faer/linalg/qr/no_pivoting/

factor.rs

use crate::assert;
use crate::internal_prelude::*;
use linalg::householder::{self};
/// information about the resulting $QR$ factorization.
#[derive(Copy, Clone, Debug)]
pub struct QrInfo {
	/// estimated rank of the matrix.
	pub rank: usize,
}

fn qr_in_place_unblocked<T: ComplexField>(
	A: MatMut<'_, T>,
	H: RowMut<'_, T>,
	row_start: usize,
	col_start: usize,
) -> usize {
	let mut A = A;
	let mut H = H;
	let (m, n) = A.shape();
	let size = H.ncols();
	let mut col = col_start;
	let mut row = row_start;

	while row < Ord::min(size, m) && col < n {
		let (mut A00, A01, A10, A11) =
			A.rb_mut().split_at_mut(row + 1, col + 1);

		let (mut A10l, A10r) = A10.split_at_col_mut(col);
		let mut A10r = A10r.col_mut(0);

		let A01 = A01.row_mut(row);
		let A00 = &mut A00[(row, col)];

		let (info, v) = if row == col {
			let info =
				householder::make_householder_in_place(A00, A10r.rb_mut());
			(info, A10r.rb())
		} else {
			let info = householder::make_householder_out_of_place(
				A00,
				A10l.rb_mut().col_mut(row),
				A10r.rb(),
			);
			let nrows = A10r.nrows();
			A10r.rb_mut()
				.get_mut(..Ord::min(nrows, col - row))
				.fill(zero());
			(info, A10l.rb().col(row))
		};

		let tau_inv = &info.tau.recip();
		H[row] = info.tau.to_cplx();

		if *tau_inv < min_positive() {
			if info.norm > zero() {
				row += 1;
			}
		} else if info.norm > zero() {
			for (head, tail) in
				core::iter::zip(A01.iter_mut(), A11.col_iter_mut())
			{
				let dot = &*head
					+ linalg::matmul::dot::inner_prod(
						v.transpose(),
						Conj::Yes,
						tail.rb(),
						Conj::No,
					);
				let k = &-dot.mul_real(tau_inv);
				*head += k;
				z!(tail, v).for_each(|uz!(dst, src)| {
					*dst += k * src;
				});
			}
			row += 1;
		}
		col += 1;
	}
	row
}

/// the recommended block size to use for a $QR$ decomposition of a matrix with
/// the given shape.
#[inline]
pub fn recommended_block_size<T: ComplexField>(
	nrows: usize,
	ncols: usize,
) -> usize {
	let prod = nrows * ncols;
	let size = nrows.min(ncols);
	(if prod > 8192 * 8192 {
		256
	} else if prod > 2048 * 2048 {
		128
	} else if prod > 1024 * 1024 {
		64
	} else if prod > 512 * 512 {
		48
	} else if prod > 128 * 128 {
		32
	} else if prod > 32 * 32 {
		8
	} else if prod > 16 * 16 {
		4
	} else {
		1
	})
	.min(size)
	.max(1)
}
/// $QR$ factorization tuning parameters.
#[derive(Debug, Copy, Clone)]
pub struct QrParams {
	/// threshold at which blocking algorithms should be disabled
	pub blocking_threshold: usize,
	#[doc(hidden)]
	pub non_exhaustive: NonExhaustive,
}
impl<T: ComplexField> Auto<T> for QrParams {
	#[inline]
	fn auto() -> Self {
		Self {
			blocking_threshold: 48 * 48,
			non_exhaustive: NonExhaustive(()),
		}
	}
}
fn qr_in_place_blocked<T: ComplexField>(
	A: MatMut<'_, T>,
	H: MatMut<'_, T>,
	row_start: usize,
	col_start: usize,
	mut par: Par,
	stack: &mut MemStack,
	params: Spec<QrParams, T>,
) -> usize {
	let params = params.config;
	let (m, n) = A.shape();
	let size = Ord::min(m, n);
	let block_size = H.nrows();
	assert!(block_size > 0);
	if block_size == 1 {
		return qr_in_place_unblocked(A, H.row_mut(0), row_start, col_start);
	}
	let sub_block_size = if m * n < params.blocking_threshold {
		1
	} else {
		block_size / 2
	};
	let mut A = A;
	let mut H = H;
	let mut col = col_start;
	let mut row = row_start;
	while row < size && col < n {
		let block_size = Ord::min(block_size, Ord::min(size - row, n - col));
		let sub_block_size = Ord::min(block_size, sub_block_size);
		let mut A = A.rb_mut();
		let mut H = H.rb_mut();
		let start = row;
		let mut offset = 0;
		while offset < block_size && col < n {
			let block_size = Ord::min(n - col, block_size - offset);
			let sub_block_size = Ord::min(block_size, sub_block_size);
			let new_row = qr_in_place_blocked(
				A.rb_mut().subcols_mut(0, col + block_size),
				H.rb_mut().subrows_mut(offset, sub_block_size),
				row,
				col,
				par.reborrow(),
				stack,
				params.into(),
			);
			let local = new_row - row;
			if local > 0 {
				let mut k = 0;
				while k < local {
					let sub_block_size = Ord::min(sub_block_size, local - k);
					if k > 0 {
						let mut H = H
							.rb_mut()
							.get_mut(offset.., ..)
							.subcols_mut(row + k, sub_block_size);
						let (H0, H1) = H.rb_mut().split_at_row_mut(k);
						let H0 = H0.rb().subrows(0, sub_block_size);
						let H1 = H1.subrows_mut(0, sub_block_size);
						{ H1 }.copy_from_triangular_upper(H0);
					}
					k += sub_block_size;
				}
				householder::upgrade_householder_factor(
					H.rb_mut()
						.get_mut(offset..offset + local, row..row + local),
					A.rb().get(row.., row..row + local),
					local,
					sub_block_size,
					par.reborrow(),
				);
				if offset > 0 {
					use crate::linalg::matmul::triangular::{
						self, BlockStructure,
					};
					let mut H = H.rb_mut().get_mut(
						..offset + local,
						start..start + offset + local,
					);
					let A = A.rb().get(start.., start..start + offset + local);
					let (A0, A1) = A.split_at_row(offset + local);
					triangular::matmul(
						H.rb_mut(),
						BlockStructure::UnitTriangularUpper,
						Accum::Replace,
						A0.adjoint(),
						BlockStructure::UnitTriangularUpper,
						A0,
						BlockStructure::UnitTriangularLower,
						one(),
						par.reborrow(),
					);
					triangular::matmul(
						H.rb_mut(),
						BlockStructure::UnitTriangularUpper,
						Accum::Add,
						A1.adjoint(),
						BlockStructure::Rectangular,
						A1,
						BlockStructure::Rectangular,
						one(),
						par.reborrow(),
					);
				}
			}
			let (Q0, A1) = A
				.rb_mut()
				.get_mut(row.., ..)
				.split_at_col_mut(col + block_size);
			let Q0 = Q0.rb().get(.., row..row + local);
			let H = H.rb().get(offset..offset + local, row..row + local);
			if A1.ncols() > 0 {
				householder::apply_block_householder_transpose_on_the_left_in_place_with_conj(Q0.rb(), H.rb(), Conj::Yes, A1, par.reborrow(), stack);
			}
			offset += local;
			row += local;
			col += block_size;
		}
	}
	row
}
/// Factors a matrix whose columns are not the contiguous direction by copying
/// it into a column-major buffer, factoring that, and copying the factor back.
///
/// Nothing in the panel is layout-aware: `make_householder_in_place` reads the
/// pivot column and then scales it, and a row-major matrix walks that column
/// with a stride of `ncols`, so every element it touches costs its own cache
/// line; the trailing block-apply is a `matmul` and does not care. LAPACK's
/// row-major interface pays the same copy internally. The arithmetic and its
/// order are the copy's, so the result is the same factorization, bit for bit;
/// what the copy costs and what it buys is on the worklog for this row.
///
/// The buffer is allocated here rather than taken from `stack` because
/// `qr_in_place_scratch` is given only the shape and cannot know whether the
/// caller will pass a row-major matrix; reserving `m * n` for every call would
/// tax the column-major majority. This is the same trade the fork's two row
/// splits make for their partial buffers.
fn qr_in_place_adapted<T: ComplexField>(
	A: MatMut<'_, T>,
	Q_coeff: MatMut<'_, T>,
	mut par: Par,
	stack: &mut MemStack,
	params: Spec<QrParams, T>,
) -> QrInfo {
	let (m, n) = A.shape();
	let mut work = alloc::vec![zero::<T>(); m * n];
	let mut work = MatMut::from_column_major_slice_mut(&mut work, m, n);
	linalg::copy_across_layouts(work.rb_mut(), A.rb(), par.reborrow());
	let info = qr_in_place(work.rb_mut(), Q_coeff, par.reborrow(), stack, params);
	linalg::copy_across_layouts(A, work.rb(), par);
	info
}
#[track_caller]
pub fn qr_in_place<T: ComplexField>(
	A: MatMut<'_, T>,
	Q_coeff: MatMut<'_, T>,
	par: Par,
	stack: &mut MemStack,
	params: Spec<QrParams, T>,
) -> QrInfo {
	let block_size = Q_coeff.nrows();
	assert!(all(
		block_size > 0,
		Q_coeff.ncols() == Ord::min(A.nrows(), A.ncols()),
	));
	// Cheap only when the panel actually walks this matrix column by column for
	// its whole length: the panel touches about `size * (m - size / 2)` elements
	// that way, one cache line each, against `m * n` elements the copy walks
	// contiguous twice. A tall or square matrix makes those within a factor of
	// four of each other, and the stride is worth more than that; a wide one has
	// `size == m` over `n` columns, so `m x 4096` would copy eight megabytes to
	// save a few per cent of the panel and measurably loses (256x4096 f64 at one
	// thread: 16.9 -> 20.7 ms). A single row is the limit case of the same thing:
	// `qr_in_place` eliminates at most one column of it.
	if A.row_stride().unsigned_abs() != 1
		&& A.ncols() > 1
		&& A.nrows() >= A.ncols()
		&& linalg::row_major_copy_pays(A.nrows(), A.ncols())
	{
		return qr_in_place_adapted(A, Q_coeff, par, stack, params);
	}
	let mut Q_coeff = Q_coeff;
	let rank =
		qr_in_place_blocked(A, Q_coeff.rb_mut(), 0, 0, par, stack, params);
	Q_coeff.rb_mut().get_mut(.., rank..).fill(zero());
	let mut col = rank / block_size * block_size;
	let n = Q_coeff.ncols();
	while col < n {
		let block_size = Ord::min(block_size, n - col);
		let start = Ord::max(rank, col);
		Q_coeff
			.rb_mut()
			.get_mut(start - col.., start..col + block_size)
			.diagonal_mut()
			.fill(infinity());
		col += block_size;
	}
	QrInfo { rank }
}
/// computes the layout of required workspace for performing a qr
/// decomposition with no pivoting
#[inline]
pub fn qr_in_place_scratch<T: ComplexField>(
	nrows: usize,
	ncols: usize,
	block_size: usize,
	par: ParShape,
	params: Spec<QrParams, T>,
) -> StackReq {
	let _ = par;
	let _ = nrows;
	let _ = &params;
	temp_mat_scratch::<T>(block_size, ncols)
}
#[cfg(test)]
mod tests {
	use super::*;
	use crate::linalg::solvers::DenseSolveCore;
	use crate::stats::prelude::*;
	use crate::utils::approx::*;
	use crate::{Mat, Row, assert, c64};
	use dyn_stack::MemBuffer;

