Another update from ingydotnet^djgoku

Task/QR-decomposition/00DESCRIPTION

@@ -1,4 +1,4 @@
-Any rectangular <math>m \times n</math> matrix <math>\mathit A</math> can be decomposed to a product of a orthogonal matrix <math>\mathit Q</math> and a upper (right) triangular matrix <math>\mathit R</math>, as described in [[wp:QR decomposition|QR decomposition]].
+Any rectangular <math>m \times n</math> matrix <math>\mathit A</math> can be decomposed to a product of an orthogonal matrix <math>\mathit Q</math> and an upper (right) triangular matrix <math>\mathit R</math>, as described in [[wp:QR decomposition|QR decomposition]].
 
 '''Task'''
 

Task/QR-decomposition/Go/qr-decomposition-1.go

@@ -1,9 +1,10 @@
 package main
 
 import (
-    "code.google.com/p/gomatrix/matrix"
     "fmt"
     "math"
+
+    "github.com/skelterjohn/go.matrix"
 )
 
 func sign(s float64) float64 {

Task/QR-decomposition/Go/qr-decomposition-2.go

@@ -0,0 +1,44 @@
+package main
+
+import (
+    "fmt"
+
+    "github.com/gonum/matrix/mat64"
+)
+
+func main() {
+    // task 1: show qr decomp of wp example
+    a := mat64.NewDense(3, 3, []float64{
+        12, -51, 4,
+        6, 167, -68,
+        -4, 24, -41,
+    })
+    var qr mat64.QR
+    qr.Factorize(a)
+    var q, r mat64.Dense
+    q.QFromQR(&qr)
+    r.RFromQR(&qr)
+    fmt.Printf("q: %.3f\n\n", mat64.Formatted(&q, mat64.Prefix("   ")))
+    fmt.Printf("r: %.3f\n\n", mat64.Formatted(&r, mat64.Prefix("   ")))
+
+    // task 2: use qr decomp for polynomial regression example
+    x := []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
+    y := []float64{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}
+    a = Vandermonde(x, 2)
+    b := mat64.NewDense(11, 1, y)
+    qr.Factorize(a)
+    var f mat64.Dense
+    f.SolveQR(&qr, false, b)
+    fmt.Printf("polyfit: %.3f\n",
+        mat64.Formatted(&f, mat64.Prefix("         ")))
+}
+
+func Vandermonde(a []float64, degree int) *mat64.Dense {
+    x := mat64.NewDense(len(a), degree+1, nil)
+    for i := range a {
+        for j, p := 0, 1.; j <= degree; j, p = j+1, p*a[i] {
+            x.Set(i, j, p)
+        }
+    }
+    return x
+}

Task/QR-decomposition/Python/qr-decomposition.py

@@ -0,0 +1,45 @@
+#!/usr/bin/env python3
+
+import numpy as np
+
+def qr(A):
+    m, n = A.shape
+    Q = np.eye(m)
+    for i in range(n - (m == n)):
+        H = np.eye(m)
+        H[i:, i:] = make_householder(A[i:, i])
+        Q = np.dot(Q, H)
+        A = np.dot(H, A)
+    return Q, A
+
+def make_householder(a):
+    v = a / (a[0] + np.copysign(np.linalg.norm(a), a[0]))
+    v[0] = 1
+    H = np.eye(a.shape[0])
+    H -= (2 / np.dot(v, v)) * np.dot(v[:, None], v[None, :])
+    return H
+
+# task 1: show qr decomp of wp example
+a = np.array(((
+    (12, -51,   4),
+    ( 6, 167, -68),
+    (-4,  24, -41),
+)))
+
+q, r = qr(a)
+print('q:\n', q.round(6))
+print('r:\n', r.round(6))
+
+# task 2: use qr decomp for polynomial regression example
+def polyfit(x, y, n):
+    return lsqr(x[:, None]**np.arange(n + 1), y.T)
+
+def lsqr(a, b):
+    q, r = qr(a)
+    _, n = r.shape
+    return np.linalg.solve(r[:n, :], np.dot(q.T, b)[:n])
+
+x = np.array((0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10))
+y = np.array((1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321))
+
+print('\npolyfit:\n', polyfit(x, y, 2))