = h.clone();
        lahqr(
            true,
            t.as_mut(),
            Some(q.as_mut()),
            w.as_mut(),
            0,
            n,
            f64::EPSILON,
            f64::MIN_POSITIVE,
        );

        let h_reconstructed = &q * &t * q.adjoint();

        for i in 0..n {
            for j in 0..n {
                assert_approx_eq!(h_reconstructed.read(i, j), h.read(i, j));
            }
        }
    }

    #[test]
    fn test_n() {
        for n in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 32, 128] {
            for _ in 0..10 {
                let mut h = Mat::<c64>::zeros(n, n);
                for j in 0..n {
                    for i in 0..n {
                        if i <= j + 1 {
                            h.write(i, j, c64::new(rand::random(), rand::random()));
                        }
                    }
                }

                let mut q = Mat::from_fn(n, n, |i, j| {
                    if i == j {
                        c64::faer_one()
                    } else {
                        c64::faer_zero()
                    }
                });

                let mut w = Mat::zeros(n, 1);

                let mut t = h.clone();
                lahqr(
                    true,
                    t.as_mut(),
                    Some(q.as_mut()),
                    w.as_mut(),
                    0,
                    n,
                    f64::EPSILON,
                    f64::MIN_POSITIVE,
                );
                dbgf::dbgf!("6.6?", &t, &h);

                let h_reconstructed = &q * &t * q.adjoint();

                for i in 0..n {
                    for j in 0..n {
                        assert_approx_eq!(h_reconstructed.read(i, j), h.read(i, j));
                    }
                }
            }
        }
    }

    #[test]
    fn test_multi_n() {
        for n in [
            1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 32, 63, 64, 65, 128, 256,
        ] {
            for _ in 0..10 {
                let mut h = Mat::<c64>::zeros(n, n);
                for j in 0..n {
                    for i in 0..n {
                        if i <= j + 1 {
                            h.write(i, j, c64::new(rand::random(), rand::random()));
                        }
                    }
                }

                let mut q = Mat::from_fn(n, n, |i, j| {
                    if i == j {
                        c64::faer_one()
                    } else {
                        c64::faer_zero()
                    }
                });

                let mut w = Mat::zeros(n, 1);

                let mut t = h.clone();
                let params = EvdParams {
                    recommended_shift_count: None,
                    recommended_deflation_window: None,
                    blocking_threshold: Some(15),
                    nibble_threshold: Some(14),
                };
                multishift_qr(
                    true,
                    t.as_mut(),
                    Some(q.as_mut()),
                    w.as_mut(),
                    0,
                    n,
                    f64::EPSILON,
                    f64::MIN_POSITIVE,
                    Parallelism::None,
                    make_stack!(multishift_qr_req::<c64>(
                        n,
                        n,
                        true,
                        true,
                        Parallelism::None,
                        params,
                    )),
                    params,
                );
                for j in 0..n {
                    for i in j + 1..n {
                        t.write(i, j, c64::faer_zero());
                    }
                }
                dbgf::dbgf!("6.6?", &t, &h);

                let h_reconstructed = &q * &t * q.adjoint();

                for i in 0..n {
                    for j in 0..n {
                        assert_approx_eq!(h_reconstructed.read(i, j), h.read(i, j));
                    }
                }
            }
        }
    }

    #[test]
    fn test_multi_100() {
        let n = 100;