	#[test]
	fn test_qr() {
		let rng = &mut StdRng::seed_from_u64(0);
		for rank in [1, 2, 3, 4, 5, 100, usize::MAX] {
			for mut par in [Par::Seq, Par::rayon(8)] {
			let par_shape = par.shape();
				for n in [2, 4, 8, 16, 24, 32, 127, 128, 257] {
					let rank = Ord::min(n, rank);
					let approx_eq = CwiseMat(ApproxEq {
						abs_tol: 1e-10,
						rel_tol: 1e-10,
					});
					let A0 = CwiseMatDistribution {
						nrows: n,
						ncols: rank,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A1 = CwiseMatDistribution {
						nrows: rank,
						ncols: n,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A = &A0 * &A1;
					let A = A.as_ref();
					let mut QR = A.cloned();
					let mut H = Row::zeros(n);
					let mut params: QrParams = auto!(c64);
					params.blocking_threshold = usize::MAX;
					let params = params.into();
					let computed_rank = qr_in_place(
						QR.as_mut(),
						H.as_mat_mut(),
						Par::Seq,
						MemStack::new(&mut MemBuffer::new(
							qr_in_place_scratch::<c64>(
								n,
								n,
								1,
								ParShape::Seq,
								params,
							),
						)),
						params,
					);
					assert!(computed_rank.rank >= rank);
					let mut Q = Mat::<c64>::zeros(n, n);
					let mut R = QR.as_ref().cloned();
					for j in 0..n {
						Q[(j, j)] = c64::ONE;
					}
					householder::apply_block_householder_sequence_on_the_left_in_place_with_conj(
						QR.as_ref(),
						H.as_mat(),
						Conj::No,
						Q.as_mut(),
						Par::Seq,
						MemStack::new(&mut MemBuffer::new(
							householder::apply_block_householder_sequence_transpose_on_the_left_in_place_scratch::<c64>(n, 1, n),
						)),
					);
					for j in 0..n {
						for i in j + 1..n {
							R[(i, j)] = c64::ZERO;
						}
					}
					assert!(Q * R ~ A);
				}
				for n in [2, 3, 4, 8, 16, 24, 32, 128, 255, 256, 257, 512] {
					let bs = 15;
					let rank = Ord::min(n, rank);
					let approx_eq = CwiseMat(ApproxEq {
						abs_tol: 1e-10,
						rel_tol: 1e-10,
					});
					let A0 = CwiseMatDistribution {
						nrows: n,
						ncols: rank,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A1 = CwiseMatDistribution {
						nrows: rank,
						ncols: n,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A = &A0 * &A1;
					let A = A.as_ref();
					let mut QR = A.cloned();
					let mut H = Mat::zeros(bs, n);
					let computed_rank = qr_in_place(
						QR.as_mut(),
						H.as_mut(),
						par.reborrow(),
						MemStack::new(&mut MemBuffer::new(
							qr_in_place_scratch::<c64>(
								n,
								n,
								bs,
								par_shape,
								default(),
							),
						)),
						default(),
					);
					assert!(computed_rank.rank >= rank);
					let mut Q = Mat::<c64>::zeros(n, n);
					let mut R = QR.as_ref().cloned();
					for j in 0..n {
						Q[(j, j)] = c64::ONE;
					}
					householder::apply_block_householder_sequence_on_the_left_in_place_with_conj(
						QR.as_ref(),
						H.as_ref(),
						Conj::No,
						Q.as_mut(),
						Par::Seq,
						MemStack::new(&mut MemBuffer::new(
							householder::apply_block_householder_sequence_on_the_left_in_place_scratch::<c64>(n, bs, n),
						)),
					);
					for j in 0..n {
						for i in j + 1..n {
							R[(i, j)] = c64::ZERO;
						}
					}
					assert!(Q * R ~ A);
				}
				let n = 20;
				for m in [2, 3, 4, 8, 16, 24, 32, 128, 255, 256, 257, 512] {
					let size = Ord::min(m, n);
					let bs = 15;
					let rank = Ord::min(size, rank);
					let approx_eq = CwiseMat(ApproxEq {
						abs_tol: 1e-10,
						rel_tol: 1e-10,
					});
					let A0 = CwiseMatDistribution {
						nrows: m,
						ncols: rank,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A1 = CwiseMatDistribution {
						nrows: rank,
						ncols: n,
						dist: ComplexDistribution::new(
							StandardNormal,
							StandardNormal,
						),
					}
					.rand::<Mat<c64>>(rng);
					let A = &A0 * &A1;
					let A = A.as_ref();
					let mut QR = A.cloned();
					let mut H = Mat::zeros(bs, size);
					let computed_rank = qr_in_place(
						QR.as_mut(),
						H.as_mut(),
						par.reborrow(),
						MemStack::new(&mut MemBuffer::new(
							qr_in_place_scratch::<c64>(
								m,
								n,
								bs,
								par_shape,
								default(),
							),
						)),
						default(),
					);
					assert!(computed_rank.rank >= rank);
					let mut Q = Mat::<c64, _, _>::zeros(m, m);
					let mut R = QR.as_ref().cloned();
					for j in 0..m {
						Q[(j, j)] = c64::ONE;
					}
					householder::apply_block_householder_sequence_on_the_left_in_place_with_conj(
						QR.as_ref().subcols(0, size),
						H.as_ref(),
						Conj::No,
						Q.as_mut(),
						Par::Seq,
						MemStack::new(&mut MemBuffer::new(
							householder::apply_block_householder_sequence_on_the_left_in_place_scratch::<c64>(m, bs, m),
						)),
					);
					for j in 0..n {
						for i in j + 1..m {
							R[(i, j)] = c64::ZERO;
						}
					}
					assert!(Q * R ~ A);
				}
			}
		}
	}
	/// A matrix whose columns are not the contiguous direction is factored through
	/// a column-major copy, so the factor, the `tau` matrix and the rank it
	/// leaves behind have to be the column-major ones for the same matrix.
	/// Anything less -- a copy that lands an element in the wrong place, a shape
	/// the copy's tiling clips, a partition that overlaps -- is a wrong
	/// factorization, and this is what says so.
	#[test]
	fn test_row_major_factor_is_the_column_major_one() {
		/// A well-conditioned fixture with no structure for the two orders to
		/// exploit: a `(i, j)`-hashed value in `[-5, 5)`. The structured
		/// alternative (`(i * 31 + j * 17) % 101`) makes the leading block
		/// nearly singular, which amplifies a rounding difference between the
		/// two orders into the `1e-8` range and would drown the comparison this
		/// test is not making.
		fn val(i: usize, j: usize) -> f64 {
			let mut h = (i as u64).wrapping_mul(0x9E37_79B9_7F4A_7C15)
				^ (j as u64).wrapping_mul(0xBF58_476D_1CE4_E5B9);
			h ^= h >> 31;
			h = h.wrapping_mul(0x94D0_49BB_1331_11EB);
			h ^= h >> 29;
			((h >> 11) as f64 / (1u64 << 53) as f64 - 0.5) * 10.0
		}
		trait Entry: ComplexField + core::fmt::Debug {
			/// The matrix' `(i, j)` entry, as the flat buffer at storage row `i`
			/// and column `j` holds it.
			fn at(i: usize, j: usize, zero_col: bool) -> Self;
			fn bits(&self) -> (u64, u64);
		}
		impl Entry for f64 {
			fn at(i: usize, j: usize, zero_col: bool) -> Self {
				if zero_col && j == 0 {
					0.0
				} else {
					val(i, j)
				}
			}
			fn bits(&self) -> (u64, u64) {
				(self.to_bits(), 0)
			}
		}
		impl Entry for c64 {
			fn at(i: usize, j: usize, zero_col: bool) -> Self {
				if zero_col && j == 0 {
					c64::new(0.0, 0.0)
				} else {
					c64::new(val(i, j), val(i + 7, j + 3))
				}
			}
			fn bits(&self) -> (u64, u64) {
				(self.re.to_bits(), self.im.to_bits())
			}
		}
		fn check<T: Entry>(
			m: usize,
			n: usize,
			zero_col: bool,
			// `true` factors the row-major side through a view whose rows are
			// walked backwards, so its strides are both negative. The
			// column-major side stays a plain view of the same matrix, because
			// a negative row stride is *not* the layout the adaptation is for.
			flipped: bool,
			// Whether the adaptation applies to this shape. Where it does, the
			// two orders run the same arithmetic on the same values, so they
			// must agree bit for bit; where it does not, they are two different
			// kernels over the same values and only the rounding may differ.
			exact: bool,
			mut par: Par,
		) {
			// An empty matrix has no row to start a backwards view from.
			let flipped = flipped && m > 0 && n > 0;
			let size = Ord::min(m, n);
			let block_size = recommended_block_size::<T>(m, n);
			let qr_scratch =
				qr_in_place_scratch::<T>(m, n, block_size, par.shape(), default());
			// The same matrix, flat in each order. The row-major side is read
			// back through the same view, so it is compared at the storage row
			// the view puts there.
			let row_of = |i: usize| if flipped { m - 1 - i } else { i };
			let mut cm = Vec::with_capacity(m * n);
			for j in 0..n {
				for i in 0..m {
					cm.push(T::at(row_of(i), j, zero_col));
				}
			}
			let mut rm = Vec::with_capacity(m * n);
			for i in 0..m {
				for j in 0..n {
					rm.push(T::at(i, j, zero_col));
				}
			}
			let mut h_cm = vec![T::zero(); block_size * size];
			let mut h_rm = vec![T::zero(); block_size * size];
			let mut run = |a: &mut [T],
			           h: &mut [T],
			           base: usize,
			           (rs, cs): (isize, isize)|
			 -> QrInfo {
				unsafe {
					qr_in_place(
						MatMut::from_raw_parts_mut(
							a.as_mut_ptr().add(base),
							m,
							n,
							rs,
							cs,
						),
						MatMut::from_raw_parts_mut(
							h.as_mut_ptr(),
							block_size,
							size,
							1,
							block_size as isize,
						),
						par.reborrow(),
						MemStack::new(&mut MemBuffer::new(qr_scratch)),
						default(),
					)
				}
			};
			let rm_base = if flipped { (m - 1) * n } else { 0 };
			let rm_strides = if flipped {
				(-(n as isize), 1)
			} else {
				(n as isize, 1)
			};
			let cm_info = run(&mut cm, &mut h_cm, 0, (1, m as isize));
			let rm_info = run(&mut rm, &mut h_rm, rm_base, rm_strides);
			if cm_info.rank != rm_info.rank {
				panic!(
					"{m}x{n} flipped={flipped}: rank {} against {}",
					cm_info.rank, rm_info.rank,
				);
			}
			let differs = |a: &T, b: &T| {
				if exact {
					a.bits() != b.bits()
				} else {
					abs1(&(a - b).clone()) > from_f64::<T::Real>(1e-10)
				}
			};
			for j in 0..n {
				for i in 0..m {
					let k = row_of(i) * n + j;
					if differs(&cm[i + j * m], &rm[k]) {
						panic!(
							"{m}x{n} flipped={flipped}: entry ({i}, {j}) \
							 differs between the two orders"
						);
					}
				}
			}
			for k in 0..block_size * size {
				if differs(&h_cm[k], &h_rm[k]) {
					panic!(
						"{m}x{n} flipped={flipped}: tau entry {k} differs \
						 between the two orders"
					);
				}
			}
		}
		// Shapes the adaptation applies to: a tall or square matrix with more
		// than one column, and at least `64 * 64` elements so the copy floor
		// lets it through. These must match the column-major factor exactly.
		for (m, n) in [(64, 64), (128, 128), (256, 64), (300, 64), (512, 64)] {
			for mut par in [Par::Seq, Par::rayon(8)] {
				check::<f64>(m, n, false, false, true, par.reborrow());
				check::<c64>(m, n, false, false, true, par);
			}
			check::<f64>(m, n, false, true, true, Par::Seq);
			check::<c64>(m, n, false, true, true, Par::rayon(4));
		}
		// An exactly zero leading column is the one place the unblocked core
		// stores a reflector's essential part in a different column than the one
		// it is eliminating, so the copy has to carry that column too.
		for (m, n) in [(64, 64), (128, 128), (256, 64)] {
			check::<f64>(m, n, true, false, true, Par::Seq);
			check::<c64>(m, n, true, false, true, Par::Seq);
		}
		// Shapes it deliberately leaves alone -- empty, one row or column, wide
		// (the panel is `m` reflectors over `n` columns and the copy would cost
		// more than it saved), and small, where the copy and the thread-region
		// start-up cost more than the strided panel they replace. Still the same
		// matrix, so still the same factor up to rounding.
		for (m, n) in [
			(0, 0),
			(3, 0),
			(0, 3),
			(1, 7),
			(7, 1),
			(2, 3),
			(2, 2),
			(3, 2),
			(3, 3),
			(4, 4),
			(17, 15),
			(16, 16),
			(32, 32),
			(300, 17),
			(512, 17),
			(8, 64),
			(17, 300),
			(64, 256),
		] {
			check::<f64>(m, n, false, false, false, Par::Seq);
			check::<c64>(m, n, false, false, false, Par::rayon(4));
			check::<f64>(m, n, false, true, false, Par::Seq);
		}
	}
	/// Factor `A`, rebuild `Q` and `R`, and return `||A - QR|| / ||A||`.
	fn qr_reconstruction_error(A: MatRef<'_, f64>, block_size: usize) -> f64 {
		let (m, n) = (A.nrows(), A.ncols());
		let k = Ord::min(m, n);
		let mut QR = A.cloned();
		let mut H = Mat::zeros(block_size, k);
		qr_in_place(
			QR.as_mut(),
			H.as_mut(),
			Par::Seq,
			MemStack::new(&mut MemBuffer::new(qr_in_place_scratch::<f64>(
				m,
				n,
				block_size,
				ParShape::Seq,
				default(),
			))),
			default(),
		);

