Use Complex instead of Real whenever possible on the base/ module.

src/base/blas.rs

@@ -1,4 +1,4 @@
-use alga::general::{ClosedAdd, ClosedMul};
+use alga::general::{ClosedAdd, ClosedMul, Complex};
 #[cfg(feature = "std")]
 use matrixmultiply;
 use num::{One, Signed, Zero};
@@ -190,6 +190,111 @@ impl<N: Scalar + PartialOrd + Signed, R: Dim, C: Dim, S: Storage<N, R, C>> Matri
     }
 }
 
+impl<N: Complex, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+    /// The dot product between two complex or real vectors or matrices (seen as vectors).
+    ///
+    /// This is the same as `.dot` except that the conjugate of each component of `self` is taken
+    /// before performing the products.
+    #[inline]
+    pub fn cdot<R2: Dim, C2: Dim, SB>(&self, rhs: &Matrix<N, R2, C2, SB>) -> N
+        where
+            SB: Storage<N, R2, C2>,
+            ShapeConstraint: DimEq<R, R2> + DimEq<C, C2>,
+    {
+        assert!(
+            self.nrows() == rhs.nrows(),
+            "Dot product dimensions mismatch."
+        );
+
+        // So we do some special cases for common fixed-size vectors of dimension lower than 8
+        // because the `for` loop below won't be very efficient on those.
+        if (R::is::<U2>() || R2::is::<U2>()) && (C::is::<U1>() || C2::is::<U1>()) {
+            unsafe {
+                let a = self.get_unchecked((0, 0)).conjugate() * *rhs.get_unchecked((0, 0));
+                let b = self.get_unchecked((1, 0)).conjugate() * *rhs.get_unchecked((1, 0));
+
+                return a + b;
+            }
+        }
+        if (R::is::<U3>() || R2::is::<U3>()) && (C::is::<U1>() || C2::is::<U1>()) {
+            unsafe {
+                let a = self.get_unchecked((0, 0)).conjugate() * *rhs.get_unchecked((0, 0));
+                let b = self.get_unchecked((1, 0)).conjugate() * *rhs.get_unchecked((1, 0));
+                let c = self.get_unchecked((2, 0)).conjugate() * *rhs.get_unchecked((2, 0));
+
+                return a + b + c;
+            }
+        }
+        if (R::is::<U4>() || R2::is::<U4>()) && (C::is::<U1>() || C2::is::<U1>()) {
+            unsafe {
+                let mut a = self.get_unchecked((0, 0)).conjugate() * *rhs.get_unchecked((0, 0));
+                let mut b = self.get_unchecked((1, 0)).conjugate() * *rhs.get_unchecked((1, 0));
+                let c = self.get_unchecked((2, 0)).conjugate() * *rhs.get_unchecked((2, 0));
+                let d = self.get_unchecked((3, 0)).conjugate() * *rhs.get_unchecked((3, 0));
+
+                a += c;
+                b += d;
+
+                return a + b;
+            }
+        }
+
+        // All this is inspired from the "unrolled version" discussed in:
+        // http://blog.theincredibleholk.org/blog/2012/12/10/optimizing-dot-product/
+        //
+        // And this comment from bluss:
+        // https://users.rust-lang.org/t/how-to-zip-two-slices-efficiently/2048/12
+        let mut res = N::zero();
+
+        // We have to define them outside of the loop (and not inside at first assignment)
+        // otherwise vectorization won't kick in for some reason.
+        let mut acc0;
+        let mut acc1;
+        let mut acc2;
+        let mut acc3;
+        let mut acc4;
+        let mut acc5;
+        let mut acc6;
+        let mut acc7;
+
+        for j in 0..self.ncols() {
+            let mut i = 0;
+
+            acc0 = N::zero();
+            acc1 = N::zero();
+            acc2 = N::zero();
+            acc3 = N::zero();
+            acc4 = N::zero();
+            acc5 = N::zero();
+            acc6 = N::zero();
+            acc7 = N::zero();
+
+            while self.nrows() - i >= 8 {
+                acc0 += unsafe { self.get_unchecked((i + 0, j)).conjugate() * *rhs.get_unchecked((i + 0, j)) };
+                acc1 += unsafe { self.get_unchecked((i + 1, j)).conjugate() * *rhs.get_unchecked((i + 1, j)) };
+                acc2 += unsafe { self.get_unchecked((i + 2, j)).conjugate() * *rhs.get_unchecked((i + 2, j)) };
+                acc3 += unsafe { self.get_unchecked((i + 3, j)).conjugate() * *rhs.get_unchecked((i + 3, j)) };
+                acc4 += unsafe { self.get_unchecked((i + 4, j)).conjugate() * *rhs.get_unchecked((i + 4, j)) };
+                acc5 += unsafe { self.get_unchecked((i + 5, j)).conjugate() * *rhs.get_unchecked((i + 5, j)) };
+                acc6 += unsafe { self.get_unchecked((i + 6, j)).conjugate() * *rhs.get_unchecked((i + 6, j)) };
+                acc7 += unsafe { self.get_unchecked((i + 7, j)).conjugate() * *rhs.get_unchecked((i + 7, j)) };
+                i += 8;
+            }
+
+            res += acc0 + acc4;
+            res += acc1 + acc5;
+            res += acc2 + acc6;
+            res += acc3 + acc7;
+
+            for k in i..self.nrows() {
+                res += unsafe { self.get_unchecked((k, j)).conjugate() * *rhs.get_unchecked((k, j)) }
+            }
+        }
+
+        res
+    }
+}
+
 impl<N, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S>
 where N: Scalar + Zero + ClosedAdd + ClosedMul
 {

