From 6ad11edf9b453472806112258ad986be87a2a30e Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Sun, 11 May 2014 19:46:04 +0200
Subject: [PATCH 1/8] more generic QR: generalize the impl of the Indexable
 trait

This allows the implementation of householder reflection without relying
on knowledge of DVec. This required a new member in the Indexable trait:
the shape() function, which returns the maximum index available.
---
 src/linalg/decompositions.rs | 55 ++++++++++++---------
 src/na.rs                    |  3 +-
 src/structs/dmat.rs          | 93 ++++++++++++++++++++++--------------
 src/structs/dvec.rs          | 48 ++++++++++++++-----
 src/structs/mat.rs           | 15 +++++-
 src/structs/mat_macros.rs    | 19 ++++++++
 src/structs/spec/vec0.rs     |  5 ++
 src/structs/vec_macros.rs    |  5 ++
 src/traits/mod.rs            |  2 +-
 src/traits/structure.rs      |  9 ++++
 10 files changed, 182 insertions(+), 72 deletions(-)

diff --git a/src/linalg/decompositions.rs b/src/linalg/decompositions.rs
index 47409224..3cc1a0f5 100644
--- a/src/linalg/decompositions.rs
+++ b/src/linalg/decompositions.rs
@@ -1,11 +1,36 @@
 use std::num::{Zero, Float};
-use na::DVec;
 use na::DMat;
 use traits::operations::Transpose;
-use traits::structure::ColSlice;
+use traits::structure::{ColSlice, Eye, Indexable};
 use traits::geometry::Norm;
 use std::cmp::min;
 
+/// Get the householder matrix corresponding to a reflexion to the hyperplane
+/// defined by `vec̀ . It can be a reflexion contained in a subspace.
+///
+/// # Arguments
+/// * `dim` - the dimension of the space the resulting matrix operates in
+/// * `start` - the starting dimension of the subspace of the reflexion
+/// * `vec` - the vector defining the reflection.
+fn householder_matrix<N: Float,
+                      Mat: Eye + Indexable<(uint, uint), N>,
+                      V: Indexable<uint, N>>
+                      (dim: uint, start: uint, vec: V) -> Mat {
+    let mut qk : Mat = Eye::new_identity(dim);
+    let stop = start + vec.shape();
+    assert!(stop <= dim);
+    for j in range(start, stop) {
+        for i in range(start, stop) {
+            unsafe {
+                let vv = vec.unsafe_at(i) * vec.unsafe_at(j);
+                let qkij = qk.unsafe_at((i, j));
+                qk.unsafe_set((i, j), qkij - vv - vv);
+            }
+        }
+    }
+    qk
+}
+
 /// QR decomposition using Householder reflections
 /// # Arguments
 /// * `m` - matrix to decompose
@@ -13,42 +38,26 @@ pub fn decomp_qr<N: Clone + Float>(m: &DMat<N>) -> (DMat<N>, DMat<N>) {
     let rows = m.nrows();
     let cols = m.ncols();
     assert!(rows >= cols);
-    let mut q : DMat<N> = DMat::new_identity(rows);
+    let mut q : DMat<N> = Eye::new_identity(rows);
     let mut r = m.clone();
 
-    let subtract_reflection = |vec: DVec<N>| -> DMat<N> {
-        // FIXME: we don't handle the complex case here
-        let mut qk : DMat<N> = DMat::new_identity(rows);
-        let start = rows - vec.at.len();
-        for j in range(start, rows) {
-            for i in range(start, rows) {
-                unsafe {
-                    let vv = vec.at_fast(i - start) * vec.at_fast(j - start);
-                    let qkij = qk.at_fast(i, j);
-                    qk.set_fast(i, j, qkij - vv - vv);
-                }
-            }
-        }
-        qk
-    };
-
     let iterations = min(rows - 1, cols);
 
     for ite in range(0u, iterations) {
         let mut v = r.col_slice(ite, ite, rows);
         let alpha =
-            if unsafe { v.at_fast(ite) } >= Zero::zero() {
+            if unsafe { v.unsafe_at(ite) } >= Zero::zero() {
                 -Norm::norm(&v)
             }
             else {
                 Norm::norm(&v)
             };
         unsafe {
-            let x = v.at_fast(0);
-            v.set_fast(0, x - alpha);
+            let x = v.unsafe_at(0);
+            v.unsafe_set(0, x - alpha);
         }
         let _ = v.normalize();
-        let qk = subtract_reflection(v);
+        let qk: DMat<N> = householder_matrix(rows, 0, v);
         r = qk * r;
         q = q * Transpose::transpose_cpy(&qk);
     }
diff --git a/src/na.rs b/src/na.rs
index 415eb8f2..6832fb18 100644
--- a/src/na.rs
+++ b/src/na.rs
@@ -40,7 +40,8 @@ pub use traits::{
     UniformSphereSample,
     AnyVec,
     VecExt,
-    ColSlice, RowSlice
+    ColSlice, RowSlice,
+    Eye
 };
 
 pub use structs::{
diff --git a/src/structs/dmat.rs b/src/structs/dmat.rs
index 8fac029c..40a3dea3 100644
--- a/src/structs/dmat.rs
+++ b/src/structs/dmat.rs
@@ -9,7 +9,7 @@ use traits::operations::ApproxEq;
 use std::mem;
 use structs::dvec::{DVec, DVecMulRhs};
 use traits::operations::{Inv, Transpose, Mean, Cov};
-use traits::structure::{Cast, ColSlice, RowSlice};
+use traits::structure::{Cast, ColSlice, RowSlice, Eye, Indexable};
 use std::fmt::{Show, Formatter, Result};
 
 #[doc(hidden)]
@@ -181,19 +181,19 @@ impl<N> DMat<N> {
 
 // FIXME: add a function to modify the dimension (to avoid useless allocations)?
 
-impl<N: One + Zero + Clone> DMat<N> {
+impl<N: One + Zero + Clone> Eye for DMat<N> {
     /// Builds an identity matrix.
     /// 
     /// # Arguments
     /// * `dim` - The dimension of the matrix. A `dim`-dimensional matrix contains `dim * dim`
     /// components.
     #[inline]
-    pub fn new_identity(dim: uint) -> DMat<N> {
+    fn new_identity(dim: uint) -> DMat<N> {
         let mut res = DMat::new_zeros(dim, dim);
 
         for i in range(0u, dim) {
             let _1: N  = One::one();
-            res.set(i, i, _1);
+            res.set((i, i), _1);
         }
 
         res
@@ -206,13 +206,16 @@ impl<N: Clone> DMat<N> {
         i + j * self.nrows
     }
 
+}
+
+impl<N: Clone> Indexable<(uint, uint), N> for DMat<N> {
     /// Changes the value of a component of the matrix.
     ///
     /// # Arguments
-    ///   * `row` - 0-based index of the line to be changed
-    ///   * `col` - 0-based index of the column to be changed
+    /// * `rowcol` - 0-based tuple (row, col)  to be changed
     #[inline]
-    pub fn set(&mut self, row: uint, col: uint, val: N) {
+    fn set(&mut self, rowcol: (uint,  uint), val: N) {
+        let (row, col) = rowcol;
         assert!(row < self.nrows);
         assert!(col < self.ncols);
 