		let mut Q = Mat::<f64>::zeros(m, m);
		let mut R = QR.as_ref().cloned();
		for j in 0..m {
			Q[(j, j)] = 1.0;
		}
		for j in 0..k {
			for i in j + 1..m {
				R[(i, j)] = 0.0;
			}
		}
		householder::apply_block_householder_sequence_on_the_left_in_place_with_conj(
			QR.as_ref().subcols(0, k),
			H.as_ref(),
			Conj::No,
			Q.as_mut(),
			Par::Seq,
			MemStack::new(&mut MemBuffer::new(
				householder::apply_block_householder_sequence_on_the_left_in_place_scratch::<f64>(
					m, block_size, m,
				),
			)),
		);
		(&A - &Q * &R).norm_l2() / A.norm_l2()
	}

	/// `A = [a, a + 1e-10 e_0, a + 1e-6 e_1]`, `a = (1, ..., 1)`, tall `m x 3`.
	#[test]
	fn test_near_dependent_columns_are_not_skipped() {
		for m in [100, 1000, 8192] {
			let A = Mat::from_fn(m, 3, |i, j| match j {
				0 => 1.0,
				1 => 1.0 + if i == 0 { 1e-10 } else { 0.0 },
				2 => 1.0 + if i == 1 { 1e-6 } else { 0.0 },
				_ => unreachable!(),
			});
			let relative_error = qr_reconstruction_error(
				A.as_ref(),
				recommended_block_size::<f64>(m, 3),
			);
			assert!(
				relative_error <= 1e-13,
				"m={m}: relative reconstruction error {relative_error:e}"
			);
		}
	}

	/// The same fixture transposed, so the nearly dependent columns are the
	/// rows: wide `3 x n`. This is the shape that a caller sees as "wide", and
	/// it exercises the blocked path's recursion into the unblocked core.
	#[test]
	fn test_near_dependent_rows_are_not_skipped() {
		for n in [100, 1000, 8192] {
			// W = A^T, i.e. W[(i, j)] = A[(j, i)] for the fixture above.
			let W = Mat::from_fn(3, n, |i, j| match i {
				0 => 1.0,
				1 => 1.0 + if j == 0 { 1e-10 } else { 0.0 },
				2 => 1.0 + if j == 1 { 1e-6 } else { 0.0 },
				_ => unreachable!(),
			});
			let relative_error = qr_reconstruction_error(
				W.as_ref(),
				recommended_block_size::<f64>(3, n),
			);
			assert!(
				relative_error <= 1e-13,
				"3x{n}: relative reconstruction error {relative_error:e}"
			);
		}
	}

