faer/mat/

matref.rs

use super::{Mat, MatMut, MatRef, *};
use crate::col::ColRef;
use crate::internal_prelude::*;
use crate::row::RowRef;
use crate::utils::bound::{Dim, Partition};
use crate::{ContiguousFwd, Idx, IdxInc};
use equator::{assert, debug_assert};
use faer_traits::Real;
use generativity::Guard;
/// see [`super::MatRef`]
pub struct Ref<
	'a,
	T,
	Rows = usize,
	Cols = usize,
	RStride = isize,
	CStride = isize,
> {
	pub(super) imp: MatView<T, Rows, Cols, RStride, CStride>,
	pub(super) __marker: PhantomData<&'a T>,
}
impl<T, Rows: Copy, Cols: Copy, RStride: Copy, CStride: Copy> Copy
	for Ref<'_, T, Rows, Cols, RStride, CStride>
{
}
impl<T, Rows: Copy, Cols: Copy, RStride: Copy, CStride: Copy> Clone
	for Ref<'_, T, Rows, Cols, RStride, CStride>
{
	#[inline]
	fn clone(&self) -> Self {
		*self
	}
}
impl<'short, T, Rows: Copy, Cols: Copy, RStride: Copy, CStride: Copy>
	Reborrow<'short> for Ref<'_, T, Rows, Cols, RStride, CStride>
{
	type Target = Ref<'short, T, Rows, Cols, RStride, CStride>;

	#[inline]
	fn rb(&'short self) -> Self::Target {
		*self
	}
}
impl<'short, T, Rows: Copy, Cols: Copy, RStride: Copy, CStride: Copy>
	ReborrowMut<'short> for Ref<'_, T, Rows, Cols, RStride, CStride>
{
	type Target = Ref<'short, T, Rows, Cols, RStride, CStride>;

	#[inline]
	fn rb_mut(&'short mut self) -> Self::Target {
		*self
	}
}
impl<'a, T, Rows: Copy, Cols: Copy, RStride: Copy, CStride: Copy> IntoConst
	for Ref<'a, T, Rows, Cols, RStride, CStride>
{
	type Target = Ref<'a, T, Rows, Cols, RStride, CStride>;

	#[inline]
	fn into_const(self) -> Self::Target {
		self
	}
}
unsafe impl<T: Sync, Rows: Sync, Cols: Sync, RStride: Sync, CStride: Sync> Sync
	for Ref<'_, T, Rows, Cols, RStride, CStride>
{
}
unsafe impl<T: Sync, Rows: Send, Cols: Send, RStride: Send, CStride: Send> Send
	for Ref<'_, T, Rows, Cols, RStride, CStride>
{
}
#[track_caller]
#[inline]
fn from_strided_column_major_slice_assert(
	nrows: usize,
	ncols: usize,
	col_stride: usize,
	len: usize,
) {
	if nrows > 0 && ncols > 0 {
		let last = usize::checked_mul(col_stride, ncols - 1)
			.and_then(|last_col| last_col.checked_add(nrows - 1));
		let Some(last) = last else {
			panic!("address computation of the last matrix element overflowed");
		};
		assert!(last < len);
	}
}
impl<'a, T> MatRef<'a, T> {
	/// equivalent to `MatRef::from_row_major_slice(array.as_flattened(), ROWS,
	/// COLS)`
	#[inline]
	pub fn from_row_major_array<const ROWS: usize, const COLS: usize>(
		array: &'a [[T; COLS]; ROWS],
	) -> Self {
		unsafe {
			Self::from_raw_parts(
				array as *const _ as *const T,
				ROWS,
				COLS,
				COLS as isize,
				1,
			)
		}
	}

	/// equivalent to `MatRef::from_column_major_slice(array.as_flattened(),
	/// ROWS, COLS)`
	#[inline]
	pub fn from_column_major_array<const ROWS: usize, const COLS: usize>(
		array: &'a [[T; ROWS]; COLS],
	) -> Self {
		unsafe {
			Self::from_raw_parts(
				array as *const _ as *const T,
				ROWS,
				COLS,
				1,
				ROWS as isize,
			)
		}
	}

	/// creates a `1×1` view over the given element
	#[inline]
	pub fn from_ref(value: &'a T) -> Self {
		unsafe { MatRef::from_raw_parts(value as *const T, 1, 1, 0, 0) }
	}
}

impl<'a, T, Rows: Shape> ColRef<'a, T, Rows> {
	/// creates a `ColRef` from a view over a single element, repeated
	/// `nrows` times
	#[inline]
	pub fn from_repeated_ref(value: &'a T, nrows: Rows) -> Self {
		unsafe { ColRef::from_raw_parts(value as *const T, nrows, 0) }
	}
}

impl<'a, T, Cols: Shape> RowRef<'a, T, Cols> {
	/// creates a `RowRef` from a view over a single element, repeated
	/// `ncols` times
	#[inline]
	pub fn from_repeated_ref(value: &'a T, ncols: Cols) -> Self {
		unsafe { RowRef::from_raw_parts(value as *const T, ncols, 0) }
	}
}

impl<'a, T, Rows: Shape, Cols: Shape> MatRef<'a, T, Rows, Cols> {
	/// creates a `MatRef` from a view over a single element, repeated
	/// `nrows×ncols` times
	#[inline]
	pub fn from_repeated_ref(value: &'a T, nrows: Rows, ncols: Cols) -> Self {
		unsafe { MatRef::from_raw_parts(value as *const T, nrows, ncols, 0, 0) }
	}

	/// creates a `MatRef` from a view over a single column, repeated
	/// `ncols` times
	#[inline]
	pub fn from_repeated_col(col: ColRef<'a, T, Rows>, ncols: Cols) -> Self {
		unsafe {
			MatRef::from_raw_parts(
				col.as_ptr(),
				col.nrows(),
				ncols,
				col.row_stride(),
				0,
			)
		}
	}

	/// creates a `MatRef` from a view over a single row, repeated
	/// `nrows` times
	#[inline]
	pub fn from_repeated_row(row: RowRef<'a, T, Cols>, nrows: Rows) -> Self {
		unsafe {
			MatRef::from_raw_parts(
				row.as_ptr(),
				nrows,
				row.ncols(),
				0,
				row.col_stride(),
			)
		}
	}

	/// creates a `MatRef` from slice views over the matrix data, and the matrix
	/// dimensions. the data is interpreted in a column-major format, so that
	/// the first chunk of `nrows` values from the slices goes in the first
	/// column of the matrix, the second chunk of `nrows` values goes in the
	/// second column, and so on
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `nrows * ncols == slice.len()`
	///
	/// # example
	/// ```
	/// use faer::{MatRef, mat};
	/// let slice = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0_f64];
	/// let view = MatRef::from_column_major_slice(&slice, 3, 2);
	/// let expected = mat![[1.0, 4.0], [2.0, 5.0], [3.0, 6.0]];
	/// assert_eq!(expected, view);
	/// ```
	#[inline]
	#[track_caller]
	pub fn from_column_major_slice(
		slice: &'a [T],
		nrows: Rows,
		ncols: Cols,
	) -> Self {
		from_slice_assert(nrows.unbound(), ncols.unbound(), slice.len());
		unsafe {
			MatRef::from_raw_parts(
				slice.as_ptr(),
				nrows,
				ncols,
				1,
				nrows.unbound() as isize,
			)
		}
	}

	/// creates a `MatRef` from slice views over the matrix data, and the matrix
	/// dimensions. the data is interpreted in a column-major format, where the
	/// beginnings of two consecutive columns are separated by `col_stride`
	/// elements
	#[inline]
	#[track_caller]
	pub fn from_column_major_slice_with_stride(
		slice: &'a [T],
		nrows: Rows,
		ncols: Cols,
		col_stride: usize,
	) -> Self {
		from_strided_column_major_slice_assert(
			nrows.unbound(),
			ncols.unbound(),
			col_stride,
			slice.len(),
		);
		unsafe {
			MatRef::from_raw_parts(
				slice.as_ptr(),
				nrows,
				ncols,
				1,
				col_stride as isize,
			)
		}
	}