src/base/matrix.rs

@@ -1,5 +1,4 @@
 use num::{One, Zero};
-use num_complex::Complex;
 #[cfg(feature = "abomonation-serialize")]
 use std::io::{Result as IOResult, Write};
 
@@ -17,7 +16,7 @@ use serde::{Deserialize, Deserializer, Serialize, Serializer};
 #[cfg(feature = "abomonation-serialize")]
 use abomonation::Abomonation;
 
-use alga::general::{ClosedAdd, ClosedMul, ClosedSub, Real, Ring};
+use alga::general::{ClosedAdd, ClosedMul, ClosedSub, Real, Ring, Complex, Field};
 
 use base::allocator::{Allocator, SameShapeAllocator, SameShapeC, SameShapeR};
 use base::constraint::{DimEq, SameNumberOfColumns, SameNumberOfRows, ShapeConstraint};
@@ -906,14 +905,14 @@ impl<N: Scalar, D: Dim, S: StorageMut<N, D, D>> Matrix<N, D, D, S> {
     }
 }
 
-impl<N: Real, R: Dim, C: Dim, S: Storage<Complex<N>, R, C>> Matrix<Complex<N>, R, C, S> {
+impl<N: Complex, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// Takes the conjugate and transposes `self` and store the result into `out`.
     #[inline]
-    pub fn conjugate_transpose_to<R2, C2, SB>(&self, out: &mut Matrix<Complex<N>, R2, C2, SB>)
+    pub fn conjugate_transpose_to<R2, C2, SB>(&self, out: &mut Matrix<N, R2, C2, SB>)
     where
         R2: Dim,
         C2: Dim,
-        SB: StorageMut<Complex<N>, R2, C2>,
+        SB: StorageMut<N, R2, C2>,
         ShapeConstraint: SameNumberOfRows<R, C2> + SameNumberOfColumns<C, R2>,
     {
         let (nrows, ncols) = self.shape();
@@ -926,7 +925,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<Complex<N>, R, C>> Matrix<Complex<N>, R
         for i in 0..nrows {
             for j in 0..ncols {
                 unsafe {
-                    *out.get_unchecked_mut((j, i)) = self.get_unchecked((i, j)).conj();
+                    *out.get_unchecked_mut((j, i)) = self.get_unchecked((i, j)).conjugate();
                 }
             }
         }
@@ -934,8 +933,8 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<Complex<N>, R, C>> Matrix<Complex<N>, R
 
     /// The conjugate transposition of `self`.
     #[inline]
-    pub fn conjugate_transpose(&self) -> MatrixMN<Complex<N>, C, R>
+    pub fn conjugate_transpose(&self) -> MatrixMN<N, C, R>
-    where DefaultAllocator: Allocator<Complex<N>, C, R> {
+    where DefaultAllocator: Allocator<N, C, R> {
         let (nrows, ncols) = self.data.shape();
 
         unsafe {
@@ -947,7 +946,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<Complex<N>, R, C>> Matrix<Complex<N>, R
     }
 }
 