	#[test]
	fn test_rank_deficient() {
		let i = c64::new(0.0, 1.0);
		let A = mat![
			[
				-42.63301571125646 + 12.332799004481444 * i,
				22.14792539952694 + 34.88980973191775 * i,
				17.08833097340157 - 35.73642756638833 * i,
				16.46359410465419 - 42.92839086211859 * i,
				-51.764730444742824 + 81.65328924776695 * i,
				16.26315772194988 + 101.5902862344418 * i,
				32.77410298966414 - 29.760957827435675 * i,
				54.439294321277174 - 19.079738442315318 * i,
				-13.472034925473565 - 13.943573427112455 * i,
				-37.92130013603088 - 5.7926798723257775 * i,
				11.194976295897504 - 31.928468357395335 * i,
				22.006669976718612 + 3.3819587127690625 * i,
				-6.674280620224554 + 3.387564723853501 * i,
				-27.293486814495225 + 15.784839382140262 * i,
				-70.1443517961362 + 45.51634877173214 * i,
				8.181229285208616 + 11.577617946683347 * i,
				91.36331153456992 + 5.978813469385237 * i,
				9.984370037964844 - 46.821962088062065 * i,
				-7.961599724739131 + 24.482176731342236 * i,
				-2.3944781784861675 + 59.88090585477201 * i,
				-30.59313953515538 + 26.04745441455745 * i,
				-23.2138559332029 + 71.42630235367764 * i,
				22.210550339117574 - 8.41265667708069 * i,
				-11.395416853480953 - 21.12459074171425 * i,
				53.214674012253 - 115.94154857087852 * i,
				-73.57661533181208 - 26.388545645959155 * i,
				-7.090193099053337 + 25.72478190092928 * i,
				69.88086414991596 - 61.87734406117171 * i,
				1.7746488531197873 + 22.338809391772543 * i,
				53.17879024337897 + 3.0434791244725403 * i,
				17.11701506186965 + 8.310347464005673 * i,
				-1.4446985641791343 - 8.302085605445917 * i,
				31.01042486753896 - 7.032023465838279 * i,
				50.529023639015065 - 30.432756878940538 * i,
				31.20832359612237 - 18.42527233678696 * i,
				-11.219592741413372 + 7.966456895330857 * i,
				-29.972014970412957 + 14.566825708553516 * i,
				0.29786829050329544 - 60.43731329910441 * i,
				68.20619173666897 + 61.4177082555907 * i,
				38.18522457770308 + 39.84442487790024 * i,
			],
			[
				12.392666547455747 + 8.598469521105585 * i,
				-4.989137989279129 - 33.00917260871823 * i,
				-36.3778560847675 + 33.965754531979854 * i,
				-14.186892703037756 + 7.8005444248189235 * i,
				40.14774343578466 - 37.073977200615204 * i,
				20.515112381381726 - 39.67019203209179 * i,
				-11.247367898035563 - 24.867001222923307 * i,
				-49.7400862432402 + 41.41428057951588 * i,
				-3.7213275390216154 + 51.57768254492541 * i,
				-6.345081010580628 + 53.304858550561896 * i,
				-11.529781103419339 + 2.902201112665864 * i,
				-19.346649644440838 + 7.461276014085186 * i,
				-28.718472256689964 - 2.8043513041184553 * i,
				-5.691563912437118 + 48.35693831223085 * i,
				-34.299545543675215 + 6.545395352640526 * i,
				7.438830278671389 - 10.739267797900883 * i,
				24.11439247436401 - 29.701199086689808 * i,
				-54.93729078780896 + 17.091416018852897 * i,
				15.68845749746022 - 2.188338155362544 * i,
				60.97624713742826 + 11.795504838770498 * i,
				53.39361626612977 + 11.614849968126038 * i,
				47.112022139988945 - 47.422312498326974 * i,
				-31.731750386335847 - 13.013784865796737 * i,
				-12.242942262135827 - 23.39632386635281 * i,
				-20.99632660387538 - 50.155831920264006 * i,
				32.853343930391404 + 39.33730823584699 * i,
				25.191258444072332 + 23.926919559757664 * i,
				-34.18728556598965 + 52.5998197331729 * i,
				64.80196305869202 - 10.491798137124405 * i,
				-3.7973985569930306 - 34.25408107334158 * i,
				31.90902882064414 - 30.56021961088784 * i,
				-7.840305921418871 + 51.263365311660365 * i,
				5.999217219970496 + 60.657097999788675 * i,
				-56.00257012089648 + 32.307128304063674 * i,
				1.6182839905382513 - 30.670206400422288 * i,
				-29.228061063708484 + 4.618529965360111 * i,
				-1.841921635134554 - 9.941332210308186 * i,
				-28.83749752522963 + 23.48822343787033 * i,
				31.33927120630301 - 71.58546530411547 * i,
				8.181832440934743 + 82.64255143443799 * i,
			],
			[
				-12.799030330740077 + 66.72947765658043 * i,
				-10.231934039979912 + 12.493297096248554 * i,
				19.285109623888076 + 16.56539210325027 * i,
				-35.97113317387823 + 12.77732345394736 * i,
				-42.753078264350506 + 31.109302328632026 * i,
				-92.5918702738831 + 13.20717885481352 * i,
				-5.444744981663884 + 48.16106562407031 * i,
				48.84587551022895 + 26.251595375317834 * i,
				32.933710253207224 + 14.718356097405149 * i,
				9.648128610518484 - 3.9457537091127026 * i,
				19.03552788630428 - 19.94207328846148 * i,
				17.638202950481865 + 18.33026376219135 * i,
				2.5078899478693097 - 45.70267635206393 * i,
				65.38384334644981 - 25.199272871591987 * i,
				-63.467061790245424 + 106.98211601738333 * i,
				-55.62393136607952 - 45.51555918892136 * i,
				19.15778634967591 - 33.028235082995124 * i,
				-16.336567724656685 - 10.44522106845141 * i,
				6.666994028110054 + 32.92591049296345 * i,
				-6.047530207659875 - 27.439036726043298 * i,
				39.52962940543614 - 42.64379506697883 * i,
				-27.573007769719624 - 9.75081496679099 * i,
				-7.684664374667875 + 23.339026825258244 * i,
				22.58119786377813 - 4.454197784068818 * i,
				45.29146872103806 - 103.15857753791218 * i,
				-31.25659917861597 + 13.70164575929067 * i,
				13.660916345380933 - 19.46889545971939 * i,
				-21.96183310377741 - 23.31280166325685 * i,
				4.5653695184317655 + 10.616453681351558 * i,
				-0.25434597828302685 + 47.167740733214515 * i,
				33.82461168808063 - 20.417606401735725 * i,
				21.40415980541069 - 27.931611547777152 * i,
				49.94594590735055 - 49.56309902061 * i,
				27.886633708526702 + 29.70147478184972 * i,
				-18.16905152235941 - 10.046271334081538 * i,
				-13.32135487551704 - 60.68677560840839 * i,
				-15.237503754455295 - 6.588689364973661 * i,
				-15.481784996249857 - 46.99021467938253 * i,
				-16.136673443184375 + 13.902154631685766 * i,
				25.194569063162533 + 34.5621640946647 * i,
			],
			[