	/// creates a `MatRef` from slice views over the matrix data, and the matrix
	/// dimensions. the data is interpreted in a row-major format, so that the
	/// first chunk of `ncols` values from the slices goes in the first column
	/// of the matrix, the second chunk of `ncols` values goes in the second
	/// column, and so on
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `nrows * ncols == slice.len()`
	///
	/// # example
	/// ```
	/// use faer::{MatRef, mat};
	/// let slice = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0_f64];
	/// let view = MatRef::from_row_major_slice(&slice, 3, 2);
	/// let expected = mat![[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]];
	/// assert_eq!(expected, view);
	/// ```
	#[inline]
	#[track_caller]
	pub fn from_row_major_slice(
		slice: &'a [T],
		nrows: Rows,
		ncols: Cols,
	) -> Self {
		MatRef::from_column_major_slice(slice, ncols, nrows).transpose()
	}

	/// creates a `MatRef` from slice views over the matrix data, and the matrix
	/// dimensions. the data is interpreted in a row-major format, where the
	/// beginnings of two consecutive rows are separated by `row_stride`
	/// elements
	#[inline]
	#[track_caller]
	pub fn from_row_major_slice_with_stride(
		slice: &'a [T],
		nrows: Rows,
		ncols: Cols,
		row_stride: usize,
	) -> Self {
		MatRef::from_column_major_slice_with_stride(
			slice, ncols, nrows, row_stride,
		)
		.transpose()
	}
}
impl<'a, T, Rows: Shape, Cols: Shape, RStride: Stride, CStride: Stride>
	MatRef<'a, T, Rows, Cols, RStride, CStride>
{
	/// creates a `MatRef` from a pointer to the matrix data, dimensions, and
	/// strides
	///
	/// the row (resp. column) stride is the offset from the memory address of a
	/// given matrix element at index `(row: i, col: j)`, to the memory address
	/// of the matrix element at index `(row: i + 1, col: 0)` (resp. `(row: 0,
	/// col: i + 1)`). this offset is specified in number of elements, not in
	/// bytes
	///
	/// # safety
	/// the behavior is undefined if any of the following conditions are
	/// violated:
	/// * for each matrix unit, the entire memory region addressed by the matrix
	///   must be contained
	/// within a single allocation, accessible in its entirety by the
	/// corresponding pointer in `ptr`
	/// * for each matrix unit, the corresponding pointer must be properly
	///   aligned,
	/// even for a zero-sized matrix
	/// * the values accessible by the matrix must be initialized at some point
	///   before they are
	/// read, or references to them are formed
	/// * no mutable aliasing is allowed. in other words, none of the elements
	///   accessible by any
	/// matrix unit may be accessed for writes by any other means for the
	/// duration of the lifetime `'a`
	///
	/// # example
	///
	/// ```
	/// use faer::{MatRef, mat};
	/// // row major matrix with 2 rows, 3 columns, with a column at the end that we want to skip.
	/// // the row stride is the pointer offset from the address of 1.0 to the address of 4.0,
	/// // which is 4
	/// // the column stride is the pointer offset from the address of 1.0 to the address of 2.0,
	/// // which is 1
	/// let data = [[1.0, 2.0, 3.0, f64::NAN], [4.0, 5.0, 6.0, f64::NAN]];
	/// let matrix = unsafe { MatRef::from_raw_parts(data.as_ptr() as *const f64, 2, 3, 4, 1) };
	/// let expected = mat![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];
	/// assert_eq!(expected.as_ref(), matrix);
	/// ```
	#[inline]
	#[track_caller]
	pub const unsafe fn from_raw_parts(
		ptr: *const T,
		nrows: Rows,
		ncols: Cols,
		row_stride: RStride,
		col_stride: CStride,
	) -> Self {
		Self(Ref {
			imp: MatView {
				ptr: NonNull::new_unchecked(ptr as *mut T),
				nrows,
				ncols,
				row_stride,
				col_stride,
			},
			__marker: PhantomData,
		})
	}

	/// returns a pointer to the matrix data
	#[inline]
	pub fn as_ptr(&self) -> *const T {
		self.imp.ptr.as_ptr() as *const T
	}

	/// returns the number of rows of the matrix
	#[inline]
	pub fn nrows(&self) -> Rows {
		self.imp.nrows
	}

	/// returns the number of columns of the matrix
	#[inline]
	pub fn ncols(&self) -> Cols {
		self.imp.ncols
	}

	/// returns the number of rows and columns of the matrix
	#[inline]
	pub fn shape(&self) -> (Rows, Cols) {
		(self.nrows(), self.ncols())
	}

	/// returns the row stride of the matrix, specified in number of elements,
	/// not in bytes
	#[inline]
	pub fn row_stride(&self) -> RStride {
		self.imp.row_stride
	}

	/// returns the column stride of the matrix, specified in number of
	/// elements, not in bytes
	#[inline]
	pub fn col_stride(&self) -> CStride {
		self.imp.col_stride
	}

	#[inline]
	pub fn map<U>(&self, f: impl FnMut(&T) -> U) -> Mat<U, Rows, Cols> {
		let mut f = f;
		zip!(self).map(|unzip!(x)| f(x))
	}

	#[inline]
	pub fn for_each(&self, f: impl FnMut(&T)) {
		let mut f = f;
		zip!(self).for_each(|unzip!(x)| f(x))
	}

	/// returns a raw pointer to the element at the given index
	#[inline]
	pub fn ptr_at(&self, row: IdxInc<Rows>, col: IdxInc<Cols>) -> *const T {
		let ptr = self.as_ptr();
		if row >= self.nrows() || col >= self.ncols() {
			ptr
		} else {
			ptr.wrapping_offset(
				row.unbound() as isize * self.row_stride().element_stride(),
			)
			.wrapping_offset(
				col.unbound() as isize * self.col_stride().element_stride(),
			)
		}
	}

	/// returns a raw pointer to the element at the given index, assuming the
	/// provided index is within the matrix bounds
	///
	/// # safety
	/// the behavior is undefined if any of the following conditions are
	/// violated:
	/// * `row < self.nrows()`
	/// * `col < self.ncols()`
	#[inline]
	#[track_caller]
	pub unsafe fn ptr_inbounds_at(
		&self,
		row: Idx<Rows>,
		col: Idx<Cols>,
	) -> *const T {
		debug_assert!(all(row < self.nrows(), col < self.ncols()));
		self.as_ptr()
			.offset(row.unbound() as isize * self.row_stride().element_stride())
			.offset(col.unbound() as isize * self.col_stride().element_stride())
	}

	/// splits the matrix horizontally and vertically at the given index into
	/// four corners and returns an array of each submatrix, in the following
	/// order:
	/// * top left
	/// * top right
	/// * bottom left
	/// * bottom right
	///
	/// # safety
	/// the function panics if any of the following conditions are violated:
	/// * `row <= self.nrows()`
	/// * `col <= self.ncols()`
	#[inline]
	#[track_caller]
	pub fn split_at(
		self,
		row: IdxInc<Rows>,
		col: IdxInc<Cols>,
	) -> (
		MatRef<'a, T, usize, usize, RStride, CStride>,
		MatRef<'a, T, usize, usize, RStride, CStride>,
		MatRef<'a, T, usize, usize, RStride, CStride>,
		MatRef<'a, T, usize, usize, RStride, CStride>,
	) {
		assert!(all(row <= self.nrows(), col <= self.ncols()));
		let rs = self.row_stride();
		let cs = self.col_stride();
		let top_left = self.ptr_at(Rows::start(), Cols::start());
		let top_right = self.ptr_at(Rows::start(), col);
		let bot_left = self.ptr_at(row, Cols::start());
		let bot_right = self.ptr_at(row, col);
		unsafe {
			(
				MatRef::from_raw_parts(
					top_left,
					row.unbound(),
					col.unbound(),
					rs,
					cs,
				),
				MatRef::from_raw_parts(
					top_right,
					row.unbound(),
					self.ncols().unbound() - col.unbound(),
					rs,
					cs,
				),
				MatRef::from_raw_parts(
					bot_left,
					self.nrows().unbound() - row.unbound(),
					col.unbound(),
					rs,
					cs,
				),
				MatRef::from_raw_parts(
					bot_right,
					self.nrows().unbound() - row.unbound(),
					self.ncols().unbound() - col.unbound(),
					rs,
					cs,
				),
			)
		}
	}

