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>,
	lookahead: bool,
) -> 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;
	// the rank of the next block's panel when the look-ahead below has already factored it (and
	// completed its factor), so that block starts at its trailing apply
	let mut prefactored: Option<usize> = None;
	let full_block_size = block_size;
	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 prefactored_rank = prefactored.take();
			let new_row = match prefactored_rank {
				Some(local) => row + local,
				None => 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(),
					false,
				),
			};
			let local = new_row - row;
			if local > 0 && prefactored_rank.is_none() {
				complete_factor(
					H.rb_mut().get_mut(offset.., row..row + local),
					A.rb().get(row.., row..row + 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(),
					);
				}
			}
			// Look-ahead (top level only): when this chunk completes a full-rank block (`row == col`
			// throughout, so the next panel's reflectors are stored in its own columns) and another
			// block follows, apply this block to the next block's columns first, then factor that
			// panel on a small group of participants while the others apply this block to the rest
			// of the trailing matrix. The panel is a recursion of short regions that the team speeds
			// up by a third at most (faer#160: 185 ms on one thread against 124 ms on 4 to 32 at
			// 4096), so it is the stage that is hidden, not split further; `lookahead_width` decides
			// per step whether hiding it pays and on how many participants.
			let next_col = col + block_size;
			let row2 = row + local;
			let bs2 = Ord::min(full_block_size, Ord::min(size.saturating_sub(row2), n.saturating_sub(next_col)));
			let far = n.saturating_sub(next_col + bs2);
			let sub2 = Ord::min(bs2, sub_block_size);
			let panel_req = || temp_mat_scratch::<T>(bs2, bs2);
			let ahead = if lookahead
				&& offset == 0
				&& local == block_size
				&& row == col
				&& bs2 > 0
				&& far > 0
				&& sub2 > 1
				&& (m - row2) * bs2 >= params.blocking_threshold
				&& stack.can_hold(StackReq::all_of(&[panel_req(), temp_mat_scratch::<T>(local, far)]))
			{
				lookahead_width::<T>(&par, m - row, local, bs2, far)
			} else {
				None
			};
			match ahead {
				#[cfg(feature = "rayon")]
				Some(width) => {
					let (Q0, A1) = A.rb_mut().get_mut(row.., ..).split_at_col_mut(next_col);
					let Q0 = Q0.rb().get(.., row..row + local);
					let (H_done, H_next) = H.rb_mut().split_at_col_mut(row2);
					let H_j = H_done.rb().get(offset..offset + local, row..row + local);
					let (mut A_next, A_far) = A1.split_at_col_mut(bs2);
					householder::apply_block_householder_transpose_on_the_left_in_place_with_conj(Q0, H_j, Conj::Yes, A_next.rb_mut(), par.reborrow(), stack);
					// the next panel in its own coordinates: rows from its diagonal down, its columns,
					// its factor columns; disjoint from what the trailing apply reads and writes
					let A_panel = A_next.get_mut(local.., ..);
					let H_panel = H_next.subcols_mut(0, bs2);
					let panel_req = panel_req();
					let (panel_mem, far_stack) = stack.make_aligned_uninit::<u8>(panel_req.size_bytes(), panel_req.align_bytes());
					let mut rank2 = 0usize;
					crate::utils::thread::join_sized(
						par.reborrow(),
						width,
						|par| {
							rank2 = factor_panel(A_panel, H_panel, sub2, par, MemStack::new(panel_mem), params.into());
						},
						|par| {
							householder::apply_block_householder_transpose_on_the_left_in_place_with_conj(Q0, H_j, Conj::Yes, A_far, par, far_stack);
						},
					);
					prefactored = Some(rank2);
					#[cfg(test)]
					LOOKAHEAD_STEPS.fetch_add(1, core::sync::atomic::Ordering::Relaxed);
				},
				_ => {
					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 one panel in its own coordinates (`A`: its rows from its diagonal down and its
/// columns; `H`: its block's factor rows and its columns) and completes its block's factor: what
/// the blocked loop does for a block at `offset == 0`. Returns the panel's rank.
#[cfg(feature = "rayon")]
fn factor_panel<T: ComplexField>(
	A: MatMut<'_, T>,
	H: MatMut<'_, T>,
	sub_block_size: usize,
	mut par: Par,
	stack: &mut MemStack,
	params: Spec<QrParams, T>,
) -> usize {
	let mut A = A;
	let mut H = H;
	let local = qr_in_place_blocked(A.rb_mut(), H.rb_mut().subrows_mut(0, sub_block_size), 0, 0, par.reborrow(), stack, params, false);
	if local > 0 {
		complete_factor(H.subcols_mut(0, local), A.rb().subcols(0, local), sub_block_size, par);
	}
	local
}

/// Completes the factor of a block whose `local = basis.ncols()` reflectors were factored in
/// sub-blocks of `sub_block_size` (`factor`: the block's factor rows from its first, and its
/// `local` columns; `basis`: the reflectors from the block's diagonal down): moves each
/// sub-block's factor to its diagonal position and merges them.
fn complete_factor<T: ComplexField>(factor: MatMut<'_, T>, basis: MatRef<'_, T>, sub_block_size: usize, par: Par) {
	let mut H = factor;
	let local = basis.ncols();
	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().subcols_mut(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.get_mut(0..local, 0..local), basis, local, sub_block_size, par);
}