        let h = [
            [
                c64::new(0.247362, 0.915457),
                c64::new(0.729822, 0.217719),
                c64::new(0.557421, 0.288464),
                c64::new(0.328797, 0.239523),
                c64::new(0.777550, 0.831544),
                c64::new(0.987836, 0.172338),
                c64::new(0.398255, 0.340496),
                c64::new(0.895207, 0.209341),
                c64::new(0.514276, 0.133539),
                c64::new(0.112494, 0.772690),
                c64::new(0.794636, 0.624804),
                c64::new(0.764424, 0.189969),
                c64::new(0.448585, 0.845465),
                c64::new(0.648511, 0.998682),
                c64::new(0.761988, 0.891177),
                c64::new(0.248977, 0.098031),
                c64::new(0.251652, 0.303864),
                c64::new(0.341391, 0.737074),
                c64::new(0.724391, 0.632790),
                c64::new(0.644896, 0.231057),
                c64::new(0.824051, 0.613854),
                c64::new(0.716392, 0.677729),
                c64::new(0.098836, 0.500676),
                c64::new(0.011666, 0.426799),
                c64::new(0.879832, 0.626437),
                c64::new(0.898294, 0.204360),
                c64::new(0.305867, 0.437079),
                c64::new(0.335694, 0.013785),
                c64::new(0.963588, 0.221925),
                c64::new(0.572279, 0.201790),
                c64::new(0.157022, 0.485770),
                c64::new(0.002149, 0.028072),
                c64::new(0.464055, 0.223266),
                c64::new(0.483257, 0.789599),
                c64::new(0.443072, 0.964070),
                c64::new(0.231826, 0.472352),
                c64::new(0.059165, 0.681934),
                c64::new(0.295935, 0.702026),
                c64::new(0.049729, 0.483953),
                c64::new(0.442269, 0.661843),
                c64::new(0.774934, 0.140070),
                c64::new(0.102210, 0.834111),
                c64::new(0.326798, 0.629013),
                c64::new(0.724024, 0.877097),
                c64::new(0.631407, 0.953489),
                c64::new(0.108545, 0.371405),
                c64::new(0.937458, 0.968946),
                c64::new(0.160439, 0.886438),
                c64::new(0.347411, 0.151744),
                c64::new(0.108679, 0.783657),
                c64::new(0.862349, 0.605088),
                c64::new(0.071644, 0.888313),
                c64::new(0.167642, 0.853602),
                c64::new(0.971924, 0.432988),
                c64::new(0.265114, 0.244212),
                c64::new(0.165003, 0.412567),
                c64::new(0.930197, 0.307039),
                c64::new(0.013761, 0.187137),
                c64::new(0.439960, 0.553093),
                c64::new(0.909387, 0.800205),
                c64::new(0.301502, 0.869243),
                c64::new(0.347811, 0.384708),
                c64::new(0.941104, 0.925138),
                c64::new(0.576176, 0.184279),
                c64::new(0.568036, 0.388525),
                c64::new(0.566769, 0.536556),
                c64::new(0.058159, 0.243945),
                c64::new(0.221479, 0.495539),
                c64::new(0.111847, 0.004560),
                c64::new(0.781100, 0.170008),
                c64::new(0.039433, 0.479257),
                c64::new(0.850582, 0.896525),
                c64::new(0.219405, 0.923765),
                c64::new(0.725664, 0.321608),
                c64::new(0.252339, 0.069854),
                c64::new(0.674924, 0.327177),
                c64::new(0.735469, 0.565386),
                c64::new(0.568203, 0.177555),
                c64::new(0.842474, 0.661287),
                c64::new(0.060288, 0.816681),
                c64::new(0.255325, 0.462335),
                c64::new(0.856666, 0.234204),
                c64::new(0.157914, 0.342594),
                c64::new(0.191756, 0.858922),
                c64::new(0.761656, 0.116901),
                c64::new(0.199292, 0.477377),
                c64::new(0.435394, 0.823778),
                c64::new(0.363253, 0.843141),
                c64::new(0.618471, 0.196208),
                c64::new(0.571452, 0.087156),
                c64::new(0.967526, 0.131176),
                c64::new(0.053565, 0.173195),
                c64::new(0.514817, 0.215551),
                c64::new(0.197349, 0.887501),
                c64::new(0.678657, 0.991695),
                c64::new(0.081140, 0.604964),
                c64::new(0.675069, 0.545161),
                c64::new(0.322368, 0.150800),
                c64::new(0.272257, 0.873348),
                c64::new(0.735019, 0.653493),
            ],
            [
                c64::new(0.315191, 0.478662),
                c64::new(0.833807, 0.697483),
                c64::new(0.310534, 0.396302),
                c64::new(0.839271, 0.949712),
                c64::new(0.325344, 0.235542),