-impl<N: Real, D: Dim, S: StorageMut<Complex<N>, D, D>> Matrix<Complex<N>, D, D, S> {
+impl<N: Complex, D: Dim, S: StorageMut<N, D, D>> Matrix<N, D, D, S> {
     /// Sets `self` to its conjugate transpose.
     pub fn conjugate_transpose_mut(&mut self) {
         assert!(
@@ -960,10 +959,10 @@ impl<N: Real, D: Dim, S: StorageMut<Complex<N>, D, D>> Matrix<Complex<N>, D, D,
         for i in 1..dim {
             for j in 0..i {
                 unsafe {
-                    let ref_ij = self.get_unchecked_mut((i, j)) as *mut Complex<N>;
+                    let ref_ij = self.get_unchecked_mut((i, j)) as *mut N;
-                    let ref_ji = self.get_unchecked_mut((j, i)) as *mut Complex<N>;
+                    let ref_ji = self.get_unchecked_mut((j, i)) as *mut N;
-                    let conj_ij = (*ref_ij).conj();
+                    let conj_ij = (*ref_ij).conjugate();
-                    let conj_ji = (*ref_ji).conj();
+                    let conj_ji = (*ref_ji).conjugate();
                     *ref_ij = conj_ji;
                     *ref_ji = conj_ij;
                 }
@@ -1407,7 +1406,7 @@ impl<N: Scalar + Ring, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     }
 }
 
-impl<N: Real, S: Storage<N, U3>> Vector<N, U3, S>
+impl<N: Scalar + Field, S: Storage<N, U3>> Vector<N, U3, S>
 where DefaultAllocator: Allocator<N, U3>
 {
     /// Computes the matrix `M` such that for all vector `v` we have `M * v == self.cross(&v)`.
@@ -1427,27 +1426,27 @@ where DefaultAllocator: Allocator<N, U3>
     }
 }
 
-impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+impl<N: Complex, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// The smallest angle between two vectors.
     #[inline]
-    pub fn angle<R2: Dim, C2: Dim, SB>(&self, other: &Matrix<N, R2, C2, SB>) -> N
+    pub fn angle<R2: Dim, C2: Dim, SB>(&self, other: &Matrix<N, R2, C2, SB>) -> N::Real
     where
         SB: Storage<N, R2, C2>,
         ShapeConstraint: DimEq<R, R2> + DimEq<C, C2>,
     {
-        let prod = self.dot(other);
+        let prod = self.cdot(other);
         let n1 = self.norm();
         let n2 = other.norm();
 
         if n1.is_zero() || n2.is_zero() {
-            N::zero()
+            N::Real::zero()
         } else {
-            let cang = prod / (n1 * n2);
+            let cang = prod.real() / (n1 * n2);
 
-            if cang > N::one() {
+            if cang > N::Real::one() {
-                N::zero()
+                N::Real::zero()
-            } else if cang < -N::one() {
+            } else if cang < -N::Real::one() {
-                N::pi()
+                N::Real::pi()
             } else {
                 cang.acos()
             }
@@ -1478,18 +1477,18 @@ impl<N: Scalar + Zero + One + ClosedAdd + ClosedSub + ClosedMul, D: Dim, S: Stor
     }
 }
 
-impl<N: Real, D: Dim, S: Storage<N, D>> Unit<Vector<N, D, S>> {
+impl<N: Complex, D: Dim, S: Storage<N, D>> Unit<Vector<N, D, S>> {
     /// Computes the spherical linear interpolation between two unit vectors.
     pub fn slerp<S2: Storage<N, D>>(
         &self,
         rhs: &Unit<Vector<N, D, S2>>,
-        t: N,
+        t: N::Real,
     ) -> Unit<VectorN<N, D>>
     where
         DefaultAllocator: Allocator<N, D>,
     {
         // FIXME: the result is wrong when self and rhs are collinear with opposite direction.
-        self.try_slerp(rhs, t, N::default_epsilon())
+        self.try_slerp(rhs, t, N::Real::default_epsilon())
             .unwrap_or(Unit::new_unchecked(self.clone_owned()))
     }
 