/// Look-ahead steps taken, for the tests that check the path ran.
#[cfg(test)]
static LOOKAHEAD_STEPS: core::sync::atomic::AtomicUsize = core::sync::atomic::AtomicUsize::new(0);
/// Tests: overlap every eligible step with a panel group of this width (0: the step model).
#[cfg(test)]
static LOOKAHEAD_FORCE: core::sync::atomic::AtomicUsize = core::sync::atomic::AtomicUsize::new(0);

/// The panel group of a QR look-ahead step, or `None` when overlapping does not pay: the step
/// model of `cost::lookahead_width`, with `F` the far columns' block apply and `P` the panel, both
/// serial estimates at the block apply rate (`cost::block_apply_estimate`; the panel as a block
/// apply of its own width on itself, which matches its measured serial time at block 128), and
/// the panel's measured speed-up on a team (`PANEL_TEAM`).
fn lookahead_width<T: ComplexField>(par: &Par, rows: usize, basis_cols: usize, panel_cols: usize, far_cols: usize) -> Option<usize> {
	// time of the panel on `a` participants over its time on one: 4096 at block 128 on broccoli
	// (EPYC 7702P), faer#160 `panel2`: 184-186 ms on one, 124-131 ms on 4 to 32, 182 ms on 64
	// (the full host: short regions at width 64). Two participants are not measured and are
	// not offered.
	const PANEL_TEAM: crate::utils::cost::PanelTeam = crate::utils::cost::PanelTeam { from: 4, to: 32, ratio: 0.68 };
	#[cfg(test)]
	match LOOKAHEAD_FORCE.load(core::sync::atomic::Ordering::Relaxed) {
		0 => {},
		a if par.degree() >= 2 => return Some(Ord::min(a, par.degree() - 1)),
		_ => return None,
	}
	if par.degree() < 2 {
		return None;
	}
	let panel = crate::utils::cost::block_apply_estimate::<T>(rows - basis_cols, panel_cols, panel_cols);
	let far = crate::utils::cost::block_apply_estimate::<T>(rows, basis_cols, far_cols);
	crate::utils::cost::lookahead_width(par, panel, far, PANEL_TEAM)
}
/// 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 = linalg::column_major_copy(A.rb(), par.reborrow());
	let mut work = MatMut::from_column_major_slice_mut(&mut work, m, n);
	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, true);
	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] {
			let mut run = |mut par: Par<'_>| {
			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);
				}
			};
			run(Par::Seq);
			crate::team::with_team(8, |par| run(par));
		}
	}
	/// 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)] {
			let run = |mut par: Par<'_>| {
				check::<f64>(m, n, false, false, true, par.reborrow());
				check::<c64>(m, n, false, false, true, par);
			};
			run(Par::Seq);
			crate::team::with_team(8, |par| run(par));
			check::<f64>(m, n, false, true, true, Par::Seq);
			crate::team::with_team(4, |par| check::<c64>(m, n, false, true, true, par));
		}
		// 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);
			crate::team::with_team(4, |par| check::<c64>(m, n, false, false, false, par));
			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}"
			);
		}
	}