	/// splits the matrix horizontally at the given row into two parts and
	/// returns an array of each submatrix, in the following order:
	/// * top
	/// * bottom
	///
	/// # panics
	/// the function panics if the following condition is violated:
	/// * `row <= self.nrows()`
	#[inline]
	#[track_caller]
	pub fn split_at_row(
		self,
		row: IdxInc<Rows>,
	) -> (
		MatRef<'a, T, usize, Cols, RStride, CStride>,
		MatRef<'a, T, usize, Cols, RStride, CStride>,
	) {
		assert!(all(row <= self.nrows()));
		let rs = self.row_stride();
		let cs = self.col_stride();
		let top = self.ptr_at(Rows::start(), Cols::start());
		let bot = self.ptr_at(row, Cols::start());
		unsafe {
			(
				MatRef::from_raw_parts(
					top,
					row.unbound(),
					self.ncols(),
					rs,
					cs,
				),
				MatRef::from_raw_parts(
					bot,
					self.nrows().unbound() - row.unbound(),
					self.ncols(),
					rs,
					cs,
				),
			)
		}
	}

	/// splits the matrix vertically at the given column into two parts and
	/// returns an array of each submatrix, in the following order:
	/// * left
	/// * right
	///
	/// # panics
	/// the function panics if the following condition is violated:
	/// * `col <= self.ncols()`
	#[inline]
	#[track_caller]
	pub fn split_at_col(
		self,
		col: IdxInc<Cols>,
	) -> (
		MatRef<'a, T, Rows, usize, RStride, CStride>,
		MatRef<'a, T, Rows, usize, RStride, CStride>,
	) {
		assert!(all(col <= self.ncols()));
		let rs = self.row_stride();
		let cs = self.col_stride();
		let left = self.ptr_at(Rows::start(), Cols::start());
		let right = self.ptr_at(Rows::start(), col);
		unsafe {
			(
				MatRef::from_raw_parts(
					left,
					self.nrows(),
					col.unbound(),
					rs,
					cs,
				),
				MatRef::from_raw_parts(
					right,
					self.nrows(),
					self.ncols().unbound() - col.unbound(),
					rs,
					cs,
				),
			)
		}
	}

	/// returns a view over the transpose of `self`
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];
	/// let view = matrix.as_ref();
	/// let transpose = view.transpose();
	/// let expected = mat![[1.0, 4.0], [2.0, 5.0], [3.0, 6.0]];
	/// assert_eq!(expected.as_ref(), transpose);
	/// ```
	#[inline]
	pub fn transpose(self) -> MatRef<'a, T, Cols, Rows, CStride, RStride> {
		MatRef {
			0: Ref {
				imp: MatView {
					ptr: self.imp.ptr,
					nrows: self.imp.ncols,
					ncols: self.imp.nrows,
					row_stride: self.imp.col_stride,
					col_stride: self.imp.row_stride,
				},
				__marker: PhantomData,
			},
		}
	}

	/// returns a view over the conjugate of `self`
	#[inline]
	pub fn conjugate(self) -> MatRef<'a, T::Conj, Rows, Cols, RStride, CStride>
	where
		T: Conjugate,
	{
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr() as *const T::Conj,
				self.nrows(),
				self.ncols(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns an unconjugated view over `self`
	#[inline]
	pub fn canonical(
		self,
	) -> MatRef<'a, T::Canonical, Rows, Cols, RStride, CStride>
	where
		T: Conjugate,
	{
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr() as *const T::Canonical,
				self.nrows(),
				self.ncols(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	#[inline]
	#[doc(hidden)]
	pub fn __canonicalize(
		self,
	) -> (MatRef<'a, T::Canonical, Rows, Cols, RStride, CStride>, Conj)
	where
		T: Conjugate,
	{
		(self.canonical(), Conj::get::<T>())
	}

	/// returns a view over the conjugate transpose of `self`.
	#[inline]
	pub fn adjoint(self) -> MatRef<'a, T::Conj, Cols, Rows, CStride, RStride>
	where
		T: Conjugate,
	{
		self.conjugate().transpose()
	}

	#[inline]
	#[track_caller]
	pub(crate) fn at(self, row: Idx<Rows>, col: Idx<Cols>) -> &'a T {
		assert!(all(row < self.nrows(), col < self.ncols()));
		unsafe { self.at_unchecked(row, col) }
	}

	#[inline]
	#[track_caller]
	pub(crate) unsafe fn at_unchecked(
		self,
		row: Idx<Rows>,
		col: Idx<Cols>,
	) -> &'a T {
		&*self.ptr_inbounds_at(row, col)
	}

	/// returns a view over the `self`, with the rows in reversed order
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];
	/// let view = matrix.as_ref();
	/// let reversed_rows = view.reverse_rows();
	/// let expected = mat![[4.0, 5.0, 6.0], [1.0, 2.0, 3.0]];
	/// assert_eq!(expected.as_ref(), reversed_rows);
	/// ```
	#[inline]
	pub fn reverse_rows(
		self,
	) -> MatRef<'a, T, Rows, Cols, RStride::Rev, CStride> {
		let row = unsafe {
			IdxInc::<Rows>::new_unbound(
				self.nrows().unbound().saturating_sub(1),
			)
		};
		let ptr = self.ptr_at(row, Cols::start());
		unsafe {
			MatRef::from_raw_parts(
				ptr,
				self.nrows(),
				self.ncols(),
				self.row_stride().rev(),
				self.col_stride(),
			)
		}
	}

	/// returns a view over the `self`, with the columns in reversed order
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];
	/// let view = matrix.as_ref();
	/// let reversed_cols = view.reverse_cols();
	/// let expected = mat![[3.0, 2.0, 1.0], [6.0, 5.0, 4.0]];
	/// assert_eq!(expected.as_ref(), reversed_cols);
	/// ```
	#[inline]
	pub fn reverse_cols(
		self,
	) -> MatRef<'a, T, Rows, Cols, RStride, CStride::Rev> {
		let col = unsafe {
			IdxInc::<Cols>::new_unbound(
				self.ncols().unbound().saturating_sub(1),
			)
		};
		let ptr = self.ptr_at(Rows::start(), col);
		unsafe {
			MatRef::from_raw_parts(
				ptr,
				self.nrows(),
				self.ncols(),
				self.row_stride(),
				self.col_stride().rev(),
			)
		}
	}

	/// returns a view over the `self`, with the rows and the columns in
	/// reversed order
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];
	/// let view = matrix.as_ref();
	/// let reversed = view.reverse_rows_and_cols();
	/// let expected = mat![[6.0, 5.0, 4.0], [3.0, 2.0, 1.0]];
	/// assert_eq!(expected.as_ref(), reversed);
	/// ```
	#[inline]
	pub fn reverse_rows_and_cols(
		self,
	) -> MatRef<'a, T, Rows, Cols, RStride::Rev, CStride::Rev> {
		self.reverse_rows().reverse_cols()
	}