@@ -222,7 +225,8 @@ impl<N: Clone> DMat<N> {
 
     /// Just like `set` without bounds checking.
     #[inline]
-    pub unsafe fn set_fast(&mut self, row: uint, col: uint, val: N) {
+    unsafe fn unsafe_set(&mut self, rowcol: (uint, uint), val: N) {
+        let (row, col) = rowcol;
         let offset = self.offset(row, col);
         *self.mij.as_mut_slice().unsafe_mut_ref(offset) = val
     }
@@ -230,20 +234,38 @@ impl<N: Clone> DMat<N> {
     /// Reads the value of a component of the matrix.
     ///
     /// # Arguments
-    /// * `row` - 0-based index of the line to be read
-    /// * `col` - 0-based index of the column to be read
+    /// * `rowcol` - 0-based tuple (row, col)  to be read
     #[inline]
-    pub fn at(&self, row: uint, col: uint) -> N {
+    fn at(&self, rowcol: (uint, uint)) -> N {
+        let (row, col) = rowcol;
         assert!(row < self.nrows);
         assert!(col < self.ncols);
-        unsafe { self.at_fast(row, col) }
+        unsafe { self.unsafe_at((row, col)) }
     }
 
     /// Just like `at` without bounds checking.
     #[inline]
-    pub unsafe fn at_fast(&self, row: uint, col: uint) -> N {
+    unsafe fn unsafe_at(&self, rowcol: (uint,  uint)) -> N {
+        let (row, col) = rowcol;
         (*self.mij.as_slice().unsafe_ref(self.offset(row, col))).clone()
     }
+
+    #[inline]
+    fn swap(&mut self, rowcol1: (uint, uint), rowcol2: (uint, uint)) {
+        let (row1, col1) = rowcol1;
+        let (row2, col2) = rowcol2;
+        let offset1 = self.offset(row1, col1);
+        let offset2 = self.offset(row2, col2);
+        let count = self.mij.len();
+        assert!(offset1 < count);
+        assert!(offset1 < count);
+        self.mij.as_mut_slice().swap(offset1, offset2);
+    }
+
+    fn shape(&self) -> (uint, uint) {
+        (self.nrows, self.ncols)
+    }
+
 }
 
 impl<N: Clone + Mul<N, N> + Add<N, N> + Zero> DMatMulRhs<N, DMat<N>> for DMat<N> {
@@ -258,10 +280,11 @@ impl<N: Clone + Mul<N, N> + Add<N, N> + Zero> DMatMulRhs<N, DMat<N>> for DMat<N>
 
                 unsafe {
                     for k in range(0u, left.ncols) {
-                        acc = acc + left.at_fast(i, k) * right.at_fast(k, j);
+                        acc = acc
+                            + left.unsafe_at((i, k)) * right.unsafe_at((k, j));
                     }
 
-                    res.set_fast(i, j, acc);
+                    res.unsafe_set((i, j), acc);
                 }
             }
         }
@@ -282,7 +305,7 @@ DMatMulRhs<N, DVec<N>> for DVec<N> {
 
             for j in range(0u, left.ncols) {
                 unsafe {
-                    acc = acc + left.at_fast(i, j) * right.at_fast(j);
+                    acc = acc + left.unsafe_at((i, j)) * right.unsafe_at(j);
                 }
             }
 
@@ -306,7 +329,7 @@ DVecMulRhs<N, DVec<N>> for DMat<N> {
 
             for j in range(0u, right.nrows) {
                 unsafe {
-                    acc = acc + left.at_fast(j) * right.at_fast(j, i);
+                    acc = acc + left.unsafe_at(j) * right.unsafe_at((j, i));
                 }
             }
 
@@ -335,7 +358,7 @@ Inv for DMat<N> {
         assert!(self.nrows == self.ncols);
 
         let dim              = self.nrows;
-        let mut res: DMat<N> = DMat::new_identity(dim);
+        let mut res: DMat<N> = Eye::new_identity(dim);
         let     _0T: N       = Zero::zero();
 
         // inversion using Gauss-Jordan elimination
@@ -347,7 +370,7 @@ Inv for DMat<N> {
             let mut n0 = k; // index of a non-zero entry
 
             while n0 != dim {
-                if unsafe { self.at_fast(n0, k) } != _0T {
+                if unsafe { self.unsafe_at((n0, k)) } != _0T {
                     break;
                 }
 
@@ -370,30 +393,30 @@ Inv for DMat<N> {
             }
 
             unsafe {
-                let pivot = self.at_fast(k, k);
+                let pivot = self.unsafe_at((k, k));
 
                 for j in range(k, dim) {
-                    let selfval = self.at_fast(k, j) / pivot;
-                    self.set_fast(k, j, selfval);
+                    let selfval = self.unsafe_at((k, j)) / pivot;
+                    self.unsafe_set((k, j), selfval);
                 }
 
                 for j in range(0u, dim) {
-                    let resval = res.at_fast(k, j) / pivot;
-                    res.set_fast(k, j, resval);
+                    let resval = res.unsafe_at((k, j)) / pivot;
+                    res.unsafe_set((k, j), resval);
                 }
 
                 for l in range(0u, dim) {
                     if l != k {
-                        let normalizer = self.at_fast(l, k);
+                        let normalizer = self.unsafe_at((l, k));
 
                         for j in range(k, dim) {
-                            let selfval = self.at_fast(l, j) - self.at_fast(k, j) * normalizer;
-                            self.set_fast(l, j, selfval);
+                            let selfval = self.unsafe_at((l, j)) - self.unsafe_at((k, j)) * normalizer;
+                            self.unsafe_set((l, j), selfval);
                         }
 
                         for j in range(0u, dim) {
-                            let resval = res.at_fast(l, j) - res.at_fast(k, j) * normalizer;
-                            res.set_fast(l, j, resval);
+                            let resval = res.unsafe_at((l, j)) - res.unsafe_at((k, j)) * normalizer;
+                            res.unsafe_set((l, j), resval);
                         }
                     }
                 }
@@ -422,7 +445,7 @@ impl<N: Clone> Transpose for DMat<N> {
             for i in range(0u, m.nrows) {
                 for j in range(0u, m.ncols) {
                     unsafe {
-                        res.set_fast(j, i, m.at_fast(i, j))
+                        res.unsafe_set((j, i), m.unsafe_at((i, j)))
                     }
                 }
             }
@@ -460,8 +483,8 @@ impl<N: Num + Cast<f32> + Clone> Mean<DVec<N>> for DMat<N> {
         for i in range(0u, m.nrows) {
             for j in range(0u, m.ncols) {
                 unsafe {
-                    let acc = res.at_fast(j) + m.at_fast(i, j) * normalizer;
-                    res.set_fast(j, acc);
+                    let acc = res.unsafe_at(j) + m.unsafe_at((i, j)) * normalizer;
+                    res.unsafe_set(j, acc);
                 }
             }
         }
@@ -482,7 +505,7 @@ impl<N: Clone + Num + Cast<f32> + DMatDivRhs<N, DMat<N>> + ToStr > Cov<DMat<N>>
         for i in range(0u, m.nrows) {
             for j in range(0u, m.ncols) {
                 unsafe {
-                    centered.set_fast(i, j, m.at_fast(i, j) - mean.at_fast(j));
+                    centered.unsafe_set((i, j), m.unsafe_at((i, j)) - mean.unsafe_at(j));
                 }
             }
         }
@@ -520,7 +543,7 @@ impl<N: Clone> RowSlice<DVec<N>> for DMat<N> {
         let mut slice_idx = 0u;
         for col_id in range(col_start, col_end) {
             unsafe {
-                slice.set_fast(slice_idx, self.at_fast(row_id, col_id));
+                slice.unsafe_set(slice_idx, self.unsafe_at((row_id, col_id)));
             }
             slice_idx += 1;
         }
@@ -553,7 +576,7 @@ impl<N: Show + Clone> Show for DMat<N> {
     fn fmt(&self, form:&mut Formatter) -> Result {
         for i in range(0u, self.nrows()) {
             for j in range(0u, self.ncols()) {
-                let _ = write!(form.buf, "{} ", self.at(i, j));
+                let _ = write!(form.buf, "{} ", self.at((i, j)));
             }
             let _ = write!(form.buf, "\n");
         }
diff --git a/src/structs/dvec.rs b/src/structs/dvec.rs
index b45a1a47..4da43229 100644
--- a/src/structs/dvec.rs
+++ b/src/structs/dvec.rs
@@ -9,7 +9,7 @@ use std::slice::{Items, MutItems};
 use traits::operations::ApproxEq;
 use std::iter::FromIterator;
 use traits::geometry::{Dot, Norm};
-use traits::structure::{Iterable, IterableMut};
+use traits::structure::{Iterable, IterableMut, Indexable};
 
