nalgebra/src/base/properties.rs

// Matrix properties checks.
use approx::RelativeEq;
use num::{One, Zero};

use simba::scalar::{ClosedAdd, ClosedMul, ComplexField, RealField};

use crate::base::allocator::Allocator;
use crate::base::dimension::{Dim, DimMin};
use crate::base::storage::Storage;
use crate::base::{DefaultAllocator, Matrix, Scalar, SquareMatrix};

impl<T: Scalar, R: Dim, C: Dim, S: Storage<T, R, C>> Matrix<T, R, C, S> {
    /// The total number of elements of this matrix.
    ///
    /// # Examples:
    ///
    /// ```
    /// # use nalgebra::Matrix3x4;
    /// let mat = Matrix3x4::<f32>::zeros();
    /// assert_eq!(mat.len(), 12);
    /// ```
    #[inline]
    #[must_use]
    pub fn len(&self) -> usize {
        let (nrows, ncols) = self.shape();
        nrows * ncols
    }

    /// Returns true if the matrix contains no elements.
    ///
    /// # Examples:
    ///
    /// ```
    /// # use nalgebra::Matrix3x4;
    /// let mat = Matrix3x4::<f32>::zeros();
    /// assert!(!mat.is_empty());
    /// ```
    #[inline]
    #[must_use]
    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }

    /// Indicates if this is a square matrix.
    #[inline]
    #[must_use]
    pub fn is_square(&self) -> bool {
        let (nrows, ncols) = self.shape();
        nrows == ncols
    }

    // TODO: RelativeEq prevents us from using those methods on integer matrices…
    /// Indicated if this is the identity matrix within a relative error of `eps`.
    ///
    /// If the matrix is diagonal, this checks that diagonal elements (i.e. at coordinates `(i, i)`
    /// for i from `0` to `min(R, C)`) are equal one; and that all other elements are zero.
    #[inline]
    #[must_use]
    pub fn is_identity(&self, eps: T::Epsilon) -> bool
    where
        T: Zero + One + RelativeEq,
        T::Epsilon: Copy,
    {
        let (nrows, ncols) = self.shape();
        let d;

        if nrows > ncols {
            d = ncols;

            for i in d..nrows {
                for j in 0..ncols {
                    if !relative_eq!(self[(i, j)], T::zero(), epsilon = eps) {
                        return false;
                    }
                }
            }
        } else {
            // nrows <= ncols
            d = nrows;

            for i in 0..nrows {
                for j in d..ncols {
                    if !relative_eq!(self[(i, j)], T::zero(), epsilon = eps) {
                        return false;
                    }
                }
            }
        }

        // Off-diagonal elements of the sub-square matrix.
        for i in 1..d {
            for j in 0..i {
                // TODO: use unsafe indexing.
                if !relative_eq!(self[(i, j)], T::zero(), epsilon = eps)
                    || !relative_eq!(self[(j, i)], T::zero(), epsilon = eps)
                {
                    return false;
                }
            }
        }

        // Diagonal elements of the sub-square matrix.
        for i in 0..d {
            if !relative_eq!(self[(i, i)], T::one(), epsilon = eps) {
                return false;
            }
        }

        true
    }
}

impl<T: ComplexField, R: Dim, C: Dim, S: Storage<T, R, C>> Matrix<T, R, C, S> {
    /// Checks that `Mᵀ × M = Id`.
    ///
    /// In this definition `Id` is approximately equal to the identity matrix with a relative error
    /// equal to `eps`.
    #[inline]
    #[must_use]
    pub fn is_orthogonal(&self, eps: T::Epsilon) -> bool
    where
        T: Zero + One + ClosedAdd + ClosedMul + RelativeEq,
        S: Storage<T, R, C>,
        T::Epsilon: Copy,
        DefaultAllocator: Allocator<T, R, C> + Allocator<T, C, C>,
    {
        (self.ad_mul(self)).is_identity(eps)
    }
}

impl<T: RealField, D: Dim, S: Storage<T, D, D>> SquareMatrix<T, D, S>
where
    DefaultAllocator: Allocator<T, D, D>,
{
    /// Checks that this matrix is orthogonal and has a determinant equal to 1.
    #[inline]
    #[must_use]
    pub fn is_special_orthogonal(&self, eps: T) -> bool
    where
        D: DimMin<D, Output = D>,
        DefaultAllocator: Allocator<(usize, usize), D>,
    {
        self.is_square() && self.is_orthogonal(eps) && self.determinant() > T::zero()
    }

    /// Returns `true` if this matrix is invertible.
    #[inline]
    #[must_use]
    pub fn is_invertible(&self) -> bool {
        // TODO: improve this?
        self.clone_owned().try_inverse().is_some()
    }
}