	/// The look-ahead factors each next panel on its own participants while the trailing apply runs.
	/// Against `Par::Seq`: R and the rank to rounding, the compact factors on full-rank input, and
	/// `Q R = A` (the reflector tails and the factor layout the sequence consumers read). Shapes:
	/// tall, square, wide, a block that does not divide the width; f32/f64/c64; the step model's
	/// own choice on 4 participants, and every eligible step forced with a panel group of 1 and of
	/// 4 (8 participants); rank loss inside a prefactored panel (one zero column, and a whole zero
	/// panel). Asserts the path ran, counted inside the team's turn.
	#[cfg(feature = "threadteam")]
	#[test]
	fn test_lookahead_matches_sequential() {
		use core::sync::atomic::Ordering::Relaxed;
		use crate::linalg::qr::no_pivoting::reconstruct;
		fn case<T: ComplexField>(m: usize, n: usize, bs: usize, threads: usize, force: usize, zero_cols: core::ops::Range<usize>, gen: &dyn Fn(usize, usize) -> T, mag: fn(&T) -> f64, eps: f64) -> bool {
			let size = Ord::min(m, n);
			let a = Mat::<T>::from_fn(m, n, |i, j| if zero_cols.contains(&j) { T::zero_impl() } else { gen(i, j) });
			let full_rank = zero_cols.is_empty();
			let mut steps = 0;
			let run = |team: bool, steps: &mut usize| {
				let mut f = a.clone();
				let mut h = Mat::<T>::zeros(bs, size);
				let mut mem = MemBuffer::new(qr_in_place_scratch::<T>(m, n, bs, crate::ParShape::team(threads), default()));
				let rank = if team {
					crate::team::with_team(threads, |par| {
						LOOKAHEAD_FORCE.store(force, Relaxed);
						let before = LOOKAHEAD_STEPS.load(Relaxed);
						let rank = qr_in_place(f.as_mut(), h.as_mut(), par, MemStack::new(&mut mem), default()).rank;
						*steps = LOOKAHEAD_STEPS.load(Relaxed) - before;
						LOOKAHEAD_FORCE.store(0, Relaxed);
						rank
					})
				} else {
					qr_in_place(f.as_mut(), h.as_mut(), Par::Seq, MemStack::new(&mut mem), default()).rank
				};
				(f, h, rank)
			};
			let (a0, h0, r0) = run(false, &mut 0);
			let (a1, h1, r1) = run(true, &mut steps);
			let tag = format!("{m}x{n} block {bs} threads {threads} force {force} zero {zero_cols:?}");
			std::assert!(r0 == r1, "{tag}: rank {r0} against {r1}");
			let mx = |x: &Mat<T>| {
				let mut v = 1.0f64;
				for j in 0..x.ncols() {
					for i in 0..x.nrows() {
						let e = mag(&x[(i, j)]);
						if e.is_finite() {
							v = v.max(e);
						}
					}
				}
				v
			};
			let tol = (m + n) as f64 * eps * mx(&a);
			for j in 0..n {
				for i in 0..Ord::min(m, j + 1) {
					let d = mag(&(a1[(i, j)].clone() - a0[(i, j)].clone()));
					std::assert!(d <= tol, "{tag}: R({i}, {j}) differs by {d}");
				}
			}
			if full_rank {
				for j in 0..size {
					for i in 0..bs {
						// a reflector with an empty tail stores an infinite `tau` on both sides
						let (x, y) = (mag(&h0[(i, j)]), mag(&h1[(i, j)]));
						if x == f64::INFINITY && y == f64::INFINITY {
							continue;
						}
						let d = mag(&(h1[(i, j)].clone() - h0[(i, j)].clone()));
						std::assert!(d <= (m as f64) * eps * 10.0 * mx(&h0), "{tag}: H({i}, {j}) differs by {d}");
					}
				}
			}
			// Q R = A through the factor layout the sequence consumers read
			let mut qr = Mat::<T>::zeros(m, n);
			let mut mem = MemBuffer::new(reconstruct::reconstruct_scratch::<T>(m, n, bs, crate::ParShape::Seq));
			reconstruct::reconstruct(qr.as_mut(), a1.as_ref().subcols(0, size), h1.as_ref(), a1.as_ref().subrows(0, size), Par::Seq, MemStack::new(&mut mem));
			for j in 0..n {
				for i in 0..m {
					let d = mag(&(qr[(i, j)].clone() - a[(i, j)].clone()));
					std::assert!(d <= tol, "{tag}: (QR - A)({i}, {j}) = {d}");
				}
			}
			steps > 0
		}
		let r = |i: usize, j: usize| ((i * 7 + j * 13) % 23) as f64 / 23.0 - 0.5 + if i == j { 2.0 } else { 0.0 };
		let r32 = |i: usize, j: usize| r(i, j) as f32;
		let c = |i: usize, j: usize| c64::new(r(i, j), r(j + 3, i + 1));
		let (f64m, f32m, c64m): (fn(&f64) -> f64, fn(&f32) -> f64, fn(&c64) -> f64) = (|x| x.abs(), |x| x.abs() as f64, |x| x.norm());
		// the step model's own choice on 4 participants: it overlaps on tall and square shapes
		for &(m, n, bs, must) in &[(800, 600, 32, true), (600, 600, 48, true), (500, 800, 32, false), (1000, 400, 64, true)] {
			std::assert!(case::<f64>(m, n, bs, 4, 0, 0..0, &r, f64m, 1e-13) || !must, "{m}x{n}: the look-ahead did not run");
			std::assert!(case::<c64>(m, n, bs, 4, 0, 0..0, &c, c64m, 1e-13) || !must, "{m}x{n} c64: the look-ahead did not run");
		}
		std::assert!(case::<f32>(600, 500, 32, 4, 1, 0..0, &r32, f32m, 1e-5));
		// every eligible step, panel groups of 1 and 4
		std::assert!(case::<f64>(600, 500, 32, 8, 1, 0..0, &r, f64m, 1e-13));
		std::assert!(case::<f64>(600, 500, 32, 8, 4, 0..0, &r, f64m, 1e-13));
		std::assert!(case::<c64>(500, 500, 48, 8, 4, 0..0, &c, c64m, 1e-13));
		// rank loss inside prefactored panels: one zero column in the second block's panel, and a
		// whole zero panel (the third block), with the look-ahead forced on every eligible step
		std::assert!(case::<f64>(600, 500, 32, 4, 1, 40..41, &r, f64m, 1e-13));
		std::assert!(case::<f64>(600, 500, 32, 8, 4, 64..96, &r, f64m, 1e-13));
	}

	#[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,
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				-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 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 (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) = crate::team::with_team(2, |par| factor(par));
		assert!(info_seq.rank == n - zero);
		assert!(info_par.rank == info_seq.rank);
		assert!(qr_par.as_ref() ~ qr_seq.as_ref());
	}
}