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    /*!
    # nalgebra

    **nalgebra** is a linear algebra library written for Rust targeting:

    * General-purpose linear algebra (still lacks a lot of features…)
    * RealField time computer graphics.
    * RealField time computer physics.

    ## Using **nalgebra**
    You will need the last stable build of the [rust compiler](http://www.rust-lang.org)
    and the official package manager: [cargo](https://github.com/rust-lang/cargo).

    Simply add the following to your `Cargo.toml` file:

    ```.ignore
    [dependencies]
    nalgebra = "0.18"
    ```

    Most useful functionalities of **nalgebra** are grouped in the root module `nalgebra::`.

    However, the recommended way to use **nalgebra** is to import types and traits
    explicitly, and call free-functions using the `na::` prefix:

    ```.rust
    #[macro_use]
    extern crate approx; // For the macro relative_eq!
    extern crate nalgebra as na;
    use na::{Vector3, Rotation3};

    fn main() {
        let axis  = Vector3::x_axis();
        let angle = 1.57;
        let b     = Rotation3::from_axis_angle(&axis, angle);

        relative_eq!(b.axis().unwrap(), axis);
        relative_eq!(b.angle(), angle);
    }
    ```

    ## Features
    **nalgebra** is meant to be a general-purpose, low-dimensional, linear algebra library, with
    an optimized set of tools for computer graphics and physics. Those features include:

    * A single parametrizable type `Matrix` for vectors, (square or rectangular) matrices, and slices
      with dimensions known either at compile-time (using type-level integers) or at runtime.
    * Matrices and vectors with compile-time sizes are statically allocated while dynamic ones are
      allocated on the heap.
    * Convenient aliases for low-dimensional matrices and vectors: `Vector1` to `Vector6` and
      `Matrix1x1` to `Matrix6x6`, including rectangular matrices like `Matrix2x5`.
    * Points sizes known at compile time, and convenience aliases:: `Point1` to `Point6`.
    * Translation (seen as a transformation that composes by multiplication): `Translation2`,
      `Translation3`.
    * Rotation matrices: `Rotation2`, `Rotation3`.
    * Quaternions: `Quaternion`, `UnitQuaternion` (for 3D rotation).
    * Unit complex numbers can be used for 2D rotation: `UnitComplex`.
    * Algebraic entities with a norm equal to one: `Unit<T>`, e.g., `Unit<Vector3<f32>>`.
    * Isometries (translation ⨯ rotation): `Isometry2`, `Isometry3`
    * Similarity transformations (translation ⨯ rotation ⨯ uniform scale): `Similarity2`, `Similarity3`.
    * Affine transformations stored as an homogeneous matrix: `Affine2`, `Affine3`.
    * Projective (i.e. invertible) transformations stored as an homogeneous matrix: `Projective2`,
      `Projective3`.
    * General transformations that does not have to be invertible, stored as an homogeneous matrix:
      `Transform2`, `Transform3`.
    * 3D projections for computer graphics: `Perspective3`, `Orthographic3`.
    * Matrix factorizations: `Cholesky`, `QR`, `LU`, `FullPivLU`, `SVD`, `Schur`, `Hessenberg`, `SymmetricEigen`.
    * Insertion and removal of rows of columns of a matrix.
    * Implements traits from the [alga](https://crates.io/crates/alga) crate for
      generic programming.
    */

    // #![feature(plugin)]
    //
    // #![plugin(clippy)]

    #![deny(non_camel_case_types)]
    #![deny(unused_parens)]
    #![deny(non_upper_case_globals)]
    #![deny(unused_qualifications)]
    #![deny(unused_results)]
    #![deny(missing_docs)]
    #![warn(incoherent_fundamental_impls)]
    #![doc(
        html_favicon_url = "http://nalgebra.org/img/favicon.ico",
        html_root_url = "http://nalgebra.org/rustdoc"
    )]
    #![cfg_attr(not(feature = "std"), no_std)]
    #![cfg_attr(all(feature = "alloc", not(feature = "std")), feature(alloc))]

    #[cfg(feature = "arbitrary")]
    extern crate quickcheck;

    #[cfg(feature = "serde")]
    extern crate serde;
    #[cfg(feature = "serde")]
    #[macro_use]
    extern crate serde_derive;

    #[cfg(feature = "abomonation-serialize")]
    extern crate abomonation;

    #[cfg(feature = "mint")]
    extern crate mint;

