tasks a-s

Task/QR-decomposition/Go/qr-decomposition.go

@@ -0,0 +1,106 @@
+package main
+
+import (
+    "code.google.com/p/gomatrix/matrix"
+    "fmt"
+    "math"
+)
+
+func sign(s float64) float64 {
+    if s > 0 {
+        return 1
+    } else if s < 0 {
+        return -1
+    }
+    return 0
+}
+
+func unitVector(n int) *matrix.DenseMatrix {
+    vec := matrix.Zeros(n, 1)
+    vec.Set(0, 0, 1)
+    return vec
+}
+
+func householder(a *matrix.DenseMatrix) *matrix.DenseMatrix {
+    m := a.Rows()
+    s := sign(a.Get(0, 0))
+    e := unitVector(m)
+    u := matrix.Sum(a, matrix.Scaled(e, a.TwoNorm()*s))
+    v := matrix.Scaled(u, 1/u.Get(0, 0))
+    // (error checking skipped in this solution)
+    prod, _ := v.Transpose().TimesDense(v)
+    β := 2 / prod.Get(0, 0)
+
+    prod, _ = v.TimesDense(v.Transpose())
+    return matrix.Difference(matrix.Eye(m), matrix.Scaled(prod, β))
+}
+
+func qr(a *matrix.DenseMatrix) (q, r *matrix.DenseMatrix) {
+    m := a.Rows()
+    n := a.Cols()
+    q = matrix.Eye(m)
+
+    last := n - 1
+    if m == n {
+        last--
+    }
+    for i := 0; i <= last; i++ {
+        // (copy is only for compatibility with an older version of gomatrix)
+        b := a.GetMatrix(i, i, m-i, n-i).Copy()
+        x := b.GetColVector(0)
+        h := matrix.Eye(m)
+        h.SetMatrix(i, i, householder(x))
+        q, _ = q.TimesDense(h)
+        a, _ = h.TimesDense(a)
+    }
+    return q, a
+}
+
+func main() {
+    // task 1: show qr decomp of wp example
+    a := matrix.MakeDenseMatrixStacked([][]float64{
+        {12, -51, 4},
+        {6, 167, -68},
+        {-4, 24, -41}})
+    q, r := qr(a)
+    fmt.Println("q:\n", q)
+    fmt.Println("r:\n", r)
+
+    // task 2: use qr decomp for polynomial regression example
+    x := matrix.MakeDenseMatrixStacked([][]float64{
+        {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}})
+    y := matrix.MakeDenseMatrixStacked([][]float64{
+        {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}})
+    fmt.Println("\npolyfit:\n", polyfit(x, y, 2))
+}
+
+func polyfit(x, y *matrix.DenseMatrix, n int) *matrix.DenseMatrix {
+    m := x.Cols()
+    a := matrix.Zeros(m, n+1)
+    for i := 0; i < m; i++ {
+        for j := 0; j <= n; j++ {
+            a.Set(i, j, math.Pow(x.Get(0, i), float64(j)))
+        }
+    }
+    return lsqr(a, y.Transpose())
+}
+
+func lsqr(a, b *matrix.DenseMatrix) *matrix.DenseMatrix {
+    q, r := qr(a)
+    n := r.Cols()
+    prod, _ := q.Transpose().TimesDense(b)
+    return solveUT(r.GetMatrix(0, 0, n, n), prod.GetMatrix(0, 0, n, 1))
+}
+
+func solveUT(r, b *matrix.DenseMatrix) *matrix.DenseMatrix {
+    n := r.Cols()
+    x := matrix.Zeros(n, 1)
+    for k := n - 1; k >= 0; k-- {
+        sum := 0.
+        for j := k + 1; j < n; j++ {
+            sum += r.Get(k, j) * x.Get(j, 0)
+        }
+        x.Set(k, 0, (b.Get(k, 0)-sum)/r.Get(k, k))
+    }
+    return x
+}