 #[doc(hidden)]
 mod metal;
@@ -48,11 +48,42 @@ impl<N: Zero + Clone> DVec<N> {
     }
 }
 
-impl<N: Clone> DVec<N> {
-    /// Indexing without bounds checking.
-    pub unsafe fn at_fast(&self, i: uint) -> N {
+impl<N: Clone> Indexable<uint, N> for DVec<N> {
+
+    fn at(&self, i: uint) -> N {
+        assert!(i < self.at.len());
+        unsafe {
+            self.unsafe_at(i)
+        }
+    }
+
+    fn set(&mut self, i: uint, val: N) {
+        assert!(i < self.at.len());
+        unsafe {
+            self.unsafe_set(i, val);
+        }
+    }
+
+    fn swap(&mut self, i: uint, j: uint) {
+        assert!(i < self.at.len());
+        assert!(j < self.at.len());
+        self.at.as_mut_slice().swap(i, j);
+    }
+
+    fn shape(&self) -> uint {
+        self.at.len()
+    }
+
+    #[inline]
+    unsafe fn unsafe_at(&self, i: uint) -> N {
         (*self.at.as_slice().unsafe_ref(i)).clone()
     }
+
+    #[inline]
+    unsafe fn unsafe_set(&mut self, i: uint, val: N) {
+        *self.at.as_mut_slice().unsafe_mut_ref(i) = val
+    }
+
 }
 
 impl<N: One + Clone> DVec<N> {
@@ -86,11 +117,6 @@ impl<N> DVec<N> {
         }
     }
 
-    #[inline]
-    pub unsafe fn set_fast(&mut self, i: uint, val: N) {
-        *self.at.as_mut_slice().unsafe_mut_ref(i) = val
-    }
-
     /// Gets a reference to of this vector data.
     #[inline]
     pub fn as_vec<'r>(&'r self) -> &'r [N] {
@@ -261,7 +287,7 @@ impl<N: Num + Clone> Dot<N> for DVec<N> {
         let mut res: N = Zero::zero();
 
         for i in range(0u, a.at.len()) {
-            res = res + unsafe { a.at_fast(i) * b.at_fast(i) };
+            res = res + unsafe { a.unsafe_at(i) * b.unsafe_at(i) };
         }
 
         res
@@ -272,7 +298,7 @@ impl<N: Num + Clone> Dot<N> for DVec<N> {
         let mut res: N = Zero::zero();
 
         for i in range(0u, a.at.len()) {
-            res = res + unsafe { (a.at_fast(i) - b.at_fast(i)) * c.at_fast(i) };
+            res = res + unsafe { (a.unsafe_at(i) - b.unsafe_at(i)) * c.unsafe_at(i) };
         }
 
         res
diff --git a/src/structs/mat.rs b/src/structs/mat.rs
index c17831cf..0d180fcf 100644
--- a/src/structs/mat.rs
+++ b/src/structs/mat.rs
@@ -9,7 +9,8 @@ use std::slice::{Items, MutItems};
 use structs::vec::{Vec1, Vec2, Vec3, Vec4, Vec5, Vec6, Vec1MulRhs, Vec4MulRhs,
                    Vec5MulRhs, Vec6MulRhs};
 
-use traits::structure::{Cast, Row, Col, Iterable, IterableMut, Dim, Indexable};
+use traits::structure::{Cast, Row, Col, Iterable, IterableMut, Dim, Indexable,
+                        Eye};
 use traits::operations::{Absolute, Transpose, Inv, Outer};
 use traits::geometry::{ToHomogeneous, FromHomogeneous};
 
@@ -34,6 +35,8 @@ pub struct Mat1<N> {
     pub m11: N
 }
 
+eye_impl!(Mat1, 1, m11)
+
 double_dispatch_binop_decl_trait!(Mat1, Mat1MulRhs)
 double_dispatch_binop_decl_trait!(Mat1, Mat1DivRhs)
 double_dispatch_binop_decl_trait!(Mat1, Mat1AddRhs)
@@ -127,6 +130,8 @@ pub struct Mat2<N> {
     pub m12: N, pub m22: N
 }
 
+eye_impl!(Mat2, 2, m11, m22)
+
 double_dispatch_binop_decl_trait!(Mat2, Mat2MulRhs)
 double_dispatch_binop_decl_trait!(Mat2, Mat2DivRhs)
 double_dispatch_binop_decl_trait!(Mat2, Mat2AddRhs)
@@ -225,6 +230,8 @@ pub struct Mat3<N> {
     pub m13: N, pub m23: N, pub m33: N
 }
 
+eye_impl!(Mat3, 3, m11, m22, m33)
+
 double_dispatch_binop_decl_trait!(Mat3, Mat3MulRhs)
 double_dispatch_binop_decl_trait!(Mat3, Mat3DivRhs)
 double_dispatch_binop_decl_trait!(Mat3, Mat3AddRhs)
@@ -337,6 +344,8 @@ pub struct Mat4<N> {
     pub m14: N, pub m24: N, pub m34: N, pub m44: N
 }
 
+eye_impl!(Mat4, 4, m11, m22, m33, m44)
+
 double_dispatch_binop_decl_trait!(Mat4, Mat4MulRhs)
 double_dispatch_binop_decl_trait!(Mat4, Mat4DivRhs)
 double_dispatch_binop_decl_trait!(Mat4, Mat4AddRhs)
@@ -501,6 +510,8 @@ pub struct Mat5<N> {
     pub m15: N, pub m25: N, pub m35: N, pub m45: N, pub m55: N
 }
 
+eye_impl!(Mat5, 5, m11, m22, m33, m44, m55)
+
 double_dispatch_binop_decl_trait!(Mat5, Mat5MulRhs)
 double_dispatch_binop_decl_trait!(Mat5, Mat5DivRhs)
 double_dispatch_binop_decl_trait!(Mat5, Mat5AddRhs)
@@ -681,6 +692,8 @@ pub struct Mat6<N> {
     pub m16: N, pub m26: N, pub m36: N, pub m46: N, pub m56: N, pub m66: N
 }
 
+eye_impl!(Mat6, 6, m11, m22, m33, m44, m55, m66)
+
 double_dispatch_binop_decl_trait!(Mat6, Mat6MulRhs)
 double_dispatch_binop_decl_trait!(Mat6, Mat6DivRhs)
 double_dispatch_binop_decl_trait!(Mat6, Mat6AddRhs)
diff --git a/src/structs/mat_macros.rs b/src/structs/mat_macros.rs
index 2c13926d..6d08435b 100644
--- a/src/structs/mat_macros.rs
+++ b/src/structs/mat_macros.rs
@@ -98,6 +98,20 @@ macro_rules! scalar_add_impl(
     )
 )
 