				65.48222712345446 - 4.138391495079304 * i,
				8.306462492672111 - 15.140617871481396 * i,
				-28.25587078248205 + 50.32663902684051 * i,
				-42.955896042037864 + 41.46713382497833 * i,
				75.13916830569464 - 64.90394781293571 * i,
				3.4250892035273566 - 55.122759708615945 * i,
				-38.67812775272067 + 21.094268967787286 * i,
				-39.40657764104256 - 3.733222413764196 * i,
				0.9695375254205487 + 3.876176033829374 * i,
				45.749414055844625 + 18.586694744497834 * i,
				11.935303844620492 + 39.477261236067676 * i,
				-55.36026338025768 - 19.755621035097477 * i,
				12.205216270291857 + 18.307651510095745 * i,
				-17.102914926687376 - 5.8964800477324815 * i,
				73.92643576109086 - 26.358635886040645 * i,
				18.62547989733079 + 21.445184756989846 * i,
				-56.327314013975666 - 24.534245355235583 * i,
				1.8624471836432406 + 30.085648450846378 * i,
				15.302535749118924 + 7.674212804643901 * i,
				25.8637235046063 + 0.9361225038480256 * i,
				15.603583412146435 - 1.6612068829250664 * i,
				18.47432197641083 - 63.34015823189532 * i,
				5.816489465444917 - 23.75750418328684 * i,
				12.548375804510385 + 14.79286301101477 * i,
				-66.90207518361544 + 63.910915160309486 * i,
				83.37918838614108 + 40.05271581472445 * i,
				-37.36994071626608 + 6.558073050359511 * i,
				-75.67731562330931 + 70.34100920967161 * i,
				17.123194674315357 + 2.6912987746509813 * i,
				-56.13906743564637 - 9.747808550411886 * i,
				-32.0719278876096 + 0.035416915023617435 * i,
				7.453294267488758 + 31.840840443445664 * i,
				-27.326428632546204 + 16.404546707806446 * i,
				-85.79400760768401 + 51.86531366733967 * i,
				-47.38136649846963 + 2.067569048796164 * i,
				22.65699895650813 - 26.500540841293287 * i,
				35.74882215661379 + 0.4279576089606252 * i,
				-76.88759162135085 + 61.729308548926745 * i,
				-37.44977437439012 - 56.87002751956335 * i,
				-35.08304533233703 - 4.462986912180606 * i,
			],
			[
				31.583724748998776 - 51.249777560521466 * i,
				22.65843906393765 + 17.313137285238554 * i,
				10.382748046036994 - 42.735528014733234 * i,
				20.724646012317805 + 20.900719508178582 * i,
				16.376566756962994 + 53.89212191787085 * i,
				10.066813288632034 + 52.523949880976545 * i,
				43.579591284487336 + 31.252883531054323 * i,
				28.184699679323813 - 82.13512631377462 * i,
				25.440457311415628 + 31.849880177745945 * i,
				-24.151628096605307 - 23.32888438479003 * i,
				-73.6350009853953 - 81.1777776721621 * i,
				28.364800957876895 - 3.6250014074372885 * i,
				38.396317055131306 - 69.546957049437 * i,
				-1.4874584247683202 + 6.860653145887976 * i,
				99.22080146828978 + 17.543861995516416 * i,
				-10.46250932797625 + 22.241347373245674 * i,
				-23.62190472546816 - 70.59350835314189 * i,
				26.34172971210808 - 60.22547306196421 * i,
				37.90974516173529 - 7.695380437440619 * i,
				-35.43744520165066 + 42.51167883371055 * i,
				-18.775026327486785 - 16.956301595807304 * i,
				21.840451013195377 - 7.788376505976714 * i,
				33.09952105067807 + 43.20960076304959 * i,
				19.34930634141674 + 21.351881549098977 * i,
				-143.72559656621607 - 48.39026005448909 * i,
				-84.861170978867 + 2.7837442780206416 * i,
				6.07739013993995 - 52.6344486015943 * i,
				-21.33187479671998 - 35.95542921378066 * i,
				0.08757059072188866 - 38.46847267117025 * i,
				53.77485342607943 + 49.47511865583735 * i,
				-76.81015945487538 - 18.43833465675783 * i,
				-31.308131660283053 - 39.82604917803266 * i,
				-79.42584088986062 - 36.85873541440512 * i,
				39.063233310761746 - 49.75991465666547 * i,
				-23.590897533452015 + 24.69942294209421 * i,
				-55.00539490222253 + 2.186956953950137 * i,
				-26.638771613402398 - 65.28585892133748 * i,
				11.968149530543663 + 15.74074558115147 * i,
				46.924712251106854 + 90.83370071059912 * i,
				76.4747123007834 - 29.289927777420857 * i,
			],
			[
				-25.578557076310645 - 26.766359674243205 * i,
				-31.644839606419964 - 5.617473287649716 * i,
				9.823483731434163 + 43.248069608313386 * i,
				2.672375889300838 - 4.755961211633132 * i,
				-76.9263312976145 - 50.17566661665678 * i,
				-64.95093839016988 - 45.4578250269475 * i,
				-35.560543630021414 - 3.5888284627331757 * i,
				-41.06467435210163 + 91.17075079127582 * i,
				10.85492097775365 + 31.427121748171828 * i,
				41.19076647688331 + 14.935860258180696 * i,
				45.83767858930871 + 65.30292213636108 * i,
				5.475162771136581 + 23.807507454288483 * i,
				29.287510839124877 + 41.159672576346225 * i,
				16.280362143178394 - 33.78524201973386 * i,
				-75.50032717807619 + 13.18905520206016 * i,
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				6.338839751387681 + 3.19971005519956 * i,
				6.922972512095463 + 32.80910049481056 * i,
				-22.01037644984361 + 50.509238396198924 * i,
				-18.958833732673924 + 10.276005903325881 * i,
				-30.891643978002413 + 35.31956533955065 * i,
				9.212918917981337 - 22.269422801901676 * i,
				2.5566393877229254 + 37.595103958950354 * i,
				-67.0685311723968 - 44.019006431020074 * i,
				-24.00785268016788 - 25.36034547703837 * i,
			],
			[
				-13.211416135469996 + 41.951520937157966 * i,
				29.287101673636634 - 9.162628305468985 * i,
				5.108397345052012 - 43.57043028843542 * i,
				-5.719379757654053 - 32.25876860224872 * i,
				35.31355844261981 + 34.85727439321789 * i,
				53.5644965212112 + 54.72812003923711 * i,
				13.892272937320357 - 38.5086132357946 * i,
				22.41596128098773 - 65.01076715828856 * i,
				27.578998281957013 - 55.312355600985825 * i,
				-29.143233640630974 - 2.412811321612992 * i,
				-56.55638117023184 + 15.792954396317537 * i,
				2.927518517416928 - 35.92936463906324 * i,
				-61.745766864030664 - 52.096549246455375 * i,
				6.4154322191591575 + 11.554631665796371 * i,
				-2.209010025447924 - 19.327946979940176 * i,
				-6.926202276578692 + 6.671588917549684 * i,
				-42.953265865881974 + 23.0145825406142 * i,