                c64::new(0.693700, 0.247816),
                c64::new(0.759020, 0.118982),
                c64::new(0.160333, 0.765951),
                c64::new(0.631740, 0.843597),
                c64::new(0.554191, 0.633453),
                c64::new(0.742897, 0.788577),
                c64::new(0.840315, 0.054435),
                c64::new(0.089497, 0.589866),
                c64::new(0.563293, 0.695071),
                c64::new(0.916185, 0.138849),
                c64::new(0.067603, 0.481400),
                c64::new(0.906483, 0.980935),
                c64::new(0.715502, 0.313255),
                c64::new(0.738656, 0.268801),
                c64::new(0.839109, 0.956635),
                c64::new(0.690422, 0.335274),
                c64::new(0.488201, 0.501880),
                c64::new(0.328307, 0.260638),
                c64::new(0.291425, 0.432487),
                c64::new(0.638943, 0.328813),
                c64::new(0.984401, 0.711157),
                c64::new(0.806059, 0.405321),
                c64::new(0.599686, 0.547603),
                c64::new(0.381971, 0.484660),
                c64::new(0.992921, 0.606005),
                c64::new(0.961329, 0.893073),
                c64::new(0.694046, 0.744479),
                c64::new(0.517949, 0.141120),
                c64::new(0.666088, 0.116443),
                c64::new(0.590009, 0.707290),
                c64::new(0.397481, 0.500323),
                c64::new(0.027404, 0.757967),
                c64::new(0.624940, 0.042651),
                c64::new(0.632875, 0.116292),
                c64::new(0.224254, 0.776686),
                c64::new(0.910262, 0.616570),
                c64::new(0.277812, 0.737948),
                c64::new(0.057866, 0.784957),
                c64::new(0.152657, 0.911839),
                c64::new(0.674222, 0.041607),
                c64::new(0.511282, 0.147960),
                c64::new(0.494807, 0.331465),
                c64::new(0.228156, 0.943136),
                c64::new(0.439864, 0.301055),
                c64::new(0.272690, 0.238139),
                c64::new(0.893550, 0.363394),
                c64::new(0.272024, 0.665481),
                c64::new(0.051683, 0.055642),
                c64::new(0.059980, 0.933988),
                c64::new(0.312177, 0.931784),
                c64::new(0.692342, 0.669074),
                c64::new(0.708301, 0.746008),
                c64::new(0.654037, 0.048102),
                c64::new(0.805826, 0.440089),
                c64::new(0.269123, 0.844255),
                c64::new(0.189183, 0.907510),
                c64::new(0.422676, 0.220691),
                c64::new(0.730443, 0.605872),
                c64::new(0.922972, 0.193306),
                c64::new(0.932751, 0.640147),
                c64::new(0.415496, 0.541532),
                c64::new(0.284173, 0.456411),
                c64::new(0.112323, 0.847687),
                c64::new(0.906880, 0.909230),
                c64::new(0.629705, 0.631650),
                c64::new(0.022350, 0.197708),
                c64::new(0.716942, 0.390986),
                c64::new(0.945447, 0.269241),
                c64::new(0.745904, 0.211792),
                c64::new(0.818865, 0.295856),
                c64::new(0.657553, 0.479682),
                c64::new(0.824392, 0.485086),
                c64::new(0.506792, 0.486432),
                c64::new(0.048943, 0.563696),
                c64::new(0.917321, 0.741947),
                c64::new(0.011085, 0.625994),
                c64::new(0.511457, 0.815998),
                c64::new(0.618929, 0.165593),
                c64::new(0.200062, 0.607617),
                c64::new(0.335587, 0.993400),
                c64::new(0.088285, 0.474581),
                c64::new(0.977839, 0.850574),
                c64::new(0.080987, 0.751683),
                c64::new(0.488594, 0.731679),
                c64::new(0.489062, 0.434918),
                c64::new(0.147371, 0.659253),
                c64::new(0.585042, 0.742962),
                c64::new(0.914979, 0.911878),
                c64::new(0.938842, 0.912843),
                c64::new(0.685735, 0.695321),
                c64::new(0.335607, 0.733544),
                c64::new(0.835737, 0.400742),
                c64::new(0.250515, 0.714784),
                c64::new(0.877781, 0.755097),
                c64::new(0.446200, 0.130484),
            ],
            [
                c64::new(0.000000, 0.000000),
                c64::new(0.331657, 0.262359),
                c64::new(0.244727, 0.638396),
                c64::new(0.581313, 0.528013),
                c64::new(0.946119, 0.256675),
                c64::new(0.555156, 0.078901),
                c64::new(0.524032, 0.703333),
                c64::new(0.993889, 0.129179),
                c64::new(0.237493, 0.327509),
                c64::new(0.055600, 0.847578),