+
+macro_rules! eye_impl(
+    ($t: ident, $ndim: expr, $($comp_diagN: ident),+) => (
+        impl<N: Zero + One> Eye for $t<N> {
+            fn new_identity(dim: uint) -> $t<N> {
+                assert!(dim == $ndim);
+                let mut eye: $t<N> = Zero::zero();
+                $(eye.$comp_diagN = One::one();)+
+                eye
+            }
+        }
+    )
+)
+
 macro_rules! scalar_sub_impl(
     ($t: ident, $n: ident, $trhs: ident, $comp0: ident $(,$compN: ident)*) => (
         impl $trhs<$n, $t<$n>> for $n {
@@ -193,6 +207,11 @@ macro_rules! indexable_impl(
             }
         }
 
+        #[inline]
+        fn shape(&self) -> (uint, uint) {
+            ($dim, $dim)
+        }
+
         #[inline]
         unsafe fn unsafe_at(&self, (i, j): (uint, uint)) -> N {
             (*cast::transmute::<&$t<N>, &[N, ..$dim * $dim]>(self).unsafe_ref(i + j * $dim)).clone()
diff --git a/src/structs/spec/vec0.rs b/src/structs/spec/vec0.rs
index 4843e3e1..ce493dda 100644
--- a/src/structs/spec/vec0.rs
+++ b/src/structs/spec/vec0.rs
@@ -25,6 +25,11 @@ impl<N> Indexable<uint, N> for vec::Vec0<N> {
     fn set(&mut self, _: uint, _: N) {
     }
 
+    #[inline]
+    fn shape(&self) -> uint {
+        0
+    }
+
     #[inline]
     fn swap(&mut self, _: uint, _: uint) {
     }
diff --git a/src/structs/vec_macros.rs b/src/structs/vec_macros.rs
index 0eeafb93..90e5ca9f 100644
--- a/src/structs/vec_macros.rs
+++ b/src/structs/vec_macros.rs
@@ -165,6 +165,11 @@ macro_rules! indexable_impl(
                 }
             }
 
+            #[inline]
+            fn shape(&self) -> uint {
+                $dim
+            }
+
             #[inline]
             fn swap(&mut self, i1: uint, i2: uint) {
                 unsafe {
diff --git a/src/traits/mod.rs b/src/traits/mod.rs
index 9c774d80..1c471b45 100644
--- a/src/traits/mod.rs
+++ b/src/traits/mod.rs
@@ -6,7 +6,7 @@ pub use self::geometry::{AbsoluteRotate, Cross, CrossMatrix, Dot, FromHomogeneou
 
 pub use self::structure::{FloatVec, FloatVecExt, Basis, Cast, Col, Dim, Indexable,
                           Iterable, IterableMut, Mat, Row, AnyVec, VecExt,
-                          ColSlice, RowSlice};
+                          ColSlice, RowSlice, Eye};
 
 pub use self::operations::{Absolute, ApproxEq, Cov, Inv, LMul, Mean, Outer, PartialOrd, RMul,
                            ScalarAdd, ScalarSub, Transpose};
diff --git a/src/traits/structure.rs b/src/traits/structure.rs
index 953f13b9..0e94e6ad 100644
--- a/src/traits/structure.rs
+++ b/src/traits/structure.rs
@@ -19,6 +19,12 @@ pub trait Mat<R, C> : Row<R> + Col<C> + RMul<R> + LMul<C> { }
 impl<M: Row<R> + Col<C> + RMul<R> + LMul<C>, R, C> Mat<R, C> for M {
 }
 
+/// Trait for constructing the identity matrix
+pub trait Eye {
+    /// Return the identity matrix of specified dimension
+    fn new_identity(dim: uint) -> Self;
+}
+
 // XXX: we keep ScalarAdd and ScalarSub here to avoid trait impl conflict (overriding) between the
 // different Add/Sub traits. This is _so_ unfortunate…
 
@@ -126,6 +132,9 @@ pub trait Indexable<Index, Res> {
     /// Swaps the `i`-th element of `self` with its `j`-th element.
     fn swap(&mut self, i: Index, j: Index);
 
+    /// Returns the shape of the iterable range
+    fn shape(&self) -> Index;
+
     /// Reads the `i`-th element of `self`.
     ///
     /// `i` is not checked.

From dcf7b8ad018c074db9586bf139fba600e76a6016 Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Sun, 11 May 2014 21:20:41 +0200
Subject: [PATCH 2/8] generic implementation of QR decomposition

But static matrices can't use it yet, they need to implement the
Row/Col slicing traits.
---
 src/linalg/decompositions.rs | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/src/linalg/decompositions.rs b/src/linalg/decompositions.rs
index 3cc1a0f5..6c28b30f 100644
--- a/src/linalg/decompositions.rs
+++ b/src/linalg/decompositions.rs
@@ -1,5 +1,4 @@
 use std::num::{Zero, Float};
-use na::DMat;
 use traits::operations::Transpose;
 use traits::structure::{ColSlice, Eye, Indexable};
 use traits::geometry::Norm;
@@ -34,11 +33,14 @@ fn householder_matrix<N: Float,
 /// QR decomposition using Householder reflections
 /// # Arguments
 /// * `m` - matrix to decompose
-pub fn decomp_qr<N: Clone + Float>(m: &DMat<N>) -> (DMat<N>, DMat<N>) {
-    let rows = m.nrows();
-    let cols = m.ncols();
+pub fn decomp_qr<N: Float,
+                 V: Indexable<uint, N> + Norm<N>,
+                 Mat: Clone + Eye + ColSlice<V> + Transpose
+                    + Indexable<(uint, uint), N> + Mul<Mat, Mat>>
+                 (m: &Mat) -> (Mat, Mat) {
+    let (rows, cols) = m.shape();
     assert!(rows >= cols);
-    let mut q : DMat<N> = Eye::new_identity(rows);
+    let mut q : Mat = Eye::new_identity(rows);
     let mut r = m.clone();
 
     let iterations = min(rows - 1, cols);
@@ -57,7 +59,7 @@ pub fn decomp_qr<N: Clone + Float>(m: &DMat<N>) -> (DMat<N>, DMat<N>) {
             v.unsafe_set(0, x - alpha);
         }
         let _ = v.normalize();
-        let qk: DMat<N> = householder_matrix(rows, 0, v);
+        let qk: Mat = householder_matrix(rows, 0, v);
         r = qk * r;
         q = q * Transpose::transpose_cpy(&qk);
     }

From 76a8bc3cf158f6fbc1cdf23ed90363385b873df7 Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Mon, 12 May 2014 14:06:25 +0200
Subject: [PATCH 3/8] QR factorization for fixed size matrices.

