Add `na::eigen_qr` that performs an eigendecomposition using the qr algorithm.

src/linalg/decompositions.rs

@@ -1,6 +1,6 @@
 use std::num::{Zero, Float};
-use traits::operations::Transpose;
-use traits::structure::{ColSlice, Eye, Indexable};
+use traits::operations::{Transpose, ApproxEq};
+use traits::structure::{ColSlice, Eye, Indexable, Diag};
 use traits::geometry::Norm;
 use std::cmp::min;
 
@@ -16,12 +16,16 @@ pub fn householder_matrix<N: Float,
                           V: Indexable<uint, N>>
                           (dim: uint, start: uint, vec: V) -> M {
     let mut qk : M = Eye::new_identity(dim);
-    let stop = start + vec.shape();
-    assert!(stop <= dim);
+    let subdim = vec.shape();
+
+    let stop = subdim + start;
+
+    assert!(dim >= stop);
+
     for j in range(start, stop) {
         for i in range(start, stop) {
             unsafe {
-                let vv = vec.unsafe_at(i) * vec.unsafe_at(j);
+                let vv = vec.unsafe_at(i - start) * vec.unsafe_at(j - start);
                 let qkij = qk.unsafe_at((i, j));
                 qk.unsafe_set((i, j), qkij - vv - vv);
             }
@@ -60,11 +64,70 @@ pub fn qr<N: Float,
             let x = v.unsafe_at(0);
             v.unsafe_set(0, x - alpha);
         }
-        let _ = v.normalize();
-        let qk: M = householder_matrix(rows, 0, v);
-        r = qk * r;
-        q = q * Transpose::transpose_cpy(&qk);
+        if !v.normalize().is_zero() {
+            let qk: M = householder_matrix(rows, ite, v);
+            r = qk * r;
+            q = q * Transpose::transpose_cpy(&qk);
+        }
     }
 
     (q, r)
 }
+
+/// Eigendecomposition of a square matrix using the qr algorithm.
+pub fn eigen_qr<N:  Float,
+                V:  Indexable<uint, N> + Norm<N>,
+                V2: Zero,
+                M:  Clone + Eye + ColSlice<V> + Transpose
+                    + Indexable<(uint, uint), N> + Mul<M, M>
+                    + Diag<V2> + ApproxEq<N> + Add<M, M>
+                    + Sub<M, M>>
+                (m: &M, eps: &N, niter: uint) -> (M, V2) {
+    let (rows, cols) = m.shape();
+
+    assert!(rows == cols, "The matrix being decomposed must be square.");
+
+    let mut eigenvectors: M = Eye::new_identity(rows);
+    let mut eigenvalues = m.clone();
+    let mut shifter: M = Eye::new_identity(rows);
+
+    let mut iter = 0u;
+    for _ in range(0, niter) {
+        let mut stop = true;
+
+        for j in range(0, cols) {
+            for i in range(0, j) {
+                if unsafe { eigenvalues.unsafe_at((i, j)) }.abs() >= *eps {
+                    stop = false;
+                    break;
+                }
+            }
+
+            for i in range(j + 1, rows) {
+                if unsafe { eigenvalues.unsafe_at((i, j)) }.abs() >= *eps {
+                    stop = false;
+                    break;
+                }
+            }
+        }
+
+        if stop {
+            break;
+        }
+        iter = iter + 1;
+
+        // FIXME: This is a very naive implementation.
+        let shift = unsafe { eigenvalues.unsafe_at((rows - 1, rows - 1)) };
+
+        for i in range(0, rows) {
+            unsafe { shifter.unsafe_set((i, i), shift.clone()) }
+        }
+
+        let (q, r) = qr(&(eigenvalues - shifter));
+
+        eigenvalues = r * q + shifter;
+        eigenvectors = eigenvectors * q;
+    }
+
+    (eigenvectors, eigenvalues.diag())
+}

src/tests/mat.rs

@@ -37,34 +37,28 @@ macro_rules! test_qr_impl(
   );
 )
 
-macro_rules! test_eigen_qr_impl(
-    ($t: ty) => {
-        for _ in range(0u, 10000) {
-            let randmat : $t = random();
-
-            let (eigenvalues, eigenvectors) = na::eigen_qr(&randmat, &Float::epsilon(), 1000);
-
-            // FIXME: provide a method to initialize a matrix from its diagonal!
-            let diag: $t = na::zero();
-
-            for i in range(0, na::dim::<$t>()) {
-                diag.set((i, i), eigenvalues.at(i));
-            }
-
-            let recomp = na::transpose(&eigenvectors) * diag * eigenvectors;
-
-            println!("mat: {}", randmat);
-            println!("eigenvectors: {}", eigenvectors);
-            println!("eigenvalues: {}", eigenvalues);
-            println!("recomp: {}", recomp);
-
-            assert!(false);
-            fail!("what!");
-
-            assert!(na::approx_eq(&randmat,  &recomp));
-        }
-    }
-)
+// NOTE: deactivated untile we get a better convergence rate.
+// macro_rules! test_eigen_qr_impl(
+//     ($t: ty) => {
+//         for _ in range(0u, 10000) {
+//             let randmat : $t = random();
+//             // Make it symetric so that we can recompose the matrix to test at the end.
+//             let randmat = na::transpose(&randmat) * randmat;
+// 
+//             let (eigenvectors, eigenvalues) = na::eigen_qr(&randmat, &Float::epsilon(), 100);
+// 
+//             let diag: $t = Diag::from_diag(&eigenvalues);
+// 
+//             let recomp = eigenvectors * diag * na::transpose(&eigenvectors);
+// 
+//             println!("eigenvalues: {}", eigenvalues);
+//             println!("   mat: {}", randmat);
+//             println!("recomp: {}", recomp);
+// 
+//             assert!(na::approx_eq_eps(&randmat,  &recomp, &1.0e-2));
+//         }
+//     }
+// )
 
 #[test]
 fn test_transpose_mat1() {
@@ -295,6 +289,7 @@ fn test_qr_mat6() {
     test_qr_impl!(Mat6<f64>);
 }
 
+// NOTE: deactivated untile we get a better convergence rate.
 // #[test]
 // fn test_eigen_qr_mat1() {
 //     test_eigen_qr_impl!(Mat1<f64>);

src/traits/structure.rs

@@ -144,7 +144,7 @@ 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
+    /// Returns the shape of the iterable range.
     fn shape(&self) -> Index;
 
     /// Reads the `i`-th element of `self`.