@@ -1500,29 +1499,29 @@ impl<N: Real, D: Dim, S: Storage<N, D>> Unit<Vector<N, D, S>> {
     pub fn try_slerp<S2: Storage<N, D>>(
         &self,
         rhs: &Unit<Vector<N, D, S2>>,
-        t: N,
+        t: N::Real,
-        epsilon: N,
+        epsilon: N::Real,
     ) -> Option<Unit<VectorN<N, D>>>
     where
         DefaultAllocator: Allocator<N, D>,
     {
-        let c_hang = self.dot(rhs);
+        let c_hang = self.cdot(rhs).real();
 
         // self == other
-        if c_hang.abs() >= N::one() {
+        if c_hang.abs() >= N::Real::one() {
             return Some(Unit::new_unchecked(self.clone_owned()));
         }
 
         let hang = c_hang.acos();
-        let s_hang = (N::one() - c_hang * c_hang).sqrt();
+        let s_hang = (N::Real::one() - c_hang * c_hang).sqrt();
 
         // FIXME: what if s_hang is 0.0 ? The result is not well-defined.
-        if relative_eq!(s_hang, N::zero(), epsilon = epsilon) {
+        if relative_eq!(s_hang, N::Real::zero(), epsilon = epsilon) {
             None
         } else {
-            let ta = ((N::one() - t) * hang).sin() / s_hang;
+            let ta = ((N::Real::one() - t) * hang).sin() / s_hang;
             let tb = (t * hang).sin() / s_hang;
-            let res = &**self * ta + &**rhs * tb;
+            let res = &**self * N::from_real(ta) + &**rhs * N::from_real(tb);
 
             Some(Unit::new_unchecked(res))
         }

src/base/matrix_alga.rs

@@ -7,7 +7,7 @@ use alga::general::{
     AbstractGroup, AbstractGroupAbelian, AbstractLoop, AbstractMagma, AbstractModule,
     AbstractMonoid, AbstractQuasigroup, AbstractSemigroup, Additive, ClosedAdd, ClosedMul,
     ClosedNeg, Field, Identity, TwoSidedInverse, JoinSemilattice, Lattice, MeetSemilattice, Module,
-    Multiplicative, Real, RingCommutative,
+    Multiplicative, Real, RingCommutative, Complex
 };
 use alga::linear::{
     FiniteDimInnerSpace, FiniteDimVectorSpace, InnerSpace, NormedSpace, VectorSpace,
@@ -145,16 +145,19 @@ where
     }
 }
 
-impl<N: Real, R: DimName, C: DimName> NormedSpace for MatrixMN<N, R, C>
+impl<N: Complex, R: DimName, C: DimName> NormedSpace for MatrixMN<N, R, C>
 where DefaultAllocator: Allocator<N, R, C>
 {
+    type Real = N::Real;
+    type Complex = N;
+
     #[inline]
-    fn norm_squared(&self) -> N {
+    fn norm_squared(&self) -> N::Real {
         self.norm_squared()
     }
 
     #[inline]
-    fn norm(&self) -> N {
+    fn norm(&self) -> N::Real {
         self.norm()
     }
 
@@ -164,34 +167,32 @@ where DefaultAllocator: Allocator<N, R, C>
     }
 
     #[inline]
-    fn normalize_mut(&mut self) -> N {
+    fn normalize_mut(&mut self) -> N::Real {
         self.normalize_mut()
     }
 
     #[inline]
-    fn try_normalize(&self, min_norm: N) -> Option<Self> {
+    fn try_normalize(&self, min_norm: N::Real) -> Option<Self> {
         self.try_normalize(min_norm)
     }
 
     #[inline]
-    fn try_normalize_mut(&mut self, min_norm: N) -> Option<N> {
+    fn try_normalize_mut(&mut self, min_norm: N::Real) -> Option<N::Real> {
         self.try_normalize_mut(min_norm)
     }
 }
 
-impl<N: Real, R: DimName, C: DimName> InnerSpace for MatrixMN<N, R, C>
+impl<N: Complex, R: DimName, C: DimName> InnerSpace for MatrixMN<N, R, C>
 where DefaultAllocator: Allocator<N, R, C>
 {
-    type Real = N;
-
     #[inline]
-    fn angle(&self, other: &Self) -> N {
+    fn angle(&self, other: &Self) -> N::Real {
         self.angle(other)
     }
 
     #[inline]
     fn inner_product(&self, other: &Self) -> N {
-        self.dot(other)
+        self.cdot(other)
     }
 }
 