The ColSlice implementation for fixed size matrices returns a DVec,
while this is probably not optimal performance-wise, the dynamic nature
of the result makes this necessary. Using a data type presenting the
ImmutableVector trait would solve this, but it looks like a non-trivial
change.
---
 src/structs/mat.rs        | 15 ++++++++-
 src/structs/mat_macros.rs | 40 +++++++++++++++++++++++
 src/tests/mat.rs          | 69 ++++++++++++++++++++++++++++++---------
 3 files changed, 108 insertions(+), 16 deletions(-)

diff --git a/src/structs/mat.rs b/src/structs/mat.rs
index 0d180fcf..14dc5476 100644
--- a/src/structs/mat.rs
+++ b/src/structs/mat.rs
@@ -8,9 +8,10 @@ use traits::operations::ApproxEq;
 use std::slice::{Items, MutItems};
 use structs::vec::{Vec1, Vec2, Vec3, Vec4, Vec5, Vec6, Vec1MulRhs, Vec4MulRhs,
                    Vec5MulRhs, Vec6MulRhs};
+use structs::dvec::DVec;
 
 use traits::structure::{Cast, Row, Col, Iterable, IterableMut, Dim, Indexable,
-                        Eye};
+                        Eye, ColSlice, RowSlice};
 use traits::operations::{Absolute, Transpose, Inv, Outer};
 use traits::geometry::{ToHomogeneous, FromHomogeneous};
 
@@ -119,6 +120,8 @@ transpose_impl!(Mat1, 1)
 approx_eq_impl!(Mat1)
 row_impl!(Mat1, Vec1, 1)
 col_impl!(Mat1, Vec1, 1)
+col_slice_impl!(Mat1, Vec1, 1)
+row_slice_impl!(Mat1, Vec1, 1)
 to_homogeneous_impl!(Mat1, Mat2, 1, 2)
 from_homogeneous_impl!(Mat1, Mat2, 1, 2)
 outer_impl!(Vec1, Mat1)
@@ -218,6 +221,8 @@ transpose_impl!(Mat2, 2)
 approx_eq_impl!(Mat2)
 row_impl!(Mat2, Vec2, 2)
 col_impl!(Mat2, Vec2, 2)
+col_slice_impl!(Mat2, Vec2, 2)
+row_slice_impl!(Mat2, Vec2, 2)
 to_homogeneous_impl!(Mat2, Mat3, 2, 3)
 from_homogeneous_impl!(Mat2, Mat3, 2, 3)
 outer_impl!(Vec2, Mat2)
@@ -331,6 +336,8 @@ transpose_impl!(Mat3, 3)
 approx_eq_impl!(Mat3)
 // (specialized) row_impl!(Mat3, Vec3, 3)
 // (specialized) col_impl!(Mat3, Vec3, 3)
+col_slice_impl!(Mat3, Vec3, 3)
+row_slice_impl!(Mat3, Vec3, 3)
 to_homogeneous_impl!(Mat3, Mat4, 3, 4)
 from_homogeneous_impl!(Mat3, Mat4, 3, 4)
 outer_impl!(Vec3, Mat3)
@@ -496,6 +503,8 @@ transpose_impl!(Mat4, 4)
 approx_eq_impl!(Mat4)
 row_impl!(Mat4, Vec4, 4)
 col_impl!(Mat4, Vec4, 4)
+col_slice_impl!(Mat4, Vec4, 4)
+row_slice_impl!(Mat4, Vec4, 4)
 to_homogeneous_impl!(Mat4, Mat5, 4, 5)
 from_homogeneous_impl!(Mat4, Mat5, 4, 5)
 outer_impl!(Vec4, Mat4)
@@ -677,6 +686,8 @@ transpose_impl!(Mat5, 5)
 approx_eq_impl!(Mat5)
 row_impl!(Mat5, Vec5, 5)
 col_impl!(Mat5, Vec5, 5)
+col_slice_impl!(Mat5, Vec5, 5)
+row_slice_impl!(Mat5, Vec5, 5)
 to_homogeneous_impl!(Mat5, Mat6, 5, 6)
 from_homogeneous_impl!(Mat5, Mat6, 5, 6)
 outer_impl!(Vec5, Mat5)
@@ -910,4 +921,6 @@ transpose_impl!(Mat6, 6)
 approx_eq_impl!(Mat6)
 row_impl!(Mat6, Vec6, 6)
 col_impl!(Mat6, Vec6, 6)
+col_slice_impl!(Mat6, Vec6, 6)
+row_slice_impl!(Mat6, Vec6, 6)
 outer_impl!(Vec6, Mat6)
diff --git a/src/structs/mat_macros.rs b/src/structs/mat_macros.rs
index 6d08435b..16005379 100644
--- a/src/structs/mat_macros.rs
+++ b/src/structs/mat_macros.rs
@@ -225,6 +225,26 @@ macro_rules! indexable_impl(
   )
 )
 
+macro_rules! col_slice_impl(
+    ($t: ident, $tv: ident, $dim: expr) => (
+        impl<N: Clone + Zero> ColSlice<DVec<N>> for $t<N> {
+            fn col_slice(& self, cid: uint, rstart: uint, rend: uint) -> DVec<N> {
+                assert!(cid < $dim);
+                assert!(rstart < rend);
+                assert!(rend <= $dim);
+                let col = self.col(cid);
+                let res  = DVec::from_vec(
+                    rend - rstart,
+                    unsafe {
+                        cast::transmute::<&$tv<N>, & [N, ..$dim]>
+                        (&col).slice(rstart, rend)
+                    });
+                res
+            }
+        }
+    )
+)
+
 macro_rules! row_impl(
   ($t: ident, $tv: ident, $dim: expr) => (
     impl<N: Clone + Zero> Row<$tv<N>> for $t<N> {
@@ -254,6 +274,26 @@ macro_rules! row_impl(
   )
 )
 
+macro_rules! row_slice_impl(
+    ($t: ident, $tv: ident, $dim: expr) => (
+        impl<N: Clone + Zero> RowSlice<DVec<N>> for $t<N> {
+            fn row_slice(& self, rid: uint, cstart: uint, cend: uint) -> DVec<N> {
+                assert!(rid < $dim);
+                assert!(cstart < cend);
+                assert!(cend <= $dim);
+                let row = self.row(rid);
+                let res  = DVec::from_vec(
+                    cend - cstart,
+                    unsafe {
+                        cast::transmute::<&$tv<N>, & [N, ..$dim]>
+                        (&row).slice(cstart, cend)
+                    });
+                res
+            }
+        }
+    )
+)
+
 macro_rules! col_impl(
   ($t: ident, $tv: ident, $dim: expr) => (
     impl<N: Clone + Zero> Col<$tv<N>> for $t<N> {
diff --git a/src/tests/mat.rs b/src/tests/mat.rs
index ca2be979..305f893d 100644
--- a/src/tests/mat.rs
+++ b/src/tests/mat.rs
@@ -3,6 +3,7 @@ use rand::random;
 use na::{Vec1, Vec3, Mat1, Mat2, Mat3, Mat4, Mat5, Mat6, Rot3, DMat, DVec, Indexable};
 use na;
 use na::decomp_qr;
+use std::cmp::{min, max};
 
 macro_rules! test_inv_mat_impl(
   ($t: ty) => (
@@ -24,6 +25,19 @@ macro_rules! test_transpose_mat_impl(
   );
 )
 
+macro_rules! test_decomp_qr_impl(
+  ($t: ty) => (
+    for _ in range(0, 10000) {
+      let randmat : $t = random();
+
+      let (q, r) = decomp_qr(&randmat);
+      let recomp = q * r;
+
+      assert!(na::approx_eq(&randmat,  &recomp));
+    }
+  );
+)
+
 #[test]
 fn test_transpose_mat1() {
     test_transpose_mat_impl!(Mat1<f64>);
@@ -210,20 +224,45 @@ fn test_dmat_from_vec() {
 
 #[test]
 fn test_decomp_qr() {
-    let mat = DMat::from_row_vec(
-        5,
-        3,
-        [
-            4.0, 2.0, 0.60,
-            4.2, 2.1, 0.59,
-            3.9, 2.0, 0.58,
-            4.3, 2.1, 0.62,
-            4.1, 2.2, 0.63
-        ]
-    );
+    for _ in range(0, 10) {
+        let dim1: uint = random();
+        let dim2: uint = random();
+        let rows = min(40, max(dim1, dim2));
+        let cols = min(40, min(dim1, dim2));
+        let randmat: DMat<f64> = DMat::new_random(rows, cols);
+        let (q, r) = decomp_qr(&randmat);
+        let recomp = q * r;
 