    #[macro_use]
    extern crate approx;
    extern crate generic_array;
    #[cfg(feature = "std")]
    extern crate matrixmultiply;
    extern crate num_complex;
    extern crate num_traits as num;
    extern crate rand;
    extern crate typenum;

    extern crate alga;

    #[cfg(all(feature = "alloc", not(feature = "std")))]
    extern crate alloc;

    #[cfg(not(feature = "std"))]
    extern crate core as std;

    #[cfg(feature = "io")]
    extern crate pest;
    #[macro_use]
    #[cfg(feature = "io")]
    extern crate pest_derive;

    pub mod base;
    #[cfg(feature = "debug")]
    pub mod debug;
    pub mod geometry;
    #[cfg(feature = "io")]
    pub mod io;
    pub mod linalg;
    #[cfg(feature = "sparse")]
    pub mod sparse;

    #[cfg(feature = "std")]
    #[deprecated(
        note = "The 'core' module is being renamed to 'base' to avoid conflicts with the 'core' crate."
    )]
    pub use base as core;
    pub use crate::base::*;
    pub use crate::geometry::*;
    pub use crate::linalg::*;
    #[cfg(feature = "sparse")]
    pub use crate::sparse::*;

    use std::cmp::{self, Ordering, PartialOrd};

    use alga::general::{
        Additive, AdditiveGroup, Identity, TwoSidedInverse, JoinSemilattice, Lattice, MeetSemilattice,
        Multiplicative, SupersetOf,
    };
    use alga::linear::SquareMatrix as AlgaSquareMatrix;
    use alga::linear::{EuclideanSpace, FiniteDimVectorSpace, InnerSpace, NormedSpace};
    use num::Signed;

    pub use alga::general::{Id, RealField, ComplexField};
    #[allow(deprecated)]
    pub use alga::general::Real;
    pub use num_complex::Complex;

    /*
     *
     * Multiplicative identity.
     *
     */
    /// Gets the ubiquitous multiplicative identity element.
    ///
    /// Same as `Id::new()`.
    #[deprecated(note = "use `Id::new()` instead.")]
    #[inline]
    pub fn id() -> Id {
        Id::new()
    }

    /// Gets the multiplicative identity element.
    ///
    /// # See also:
    ///
    /// * [`origin`](../nalgebra/fn.origin.html)
    /// * [`zero`](fn.zero.html)
    #[inline]
    pub fn one<T: Identity<Multiplicative>>() -> T {
        T::identity()
    }

    /// Gets the additive identity element.
    ///
    /// # See also:
    ///
    /// * [`one`](fn.one.html)
    /// * [`origin`](../nalgebra/fn.origin.html)
    #[inline]
    pub fn zero<T: Identity<Additive>>() -> T {
        T::identity()
    }

    /// Gets the origin of the given point.
    ///
    /// # See also:
    ///
    /// * [`one`](fn.one.html)
    /// * [`zero`](fn.zero.html)
    ///
    /// # Deprecated
    /// Use [Point::origin] instead.
    ///
    /// Or, use [EuclideanSpace::origin](https://docs.rs/alga/0.7.2/alga/linear/trait.EuclideanSpace.html#tymethod.origin).
    #[deprecated(note = "use `Point::origin` instead")]
    #[inline]
    pub fn origin<P: EuclideanSpace>() -> P {
        P::origin()
    }

    /*
     *
     * Dimension
     *
     */
    /// The dimension of the given algebraic entity seen as a vector space.
    #[inline]
    pub fn dimension<V: FiniteDimVectorSpace>() -> usize {
        V::dimension()
    }

    /*
     *
     * Ordering
     *
     */
    // XXX: this is very naive and could probably be optimized for specific types.
    // XXX: also, we might just want to use divisions, but assuming `val` is usually not far from `min`
    // or `max`, would it still be more efficient?
    /// Wraps `val` into the range `[min, max]` using modular arithmetics.
    ///
    /// The range must not be empty.
    #[inline]
    pub fn wrap<T>(mut val: T, min: T, max: T) -> T
    where T: Copy + PartialOrd + AdditiveGroup {
        assert!(min < max, "Invalid wrapping bounds.");
        let width = max - min;

        if val < min {
            val += width;

            while val < min {
                val += width
            }

            val
        } else if val > max {
            val -= width;

            while val > max {
                val -= width
            }

            val
        } else {
            val
        }
    }

    /// Returns a reference to the input value clamped to the interval `[min, max]`.
    ///
    /// In particular:
    ///     * If `min < val < max`, this returns `val`.
    ///     * If `val <= min`, this returns `min`.
    ///     * If `val >= max`, this returns `max`.
    #[inline]
    pub fn clamp<T: PartialOrd>(val: T, min: T, max: T) -> T {
        if val > min {
            if val < max {
                val
            } else {
                max
            }
        } else {
            min
        }
    }