@@ -199,7 +200,7 @@ where DefaultAllocator: Allocator<N, R, C>
 // In particular:
 //   − use `x()` instead of `::canonical_basis_element`
 //   − use `::new(x, y, z)` instead of `::from_slice`
-impl<N: Real, R: DimName, C: DimName> FiniteDimInnerSpace for MatrixMN<N, R, C>
+impl<N: Complex, R: DimName, C: DimName> FiniteDimInnerSpace for MatrixMN<N, R, C>
 where DefaultAllocator: Allocator<N, R, C>
 {
     #[inline]
@@ -215,7 +216,7 @@ where DefaultAllocator: Allocator<N, R, C>
                 }
             }
 
-            if vs[i].try_normalize_mut(N::zero()).is_some() {
+            if vs[i].try_normalize_mut(N::Real::zero()).is_some() {
                 // FIXME: this will be efficient on dynamically-allocated vectors but for
                 // statically-allocated ones, `.clone_from` would be better.
                 vs.swap(nbasis_elements, i);
@@ -268,7 +269,7 @@ where DefaultAllocator: Allocator<N, R, C>
                     let v = &vs[0];
                     let mut a;
 
-                    if v[0].abs() > v[1].abs() {
+                    if v[0].modulus() > v[1].modulus() {
                         a = Self::from_column_slice(&[v[2], N::zero(), -v[0]]);
                     } else {
                         a = Self::from_column_slice(&[N::zero(), -v[2], v[1]]);
@@ -300,7 +301,7 @@ where DefaultAllocator: Allocator<N, R, C>
                             elt -= v * elt.dot(v)
                         }
 
-                        if let Some(subsp_elt) = elt.try_normalize(N::zero()) {
+                        if let Some(subsp_elt) = elt.try_normalize(N::Real::zero()) {
                             if !f(&subsp_elt) {
                                 return;
                             };

src/base/norm.rs

@@ -1,8 +1,8 @@
-use num::Signed;
+use num::{Signed, Zero};
 use std::cmp::PartialOrd;
 
 use allocator::Allocator;
-use ::{Real, Scalar};
+use ::{Real, Complex, Scalar};
 use storage::{Storage, StorageMut};
 use base::{DefaultAllocator, Matrix, Dim, MatrixMN};
 use constraint::{SameNumberOfRows, SameNumberOfColumns, ShapeConstraint};
@@ -12,12 +12,12 @@ use constraint::{SameNumberOfRows, SameNumberOfColumns, ShapeConstraint};
 /// A trait for abstract matrix norms.
 ///
 /// This may be moved to the alga crate in the future.
-pub trait Norm<N: Scalar> {
+pub trait Norm<N: Complex> {
     /// Apply this norm to the given matrix.
-    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N
+    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N::Real
         where R: Dim, C: Dim, S: Storage<N, R, C>;
     /// Use the metric induced by this norm to compute the metric distance between the two given matrices.
-    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N
+    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N::Real
         where R1: Dim, C1: Dim, S1: Storage<N, R1, C1>,
               R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2>;
@@ -30,60 +30,60 @@ pub struct LpNorm(pub i32);
 /// L-infinite norm aka. Chebytchev norm aka. uniform norm aka. suppremum norm.
 pub struct UniformNorm;
 
-impl<N: Real> Norm<N> for EuclideanNorm {
+impl<N: Complex> Norm<N> for EuclideanNorm {
     #[inline]
-    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N
+    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N::Real
         where R: Dim, C: Dim, S: Storage<N, R, C> {
-        m.norm_squared().sqrt()
+        m.cdot(m).real().sqrt()
     }
 
     #[inline]
-    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N
+    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N::Real
         where R1: Dim, C1: Dim, S1: Storage<N, R1, C1>,
               R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
-        m1.zip_fold(m2, N::zero(), |acc, a, b| {
+        m1.zip_fold(m2, N::Real::zero(), |acc, a, b| {
             let diff = a - b;
-            acc + diff * diff
+            acc + (diff.conjugate() * diff).real()
         }).sqrt()
     }
 }
 
-impl<N: Real> Norm<N> for LpNorm {
+impl<N: Complex> Norm<N> for LpNorm {
     #[inline]
-    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N
+    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N::Real
         where R: Dim, C: Dim, S: Storage<N, R, C> {
-        m.fold(N::zero(), |a, b| {
+        m.fold(N::Real::zero(), |a, b| {
-            a + b.abs().powi(self.0)
+            a + b.modulus().powi(self.0)
         }).powf(::convert(1.0 / (self.0 as f64)))
     }
 