				-32.23184179691662 - 29.161043334160297 * i,
				-20.35122194222305 - 2.6819436620743815 * i,
				9.591727193768577 + 37.04740251256568 * i,
				-30.72489779095602 + 3.295685212498114 * i,
				-28.46644402098077 + 33.15580231087127 * i,
				23.617282931071145 - 58.10847503252987 * i,
				14.929738152078983 - 2.0126692695147455 * i,
				27.00027265084705 + 37.28939256287889 * i,
				-32.75480210032184 + 78.97106904413369 * i,
				-19.909512246706193 - 14.286797688983107 * i,
				-35.19470430698671 - 44.239228263599614 * i,
				22.729726961380003 + 9.209352310110368 * i,
				54.227754957839736 - 36.836906506258174 * i,
				-41.968744203095994 + 11.906641097530922 * i,
				-21.054350649290292 - 5.35627500744134 * i,
				-21.53831614444494 - 34.79788428767433 * i,
				-29.148945175947887 - 46.29095305917917 * i,
				20.951194059556713 - 17.191483440705753 * i,
				-25.40359822630483 + 33.3207279701501 * i,
				-20.508410861296316 + 35.123345845623 * i,
				31.908848786822723 - 86.29171790195822 * i,
				64.04990903476033 + 7.139766550292478 * i,
				-53.73175092624056 - 68.0266773244694 * i,
			],
			[
				-31.07693246918978 + 22.298947857931793 * i,
				43.503435093929035 - 11.88430785516509 * i,
				13.236065490387233 - 16.162001182145644 * i,
				40.389914091923444 - 24.95630156826025 * i,
				-44.55665999561507 - 1.2890855508178163 * i,
				-5.401783260055595 + 31.247395370004767 * i,
				-7.794962284389607 - 18.939505013934422 * i,
				25.289982351683847 + 40.07997434159475 * i,
				15.731584801680143 + 20.18082952891806 * i,
				36.95965399701333 - 32.71103389245309 * i,
				2.28086742369058 - 12.86079231167895 * i,
				7.331971090941579 + 10.999894779383382 * i,
				22.958205509017617 + 12.948033720947919 * i,
				24.575766746955857 - 11.648277658567586 * i,
				18.12031366910233 + 61.23988669347962 * i,
				-46.929661022049316 + 20.099911012294214 * i,
				-17.96720343708418 - 42.53291307203543 * i,
				-31.996327482300103 + 9.373133016637919 * i,
				27.023860159767242 + 5.113869040537111 * i,
				17.029537291856464 - 82.33720769110722 * i,
				-11.420545522824675 - 67.06582542470075 * i,
				-10.180292317265497 + 20.540783000138703 * i,
				-30.345348986029425 + 56.727488500538804 * i,
				0.8035832364545286 + 20.502414144808128 * i,
				-42.48919428020813 - 23.968097440814873 * i,
				11.253595495328298 - 32.740405358950575 * i,
				26.45500638783109 - 21.9576494910674 * i,
				21.598377134333457 - 0.6405425291244597 * i,
				14.35627243011723 - 39.39628030376109 * i,
				30.920588023011756 - 52.34737650914744 * i,
				34.92881813258915 - 28.572001308435105 * i,
				40.50406759029518 - 6.594263316072144 * i,
				13.064047387769975 - 17.06519931493132 * i,
				21.9944466549845 + 7.573033513537439 * i,
				-0.5986376091863406 - 42.279821798624525 * i,
				-28.346857638440106 + 36.68691508529607 * i,
				-7.398799500691627 + 11.718026533673715 * i,
				24.760291448568754 - 0.7918836640538416 * i,
				-29.95001739590883 - 30.378495238579994 * i,
				90.98472995390726 - 29.10886695441996 * i,
			],
			[
				-40.25963601878951 - 62.64777175089944 * i,
				6.214037286628837 + 25.628326182386644 * i,
				47.23307971549795 - 23.69660799763591 * i,
				88.42929655937778 + 28.782035522328954 * i,
				-17.47282730214239 - 70.97686111921425 * i,
				-24.458949148286678 + 18.503439196107195 * i,
				30.98671950870576 - 4.554370597875746 * i,
				21.71647131852896 + 39.3763636569602 * i,
				0.08430663951475736 + 1.2224286918491316 * i,
				15.778746128625627 - 10.047068987823607 * i,
				-16.86868864966445 - 23.6569545924675 * i,
				1.798134997187674 + 42.55574017812603 * i,
				-24.06535717974631 + 14.493063095263253 * i,
				13.696492214723108 + 64.5146569530664 * i,
				-32.43654496456537 - 67.36607144674804 * i,
				-11.84860501948333 - 48.740846758974385 * i,
				-9.381429296027989 + 4.42157191149488 * i,
				30.73163858464374 - 17.346352565950582 * i,
				-24.72122121765969 - 22.06556757272523 * i,
				51.76602173320938 - 11.176510974756653 * i,
				36.867309752718306 - 37.541800605160816 * i,
				-38.956154647563025 - 29.733316627038207 * i,
				-53.185575536718474 + 6.6771986367810285 * i,
				-41.860363462219894 + 2.9865746808951883 * i,
				-18.710855165599916 + 87.06731802293524 * i,
				71.77694667728251 - 21.126775466922524 * i,
				63.827447837065115 - 15.02670877943666 * i,
				62.58444401756677 + 36.09211161804461 * i,
				49.99727272529665 - 38.86683532936699 * i,
				43.67240875925811 + 37.90266489693023 * i,
				3.491307357293053 + 23.461636436328618 * i,
				15.836838080040557 + 16.04882223161198 * i,
				20.521286972956517 + 8.384317900191432 * i,
				20.820757325911075 + 7.016190024984741 * i,
				42.896549861459526 - 4.974308517481049 * i,
				-65.64027519170588 + 59.888425961409084 * i,
				-46.94690438439747 - 37.13962589387811 * i,
				98.91952407930265 + 12.519509819791068 * i,
				-31.879155388724172 + 9.191463000504049 * i,
				71.62026804691611 + 22.27809178319523 * i,
			],
			[
				-12.04353224674183 - 8.31187930570298 * i,
				-19.358018763263487 - 21.893440110915847 * i,
				8.572414295855168 - 17.050826090022326 * i,
				32.22420645303372 + 2.713588640408304 * i,
				-6.887459524672408 + 1.7126217221695939 * i,
				39.51684899959989 - 10.571546818946272 * i,
				-6.634061729901589 - 18.45511681538524 * i,
				18.27461612242951 - 38.16432925320339 * i,
				-2.161565943400346 - 9.22608177796939 * i,
				-10.756414063240456 - 0.9584084409136722 * i,
				-0.9218215824084837 - 2.4105755073021875 * i,
				19.293858468371525 + 18.189504497999835 * i,
				-33.26498375161637 + 17.18843344694838 * i,
				48.28068231900177 + 2.5251362844498573 * i,
				15.876688933154085 - 16.563617783574337 * i,
				10.071645982925958 - 11.458875522558927 * i,
				-4.373597621973289 - 4.931293504945643 * i,
				30.18989546916572 - 22.66579070674601 * i,
				-11.682224193123801 - 10.207725389472301 * i,