    /// Same as `cmp::max`.
    #[inline]
    pub fn max<T: Ord>(a: T, b: T) -> T {
        cmp::max(a, b)
    }

    /// Same as `cmp::min`.
    #[inline]
    pub fn min<T: Ord>(a: T, b: T) -> T {
        cmp::min(a, b)
    }

    /// The absolute value of `a`.
    ///
    /// Deprecated: Use [Matrix::abs] or [RealField::abs] instead.
    #[deprecated(note = "use `Matrix::abs` or `RealField::abs` instead")]
    #[inline]
    pub fn abs<T: Signed>(a: &T) -> T {
        a.abs()
    }

    /// Returns the infimum of `a` and `b`.
    #[inline]
    pub fn inf<T: MeetSemilattice>(a: &T, b: &T) -> T {
        a.meet(b)
    }

    /// Returns the supremum of `a` and `b`.
    #[inline]
    pub fn sup<T: JoinSemilattice>(a: &T, b: &T) -> T {
        a.join(b)
    }

    /// Returns simultaneously the infimum and supremum of `a` and `b`.
    #[inline]
    pub fn inf_sup<T: Lattice>(a: &T, b: &T) -> (T, T) {
        a.meet_join(b)
    }

    /// Compare `a` and `b` using a partial ordering relation.
    #[inline]
    pub fn partial_cmp<T: PartialOrd>(a: &T, b: &T) -> Option<Ordering> {
        a.partial_cmp(b)
    }

    /// Returns `true` iff `a` and `b` are comparable and `a < b`.
    #[inline]
    pub fn partial_lt<T: PartialOrd>(a: &T, b: &T) -> bool {
        a.lt(b)
    }

    /// Returns `true` iff `a` and `b` are comparable and `a <= b`.
    #[inline]
    pub fn partial_le<T: PartialOrd>(a: &T, b: &T) -> bool {
        a.le(b)
    }

    /// Returns `true` iff `a` and `b` are comparable and `a > b`.
    #[inline]
    pub fn partial_gt<T: PartialOrd>(a: &T, b: &T) -> bool {
        a.gt(b)
    }

    /// Returns `true` iff `a` and `b` are comparable and `a >= b`.
    #[inline]
    pub fn partial_ge<T: PartialOrd>(a: &T, b: &T) -> bool {
        a.ge(b)
    }

    /// Return the minimum of `a` and `b` if they are comparable.
    #[inline]
    pub fn partial_min<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<&'a T> {
        if let Some(ord) = a.partial_cmp(b) {
            match ord {
                Ordering::Greater => Some(b),
                _ => Some(a),
            }
        } else {
            None
        }
    }

    /// Return the maximum of `a` and `b` if they are comparable.
    #[inline]
    pub fn partial_max<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<&'a T> {
        if let Some(ord) = a.partial_cmp(b) {
            match ord {
                Ordering::Less => Some(b),
                _ => Some(a),
            }
        } else {
            None
        }
    }

    /// Clamp `value` between `min` and `max`. Returns `None` if `value` is not comparable to
    /// `min` or `max`.
    #[inline]
    pub fn partial_clamp<'a, T: PartialOrd>(value: &'a T, min: &'a T, max: &'a T) -> Option<&'a T> {
        if let (Some(cmp_min), Some(cmp_max)) = (value.partial_cmp(min), value.partial_cmp(max)) {
            if cmp_min == Ordering::Less {
                Some(min)
            } else if cmp_max == Ordering::Greater {
                Some(max)
            } else {
                Some(value)
            }
        } else {
            None
        }
    }

    /// Sorts two values in increasing order using a partial ordering.
    #[inline]
    pub fn partial_sort2<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<(&'a T, &'a T)> {
        if let Some(ord) = a.partial_cmp(b) {
            match ord {
                Ordering::Less => Some((a, b)),
                _ => Some((b, a)),
            }
        } else {
            None
        }
    }

    /*
     * Inverse
     */

    /// Tries to gets an inverted copy of a square matrix.
    ///
    /// # See also:
    ///
    /// * [`inverse`](fn.inverse.html)
    #[deprecated(note = "use the `.try_inverse()` method instead")]
    #[inline]
    pub fn try_inverse<M: AlgaSquareMatrix>(m: &M) -> Option<M> {
        m.try_inverse()
    }