                c64::new(0.088908, 0.274087),
                c64::new(0.149218, 0.359951),
                c64::new(0.701262, 0.528096),
                c64::new(0.718525, 0.572322),
                c64::new(0.990188, 0.336295),
                c64::new(0.535049, 0.076191),
                c64::new(0.186912, 0.608159),
                c64::new(0.977768, 0.997542),
                c64::new(0.872098, 0.900481),
                c64::new(0.450666, 0.425147),
                c64::new(0.621391, 0.445795),
                c64::new(0.613905, 0.690011),
                c64::new(0.163519, 0.248689),
                c64::new(0.205288, 0.506597),
                c64::new(0.744933, 0.297656),
                c64::new(0.138310, 0.293064),
                c64::new(0.874282, 0.170779),
                c64::new(0.740816, 0.825517),
                c64::new(0.045324, 0.683082),
                c64::new(0.339080, 0.033500),
                c64::new(0.077581, 0.066504),
                c64::new(0.220183, 0.032719),
                c64::new(0.163831, 0.811177),
                c64::new(0.524874, 0.049641),
                c64::new(0.036958, 0.062986),
                c64::new(0.568491, 0.884327),
                c64::new(0.914892, 0.844849),
                c64::new(0.656459, 0.432534),
                c64::new(0.981313, 0.996293),
                c64::new(0.156926, 0.438136),
                c64::new(0.518909, 0.711744),
                c64::new(0.812142, 0.201055),
                c64::new(0.385009, 0.749246),
                c64::new(0.834279, 0.256328),
                c64::new(0.378963, 0.761068),
                c64::new(0.037043, 0.091670),
                c64::new(0.091428, 0.886068),
                c64::new(0.321316, 0.060991),
                c64::new(0.197774, 0.109478),
                c64::new(0.013886, 0.152236),
                c64::new(0.733118, 0.678769),
                c64::new(0.349739, 0.922512),
                c64::new(0.437076, 0.985890),
                c64::new(0.789399, 0.863788),
                c64::new(0.098481, 0.433876),
                c64::new(0.140539, 0.363167),
                c64::new(0.624408, 0.333908),
                c64::new(0.149382, 0.749237),
                c64::new(0.102141, 0.391356),
                c64::new(0.870619, 0.233448),
                c64::new(0.276125, 0.033807),
                c64::new(0.788045, 0.012740),
                c64::new(0.140971, 0.337287),
                c64::new(0.666063, 0.203683),
                c64::new(0.045933, 0.211809),
                c64::new(0.034878, 0.716547),
                c64::new(0.989171, 0.077564),
                c64::new(0.664159, 0.717327),
                c64::new(0.285891, 0.286707),
                c64::new(0.501182, 0.174097),
                c64::new(0.013578, 0.152712),
                c64::new(0.973179, 0.909657),
                c64::new(0.412261, 0.975651),
                c64::new(0.657502, 0.195360),
                c64::new(0.080765, 0.520424),
                c64::new(0.740716, 0.938439),
                c64::new(0.960119, 0.588442),
                c64::new(0.037048, 0.442824),
                c64::new(0.858198, 0.148410),
                c64::new(0.844994, 0.632135),
                c64::new(0.189434, 0.551046),
                c64::new(0.906756, 0.418544),
                c64::new(0.145521, 0.241500),
                c64::new(0.896957, 0.102364),
                c64::new(0.963494, 0.481932),
                c64::new(0.949465, 0.949494),
                c64::new(0.698473, 0.304047),
                c64::new(0.496960, 0.740081),
                c64::new(0.624386, 0.863860),
                c64::new(0.186898, 0.679316),
                c64::new(0.883993, 0.641905),
                c64::new(0.843153, 0.417276),
                c64::new(0.011741, 0.892577),
                c64::new(0.911255, 0.720138),
                c64::new(0.924570, 0.168352),
                c64::new(0.742747, 0.637457),
                c64::new(0.499916, 0.954243),
                c64::new(0.895616, 0.098343),
                c64::new(0.204341, 0.314443),
                c64::new(0.649155, 0.065885),
            ],
            [
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.279330, 0.200421),
                c64::new(0.656860, 0.846191),
                c64::new(0.078371, 0.908207),
                c64::new(0.118789, 0.598040),
                c64::new(0.886693, 0.984756),
                c64::new(0.289293, 0.632213),
                c64::new(0.326069, 0.667493),
                c64::new(0.243075, 0.405723),
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                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.000000, 0.000000),
                c64::new(0.329299, 0.333973),
                c64::new(0.626537, 0.651087),
            ],
        ];
        let h = Mat::from_fn(n, n, |i, j| h[i][j]);