-    let (q, r) = decomp_qr(&mat);
-    let mat_ = q * r;
-
-    assert!(na::approx_eq(&mat_,  &mat));
+        assert!(na::approx_eq(&randmat,  &recomp));
+    }
+}
+
+#[test]
+fn test_decomp_qr_mat1() {
+    test_decomp_qr_impl!(Mat1<f64>);
+}
+
+#[test]
+fn test_decomp_qr_mat2() {
+    test_decomp_qr_impl!(Mat2<f64>);
+}
+
+#[test]
+fn test_decomp_qr_mat3() {
+    test_decomp_qr_impl!(Mat3<f64>);
+}
+
+#[test]
+fn test_decomp_qr_mat4() {
+    test_decomp_qr_impl!(Mat4<f64>);
+}
+
+#[test]
+fn test_decomp_qr_mat5() {
+    test_decomp_qr_impl!(Mat5<f64>);
+}
+
+#[test]
+fn test_decomp_qr_mat6() {
+    test_decomp_qr_impl!(Mat6<f64>);
 }

From 3dc0e27fd2d0a3cf8cb719711891763eff7244d5 Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Mon, 12 May 2014 21:54:18 +0200
Subject: [PATCH 4/8] fix typo

---
 src/structs/dmat.rs | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/structs/dmat.rs b/src/structs/dmat.rs
index 40a3dea3..6d101698 100644
--- a/src/structs/dmat.rs
+++ b/src/structs/dmat.rs
@@ -258,7 +258,7 @@ impl<N: Clone> Indexable<(uint, uint), N> for DMat<N> {
         let offset2 = self.offset(row2, col2);
         let count = self.mij.len();
         assert!(offset1 < count);
-        assert!(offset1 < count);
+        assert!(offset2 < count);
         self.mij.as_mut_slice().swap(offset1, offset2);
     }
 

From 73c6610048cd452f8f643d759cc1bf49cb096c24 Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Mon, 12 May 2014 21:54:59 +0200
Subject: [PATCH 5/8] new_identity and housholder matrix available under na::

---
 src/linalg/decompositions.rs | 18 +++++++++---------
 src/linalg/mod.rs            |  2 +-
 src/na.rs                    |  7 ++++++-
 3 files changed, 16 insertions(+), 11 deletions(-)

diff --git a/src/linalg/decompositions.rs b/src/linalg/decompositions.rs
index 6c28b30f..6cb50153 100644
--- a/src/linalg/decompositions.rs
+++ b/src/linalg/decompositions.rs
@@ -11,11 +11,11 @@ use std::cmp::min;
 /// * `dim` - the dimension of the space the resulting matrix operates in
 /// * `start` - the starting dimension of the subspace of the reflexion
 /// * `vec` - the vector defining the reflection.
-fn householder_matrix<N: Float,
-                      Mat: Eye + Indexable<(uint, uint), N>,
+pub fn householder_matrix<N: Float,
+                      M: Eye + Indexable<(uint, uint), N>,
                       V: Indexable<uint, N>>
-                      (dim: uint, start: uint, vec: V) -> Mat {
-    let mut qk : Mat = Eye::new_identity(dim);
+                      (dim: uint, start: uint, vec: V) -> M {
+    let mut qk : M = Eye::new_identity(dim);
     let stop = start + vec.shape();
     assert!(stop <= dim);
     for j in range(start, stop) {
@@ -35,12 +35,12 @@ fn householder_matrix<N: Float,
 /// * `m` - matrix to decompose
 pub fn decomp_qr<N: Float,
                  V: Indexable<uint, N> + Norm<N>,
-                 Mat: Clone + Eye + ColSlice<V> + Transpose
-                    + Indexable<(uint, uint), N> + Mul<Mat, Mat>>
-                 (m: &Mat) -> (Mat, Mat) {
+                 M: Clone + Eye + ColSlice<V> + Transpose
+                    + Indexable<(uint, uint), N> + Mul<M, M>>
+                 (m: &M) -> (M, M) {
     let (rows, cols) = m.shape();
     assert!(rows >= cols);
-    let mut q : Mat = Eye::new_identity(rows);
+    let mut q : M = Eye::new_identity(rows);
     let mut r = m.clone();
 
     let iterations = min(rows - 1, cols);
@@ -59,7 +59,7 @@ pub fn decomp_qr<N: Float,
             v.unsafe_set(0, x - alpha);
         }
         let _ = v.normalize();
-        let qk: Mat = householder_matrix(rows, 0, v);
+        let qk: M = householder_matrix(rows, 0, v);
         r = qk * r;
         q = q * Transpose::transpose_cpy(&qk);
     }
diff --git a/src/linalg/mod.rs b/src/linalg/mod.rs
index 6c2c6d63..e6e09f2a 100644
--- a/src/linalg/mod.rs
+++ b/src/linalg/mod.rs
@@ -1,4 +1,4 @@
 
-pub use self::decompositions::decomp_qr;
+pub use self::decompositions::{decomp_qr, householder_matrix};
 
 mod decompositions;
diff --git a/src/na.rs b/src/na.rs
index 6832fb18..b3bab4e8 100644
--- a/src/na.rs
+++ b/src/na.rs
@@ -55,7 +55,8 @@ pub use structs::{
 };
 
 pub use linalg::{
-    decomp_qr
+    decomp_qr,
+    householder_matrix
 };
 
 /// Traits to work around the language limitations related to operator overloading.
@@ -721,6 +722,10 @@ pub fn mean<N, M: Mean<N>>(observations: &M) -> N {
 //
 //
 
+/// Construct the identity matrix for a given dimension
+#[inline(always)]
+pub fn new_identity<M: Eye>(dim: uint) -> M { Eye::new_identity(dim) }
+
 /*
  * Basis
  */

From f8ac86effc44dac70b576be0f25275ecdad112b8 Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Tue, 13 May 2014 08:29:56 +0200
Subject: [PATCH 6/8] update to latest rust

---
 src/structs/mat_macros.rs | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/structs/mat_macros.rs b/src/structs/mat_macros.rs
index b3fb0513..e574a770 100644
--- a/src/structs/mat_macros.rs
+++ b/src/structs/mat_macros.rs
@@ -236,7 +236,7 @@ macro_rules! col_slice_impl(
                 let res  = DVec::from_vec(
                     rend - rstart,
                     unsafe {
-                        cast::transmute::<&$tv<N>, & [N, ..$dim]>
+                        mem::transmute::<&$tv<N>, & [N, ..$dim]>
                         (&col).slice(rstart, rend)
                     });
                 res
@@ -285,7 +285,7 @@ macro_rules! row_slice_impl(
                 let res  = DVec::from_vec(
                     cend - cstart,
                     unsafe {
-                        cast::transmute::<&$tv<N>, & [N, ..$dim]>
+                        mem::transmute::<&$tv<N>, & [N, ..$dim]>
                         (&row).slice(cstart, cend)
                     });
                 res

From b3bc4c66f179b1c67e743a6dedcf0261f658de8a Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Fri, 16 May 2014 19:55:20 +0200
Subject: [PATCH 7/8] QR decomposition mentionned in README

---
 README.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/README.md b/README.md
index aaade091..904cbd01 100644
--- a/README.md
+++ b/README.md
@@ -50,6 +50,7 @@ and keeps an optimized set of tools for computational graphics and physics. Thos
 * Dynamically sized vector: `DVec`.
 * Dynamically sized (square or rectangular) matrix: `DMat`.
 * A few methods for data analysis: `Cov`, `Mean`.
+* Some matrix factorization algorithms: QR decomposition, ...
 * Almost one trait per functionality: useful for generic programming.
 * Operator overloading using the double trait dispatch
   [trick](http://smallcultfollowing.com/babysteps/blog/2012/10/04/refining-traits-slash-impls/).