     #[inline]
-    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N
+    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N::Real
         where R1: Dim, C1: Dim, S1: Storage<N, R1, C1>,
               R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
-        m1.zip_fold(m2, N::zero(), |acc, a, b| {
+        m1.zip_fold(m2, N::Real::zero(), |acc, a, b| {
             let diff = a - b;
-            acc + diff.abs().powi(self.0)
+            acc + diff.modulus().powi(self.0)
         }).powf(::convert(1.0 / (self.0 as f64)))
     }
 }
 
-impl<N: Scalar + PartialOrd + Signed> Norm<N> for UniformNorm {
+impl<N: Complex> Norm<N> for UniformNorm {
     #[inline]
-    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N
+    fn norm<R, C, S>(&self, m: &Matrix<N, R, C, S>) -> N::Real
         where R: Dim, C: Dim, S: Storage<N, R, C> {
-        m.amax()
+        m.fold(N::Real::zero(), |acc, a| acc.max(a.modulus()))
     }
 
     #[inline]
-    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N
+    fn metric_distance<R1, C1, S1, R2, C2, S2>(&self, m1: &Matrix<N, R1, C1, S1>, m2: &Matrix<N, R2, C2, S2>) -> N::Real
         where R1: Dim, C1: Dim, S1: Storage<N, R1, C1>,
               R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
-        m1.zip_fold(m2, N::zero(), |acc, a, b| {
+        m1.zip_fold(m2, N::Real::zero(), |acc, a, b| {
-            let val = (a - b).abs();
+            let val = (a - b).modulus();
             if val > acc {
                 val
             } else {
@@ -94,15 +94,15 @@ impl<N: Scalar + PartialOrd + Signed> Norm<N> for UniformNorm {
 }
 
 
-impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+impl<N: Complex, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// The squared L2 norm of this vector.
     #[inline]
-    pub fn norm_squared(&self) -> N {
+    pub fn norm_squared(&self) -> N::Real {
-        let mut res = N::zero();
+        let mut res = N::Real::zero();
 
         for i in 0..self.ncols() {
             let col = self.column(i);
-            res += col.dot(&col)
+            res += col.cdot(&col).real()
         }
 
         res
@@ -112,7 +112,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     ///
     /// Use `.apply_norm` to apply a custom norm.
     #[inline]
-    pub fn norm(&self) -> N {
+    pub fn norm(&self) -> N::Real {
         self.norm_squared().sqrt()
     }
 
@@ -120,7 +120,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     ///
     /// Use `.apply_metric_distance` to apply a custom norm.
     #[inline]
-    pub fn metric_distance<R2, C2, S2>(&self, rhs: &Matrix<N, R2, C2, S2>) -> N
+    pub fn metric_distance<R2, C2, S2>(&self, rhs: &Matrix<N, R2, C2, S2>) -> N::Real
         where R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R, R2> + SameNumberOfColumns<C, C2> {
         self.apply_metric_distance(rhs, &EuclideanNorm)
@@ -139,7 +139,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// assert_eq!(v.apply_norm(&EuclideanNorm), v.norm());
     /// ```
     #[inline]
-    pub fn apply_norm(&self, norm: &impl Norm<N>) -> N {
+    pub fn apply_norm(&self, norm: &impl Norm<N>) -> N::Real {
         norm.norm(self)
     }
 
@@ -158,16 +158,10 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// assert_eq!(v1.apply_metric_distance(&v2, &EuclideanNorm), (v1 - v2).norm());
     /// ```
     #[inline]
-    pub fn apply_metric_distance<R2, C2, S2>(&self, rhs: &Matrix<N, R2, C2, S2>, norm: &impl Norm<N>) -> N
+    pub fn apply_metric_distance<R2, C2, S2>(&self, rhs: &Matrix<N, R2, C2, S2>, norm: &impl Norm<N>) -> N::Real
         where R2: Dim, C2: Dim, S2: Storage<N, R2, C2>,
               ShapeConstraint: SameNumberOfRows<R, R2> + SameNumberOfColumns<C, C2> {
-        norm.metric_distance(self,rhs)
+        norm.metric_distance(self, rhs)
-    }
-
-    /// The Lp norm of this matrix.
-    #[inline]
-    pub fn lp_norm(&self, p: i32) -> N {
-        self.apply_norm(&LpNorm(p))
     }
 