        let mut q = Mat::from_fn(n, n, |i, j| {
            if i == j {
                c64::faer_one()
            } else {
                c64::faer_zero()
            }
        });
        let mut w = Mat::zeros(n, 1);
        let mut t = h.clone();
        let params = EvdParams {
            recommended_shift_count: None,
            recommended_deflation_window: None,
            blocking_threshold: Some(15),
            nibble_threshold: Some(14),
        };
        let (_, n_aed, n_sweep) = multishift_qr(
            true,
            t.as_mut(),
            Some(q.as_mut()),
            w.as_mut(),
            0,
            n,
            f64::EPSILON,
            f64::MIN_POSITIVE,
            Parallelism::None,
            make_stack!(multishift_qr_req::<c64>(
                n,
                n,
                true,
                true,
                Parallelism::None,
                params,
            )),
            params,
        );
        // asserts to ensure that we don't mess up the shift computation and slow down convergence
        assert!(n_aed <= 50);
        assert!(n_sweep <= 25);

        for j in 0..n {
            for i in j + 1..n {
                t.write(i, j, c64::faer_zero());
            }
        }

        let h_reconstructed = &q * &t * q.adjoint();

        for i in 0..n {
            for j in 0..n {
                assert_approx_eq!(h_reconstructed.read(i, j), h.read(i, j));
            }
        }
    }