	/// returns a view over the submatrix starting at index `(row_start,
	/// col_start)`, and with dimensions `(nrows, ncols)`
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `row_start <= self.nrows()`
	/// * `col_start <= self.ncols()`
	/// * `nrows <= self.nrows() - row_start`
	/// * `ncols <= self.ncols() - col_start`
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![
	/// 	[1.0, 5.0, 9.0], //
	/// 	[2.0, 6.0, 10.0],
	/// 	[3.0, 7.0, 11.0],
	/// 	[4.0, 8.0, 12.0f64],
	/// ];
	/// let view = matrix.as_ref();
	/// let submatrix = view.submatrix(
	/// 	/* row_start: */ 2, /* col_start: */ 1, /* nrows: */ 2,
	/// 	/* ncols: */ 2,
	/// );
	/// let expected = mat![[7.0, 11.0], [8.0, 12.0f64]];
	/// assert_eq!(expected.as_ref(), submatrix);
	/// ```
	#[inline]
	#[track_caller]
	pub fn submatrix<V: Shape, H: Shape>(
		self,
		row_start: IdxInc<Rows>,
		col_start: IdxInc<Cols>,
		nrows: V,
		ncols: H,
	) -> MatRef<'a, T, V, H, RStride, CStride> {
		assert!(all(row_start <= self.nrows(), col_start <= self.ncols()));
		{
			let nrows = nrows.unbound();
			let full_nrows = self.nrows().unbound();
			let row_start = row_start.unbound();
			let ncols = ncols.unbound();
			let full_ncols = self.ncols().unbound();
			let col_start = col_start.unbound();
			assert!(all(
				nrows <= full_nrows - row_start,
				ncols <= full_ncols - col_start,
			));
		}
		let rs = self.row_stride();
		let cs = self.col_stride();
		unsafe {
			MatRef::from_raw_parts(
				self.ptr_at(row_start, col_start),
				nrows,
				ncols,
				rs,
				cs,
			)
		}
	}

	/// returns a view over the submatrix starting at row `row_start`, and with
	/// number of rows `nrows`
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `row_start <= self.nrows()`
	/// * `nrows <= self.nrows() - row_start`
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![
	/// 	[1.0, 5.0, 9.0], //
	/// 	[2.0, 6.0, 10.0],
	/// 	[3.0, 7.0, 11.0],
	/// 	[4.0, 8.0, 12.0f64],
	/// ];
	/// let view = matrix.as_ref();
	/// let subrows = view.subrows(/* row_start: */ 1, /* nrows: */ 2);
	/// let expected = mat![[2.0, 6.0, 10.0], [3.0, 7.0, 11.0],];
	/// assert_eq!(expected.as_ref(), subrows);
	/// ```
	#[inline]
	#[track_caller]
	pub fn subrows<V: Shape>(
		self,
		row_start: IdxInc<Rows>,
		nrows: V,
	) -> MatRef<'a, T, V, Cols, RStride, CStride> {
		assert!(all(row_start <= self.nrows()));
		{
			let nrows = nrows.unbound();
			let full_nrows = self.nrows().unbound();
			let row_start = row_start.unbound();
			assert!(all(nrows <= full_nrows - row_start));
		}
		let rs = self.row_stride();
		let cs = self.col_stride();
		unsafe {
			MatRef::from_raw_parts(
				self.ptr_at(row_start, Cols::start()),
				nrows,
				self.ncols(),
				rs,
				cs,
			)
		}
	}

	/// returns a view over the submatrix starting at column `col_start`, and
	/// with number of columns `ncols`
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `col_start <= self.ncols()`
	/// * `ncols <= self.ncols() - col_start`
	///
	/// # example
	/// ```
	/// use faer::mat;
	/// let matrix = mat![
	/// 	[1.0, 5.0, 9.0], //
	/// 	[2.0, 6.0, 10.0],
	/// 	[3.0, 7.0, 11.0],
	/// 	[4.0, 8.0, 12.0f64],
	/// ];
	/// let view = matrix.as_ref();
	/// let subcols = view.subcols(/* col_start: */ 2, /* ncols: */ 1);
	/// let expected = mat![[9.0], [10.0], [11.0], [12.0f64]];
	/// assert_eq!(expected.as_ref(), subcols);
	/// ```
	#[inline]
	#[track_caller]
	pub fn subcols<H: Shape>(
		self,
		col_start: IdxInc<Cols>,
		ncols: H,
	) -> MatRef<'a, T, Rows, H, RStride, CStride> {
		assert!(all(col_start <= self.ncols()));
		{
			let ncols = ncols.unbound();
			let full_ncols = self.ncols().unbound();
			let col_start = col_start.unbound();
			assert!(all(ncols <= full_ncols - col_start));
		}
		let rs = self.row_stride();
		let cs = self.col_stride();
		unsafe {
			MatRef::from_raw_parts(
				self.ptr_at(Rows::start(), col_start),
				self.nrows(),
				ncols,
				rs,
				cs,
			)
		}
	}

	/// returns the input matrix with the given shape after checking that it
	/// matches the current shape
	#[inline]
	#[track_caller]
	pub fn as_shape<V: Shape, H: Shape>(
		self,
		nrows: V,
		ncols: H,
	) -> MatRef<'a, T, V, H, RStride, CStride> {
		assert!(all(
			self.nrows().unbound() == nrows.unbound(),
			self.ncols().unbound() == ncols.unbound(),
		));
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				nrows,
				ncols,
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns the input matrix with the given row shape after checking that it
	/// matches the current row shape
	#[inline]
	#[track_caller]
	pub fn as_row_shape<V: Shape>(
		self,
		nrows: V,
	) -> MatRef<'a, T, V, Cols, RStride, CStride> {
		assert!(all(self.nrows().unbound() == nrows.unbound()));
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				nrows,
				self.ncols(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns the input matrix with the given column shape after checking that
	/// it matches the current column shape
	#[inline]
	#[track_caller]
	pub fn as_col_shape<H: Shape>(
		self,
		ncols: H,
	) -> MatRef<'a, T, Rows, H, RStride, CStride> {
		assert!(all(self.ncols().unbound() == ncols.unbound()));
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows(),
				ncols,
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns the input matrix with dynamic stride
	#[inline]
	pub fn as_dyn_stride(self) -> MatRef<'a, T, Rows, Cols, isize, isize> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows(),
				self.ncols(),
				self.row_stride().element_stride(),
				self.col_stride().element_stride(),
			)
		}
	}

	/// returns the input matrix with dynamic shape
	#[inline]
	pub fn as_dyn(self) -> MatRef<'a, T, usize, usize, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows().unbound(),
				self.ncols().unbound(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns the input matrix with dynamic row shape
	#[inline]
	pub fn as_dyn_rows(self) -> MatRef<'a, T, usize, Cols, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows().unbound(),
				self.ncols(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns the input matrix with dynamic column shape
	#[inline]
	pub fn as_dyn_cols(self) -> MatRef<'a, T, Rows, usize, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows(),
				self.ncols().unbound(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	/// returns a view over the row at the given index
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `row_idx < self.nrows()`
	#[inline]
	#[track_caller]
	pub fn row(self, i: Idx<Rows>) -> RowRef<'a, T, Cols, CStride> {
		assert!(i < self.nrows());
		unsafe {
			RowRef::from_raw_parts(
				self.ptr_at(i.into(), Cols::start()),
				self.ncols(),
				self.col_stride(),
			)
		}
	}

	/// returns a view over the column at the given index
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `col_idx < self.ncols()`
	#[inline]
	#[track_caller]
	pub fn col(self, j: Idx<Cols>) -> ColRef<'a, T, Rows, RStride> {
		assert!(j < self.ncols());
		unsafe {
			ColRef::from_raw_parts(
				self.ptr_at(Rows::start(), j.into()),
				self.nrows(),
				self.row_stride(),
			)
		}
	}

	/// returns an iterator over the columns of the matrix
	#[inline]
	pub fn col_iter(
		self,
	) -> impl 'a
	+ ExactSizeIterator
	+ DoubleEndedIterator<Item = ColRef<'a, T, Rows, RStride>>
	where
		Rows: 'a,
		Cols: 'a,
	{
		Cols::indices(Cols::start(), self.ncols().end())
			.map(move |j| self.col(j))
	}