     /// A synonym for the norm of this matrix.
@@ -176,7 +170,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     ///
     /// This function is simply implemented as a call to `norm()`
     #[inline]
-    pub fn magnitude(&self) -> N {
+    pub fn magnitude(&self) -> N::Real {
         self.norm()
     }
 
@@ -186,7 +180,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     ///
     /// This function is simply implemented as a call to `norm_squared()`
     #[inline]
-    pub fn magnitude_squared(&self) -> N {
+    pub fn magnitude_squared(&self) -> N::Real {
         self.norm_squared()
     }
 
@@ -194,29 +188,36 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     #[inline]
     pub fn normalize(&self) -> MatrixMN<N, R, C>
         where DefaultAllocator: Allocator<N, R, C> {
-        self / self.norm()
+        self.map(|e| e.unscale(self.norm()))
     }
 
     /// Returns a normalized version of this matrix unless its norm as smaller or equal to `eps`.
     #[inline]
-    pub fn try_normalize(&self, min_norm: N) -> Option<MatrixMN<N, R, C>>
+    pub fn try_normalize(&self, min_norm: N::Real) -> Option<MatrixMN<N, R, C>>
         where DefaultAllocator: Allocator<N, R, C> {
         let n = self.norm();
 
         if n <= min_norm {
             None
         } else {
-            Some(self / n)
+            Some(self.map(|e| e.unscale(n)))
         }
     }
+
+    /// The Lp norm of this matrix.
+    #[inline]
+    pub fn lp_norm(&self, p: i32) -> N::Real {
+        self.apply_norm(&LpNorm(p))
+    }
 }
 
-impl<N: Real, R: Dim, C: Dim, S: StorageMut<N, R, C>> Matrix<N, R, C, S> {
+
+impl<N: Complex, R: Dim, C: Dim, S: StorageMut<N, R, C>> Matrix<N, R, C, S> {
     /// Normalizes this matrix in-place and returns its norm.
     #[inline]
-    pub fn normalize_mut(&mut self) -> N {
+    pub fn normalize_mut(&mut self) -> N::Real {
         let n = self.norm();
-        *self /= n;
+        self.apply(|e| e.unscale(n));
 
         n
     }
@@ -225,13 +226,13 @@ impl<N: Real, R: Dim, C: Dim, S: StorageMut<N, R, C>> Matrix<N, R, C, S> {
     ///
     /// If the normalization succeeded, returns the old normal of this matrix.
     #[inline]
-    pub fn try_normalize_mut(&mut self, min_norm: N) -> Option<N> {
+    pub fn try_normalize_mut(&mut self, min_norm: N::Real) -> Option<N::Real> {
         let n = self.norm();
 
         if n <= min_norm {
             None
         } else {
-            *self /= n;
+            self.apply(|e| e.unscale(n));
             Some(n)
         }
     }

src/base/statistics.rs

@@ -1,8 +1,9 @@
-use ::{Real, Dim, Matrix, VectorN, RowVectorN, DefaultAllocator, U1, VectorSliceN};
+use ::{Scalar, Dim, Matrix, VectorN, RowVectorN, DefaultAllocator, U1, VectorSliceN};
+use alga::general::{Field, SupersetOf};
 use storage::Storage;
 use allocator::Allocator;
 
-impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /// Returns a row vector where each element is the result of the application of `f` on the
     /// corresponding column of the original matrix.
     #[inline]
@@ -53,7 +54,7 @@ impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     }
 }
 
-impl<N: Real, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+impl<N: Scalar + Field + SupersetOf<f64>, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
     /*
      *
      * Sum computation.

src/base/unit.rs

@@ -10,7 +10,7 @@ use serde::{Deserialize, Deserializer, Serialize, Serializer};
 #[cfg(feature = "abomonation-serialize")]
 use abomonation::Abomonation;
 
-use alga::general::SubsetOf;
+use alga::general::{SubsetOf, Complex};
 use alga::linear::NormedSpace;
 
 use ::Real;
@@ -66,13 +66,13 @@ impl<T: NormedSpace> Unit<T> {
     ///
     /// Returns `None` if the norm was smaller or equal to `min_norm`.
     #[inline]
-    pub fn try_new(value: T, min_norm: T::Field) -> Option<Self> {
+    pub fn try_new(value: T, min_norm: T::Real) -> Option<Self> {
         Self::try_new_and_get(value, min_norm).map(|res| res.0)
     }
 