    #[test]
    fn test_gh_84_cplx_20x20() {
        let a = mat![
            [
                0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                -1.0,
                0.0,
                1.414213562373095,
                0.0,
                4.2913523532056946e-17,
                0.0,
                -1.044653549357173e-17,
                0.0,
                -3.1114138757927777e-18,
                0.0,
                -1.7041915655062705e-17,
                0.0,
                -1.5407439555097887e-33,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0,
                1.4142135623730951,
                0.0,
                1.8708286933869704,
                0.0,
                5.584290580070408e-17,
                0.0,
                4.247123568135012e-17,
                0.0,
                3.383773620463798e-17,
                0.0,
                -7.996931848726948e-17,
                0.0,
                2.137243672031505e-16,
                -2.1697413347872967e-16,
                2.1669070851885488e-16,
                8.793731080691344e-17,
                -6.366867306877532e-17,
                -2.166907085188549e-16,
                -1.5209416587779695e-16,
            ],
            [
                0.0,
                0.0,
                1.8708286933869707,
                0.0,
                0.8864052604279185,
                0.0,
                -5.4482028175548746e-18,
                0.0,
                -1.6227019814468362e-18,
                0.0,
                -8.887904793467559e-18,
                0.0,
                -1.5407439555097887e-33,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.8864052604279183,
                0.0,
                -1.780121110720045,
                0.0,
                2.4905953568839346e-17,
                0.0,
                1.9539297406873977e-17,
                0.0,
                -4.520412950660578e-17,
                0.0,
                7.136040597386633e-17,
                -3.3287408030765184e-17,
                3.9021417760079386e-17,
                2.050611271670174e-18,
                8.594289573868776e-17,
                -3.9021417760079386e-17,
                -2.225425376396652e-17,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                -1.7801211107200454,
                0.0,
                -0.27971669721424774,
                0.0,
                1.3907420351599062e-17,
                0.0,
                7.617407843277181e-17,
                0.0,
                6.162975822039155e-33,
                2.413689799097554e-35,
                0.0,
                7.173100169147405e-36,
                2.1640125834190973e-35,
                1.746648713648135e-36,
                -7.173100169147405e-36,
                -2.3028287065724625e-36,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                -0.27971669721424774,
                0.0,
                2.08179724791363,
                0.0,
                3.128025644367276e-17,
                0.0,
                1.4857837468143015e-17,
                3.0814879110195774e-33,
                -4.353679358901167e-17,
                1.575970043415495e-17,
                3.371331087010223e-17,
                -3.158189568582724e-17,
                3.363028106986343e-17,
                -3.371331087010223e-17,
                1.0436797917561959e-16,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                2.08179724791363,
                0.0,
                -0.591353947688746,
                0.0,
                -1.6942472264780397e-16,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                -0.5913539476887459,
                0.0,
                1.382991720854028,
                0.0,
                2.9339388813743063e-16,
                0.0,
                -2.5386000785908e-17,
                4.58003519909708e-17,
                -3.248053804275205e-17,
                -1.7259032483343233e-17,
                -6.634605605063898e-17,
                3.248053804275205e-17,
                6.644449342304512e-17,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                1.3829917208540286,
                0.0,
                0.5110217026036908,
                0.0,
                4.2381199337456166e-17,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.5110217026036906,
                0.0,
                -1.2687886196697642,
                0.0,
                -1.9866208356455852e-17,
                6.074934010535109e-17,
                -7.39350135280486e-17,
                1.5423322066319723e-17,
                -9.775009908321562e-17,
                7.39350135280486e-17,
                2.485877967834109e-16,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                -1.2687886196697646,
                0.0,
                -1.1161328626787337e-16,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                -4.319252649148509e-17,
                0.0,
                2.095595710655279e-17,
                1.2259068349524595e-18,
                1.2354010708239072e-17,
                -6.000254847480346e-18,
                -7.043660603033051e-18,
                -1.2354010708239072e-17,
                2.8117846829535463e-17,
            ],
            [
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                -3.3963911681598194e-17,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
                0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
            [
                0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                0.0, 0.0, 0.0, 0.0,
            ],
        ];
        let mut a = zipped!(a.as_ref()).map(|unzipped!(x)| c64::new(x.read(), 0.0));

        let mut w = Mat::zeros(20, 1);
        lahqr(
            false,
            a.as_mut(),
            None,
            w.as_mut(),
            0,
            20,
            f64::EPSILON,
            f64::MIN_POSITIVE,
        );
    }
}