	/// returns an iterator over the rows of the matrix
	#[inline]
	pub fn row_iter(
		self,
	) -> impl 'a
	+ ExactSizeIterator
	+ DoubleEndedIterator<Item = RowRef<'a, T, Cols, CStride>>
	where
		Rows: 'a,
		Cols: 'a,
	{
		Rows::indices(Rows::start(), self.nrows().end())
			.map(move |i| self.row(i))
	}

	/// returns a parallel iterator over the columns of the matrix
	#[inline]
	#[cfg(feature = "rayon")]
	pub fn par_col_iter(
		self,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = ColRef<'a, T, Rows, RStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		#[inline]
		fn col_fn<T, Rows: Shape, RStride: Stride, CStride: Stride>(
			col: MatRef<'_, T, Rows, usize, RStride, CStride>,
		) -> ColRef<'_, T, Rows, RStride> {
			col.col(0)
		}
		self.par_col_chunks(1).map(col_fn)
	}

	/// returns a parallel iterator over the rows of the matrix
	#[inline]
	#[cfg(feature = "rayon")]
	pub fn par_row_iter(
		self,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = RowRef<'a, T, Cols, CStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		self.transpose().par_col_iter().map(ColRef::transpose)
	}

	/// returns a parallel iterator that provides successive chunks of the
	/// columns of this matrix, with each having at most `chunk_size` columns
	///
	/// if the number of columns is a multiple of `chunk_size`, then all chunks
	/// have `chunk_size` columns
	///
	/// available in every build
	#[inline]
	#[track_caller]
	#[cfg(feature = "rayon")]
	pub fn par_col_chunks(
		self,
		chunk_size: usize,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = MatRef<'a, T, Rows, usize, RStride, CStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		let this = self.as_dyn_cols();
		assert!(chunk_size > 0);
		let chunk_count = this.ncols().msrv_div_ceil(chunk_size);
		(0..chunk_count).into_par_iter().map(move |chunk_idx| {
			let pos = chunk_size * chunk_idx;
			this.subcols(pos, Ord::min(chunk_size, this.ncols() - pos))
		})
	}

	/// returns a parallel iterator that provides exactly `count` successive
	/// chunks of the columns of this matrix
	///
	/// available in every build
	#[inline]
	#[track_caller]
	#[cfg(feature = "rayon")]
	pub fn par_col_partition(
		self,
		count: usize,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = MatRef<'a, T, Rows, usize, RStride, CStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		let this = self.as_dyn_cols();
		assert!(count > 0);
		(0..count).into_par_iter().map(move |chunk_idx| {
			let (start, len) = crate::utils::thread::par_split_indices(
				this.ncols(),
				chunk_idx,
				count,
			);
			this.subcols(start, len)
		})
	}

	/// returns a parallel iterator that provides successive chunks of the rows
	/// of this matrix, with each having at most `chunk_size` rows
	///
	/// if the number of rows is a multiple of `chunk_size`, then all chunks
	/// have `chunk_size` rows
	///
	/// available in every build
	#[inline]
	#[track_caller]
	#[cfg(feature = "rayon")]
	pub fn par_row_chunks(
		self,
		chunk_size: usize,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = MatRef<'a, T, usize, Cols, RStride, CStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		self.transpose()
			.par_col_chunks(chunk_size)
			.map(MatRef::transpose)
	}

	/// returns a parallel iterator that provides exactly `count` successive
	/// chunks of the rows of this matrix
	///
	/// available in every build
	#[inline]
	#[track_caller]
	#[cfg(feature = "rayon")]
	pub fn par_row_partition(
		self,
		count: usize,
	) -> impl 'a
	+ rayon::iter::IndexedParallelIterator<
		Item = MatRef<'a, T, usize, Cols, RStride, CStride>,
	>
	where
		T: Sync,
		Rows: 'a,
		Cols: 'a,
	{
		use rayon::prelude::*;
		self.transpose()
			.par_col_partition(count)
			.map(MatRef::transpose)
	}

	/// returns a reference to the first row and a view over the remaining ones
	/// if the matrix has at least one row, otherwise `None`
	#[inline]
	pub fn split_first_row(
		self,
	) -> Option<(
		RowRef<'a, T, Cols, CStride>,
		MatRef<'a, T, usize, Cols, RStride, CStride>,
	)> {
		if let Some(i0) = self.nrows().idx_inc(1) {
			let (head, tail) = self.split_at_row(i0);
			Some((head.row(0), tail))
		} else {
			None
		}
	}

	/// returns a reference to the first column and a view over the remaining
	/// ones if the matrix has at least one column, otherwise `None`
	#[inline]
	pub fn split_first_col(
		self,
	) -> Option<(
		ColRef<'a, T, Rows, RStride>,
		MatRef<'a, T, Rows, usize, RStride, CStride>,
	)> {
		if let Some(i0) = self.ncols().idx_inc(1) {
			let (head, tail) = self.split_at_col(i0);
			Some((head.col(0), tail))
		} else {
			None
		}
	}

	/// returns a reference to the last row and a view over the remaining ones
	/// if the matrix has at least one row, otherwise `None`
	#[inline]
	pub fn split_last_row(
		self,
	) -> Option<(
		RowRef<'a, T, Cols, CStride>,
		MatRef<'a, T, usize, Cols, RStride, CStride>,
	)> {
		if self.nrows().unbound() > 0 {
			let i0 = self.nrows().checked_idx_inc(self.nrows().unbound() - 1);
			let (head, tail) = self.split_at_row(i0);
			Some((tail.row(0), head))
		} else {
			None
		}
	}

	/// returns a reference to the last column and a view over the remaining
	/// ones if the matrix has at least one column, otherwise `None`
	#[inline]
	pub fn split_last_col(
		self,
	) -> Option<(
		ColRef<'a, T, Rows, RStride>,
		MatRef<'a, T, Rows, usize, RStride, CStride>,
	)> {
		if self.ncols().unbound() > 0 {
			let i0 = self.ncols().checked_idx_inc(self.ncols().unbound() - 1);
			let (head, tail) = self.split_at_col(i0);
			Some((tail.col(0), head))
		} else {
			None
		}
	}

	/// returns a view over the matrix with a static column stride equal to
	/// `+1`, or `None` otherwise
	#[inline]
	pub fn try_as_row_major(
		self,
	) -> Option<MatRef<'a, T, Rows, Cols, RStride, ContiguousFwd>> {
		if self.col_stride().element_stride() == 1 {
			Some(unsafe {
				MatRef::from_raw_parts(
					self.as_ptr(),
					self.nrows(),
					self.ncols(),
					self.row_stride(),
					ContiguousFwd,
				)
			})
		} else {
			None
		}
	}

	#[doc(hidden)]
	#[inline]
	pub fn bind<'M, 'N>(
		self,
		row: Guard<'M>,
		col: Guard<'N>,
	) -> MatRef<'a, T, Dim<'M>, Dim<'N>, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows().bind(row),
				self.ncols().bind(col),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	#[doc(hidden)]
	#[inline]
	pub fn bind_r<'M>(
		self,
		row: Guard<'M>,
	) -> MatRef<'a, T, Dim<'M>, Cols, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows().bind(row),
				self.ncols(),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	#[doc(hidden)]
	#[inline]
	pub fn bind_c<'N>(
		self,
		col: Guard<'N>,
	) -> MatRef<'a, T, Rows, Dim<'N>, RStride, CStride> {
		unsafe {
			MatRef::from_raw_parts(
				self.as_ptr(),
				self.nrows(),
				self.ncols().bind(col),
				self.row_stride(),
				self.col_stride(),
			)
		}
	}

	#[doc(hidden)]
	#[inline]
	pub unsafe fn const_cast(
		self,
	) -> MatMut<'a, T, Rows, Cols, RStride, CStride> {
		MatMut::from_raw_parts_mut(
			self.as_ptr() as *mut T,
			self.nrows(),
			self.ncols(),
			self.row_stride(),
			self.col_stride(),
		)
	}