				-1.6463836307005089 - 3.3216391457421537 * i,
				7.8306004389795305 + 19.108676304972462 * i,
				-20.677758981818073 + 9.278389513156107 * i,
				3.124101268135206 + 4.149036275712234 * i,
				-36.335459695662614 + 17.248575047880703 * i,
				-51.937241874384455 + 10.923633349041431 * i,
				-4.628861429076764 + 8.680817703812977 * i,
				21.834641980861676 - 22.835221490431692 * i,
				12.744243220084002 - 26.231317796529027 * i,
				-2.38897215556689 - 38.88707207701086 * i,
				-15.278555991400822 + 9.945012703950283 * i,
				-14.700678875786242 + 11.059241025999093 * i,
				12.340864525292261 - 14.004822334333454 * i,
				-8.044891094671073 - 19.108026457751237 * i,
				34.716113879257115 - 19.730020611071946 * i,
				-21.62478136892574 + 7.993531737025805 * i,
				35.28765006264224 + 27.03943710952273 * i,
				-23.713276920119192 + 26.02563895029239 * i,
				38.70701054111705 - 10.504807061663824 * i,
				-1.466945281101271 + 0.3779290617959692 * i,
				24.139474430708862 - 60.817958570889864 * i,
			],
			[
				51.100228727799376 + 37.22654371536027 * i,
				32.46268087256175 - 32.82422603130783 * i,
				-29.44413268485738 + 15.765319502259104 * i,
				-39.395273433537724 - 25.818110853275662 * i,
				91.1918363550727 - 24.09689836725063 * i,
				83.4549967540554 - 20.34134879703427 * i,
				14.845553062520082 - 17.329688701627873 * i,
				-57.79299873582244 - 61.6190659268681 * i,
				-11.341895639832885 - 29.389275980068188 * i,
				-6.103606129159111 + 2.6732605138042516 * i,
				-30.639674305109082 + 38.30047965862747 * i,
				-32.52405690925754 - 1.147975907681655 * i,
				-60.91464903420922 + 3.3569835263393486 * i,
				0.9465373318364527 - 12.319332957187953 * i,
				74.55664621734081 - 18.56283082736708 * i,
				17.803562763968827 - 23.82546060059542 * i,
				-51.806669208246944 - 30.915651046889167 * i,
				-62.162718932301345 - 32.517877652384755 * i,
				36.98331909299938 - 8.453357560368715 * i,
				25.670062202629836 + 14.042422507005398 * i,
				21.722934057138772 + 12.849989979431058 * i,
				40.15558451876862 + 12.329157892817022 * i,
				-9.139517533292345 - 25.743989625176226 * i,
				-4.468120791789221 - 13.755867478499477 * i,
				-69.01712528118367 + 8.639846913771777 * i,
				10.839156913536948 + 75.26051799617554 * i,
				-37.746419555778104 - 15.850150517077978 * i,
				-68.64741450018217 - 20.527021740042183 * i,
				-15.434262576161444 - 28.137900034870583 * i,
				13.210389026964918 + 4.156278279710207 * i,
				-45.62063377733667 + 31.0877877497326 * i,
				-26.1631431584863 - 12.079306214684529 * i,
				-78.11257536327938 - 2.0945118991163714 * i,
				-48.602415683300734 - 50.252561412350786 * i,
				-36.63139136867553 - 10.753264854693908 * i,
				-4.413610245696162 + 48.86992836326893 * i,
				-9.688690027055344 + 55.27630347193527 * i,
				-49.44780929458748 + 8.589562480583625 * i,
				81.94654000279007 - 97.13615598710763 * i,
				1.9811020394914598 - 53.536238319414494 * i,
			],
			[
				-52.26002756952291 - 12.420745406989852 * i,
				-48.649220026936035 + 7.357476578955215 * i,
				38.787613829805125 + 10.060178851506798 * i,
				66.73096163304089 - 29.035248735396436 * i,
				-61.64298521757553 - 18.69502418563181 * i,
				-29.453498465992812 + 17.273712641503778 * i,
				-23.978289374625334 + 5.426227955442368 * i,
				-22.187325339436626 + 7.616246279667706 * i,
				0.23983491995083472 + 29.039311850170208 * i,
				-23.98539518469839 + 22.208046420756617 * i,
				-21.949181055868166 - 6.377838450298583 * i,
				-3.7265685887609905 + 37.45221650597609 * i,
				-4.143690915425717 + 24.28814333995993 * i,
				31.38920960676393 + 12.194486215746437 * i,
				-79.51417920753015 - 16.50518165970584 * i,
				32.71507304910372 - 11.776944279561476 * i,
				17.17306580323782 - 14.977922134779789 * i,
				6.885898385994769 - 10.98778009512454 * i,
				-8.275369291448726 - 1.9605494472583729 * i,
				19.48762176513239 - 54.32936286437996 * i,
				44.357911062016996 - 3.9032778083657718 * i,
				-44.97315666672529 - 6.184203147927221 * i,
				-6.058516855245614 + 27.83425935490075 * i,
				-2.050191134621514 - 16.210824046071107 * i,
				-60.30984541807075 + 73.2259570501726 * i,
				17.61949960977492 - 35.141160821726814 * i,
				5.115244950064259 + 10.874296478175332 * i,
				55.83189663550353 + 10.363015579614597 * i,
				-24.990503094223847 - 21.874177365821854 * i,
				-10.097267728879046 - 8.658355090659887 * i,
				17.767114705732443 + 39.570479820025625 * i,
				29.577401265203108 - 25.912819592353227 * i,
				57.09539968262975 + 82.31124802224056 * i,
				30.92321422368017 - 3.8563763701881157 * i,
				8.845784787148993 + 37.3323860605405 * i,
				5.002676757558589 - 7.182096583667285 * i,
				-51.886884166861805 + 16.09726666941668 * i,
				47.074640241096716 - 32.6760384906439 * i,
				-16.303505294511297 - 13.412914880672735 * i,
				43.70182473837136 - 32.15973260778437 * i,
			],
		];
		let (m, n) = A.shape();
		let mut QR = A.clone();
		let mut H = Mat::zeros(20, A.ncols());
		qr_in_place(
			QR.rb_mut(),
			H.rb_mut(),
			Par::Seq,
			MemStack::new(&mut MemBuffer::new(qr_in_place_scratch::<c64>(
				A.nrows(),
				A.ncols(),
				20,
				ParShape::Seq,
				default(),
			))),
			default(),
		);
		let approx_eq = CwiseMat(ApproxEq {
			abs_tol: 1e-10,
			rel_tol: 1e-10,
		});
		let mut Q = Mat::<c64>::zeros(m, m);
		let mut R = QR.as_ref().cloned();
		for j in 0..m {
			Q[(j, j)] = c64::ONE;
		}
		householder::apply_block_householder_sequence_on_the_left_in_place_with_conj(
			QR.rb(),
			H.rb(),
			Conj::No,
			Q.as_mut(),
			Par::Seq,
			MemStack::new(&mut MemBuffer::new(
				householder::apply_block_householder_sequence_transpose_on_the_left_in_place_scratch::<c64>(m, 20, m),
			)),
		);
		for j in 0..n {
			for i in j + 1..m {
				R[(i, j)] = c64::ZERO;
			}
		}
		assert!(Q * R ~ A);
		let QR = A.qr(Par::Seq).reconstruct(Par::Seq);
		assert!(QR ~ A);
	}