From 987b91767acce1e2dd29cb10c01da396aafc80fa Mon Sep 17 00:00:00 2001
From: Vincent Barrielle <vincent.barrielle@polytechnique.edu>
Date: Fri, 16 May 2014 21:04:35 +0200
Subject: [PATCH 8/8] update to the latest rust: FloatMath for math functions
 (sin/exp/...)

Also removed a bunch of duplicate trait usages
---
 src/na.rs                 |  4 ++--
 src/structs/dmat.rs       |  6 +++---
 src/structs/dvec.rs       |  4 ++--
 src/structs/iso.rs        |  2 +-
 src/structs/iso_macros.rs | 16 ++++++++--------
 src/structs/rot.rs        | 16 ++++++++--------
 src/structs/rot_macros.rs |  2 +-
 src/structs/spec/vec0.rs  |  2 +-
 src/structs/vec_macros.rs |  6 +++---
 9 files changed, 29 insertions(+), 29 deletions(-)

diff --git a/src/na.rs b/src/na.rs
index b3bab4e8..49c49b8d 100644
--- a/src/na.rs
+++ b/src/na.rs
@@ -1,6 +1,6 @@
 //! **nalgebra** prelude.
 
-use std::num::{Zero, One};
+use std::num::{Zero, One, FloatMath};
 use std::cmp;
 pub use traits::{PartialLess, PartialEqual, PartialGreater, NotComparable};
 pub use traits::{
@@ -217,7 +217,7 @@ pub fn one<T: One>() -> T {
  */
 /// Computes a projection matrix given the frustrum near plane width, height, the field of
 /// view, and the distance to the clipping planes (`znear` and `zfar`).
-pub fn perspective3d<N: Float + Cast<f32> + Zero + One>(width: N, height: N, fov: N, znear: N, zfar: N) -> Mat4<N> {
+pub fn perspective3d<N: FloatMath + Cast<f32> + Zero + One>(width: N, height: N, fov: N, znear: N, zfar: N) -> Mat4<N> {
     let aspect = width / height;
 
     let _1: N = one();
diff --git a/src/structs/dmat.rs b/src/structs/dmat.rs
index 6d101698..8f905d78 100644
--- a/src/structs/dmat.rs
+++ b/src/structs/dmat.rs
@@ -576,11 +576,11 @@ impl<N: Show + Clone> Show for DMat<N> {
     fn fmt(&self, form:&mut Formatter) -> Result {
         for i in range(0u, self.nrows()) {
             for j in range(0u, self.ncols()) {
-                let _ = write!(form.buf, "{} ", self.at((i, j)));
+                let _ = write!(form, "{} ", self.at((i, j)));
             }
-            let _ = write!(form.buf, "\n");
+            let _ = write!(form, "\n");
         }
-        write!(form.buf, "\n")
+        write!(form, "\n")
     }
 }
 
diff --git a/src/structs/dvec.rs b/src/structs/dvec.rs
index 4da43229..662d3bfe 100644
--- a/src/structs/dvec.rs
+++ b/src/structs/dvec.rs
@@ -198,7 +198,7 @@ impl<N> FromIterator<N> for DVec<N> {
     }
 }
 
-impl<N: Clone + Num + Float + ApproxEq<N> + DVecMulRhs<N, DVec<N>>> DVec<N> {
+impl<N: Clone + Float + ApproxEq<N> + DVecMulRhs<N, DVec<N>>> DVec<N> {
     /// Computes the canonical basis for the given dimension. A canonical basis is a set of
     /// vectors, mutually orthogonal, with all its component equal to 0.0 except one which is equal
     /// to 1.0.
@@ -305,7 +305,7 @@ impl<N: Num + Clone> Dot<N> for DVec<N> {
     } 
 }
 
-impl<N: Num + Float + Clone> Norm<N> for DVec<N> {
+impl<N: Float + Clone> Norm<N> for DVec<N> {
     #[inline]
     fn sqnorm(v: &DVec<N>) -> N {
         Dot::dot(v, v)
diff --git a/src/structs/iso.rs b/src/structs/iso.rs
index fc543ace..de4a78e2 100644
--- a/src/structs/iso.rs
+++ b/src/structs/iso.rs
@@ -51,7 +51,7 @@ pub struct Iso4<N> {
     pub translation: Vec4<N>
 }
 
-impl<N: Clone + Num + Float> Iso3<N> {
+impl<N: Clone + Float> Iso3<N> {
     /// Reorient and translate this transformation such that its local `x` axis points to a given
     /// direction.  Note that the usually known `look_at` function does the same thing but with the
     /// `z` axis. See `look_at_z` for that.
diff --git a/src/structs/iso_macros.rs b/src/structs/iso_macros.rs
index 43c05edb..1c618df4 100644
--- a/src/structs/iso_macros.rs
+++ b/src/structs/iso_macros.rs
@@ -2,7 +2,7 @@
 
 macro_rules! iso_impl(
     ($t: ident, $submat: ident, $subvec: ident, $subrotvec: ident) => (
-        impl<N: Clone + Float + Float + Num> $t<N> {
+        impl<N: Clone + FloatMath + Num> $t<N> {
             /// Creates a new isometry from a rotation matrix and a vector.
             #[inline]
             pub fn new(translation: $subvec<N>, rotation: $subrotvec<N>) -> $t<N> {
@@ -26,7 +26,7 @@ macro_rules! iso_impl(
 
 macro_rules! rotation_matrix_impl(
     ($t: ident, $trot: ident, $tlv: ident, $tav: ident) => (
-        impl<N: Cast<f32> + Float + Float + Num + Clone>
+        impl<N: Cast<f32> + FloatMath + Num + Clone>
         RotationMatrix<$tlv<N>, $tav<N>, $trot<N>> for $t<N> {
             #[inline]
             fn to_rot_mat(&self) -> $trot<N> {
@@ -50,7 +50,7 @@ macro_rules! dim_impl(
 
 macro_rules! one_impl(
     ($t: ident) => (
-        impl<N: Float + Float + Num + Clone> One for $t<N> {
+        impl<N: FloatMath + Clone> One for $t<N> {
             #[inline]
             fn one() -> $t<N> {
                 $t::new_with_rotmat(Zero::zero(), One::one())
@@ -61,7 +61,7 @@ macro_rules! one_impl(
 
 macro_rules! iso_mul_iso_impl(
     ($t: ident, $tmul: ident) => (
-        impl<N: Num + Float + Float + Clone> $tmul<N, $t<N>> for $t<N> {
+        impl<N: FloatMath + Clone> $tmul<N, $t<N>> for $t<N> {
             #[inline]
             fn binop(left: &$t<N>, right: &$t<N>) -> $t<N> {
                 $t::new_with_rotmat(
@@ -96,7 +96,7 @@ macro_rules! vec_mul_iso_impl(
 
 macro_rules! translation_impl(
     ($t: ident, $tv: ident) => (
-        impl<N: Float + Num + Float + Clone> Translation<$tv<N>> for $t<N> {
+        impl<N: FloatMath + Clone> Translation<$tv<N>> for $t<N> {
             #[inline]
             fn translation(&self) -> $tv<N> {
                 self.translation.clone()
@@ -153,7 +153,7 @@ macro_rules! translate_impl(
 