	/// returns a view over the matrix with a static row stride equal to `+1`,
	/// or `None` otherwise
	#[inline]
	pub fn try_as_col_major(
		self,
	) -> Option<MatRef<'a, T, Rows, Cols, ContiguousFwd, CStride>> {
		if self.row_stride().element_stride() == 1 {
			Some(unsafe {
				MatRef::from_raw_parts(
					self.as_ptr(),
					self.nrows(),
					self.ncols(),
					ContiguousFwd,
					self.col_stride(),
				)
			})
		} else {
			None
		}
	}

	/// returns references to the element at the given index, or submatrices if
	/// either `row` or `col` is a range, with bound checks
	///
	/// # panics
	/// the function panics if any of the following conditions are violated:
	/// * `row` must be contained in `[0, self.nrows())`
	/// * `col` must be contained in `[0, self.ncols())`
	#[track_caller]
	#[inline]
	pub fn get<RowRange, ColRange>(
		self,
		row: RowRange,
		col: ColRange,
	) -> <MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
		RowRange,
		ColRange,
	>>::Target
	where
		MatRef<'a, T, Rows, Cols, RStride, CStride>:
			MatIndex<RowRange, ColRange>,
	{
		<MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
			RowRange,
			ColRange,
		>>::get(self, row, col)
	}

	/// equivalent to `self.get(row, ..)`
	#[track_caller]
	#[inline]
	pub fn get_r<RowRange>(
		self,
		row: RowRange,
	) -> <MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
		RowRange,
		core::ops::RangeFull,
	>>::Target
	where
		MatRef<'a, T, Rows, Cols, RStride, CStride>:
			MatIndex<RowRange, core::ops::RangeFull>,
	{
		<MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
			RowRange,
			core::ops::RangeFull,
		>>::get(self, row, ..)
	}

	/// equivalent to `self.get(.., col)`
	#[track_caller]
	#[inline]
	pub fn get_c<ColRange>(
		self,
		col: ColRange,
	) -> <MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
		core::ops::RangeFull,
		ColRange,
	>>::Target
	where
		MatRef<'a, T, Rows, Cols, RStride, CStride>:
			MatIndex<core::ops::RangeFull, ColRange>,
	{
		<MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
			core::ops::RangeFull,
			ColRange,
		>>::get(self, .., col)
	}

	/// returns references to the element at the given index, or submatrices if
	/// either `row` or `col` is a range, without bound checks
	///
	/// # safety
	/// the behavior is undefined if any of the following conditions are
	/// violated:
	/// * `row` must be contained in `[0, self.nrows())`
	/// * `col` must be contained in `[0, self.ncols())`
	#[track_caller]
	#[inline]
	pub unsafe fn get_unchecked<RowRange, ColRange>(
		self,
		row: RowRange,
		col: ColRange,
	) -> <MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
		RowRange,
		ColRange,
	>>::Target
	where
		MatRef<'a, T, Rows, Cols, RStride, CStride>:
			MatIndex<RowRange, ColRange>,
	{
		unsafe {
			<MatRef<'a, T, Rows, Cols, RStride, CStride> as MatIndex<
				RowRange,
				ColRange,
			>>::get_unchecked(self, row, col)
		}
	}

	#[inline]
	pub(crate) fn __at(self, (i, j): (Idx<Rows>, Idx<Cols>)) -> &'a T {
		self.at(i, j)
	}
}
impl<
	T,
	Rows: Shape,
	Cols: Shape,
	RStride: Stride,
	CStride: Stride,
	Inner: for<'short> Reborrow<
			'short,
			Target = Ref<'short, T, Rows, Cols, RStride, CStride>,
		>,
> generic::Mat<Inner>
{
	/// returns a view over `self`
	#[inline]
	pub fn as_ref(&self) -> MatRef<'_, T, Rows, Cols, RStride, CStride> {
		self.rb()
	}

	/// returns a newly allocated matrix holding the cloned values of `self`
	#[inline]
	pub fn cloned(&self) -> Mat<T, Rows, Cols>
	where
		T: Clone,
	{
		fn imp<'M, 'N, T: Clone, RStride: Stride, CStride: Stride>(
			this: MatRef<'_, T, Dim<'M>, Dim<'N>, RStride, CStride>,
		) -> Mat<T, Dim<'M>, Dim<'N>> {
			Mat::from_fn(this.nrows(), this.ncols(), |i, j| {
				this.at(i, j).clone()
			})
		}
		let this = self.rb();
		with_dim!(M, this.nrows().unbound());
		with_dim!(N, this.ncols().unbound());
		imp(this.as_shape(M, N)).into_shape(this.nrows(), this.ncols())
	}

	/// returns a newly allocated matrix holding the (possibly conjugated)
	/// values of `self`
	#[inline]
	pub fn to_owned(&self) -> Mat<T::Canonical, Rows, Cols>
	where
		T: Conjugate,
	{
		fn imp<'M, 'N, T, RStride: Stride, CStride: Stride>(
			this: MatRef<'_, T, Dim<'M>, Dim<'N>, RStride, CStride>,
		) -> Mat<T::Canonical, Dim<'M>, Dim<'N>>
		where
			T: Conjugate,
		{
			// `Mat::from_fn` fills column by column, so for a source whose
			// columns are not the contiguous direction every read steps by the
			// source's row stride and costs its own cache line -- on `512x512`
			// f64 that is 1.01 ms against 0.057, and this is the first thing
			// `partial_piv_lu`, `full_piv_lu` and `Qr::new` do with their
			// argument. Stage the copy in tiles instead, as the row-major
			// factorisations do.
			//
			// Only an already-canonical source: the tile copy reads the source's
			// memory as `T::Canonical`, which is a reinterpretation and not the
			// conjugation `Conj::apply` below performs, so a conjugated view (an
			// `adjoint().to_owned()`, which the sparse factorisations use) has to
			// keep the elementwise path.
			let src = this.canonical();
			if T::IS_CANONICAL
				&& src.row_stride().element_stride().unsigned_abs() != 1
			{
				let mut out = Mat::<T::Canonical, Dim<'M>, Dim<'N>>::zeros(
					this.nrows(),
					this.ncols(),
				);
				crate::linalg::copy_across_layouts(
					out.as_mut().as_dyn_stride_mut().as_dyn_mut(),
					src.as_dyn_stride().as_dyn(),
					crate::Par::Seq,
				);
				return out;
			}
			Mat::from_fn(this.nrows(), this.ncols(), |i, j| {
				Conj::apply::<T>(this.at(i, j))
			})
		}
		let this = self.rb();
		with_dim!(M, this.nrows().unbound());
		with_dim!(N, this.ncols().unbound());
		imp(this.as_shape(M, N)).into_shape(this.nrows(), this.ncols())
	}

	/// returns the maximum norm of `self`
	#[inline]
	pub fn norm_max(&self) -> Real<T>
	where
		T: Conjugate,
	{
		linalg::reductions::norm_max::norm_max(
			self.rb().canonical().as_dyn_stride().as_dyn(),
		)
	}

	/// returns the l2 norm of `self`
	#[inline]
	pub fn norm_l2(&self) -> Real<T>
	where
		T: Conjugate,
	{
		linalg::reductions::norm_l2::norm_l2(
			self.rb().canonical().as_dyn_stride().as_dyn(),
		)
	}

	/// returns the squared l2 norm of `self`
	#[inline]
	pub fn squared_norm_l2(&self) -> Real<T>
	where
		T: Conjugate,
	{
		linalg::reductions::norm_l2_sqr::norm_l2_sqr(
			self.rb().canonical().as_dyn_stride().as_dyn(),
		)
	}

	/// returns the l1 norm of `self`
	#[inline]
	pub fn norm_l1(&self) -> Real<T>
	where
		T: Conjugate,
	{
		linalg::reductions::norm_l1::norm_l1(
			self.rb().canonical().as_dyn_stride().as_dyn(),
		)
	}