	/// A column block of exactly zero columns retains no reflectors, so it has no trailing update
	/// to make. `A1.ncols() > 0` used to be the only condition there, so the block reached the
	/// block Householder apply with an empty basis, whose row split took `N * K = 0` as its chunk
	/// size and panicked in rayon -- [tensor4all/faer#51](https://github.com/tensor4all/faer/issues/51).
	/// The threaded factorization must produce the sequential one.
	#[test]
	fn zero_column_block_does_not_panic_with_threads() {
		let pool = rayon::ThreadPoolBuilder::new()
			.num_threads(2)
			.build()
			.unwrap();
		let (m, n, zero, block_size) = (256usize, 96usize, 32usize, 32usize);
		let mut s = 12345u64;
		let a = Mat::<f64>::from_fn(m, n, |_, j| {
			s = s
				.wrapping_mul(6364136223846793005)
				.wrapping_add(1442695040888963407);
			if j < zero {
				0.0
			} else {
				(s >> 11) as f64 / (1u64 << 53) as f64 - 0.5
			}
		});
		let factor = |mut par: Par| {
			let mut qr = a.cloned();
			let mut h = Mat::<f64>::zeros(block_size, n);
			let par_shape = par.shape();
			let info = qr_in_place(
				qr.as_mut(),
				h.as_mut(),
				par.reborrow(),
				MemStack::new(&mut MemBuffer::new(
					qr_in_place_scratch::<f64>(m, n, block_size, par_shape, default()),
				)),
				default(),
			);
			(info, qr)
		};
		let approx_eq = CwiseMat(ApproxEq {
			abs_tol: 1e-10,
			rel_tol: 1e-10,
		});
		let (info_seq, qr_seq) = factor(Par::Seq);
		let (info_par, qr_par) = pool.install(|| factor(Par::rayon(2)));
		assert!(info_seq.rank == n - zero);
		assert!(info_par.rank == info_seq.rank);
		assert!(qr_par.as_ref() ~ qr_seq.as_ref());
	}
}