    /// Computes the multiplicative inverse of an (always invertible) algebraic entity.
    ///
    /// # See also:
    ///
    /// * [`try_inverse`](fn.try_inverse.html)
    #[deprecated(note = "use the `.inverse()` method instead")]
    #[inline]
    pub fn inverse<M: TwoSidedInverse<Multiplicative>>(m: &M) -> M {
        m.two_sided_inverse()
    }

    /*
     * Inner vector space
     */

    /// Computes the dot product of two vectors.
    ///
    /// ## Deprecated
    /// Use these methods instead:
    ///   - [Matrix::dot]
    ///   - [Quaternion::dot]
    ///
    /// Or, use [FiniteDimVectorSpace::dot](https://docs.rs/alga/0.7.2/alga/linear/trait.FiniteDimVectorSpace.html#tymethod.dot).
    #[deprecated(note = "use `Matrix::dot` or `Quaternion::dot` instead")]
    #[inline]
    pub fn dot<V: FiniteDimVectorSpace>(a: &V, b: &V) -> V::Field {
        a.dot(b)
    }

    /// Computes the smallest angle between two vectors.
    ///
    /// ## Deprecated
    /// Use [Matrix::angle] instead.
    ///
    /// Or, use [InnerSpace::angle](https://docs.rs/alga/0.7.2/alga/linear/trait.InnerSpace.html#method.angle).
    #[deprecated(note = "use `Matrix::angle` instead")]
    #[inline]
    pub fn angle<V: InnerSpace>(a: &V, b: &V) -> V::RealField {
        a.angle(b)
    }

    /*
     * Normed space
     */

    /// Computes the L2 (Euclidean) norm of a vector.
    ///
    /// # See also:
    ///
    /// * [`magnitude`](fn.magnitude.html)
    /// * [`magnitude_squared`](fn.magnitude_squared.html)
    /// * [`norm_squared`](fn.norm_squared.html)
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::norm]
    /// * [Quaternion::norm]
    ///
    /// Or, use [NormedSpace::norm](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.norm).
    #[deprecated(note = "use `Matrix::norm` or `Quaternion::norm` instead")]
    #[inline]
    pub fn norm<V: NormedSpace>(v: &V) -> V::RealField {
        v.norm()
    }

    /// Computes the squared L2 (Euclidean) norm of the vector `v`.
    ///
    /// # See also:
    ///
    /// * [`magnitude`](fn.magnitude.html)
    /// * [`magnitude_squared`](fn.magnitude_squared.html)
    /// * [`norm`](fn.norm.html)
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::norm_squared]
    /// * [Quaternion::norm_squared]
    ///
    /// Or, use [NormedSpace::norm_squared](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.norm_squared).
    #[deprecated(note = "use `Matrix::norm_squared` or `Quaternion::norm_squared` instead")]
    #[inline]
    pub fn norm_squared<V: NormedSpace>(v: &V) -> V::RealField {
        v.norm_squared()
    }

    /// A synonym for [`norm`](fn.norm.html), aka length.
    ///
    /// # See also:
    ///
    /// * [`magnitude_squared`](fn.magnitude_squared.html)
    /// * [`norm`](fn.norm.html)
    /// * [`norm_squared`](fn.norm_squared.html)
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::magnitude]
    /// * [Quaternion::magnitude]
    ///
    /// Or, use [NormedSpace::norm](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.norm).
    #[deprecated(note = "use `Matrix::magnitude` or `Quaternion::magnitude` instead")]
    #[inline]
    pub fn magnitude<V: NormedSpace>(v: &V) -> V::RealField {
        v.norm()
    }

    /// A synonym for [`norm_squared`](fn.norm_squared.html),
    ///  aka length squared.
    ///
    /// # See also:
    ///
    /// * [`magnitude`](fn.magnitude.html)
    /// * [`norm`](fn.norm.html)
    /// * [`norm_squared`](fn.norm_squared.html)
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::magnitude_squared]
    /// * [Quaternion::magnitude_squared]
    ///
    /// Or, use [NormedSpace::norm_squared](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.norm_squared).
    #[deprecated(note = "use `Matrix::magnitude_squared` or `Quaternion::magnitude_squared` instead")]
    #[inline]
    pub fn magnitude_squared<V: NormedSpace>(v: &V) -> V::RealField {
        v.norm_squared()
    }

    /// Computes the normalized version of the vector `v`.
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::normalize]
    /// * [Quaternion::normalize]
    ///
    /// Or, use [NormedSpace::normalize](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.normalize).
    #[deprecated(note = "use `Matrix::normalize` or `Quaternion::normalize` instead")]
    #[inline]
    pub fn normalize<V: NormedSpace>(v: &V) -> V {
        v.normalize()
    }