 macro_rules! rotation_impl(
     ($t: ident, $trot: ident, $tav: ident) => (
-        impl<N: Cast<f32> + Num + Float + Float + Clone> Rotation<$tav<N>> for $t<N> {
+        impl<N: Cast<f32> + FloatMath + Clone> Rotation<$tav<N>> for $t<N> {
             #[inline]
             fn rotation(&self) -> $tav<N> {
                 self.rotation.rotation()
@@ -220,7 +220,7 @@ macro_rules! rotate_impl(
 
 macro_rules! transformation_impl(
     ($t: ident) => (
-        impl<N: Num + Float + Float + Clone> Transformation<$t<N>> for $t<N> {
+        impl<N: FloatMath + Clone> Transformation<$t<N>> for $t<N> {
             fn transformation(&self) -> $t<N> {
                 self.clone()
             }
@@ -336,7 +336,7 @@ macro_rules! approx_eq_impl(
 
 macro_rules! rand_impl(
     ($t: ident) => (
-        impl<N: Rand + Clone + Float + Float + Num> Rand for $t<N> {
+        impl<N: Rand + Clone + FloatMath> Rand for $t<N> {
             #[inline]
             fn rand<R: Rng>(rng: &mut R) -> $t<N> {
                 $t::new(rng.gen(), rng.gen())
diff --git a/src/structs/rot.rs b/src/structs/rot.rs
index 4bdcb84d..b184dd6d 100644
--- a/src/structs/rot.rs
+++ b/src/structs/rot.rs
@@ -20,7 +20,7 @@ pub struct Rot2<N> {
     submat: Mat2<N>
 }
 
-impl<N: Clone + Float + Neg<N>> Rot2<N> {
+impl<N: Clone + FloatMath + Neg<N>> Rot2<N> {
     /// Builds a 2 dimensional rotation matrix from an angle in radian.
     pub fn new(angle: Vec1<N>) -> Rot2<N> {
         let (sia, coa) = angle.x.sin_cos();
@@ -31,7 +31,7 @@ impl<N: Clone + Float + Neg<N>> Rot2<N> {
     }
 }
 
-impl<N: Float + Num + Clone>
+impl<N: FloatMath + Clone>
 Rotation<Vec1<N>> for Rot2<N> {
     #[inline]
     fn rotation(&self) -> Vec1<N> {
@@ -69,7 +69,7 @@ Rotation<Vec1<N>> for Rot2<N> {
     }
 }
 
-impl<N: Clone + Rand + Float + Neg<N>> Rand for Rot2<N> {
+impl<N: Clone + Rand + FloatMath + Neg<N>> Rand for Rot2<N> {
     #[inline]
     fn rand<R: Rng>(rng: &mut R) -> Rot2<N> {
         Rot2::new(rng.gen())
@@ -99,7 +99,7 @@ pub struct Rot3<N> {
 }
 
 
-impl<N: Clone + Float + Num + Float> Rot3<N> {
+impl<N: Clone + FloatMath> Rot3<N> {
     /// Builds a 3 dimensional rotation matrix from an axis and an angle.
     ///
     /// # Arguments
@@ -140,7 +140,7 @@ impl<N: Clone + Float + Num + Float> Rot3<N> {
     }
 }
 
-impl<N: Clone + Num + Float> Rot3<N> {
+impl<N: Clone + Float> Rot3<N> {
     /// Reorient this matrix such that its local `x` axis points to a given point. Note that the
     /// usually known `look_at` function does the same thing but with the `z` axis. See `look_at_z`
     /// for that.
@@ -180,7 +180,7 @@ impl<N: Clone + Num + Float> Rot3<N> {
     }
 }
 
-impl<N: Clone + Float + Num + Float + Cast<f32>>
+impl<N: Clone + FloatMath + Cast<f32>>
 Rotation<Vec3<N>> for Rot3<N> {
     #[inline]
     fn rotation(&self) -> Vec3<N> {
@@ -245,7 +245,7 @@ Rotation<Vec3<N>> for Rot3<N> {
     }
 }
 
-impl<N: Clone + Rand + Float + Num + Float>
+impl<N: Clone + Rand + FloatMath>
 Rand for Rot3<N> {
     #[inline]
     fn rand<R: Rng>(rng: &mut R) -> Rot3<N> {
@@ -309,7 +309,7 @@ impl<N: Signed> AbsoluteRotate<Vec4<N>> for Rot4<N> {
     }
 }
 
-impl<N: Float + Num + Clone>
+impl<N: Float + Clone>
 Rotation<Vec4<N>> for Rot4<N> {
     #[inline]
     fn rotation(&self) -> Vec4<N> {
diff --git a/src/structs/rot_macros.rs b/src/structs/rot_macros.rs
index cee3972c..1a168080 100644
--- a/src/structs/rot_macros.rs
+++ b/src/structs/rot_macros.rs
@@ -56,7 +56,7 @@ macro_rules! dim_impl(
 
 macro_rules! rotation_matrix_impl(
     ($t: ident, $tlv: ident, $tav: ident) => (
-        impl<N: Cast<f32> + Float + Float + Num + Clone>
+        impl<N: Cast<f32> + FloatMath + Clone>
         RotationMatrix<$tlv<N>, $tav<N>, $t<N>> for $t<N> {
             #[inline]
             fn to_rot_mat(&self) -> $t<N> {
diff --git a/src/structs/spec/vec0.rs b/src/structs/spec/vec0.rs
index 1ea49bc2..3ca49091 100644
--- a/src/structs/spec/vec0.rs
+++ b/src/structs/spec/vec0.rs
@@ -164,7 +164,7 @@ impl<N: Clone + Add<N, N> + Neg<N>> Translation<vec::Vec0<N>> for vec::Vec0<N> {
     }
 }
 
-impl<N: Num + Float> Norm<N> for vec::Vec0<N> {
+impl<N: Float> Norm<N> for vec::Vec0<N> {
     #[inline]
     fn sqnorm(_: &vec::Vec0<N>) -> N {
         Zero::zero()
diff --git a/src/structs/vec_macros.rs b/src/structs/vec_macros.rs
index a86fdfbf..6f4e2595 100644
--- a/src/structs/vec_macros.rs
+++ b/src/structs/vec_macros.rs
@@ -38,7 +38,7 @@ macro_rules! at_fast_impl(
 // However, f32/f64 does not implement TotalOrd…
 macro_rules! ord_impl(
     ($t: ident, $comp0: ident $(,$compN: ident)*) => (
-        impl<N: Float + Eq + Clone> PartialOrd for $t<N> {
+        impl<N: FloatMath + Eq + Clone> PartialOrd for $t<N> {
             #[inline]
             fn inf(a: &$t<N>, b: &$t<N>) -> $t<N> {
                 $t::new(a.$comp0.min(b.$comp0.clone())
@@ -255,7 +255,7 @@ macro_rules! container_impl(
 
 macro_rules! basis_impl(
     ($t: ident, $trhs: ident, $dim: expr) => (
-        impl<N: Clone + Num + Float + ApproxEq<N> + $trhs<N, $t<N>>> Basis for $t<N> {
+        impl<N: Clone + Float + ApproxEq<N> + $trhs<N, $t<N>>> Basis for $t<N> {
             #[inline]
             fn canonical_basis(f: |$t<N>| -> bool) {
                 for i in range(0u, $dim) {
@@ -465,7 +465,7 @@ macro_rules! translation_impl(
 
 macro_rules! norm_impl(
     ($t: ident, $comp0: ident $(,$compN: ident)*) => (
-        impl<N: Clone + Num + Float> Norm<N> for $t<N> {
+        impl<N: Clone + Float> Norm<N> for $t<N> {
             #[inline]
             fn sqnorm(v: &$t<N>) -> N {
                 Dot::dot(v, v)