faer/linalg/qr/no_pivoting/

factor.rs

use crate::assert;
use crate::internal_prelude::*;
use linalg::householder::{self};

// R R R R|R R R | A A A
// H R R R|R R R | A A A
// H H 0 R|R R R | A A A
//        |------| A A A
// H H H 0|R R R | A A A
// H H H H|0 R R | A A A

/// information about the resulting $QR$ factorization.
#[derive(Copy, Clone, Debug)]
pub struct QrInfo {
	/// estimated rank of the matrix.
	pub rank: usize,
}

#[math]
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 norm = A.rb().col(col).get(..row).norm_l2();

		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 norm = hypot(info.norm, norm);
		let eps = eps::<T::Real>();
		let leeway = from_f64::<T::Real>((m - row) as f64 * 16.0);
		let threshold = eps * leeway * norm;

		if info.norm > threshold {
			let tau_inv = recip(info.tau);

			H[row] = from_real(info.tau);

			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 = -mul_real(dot, tau_inv);
				*head = *head + k;
				z!(tail, v).for_each(|uz!(dst, src)| {
					*dst = *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_blocksize<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,
	/// threshold at which the parallelism should be disabled
	pub par_threshold: usize,

	#[doc(hidden)]
	pub non_exhaustive: NonExhaustive,
}

impl<T: ComplexField> Auto<T> for QrParams {
	#[inline]
	fn auto() -> Self {
		Self {
			blocking_threshold: 48 * 48,
			par_threshold: 192 * 256,
			non_exhaustive: NonExhaustive(()),
		}
	}
}

#[math]
fn qr_in_place_blocked<T: ComplexField>(
	A: MatMut<'_, T>,
	H: MatMut<'_, T>,
	row_start: usize,
	col_start: usize,
	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 blocksize = H.nrows();

	assert!(blocksize > 0);

	if blocksize == 1 {
		return qr_in_place_unblocked(A, H.row_mut(0), row_start, col_start);
	}
	let sub_blocksize = if m * n < params.blocking_threshold { 1 } else { blocksize / 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 blocksize = Ord::min(blocksize, Ord::min(size - row, n - col));
		let sub_blocksize = Ord::min(blocksize, sub_blocksize);

		let mut A = A.rb_mut();
		let mut H = H.rb_mut();

		//
		//  offset col_start
		//       v v
		// R R R R|A A|A A
		// H 0 R R|A A|A A
		// H H 0 R|A A|A A
		// H H H 0|R A|A A
		// H H H H|0 R|A A
		// H H H H|H 0|A A

		// householder coeffs
		//
		// H H H H H 0 0 0 ∞ 0 0 0
		//   H H H   ∞ 0 0   ∞ 0 0
		//     H H     ∞ 0     ∞ 0
		//       H       ∞       ∞

		let start = row;
		let mut offset = 0;
		while offset < blocksize && col < n {
			let blocksize = Ord::min(n - col, blocksize - offset);
			let sub_blocksize = Ord::min(blocksize, sub_blocksize);

			let new_row = qr_in_place_blocked(
				A.rb_mut().subcols_mut(0, col + blocksize),
				H.rb_mut().subrows_mut(offset, sub_blocksize),
				row,
				col,
				par,
				stack,
				params.into(),
			);
			let local = new_row - row;

			if local > 0 {
				let mut k = 0;
				while k < local {
					let sub_blocksize = Ord::min(sub_blocksize, local - k);

					if k > 0 {
						let mut H = H.rb_mut().get_mut(offset.., ..).subcols_mut(row + k, sub_blocksize);
						let (H0, H1) = H.rb_mut().split_at_row_mut(k);
						let H0 = H0.rb().subrows(0, sub_blocksize);
						let H1 = H1.subrows_mut(0, sub_blocksize);

						{ H1 }.copy_from_triangular_upper(H0);
					}
					k += sub_blocksize;
				}

				householder::upgrade_householder_factor(
					H.rb_mut().get_mut(offset..offset + local, row..row + local),
					A.rb().get(row.., row..row + local),
					local,
					sub_blocksize,
					par,
				);

				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,
					);
					triangular::matmul(
						H.rb_mut(),
						BlockStructure::UnitTriangularUpper,
						Accum::Add,
						A1.adjoint(),
						BlockStructure::Rectangular,
						A1,
						BlockStructure::Rectangular,
						one(),
						par,
					);
				}
			}

			let (Q0, A1) = A.rb_mut().get_mut(row.., ..).split_at_col_mut(col + blocksize);
			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, stack);
			};

			offset += local;
			row += local;
			col += blocksize;
		}
	}
	row
}

#[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 blocksize = Q_coeff.nrows();
	assert!(all(blocksize > 0, Q_coeff.ncols() == Ord::min(A.nrows(), A.ncols()),));

	#[cfg(feature = "perf-warn")]
	if A.row_stride().unsigned_abs() != 1 && crate::__perf_warn!(QR_WARN) {
		if A.col_stride().unsigned_abs() == 1 {
			log::warn!(target: "faer_perf", "QR prefers column-major matrix. Found row-major matrix.");
		} else {
			log::warn!(target: "faer_perf", "QR prefers column-major matrix. Found matrix with generic strides.");
		}
	}

	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 / blocksize * blocksize;
	let n = Q_coeff.ncols();
	while col < n {
		let blocksize = Ord::min(blocksize, n - col);

		let start = Ord::max(rank, col);

		Q_coeff
			.rb_mut()
			.get_mut(start - col.., start..col + blocksize)
			.diagonal_mut()
			.fill(infinity());

		col += blocksize;
	}

	QrInfo { rank }
}

/// computes the size and alignment of required workspace for performing a qr
/// decomposition with no pivoting
#[inline]
pub fn qr_in_place_scratch<T: ComplexField>(nrows: usize, ncols: usize, blocksize: usize, par: Par, params: Spec<QrParams, T>) -> StackReq {
	let _ = par;
	let _ = nrows;
	let _ = &params;
	temp_mat_scratch::<T>(blocksize, 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 par in [Par::Seq, Par::rayon(8)] {
				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, Par::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,
						MemStack::new(&mut MemBuffer::new(qr_in_place_scratch::<c64>(n, n, bs, par, 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,
						MemStack::new(&mut MemBuffer::new(qr_in_place_scratch::<c64>(m, n, bs, par, 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);
				}
			}
		}
	}

	#[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,
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				-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,
				Par::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().reconstruct();
		assert!(QR ~ A);
	}
}