    /// Computes the normalized version of the vector `v` or returns `None` if its norm is smaller than `min_norm`.
    ///
    /// # Deprecated
    /// Use these methods instead:
    /// * [Matrix::try_normalize]
    /// * [Quaternion::try_normalize]
    ///
    /// Or, use [NormedSpace::try_normalize](https://docs.rs/alga/0.7.2/alga/linear/trait.NormedSpace.html#tymethod.try_normalize).
    #[deprecated(note = "use `Matrix::try_normalize` or `Quaternion::try_normalize` instead")]
    #[inline]
    pub fn try_normalize<V: NormedSpace>(v: &V, min_norm: V::RealField) -> Option<V> {
        v.try_normalize(min_norm)
    }

    /*
     *
     * Point operations.
     *
     */
    /// The center of two points.
    ///
    /// # See also:
    ///
    /// * [distance](fn.distance.html)
    /// * [distance_squared](fn.distance_squared.html)
    #[inline]
    pub fn center<P: EuclideanSpace>(p1: &P, p2: &P) -> P {
        P::from_coordinates((p1.coordinates() + p2.coordinates()) * convert(0.5))
    }

    /// The distance between two points.
    ///
    /// # See also:
    ///
    /// * [center](fn.center.html)
    /// * [distance_squared](fn.distance_squared.html)
    #[inline]
    pub fn distance<P: EuclideanSpace>(p1: &P, p2: &P) -> P::RealField {
        (p2.coordinates() - p1.coordinates()).norm()
    }

    /// The squared distance between two points.
    ///
    /// # See also:
    ///
    /// * [center](fn.center.html)
    /// * [distance](fn.distance.html)
    #[inline]
    pub fn distance_squared<P: EuclideanSpace>(p1: &P, p2: &P) -> P::RealField {
        (p2.coordinates() - p1.coordinates()).norm_squared()
    }

    /*
     * Cast
     */
    /// Converts an object from one type to an equivalent or more general one.
    ///
    /// See also [`try_convert`](fn.try_convert.html) for conversion to more specific types.
    ///
    /// # See also:
    ///
    /// * [convert_ref](fn.convert_ref.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert](fn.try_convert.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub fn convert<From, To: SupersetOf<From>>(t: From) -> To {
        To::from_subset(&t)
    }

    /// Attempts to convert an object to a more specific one.
    ///
    /// See also [`convert`](fn.convert.html) for conversion to more general types.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref](fn.convert_ref.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub fn try_convert<From: SupersetOf<To>, To>(t: From) -> Option<To> {
        t.to_subset()
    }

    /// Indicates if [`try_convert`](fn.try_convert.html) will succeed without
    /// actually performing the conversion.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref](fn.convert_ref.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [try_convert](fn.try_convert.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub fn is_convertible<From: SupersetOf<To>, To>(t: &From) -> bool {
        t.is_in_subset()
    }

    /// Use with care! Same as [`try_convert`](fn.try_convert.html) but
    /// without any property checks.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref](fn.convert_ref.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert](fn.try_convert.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub unsafe fn convert_unchecked<From: SupersetOf<To>, To>(t: From) -> To {
        t.to_subset_unchecked()
    }

    /// Converts an object from one type to an equivalent or more general one.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert](fn.try_convert.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub fn convert_ref<From, To: SupersetOf<From>>(t: &From) -> To {
        To::from_subset(t)
    }

    /// Attempts to convert an object to a more specific one.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref](fn.convert_ref.html)
    /// * [convert_ref_unchecked](fn.convert_ref_unchecked.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert](fn.try_convert.html)
    #[inline]
    pub fn try_convert_ref<From: SupersetOf<To>, To>(t: &From) -> Option<To> {
        t.to_subset()
    }

    /// Use with care! Same as [`try_convert`](fn.try_convert.html) but
    /// without any property checks.
    ///
    /// # See also:
    ///
    /// * [convert](fn.convert.html)
    /// * [convert_ref](fn.convert_ref.html)
    /// * [is_convertible](../nalgebra/fn.is_convertible.html)
    /// * [try_convert](fn.try_convert.html)
    /// * [try_convert_ref](fn.try_convert_ref.html)
    #[inline]
    pub unsafe fn convert_ref_unchecked<From: SupersetOf<To>, To>(t: &From) -> To {
        t.to_subset_unchecked()
    }

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