	/// returns the sum of the elements of `self`
	#[inline]
	pub fn sum(&self) -> T::Canonical
	where
		T: Conjugate,
	{
		let val = linalg::reductions::sum::sum(
			self.rb().canonical().as_dyn_stride().as_dyn(),
		);
		if const { Conj::get::<T>().is_conj() } {
			val.conj()
		} else {
			val
		}
	}

	/// returns the determinant of `self`
	#[inline]
	pub fn determinant(&self, par: Par<'_>) -> T::Canonical
	where
		T: Conjugate,
	{
		let det = linalg::reductions::determinant::determinant(
			self.rb().canonical().as_dyn_stride().as_dyn(),
			par,
		);
		if const { T::IS_CANONICAL } {
			det
		} else {
			det.conj()
		}
	}

	/// kronecker product of two matrices
	///
	/// the kronecker product of two matrices $A$ and $B$ is a block matrix
	/// $B$ with the following structure:
	///
	/// ```text
	/// C = [ a[(0, 0)] * B    , a[(0, 1)] * B    , ... , a[(0, n-1)] * B    ]
	///     [ a[(1, 0)] * B    , a[(1, 1)] * B    , ... , a[(1, n-1)] * B    ]
	///     [ ...              , ...              , ... , ...              ]
	///     [ a[(m-1, 0)] * B  , a[(m-1, 1)] * B  , ... , a[(m-1, n-1)] * B  ]
	/// ```
	///
	/// # panics
	///
	/// panics if `dst` does not have the correct dimensions. the dimensions
	/// of `dst` must be `A.nrows() * B.nrows()` by `A.ncols() * B.ncols()`.
	///
	/// # example
	///
	/// ```
	/// use faer::linalg::kron::kron;
	/// use faer::{Mat, mat};
	/// let a = mat![[1.0, 2.0], [3.0, 4.0]];
	/// let b = mat![[0.0, 5.0], [6.0, 7.0]];
	/// let c = mat![
	/// 	[0.0, 5.0, 0.0, 10.0],
	/// 	[6.0, 7.0, 12.0, 14.0],
	/// 	[0.0, 15.0, 0.0, 20.0],
	/// 	[18.0, 21.0, 24.0, 28.0],
	/// ];
	/// let mut dst = Mat::zeros(4, 4);
	/// kron(dst.as_mut(), a.as_ref(), b.as_ref());
	/// assert_eq!(dst, c);
	/// ```
	#[inline]
	pub fn kron(
		&self,
		rhs: impl AsMatRef<T: Conjugate<Canonical = T::Canonical>>,
	) -> Mat<T::Canonical>
	where
		T: Conjugate,
	{
		fn imp<T: ComplexField>(
			lhs: MatRef<'_, impl Conjugate<Canonical = T>>,
			rhs: MatRef<'_, impl Conjugate<Canonical = T>>,
		) -> Mat<T> {
			let mut out = Mat::zeros(
				lhs.nrows() * rhs.nrows(),
				lhs.ncols() * rhs.ncols(),
			);
			linalg::kron::kron(out.rb_mut(), lhs, rhs);
			out
		}
		imp(
			self.rb().as_dyn().as_dyn_stride(),
			rhs.as_mat_ref().as_dyn().as_dyn_stride(),
		)
	}

	/// returns `true` if all of the elements of `self` are finite.
	/// otherwise returns `false`.
	#[inline]
	pub fn is_all_finite(&self) -> bool
	where
		T: Conjugate,
	{
		fn imp<T: ComplexField>(A: MatRef<'_, T>) -> bool {
			with_dim!({
				let M = A.nrows();
				let N = A.ncols();
			});
			let A = A.as_shape(M, N);
			for j in N.indices() {
				for i in M.indices() {
					if !A[(i, j)].is_finite() {
						return false;
					}
				}
			}
			true
		}
		imp(self.rb().as_dyn().as_dyn_stride().canonical())
	}

	/// returns `true` if `self` is the identity matrix.
	/// otherwise returns `false`.
	#[inline]
	pub fn is_identity(&self) -> bool
	where
		T: Conjugate,
	{
		fn imp<T: ComplexField>(A: MatRef<'_, T>) -> bool {
			with_dim!({
				let M = A.nrows();
				let N = A.ncols();
			});
			let A = A.as_shape(M, N);
			for j in N.indices() {
				for i in M.indices() {
					if *i == *j {
						if A[(i, j)] != one() {
							return false;
						}
					} else {
						if A[(i, j)] != zero() {
							return false;
						}
					}
				}
			}
			true
		}
		imp(self.rb().as_dyn().as_dyn_stride().canonical())
	}

	/// returns `true` if any of the elements of `self` is `NaN`.
	/// otherwise returns `false`.
	#[inline]
	pub fn has_nan(&self) -> bool
	where
		T: Conjugate,
	{
		fn imp<T: ComplexField>(A: MatRef<'_, T>) -> bool {
			with_dim!({
				let M = A.nrows();
				let N = A.ncols();
			});
			let A = A.as_shape(M, N);
			for j in N.indices() {
				for i in M.indices() {
					if A[(i, j)].is_nan() {
						return true;
					}
				}
			}
			false
		}
		imp(self.rb().as_dyn().as_dyn_stride().canonical())
	}
}
impl<'a, T, Dim: Shape, RStride: Stride, CStride: Stride>
	MatRef<'a, T, Dim, Dim, RStride, CStride>
{
	/// returns the diagonal of the matrix
	#[inline]
	pub fn diagonal(self) -> DiagRef<'a, T, Dim, isize> {
		let k = Ord::min(self.nrows(), self.ncols());
		DiagRef {
			0: crate::diag::Ref {
				inner: unsafe {
					ColRef::from_raw_parts(
						self.as_ptr(),
						k,
						self.row_stride().element_stride()
							+ self.col_stride().element_stride(),
					)
				},
			},
		}
	}
}
impl<'ROWS, 'COLS, 'a, T, RStride: Stride, CStride: Stride>
	MatRef<'a, T, Dim<'ROWS>, Dim<'COLS>, RStride, CStride>
{
	#[doc(hidden)]
	#[inline]
	pub fn split_with<'TOP, 'BOT, 'LEFT, 'RIGHT>(
		self,
		row: Partition<'TOP, 'BOT, 'ROWS>,
		col: Partition<'LEFT, 'RIGHT, 'COLS>,
	) -> (
		MatRef<'a, T, Dim<'TOP>, Dim<'LEFT>, RStride, CStride>,
		MatRef<'a, T, Dim<'TOP>, Dim<'RIGHT>, RStride, CStride>,
		MatRef<'a, T, Dim<'BOT>, Dim<'LEFT>, RStride, CStride>,
		MatRef<'a, T, Dim<'BOT>, Dim<'RIGHT>, RStride, CStride>,
	) {
		let (a, b, c, d) = self.split_at(row.midpoint(), col.midpoint());
		(
			a.as_shape(row.head, col.head),
			b.as_shape(row.head, col.tail),
			c.as_shape(row.tail, col.head),
			d.as_shape(row.tail, col.tail),
		)
	}
}
impl<'ROWS, 'a, T, Cols: Shape, RStride: Stride, CStride: Stride>
	MatRef<'a, T, Dim<'ROWS>, Cols, RStride, CStride>
{
	#[doc(hidden)]
	#[inline]
	pub fn split_rows_with<'TOP, 'BOT>(
		self,
		row: Partition<'TOP, 'BOT, 'ROWS>,
	) -> (
		MatRef<'a, T, Dim<'TOP>, Cols, RStride, CStride>,
		MatRef<'a, T, Dim<'BOT>, Cols, RStride, CStride>,
	) {
		let (a, b) = self.split_at_row(row.midpoint());
		(a.as_row_shape(row.head), b.as_row_shape(row.tail))
	}
}
impl<'COLS, 'a, T, Rows: Shape, RStride: Stride, CStride: Stride>
	MatRef<'a, T, Rows, Dim<'COLS>, RStride, CStride>
{
	#[doc(hidden)]
	#[inline]
	pub fn split_cols_with<'LEFT, 'RIGHT>(
		self,
		col: Partition<'LEFT, 'RIGHT, 'COLS>,
	) -> (
		MatRef<'a, T, Rows, Dim<'LEFT>, RStride, CStride>,
		MatRef<'a, T, Rows, Dim<'RIGHT>, RStride, CStride>,
	) {
		let (a, b) = self.split_at_col(col.midpoint());
		(a.as_col_shape(col.head), b.as_col_shape(col.tail))
	}
}
impl<
	'a,
	T: core::fmt::Debug,
	Rows: Shape,
	Cols: Shape,
	RStride: Stride,
	CStride: Stride,
> core::fmt::Debug for Ref<'a, T, Rows, Cols, RStride, CStride>
{
	fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
		fn imp<'M, 'N, T: core::fmt::Debug>(
			this: MatRef<'_, T, Dim<'M>, Dim<'N>>,
			f: &mut core::fmt::Formatter<'_>,
		) -> core::fmt::Result {
			writeln!(f, "[")?;
			for i in this.nrows().indices() {
				this.row(i).fmt(f)?;
				f.write_str(",\n")?;
			}
			write!(f, "]")
		}
		let this = generic::Mat::from_inner_ref(self);
		with_dim!(M, this.nrows().unbound());
		with_dim!(N, this.ncols().unbound());
		imp(this.as_shape(M, N).as_dyn_stride(), f)
	}
}
impl<'a, T: RealField> MatRef<'a, T, usize, usize> {
	pub(crate) fn internal_max(self) -> Option<T> {
		if self.nrows().unbound() == 0 || self.ncols().unbound() == 0 {
			return None;
		}
		let mut max_val = self.get(0, 0);
		let this = if self.row_stride().unsigned_abs() == 1 {
			self.transpose()
		} else {
			self
		};
		let this = if this.col_stride() > 0 {
			this
		} else {
			this.reverse_cols()
		};
		this.row_iter().for_each(|row| {
			row.iter().for_each(|val| {
				if val > max_val {
					max_val = &val;
				}
			});
		});
		Some((*max_val).clone())
	}