     /// Normalize the given value and return it wrapped on a `Unit` structure and its norm.
     #[inline]
-    pub fn new_and_get(mut value: T) -> (Self, T::Field) {
+    pub fn new_and_get(mut value: T) -> (Self, T::Real) {
         let n = value.normalize_mut();
 
         (Unit { value: value }, n)
@@ -82,7 +82,7 @@ impl<T: NormedSpace> Unit<T> {
     ///
     /// Returns `None` if the norm was smaller or equal to `min_norm`.
     #[inline]
-    pub fn try_new_and_get(mut value: T, min_norm: T::Field) -> Option<(Self, T::Field)> {
+    pub fn try_new_and_get(mut value: T, min_norm: T::Real) -> Option<(Self, T::Real)> {
         if let Some(n) = value.try_normalize_mut(min_norm) {
             Some((Unit { value: value }, n))
         } else {
@@ -96,7 +96,7 @@ impl<T: NormedSpace> Unit<T> {
     /// Returns the norm before re-normalization. See `.renormalize_fast` for a faster alternative
     /// that may be slightly less accurate if `self` drifted significantly from having a unit length.
     #[inline]
-    pub fn renormalize(&mut self) -> T::Field {
+    pub fn renormalize(&mut self) -> T::Real {
         self.value.normalize_mut()
     }
 
@@ -104,12 +104,11 @@ impl<T: NormedSpace> Unit<T> {
     /// This is useful when repeated computations might cause a drift in the norm
     /// because of float inaccuracies.
     #[inline]
-    pub fn renormalize_fast(&mut self)
+    pub fn renormalize_fast(&mut self) {
-        where T::Field: Real {
         let sq_norm = self.value.norm_squared();
-        let _3: T::Field = ::convert(3.0);
+        let _3: T::Real = ::convert(3.0);
-        let _0_5: T::Field = ::convert(0.5);
+        let _0_5: T::Real = ::convert(0.5);
-        self.value *= _0_5 * (_3 - sq_norm);
+        self.value *= T::Complex::from_real(_0_5 * (_3 - sq_norm));
     }
 }
 

src/geometry/quaternion_alga.rs

@@ -118,6 +118,9 @@ impl<N: Real> FiniteDimVectorSpace for Quaternion<N> {
 }
 
 impl<N: Real> NormedSpace for Quaternion<N> {
+    type Real = N;
+    type Complex = N;
+
     #[inline]
     fn norm_squared(&self) -> N {
         self.coords.norm_squared()

src/lib.rs

@@ -160,7 +160,7 @@ use alga::linear::SquareMatrix as AlgaSquareMatrix;
 use alga::linear::{EuclideanSpace, FiniteDimVectorSpace, InnerSpace, NormedSpace};
 use num::Signed;
 
-pub use alga::general::{Id, Real};
+pub use alga::general::{Id, Real, Complex};
 
 /*
  *
@@ -481,7 +481,7 @@ pub fn angle<V: InnerSpace>(a: &V, b: &V) -> V::Real {
 /// 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::Field {
+pub fn norm<V: NormedSpace>(v: &V) -> V::Real {
     v.norm()
 }
 
@@ -501,7 +501,7 @@ pub fn norm<V: NormedSpace>(v: &V) -> V::Field {
 /// 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::Field {
+pub fn norm_squared<V: NormedSpace>(v: &V) -> V::Real {
     v.norm_squared()
 }
 
@@ -521,7 +521,7 @@ pub fn norm_squared<V: NormedSpace>(v: &V) -> V::Field {
 /// 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::Field {
+pub fn magnitude<V: NormedSpace>(v: &V) -> V::Real {
     v.norm()
 }
 
@@ -542,7 +542,7 @@ pub fn magnitude<V: NormedSpace>(v: &V) -> V::Field {
 /// 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::Field {
+pub fn magnitude_squared<V: NormedSpace>(v: &V) -> V::Real {
     v.norm_squared()
 }
 
@@ -570,7 +570,7 @@ pub fn normalize<V: NormedSpace>(v: &V) -> V {
 /// 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::Field) -> Option<V> {
+pub fn try_normalize<V: NormedSpace>(v: &V, min_norm: V::Real) -> Option<V> {
     v.try_normalize(min_norm)
 }