Fix Vector::axpy for noncommutative cases

One example would be performing simple matrix multiplication
over a division algebra such as quaternions.

src/base/blas.rs

@@ -473,7 +473,7 @@ where N: Scalar + Zero + ClosedAdd + ClosedMul {
     for i in 0..len {
         unsafe {
             let y = y.get_unchecked_mut(i * stride1);
-            *y = a * *x.get_unchecked(i * stride2) + beta * *y;
+            *y = *x.get_unchecked(i * stride2) * a + *y * beta;
         }
     }
 }
@@ -482,7 +482,7 @@ fn array_ax<N>(y: &mut [N], a: N, x: &[N], stride1: usize, stride2: usize, len:
 where N: Scalar + Zero + ClosedAdd + ClosedMul {
     for i in 0..len {
         unsafe {
-            *y.get_unchecked_mut(i * stride1) = a * *x.get_unchecked(i * stride2);
+            *y.get_unchecked_mut(i * stride1) = *x.get_unchecked(i * stride2) * a;
         }
     }
 }
@@ -579,13 +579,13 @@ where
         // FIXME: avoid bound checks.
         let col2 = a.column(0);
         let val = unsafe { *x.vget_unchecked(0) };
-        self.axpy(alpha * val, &col2, beta);
+        self.axpy(val * alpha, &col2, beta);
 
         for j in 1..ncols2 {
             let col2 = a.column(j);
             let val = unsafe { *x.vget_unchecked(j) };
 
-            self.axpy(alpha * val, &col2, N::one());
+            self.axpy(val * alpha, &col2, N::one());
         }
     }
 
@@ -624,7 +624,7 @@ where
         // FIXME: avoid bound checks.
         let col2 = a.column(0);
         let val = unsafe { *x.vget_unchecked(0) };
-        self.axpy(alpha * val, &col2, beta);
+        self.axpy(val * alpha, &col2, beta);
         self[0] += alpha * dot(&a.slice_range(1.., 0), &x.rows_range(1..));
 
         for j in 1..dim2 {
@@ -637,7 +637,7 @@ where
                 *self.vget_unchecked_mut(j) += alpha * dot;
             }
             self.rows_range_mut(j + 1..)
-                .axpy(alpha * val, &col2.rows_range(j + 1..), N::one());
+                .axpy(val * alpha, &col2.rows_range(j + 1..), N::one());
         }
     }
 
@@ -890,7 +890,7 @@ where N: Scalar + Zero + ClosedAdd + ClosedMul
         for j in 0..ncols1 {
             // FIXME: avoid bound checks.
             let val = unsafe { conjugate(*y.vget_unchecked(j)) };
-            self.column_mut(j).axpy(alpha * val, x, beta);
+            self.column_mut(j).axpy(val * alpha, x, beta);
         }
     }
 
@@ -1256,7 +1256,7 @@ where N: Scalar + Zero + ClosedAdd + ClosedMul
             let subdim = Dynamic::new(dim1 - j);
             // FIXME: avoid bound checks.
             self.generic_slice_mut((j, j), (subdim, U1)).axpy(
-                alpha * val,
+                val * alpha,
                 &x.rows_range(j..),
                 beta,
             );

tests/core/blas.rs

@@ -1,5 +1,28 @@
-#![cfg(feature = "arbitrary")]
+use na::{geometry::Quaternion, Matrix2, Vector3};
+use num_traits::{One, Zero};
 
+#[test]
+fn gemm_noncommutative() {
+    type Qf64 = Quaternion<f64>;
+    let i = Qf64::from_imag(Vector3::new(1.0, 0.0, 0.0));
+    let j = Qf64::from_imag(Vector3::new(0.0, 1.0, 0.0));
+    let k = Qf64::from_imag(Vector3::new(0.0, 0.0, 1.0));
+
+    let m1 = Matrix2::new(k, Qf64::zero(), j, i);
+    // this is the inverse of m1
+    let m2 = Matrix2::new(-k, Qf64::zero(), Qf64::one(), -i);
+
+    let mut res: Matrix2<Qf64> = Matrix2::zero();
+    res.gemm(Qf64::one(), &m1, &m2, Qf64::zero());
+    assert_eq!(res, Matrix2::identity());
+
+    let mut res: Matrix2<Qf64> = Matrix2::identity();
+    res.gemm(k, &m1, &m2, -k);
+    assert_eq!(res, Matrix2::zero());
+}
+
+#[cfg(feature = "arbitrary")]
+mod blas_quickcheck {
     use na::{DMatrix, DVector};
     use std::cmp;
 
@@ -103,3 +126,4 @@ quickcheck! {
             relative_eq!(res, expected, epsilon = 1.0e-7)
         }
     }
+}