	pub(crate) fn internal_min(self) -> Option<T> {
		if self.nrows().unbound() == 0 || self.ncols().unbound() == 0 {
			return None;
		}
		let mut min_val = self.get(0, 0);
		let this = if self.row_stride().unsigned_abs() == 1 {
			self.transpose()
		} else {
			self
		};
		let this = if this.col_stride() > 0 {
			this
		} else {
			this.reverse_cols()
		};
		this.row_iter().for_each(|row| {
			row.iter().for_each(|val| {
				if val < min_val {
					min_val = &val;
				}
			});
		});
		Some((*min_val).clone())
	}
}
impl<'a, T: RealField, Rows: Shape, Cols: Shape> MatRef<'a, T, Rows, Cols> {
	/// Returns the maximum element in the matrix
	///
	/// # Returns
	///
	/// * `Option<T>` - The maximum element in the matrix, or `None` if the
	///   matrix is empty
	///
	/// # Examples
	///
	/// ```
	/// use faer::{Mat, mat};
	/// let m = mat![[1.0, 5.0, 3.0], [4.0, 2.0, 9.0], [7.0, 8.0, 6.0],];
	/// assert_eq!(m.max(), Some(9.0));
	/// let empty: Mat<f64> = Mat::new();
	/// assert_eq!(empty.max(), None);
	/// ```
	pub fn max(self) -> Option<T> {
		MatRef::internal_max(self.as_dyn())
	}

	/// Returns the minimum element in the matrix
	///
	/// # Returns
	///
	/// * `Option<T>` - The minimum element in the matrix, or `None` if the
	///   matrix is empty
	///
	/// # Examples
	///
	/// ```
	/// use faer::{Mat, mat};
	/// let m = mat![[1.0, 5.0, 3.0], [4.0, 2.0, 9.0], [7.0, 8.0, 6.0],];
	/// assert_eq!(m.min(), Some(1.0));
	/// let empty: Mat<f64> = Mat::new();
	/// assert_eq!(empty.min(), None);
	/// ```
	pub fn min(self) -> Option<T> {
		MatRef::internal_min(self.as_dyn())
	}
}
#[cfg(test)]
mod tests {
	use super::*;
	#[test]
	fn test_min() {
		let m = mat![[1.0, 5.0, 3.0], [4.0, 2.0, 9.0], [7.0, 8.0, 6.0],];
		assert_eq!(m.as_ref().min(), Some(1.0));
		let empty: Mat<f64> = Mat::new();
		assert_eq!(empty.as_ref().min(), None);
	}
	#[test]
	fn test_max() {
		let m = mat![[1.0, 5.0, 3.0], [4.0, 2.0, 9.0], [7.0, 8.0, 6.0],];
		assert_eq!(m.as_ref().max(), Some(9.0));
		let empty: Mat<f64> = Mat::new();
		assert_eq!(empty.as_ref().max(), None);
	}
	#[test]
	fn test_cloned_does_not_replicate_the_first_heap_value() {
		use std::{cell::Cell, rc::Rc};
		struct Value {
			text: String,
			clones: Rc<Cell<usize>>,
		}
		impl Clone for Value {
			fn clone(&self) -> Self {
				self.clones.set(self.clones.get() + 1);
				Self {
					text: self.text.clone(),
					clones: self.clones.clone(),
				}
			}
		}
		let source: Vec<_> = (0..6)
			.map(|i| Value {
				text: if i == 0 {
					"x".repeat(1024)
				} else {
					String::new()
				},
				clones: Rc::new(Cell::new(0)),
			})
			.collect();
		let view = MatRef::from_row_major_slice(&source, 2, 3);
		let copied = view.cloned();
		for i in 0..2 {
			for j in 0..3 {
				let original = &source[3 * i + j];
				assert_eq!(copied[(i, j)].text, original.text);
				assert_eq!(original.clones.get(), 1);
			}
		}
	}

	/// Cloning preserves values for column/row-major and general strides.
	#[test]
	fn test_cloned_from_a_strided_source() {
		fn check(m: usize, n: usize, (rs, cs): (isize, isize)) {
			// The buffer has to be long enough for the strides the view is
			// built with, and the view starts far enough in that a negative
			// stride stays inside it.
			let span = if m == 0 || n == 0 {
				0
			} else {
				(m - 1) * rs.unsigned_abs() + (n - 1) * cs.unsigned_abs() + 1
			};
			let raw: Vec<f64> = (0..span).map(|k| (k % 37) as f64 * 0.5 - 3.0).collect();
			let base = if m == 0 || n == 0 {
				0
			} else {
				(if rs < 0 { (m - 1) * rs.unsigned_abs() } else { 0 })
					+ (if cs < 0 { (n - 1) * cs.unsigned_abs() } else { 0 })
			};
			let src = unsafe {
				MatRef::from_raw_parts(
					raw.as_ptr().add(base),
					m,
					n,
					rs,
					cs,
				)
			};
			let expected = Mat::from_fn(m, n, |i, j| src.at(i, j).clone());
			let got = src.cloned();
			if got.nrows() != m || got.ncols() != n {
				panic!("cloned {m}x{n}: shape is {}x{}", got.nrows(), got.ncols());
			}
			for j in 0..n {
				for i in 0..m {
					if got[(i, j)] != expected[(i, j)] {
						panic!(
							"cloned {m}x{n} strides {rs}x{cs}: ({i}, {j}) is {} against {}",
							got[(i, j)],
							expected[(i, j)]
						);
					}
				}
			}
		}
		for (m, n) in [
			(0, 0),
			(1, 1),
			(1, 5),
			(5, 1),
			(16, 16),
			(17, 15),
			(15, 17),
			(64, 3),
			(3, 64),
			(65, 33),
		] {
			check(m, n, (n as isize, 1));
			check(m, n, (1, m as isize));
			check(m, n, (2 * n as isize, 3));
			check(m, n, (-(n as isize), 1));
		}
	}
}