Task: QR decomposition


C++: 
#include <cstdio>
#include <cstdlib>
#include <cstring> 
#include <limits>
#include <iostream>
#include <vector>

#include <math.h>

class Vector;

class Matrix {

public:
  
  Matrix() : m(0), n(0), data(nullptr) {}
  
  
  Matrix(int m_, int n_) : Matrix() {
    m = m_;
    n = n_;
    allocate(m_,n_);
  }

  
  Matrix(const Matrix& mat) : Matrix(mat.m,mat.n) {

    for (int i = 0; i < m; i++)
      for (int j = 0; j < n; j++)
	(*this)(i,j) = mat(i,j);
  }
  
  
  template<int rows, int cols>
  Matrix(double (&a)[rows][cols]) : Matrix(rows,cols) {

    for (int i = 0; i < m; i++)
      for (int j = 0; j < n; j++)
	(*this)(i,j) = a[i][j];
  }

  
  ~Matrix() {
    deallocate();
  }


  
  double& operator() (int i, int j) {
    return data[i+m*j]; }
  double  operator() (int i, int j) const {
    return data[i+m*j]; }

  
  Matrix& operator=(const Matrix& source) {
    
    
    if (this != &source) { 
      if ( (m*n) != (source.m * source.n) ) { 
	allocate(source.m,source.n);          
      }
      
      std::copy(source.data, source.data + source.m*source.n, data);
    }
    return *this;
  }
  
  
  void compute_minor(const Matrix& mat, int d) {

    allocate(mat.m, mat.n);
    
    for (int i = 0; i < d; i++)
      (*this)(i,i) = 1.0;
    for (int i = d; i < mat.m; i++)
      for (int j = d; j < mat.n; j++)
	(*this)(i,j) = mat(i,j);
    
  }

  
  
  
  void mult(const Matrix& a, const Matrix& b) {

    if (a.n != b.m) {
      std::cerr << "Matrix multiplication not possible, sizes don't match !\n";
      return;
    }

    
    if (a.m != m or b.n != n)
      allocate(a.m, b.n);

    memset(data,0,m*n*sizeof(double));
    
    for (int i = 0; i < a.m; i++)
      for (int j = 0; j < b.n; j++)
	for (int k = 0; k < a.n; k++)
	  (*this)(i,j) += a(i,k) * b(k,j);
    
  }

  void transpose() {
    for (int i = 0; i < m; i++) {
      for (int j = 0; j < i; j++) {
	double t = (*this)(i,j);
	(*this)(i,j) = (*this)(j,i);
	(*this)(j,i) = t;
      }
    }
  }

  
  void extract_column(Vector& v, int c);  

  
  void allocate(int m_, int n_) {

    
    deallocate();
    
    
    m = m_;
    n = n_;
    
    data = new double[m_*n_];
    memset(data,0,m_*n_*sizeof(double));

  } 

  
  void deallocate() {

    if (data)
      delete[] data;

    data = nullptr;

  }    
  
  int m, n;
  
private:
  double* data;
  
}; 


class Vector {

public:
  
  Vector() : size(0), data(nullptr) {}
  
  
  Vector(int size_) : Vector() {
    size = size_;
    allocate(size_);
  }

  
  ~Vector() {
    deallocate();
  }

  
  double& operator() (int i) {
    return data[i]; }
  double  operator() (int i) const {
    return data[i]; }

  
  Vector& operator=(const Vector& source) {
    
    
    if (this != &source) { 
      if ( size != (source.size) ) {   
	allocate(source.size);         
      }
      
      std::copy(source.data, source.data + source.size, data);
    }
    return *this;
  }

  
  void allocate(int size_) {

    deallocate();
    
    
    size = size_;
    
    data = new double[size_];
    memset(data,0,size_*sizeof(double));

  } 

  
  void deallocate() {

    if (data)
      delete[] data;

    data = nullptr;

  }    

  
  double norm() {
    double sum = 0;
    for (int i = 0; i < size; i++) sum += (*this)(i) * (*this)(i);
    return sqrt(sum);
  }

  
  void rescale(double factor) {
    for (int i = 0; i < size; i++) (*this)(i) /= factor;
  }

  void rescale_unit() {
    double factor = norm();
    rescale(factor);
  }
  
  int size;
  
private:
  double* data;

}; 


void vmadd(const Vector& a, const Vector& b, double s, Vector& c)
{
  if (c.size != a.size or c.size != b.size) {
    std::cerr << "[vmadd]: vector sizes don't match\n";
    return;
  }
  
  for (int i = 0; i < c.size; i++)
    c(i) = a(i) + s * b(i);
}



void compute_householder_factor(Matrix& mat, const Vector& v)
{

  int n = v.size;
  mat.allocate(n,n);
  for (int i = 0; i < n; i++)
    for (int j = 0; j < n; j++)
      mat(i,j) = -2 *  v(i) * v(j);
  for (int i = 0; i < n; i++)
    mat(i,i) += 1;  
}


void Matrix::extract_column(Vector& v, int c) {
  if (m != v.size) {
    std::cerr << "[Matrix::extract_column]: Matrix and Vector sizes don't match\n";
    return;
  }
  
  for (int i = 0; i < m; i++)
    v(i) = (*this)(i,c);
}

void matrix_show(const Matrix&  m, const std::string& str="")
{
  std::cout << str << "\n";
  for(int i = 0; i < m.m; i++) {
    for (int j = 0; j < m.n; j++) {
      printf(" %8.3f", m(i,j));
    }
    printf("\n");
  }
  printf("\n");
}


double matrix_compare(const Matrix& A, const Matrix& B) {
  
  if (A.m != B.m or  A.n != B.n)
    return std::numeric_limits<double>::max();

  double res=0;
  for(int i = 0; i < A.m; i++) {
    for (int j = 0; j < A.n; j++) {
      res += (A(i,j)-B(i,j)) * (A(i,j)-B(i,j));
    }
  }

  res /= A.m*A.n;
  return res;
}

void householder(Matrix& mat,
		 Matrix& R,
		 Matrix& Q)
{

  int m = mat.m;
  int n = mat.n;

  
  std::vector<Matrix> qv(m);

  
  Matrix z(mat);
  Matrix z1;
  
  for (int k = 0; k < n && k < m - 1; k++) {

    Vector e(m), x(m);
    double a;
    
    
    z1.compute_minor(z, k);
    
    
    z1.extract_column(x, k);
    
    a = x.norm();
    if (mat(k,k) > 0) a = -a;
    
    for (int i = 0; i < e.size; i++)
      e(i) = (i == k) ? 1 : 0;

    
    vmadd(x, e, a, e);

    
    e.rescale_unit();
    
    
    compute_householder_factor(qv[k], e);

    
    z.mult(qv[k], z1);

  }
  
  Q = qv[0];

  
  for (int i = 1; i < n && i < m - 1; i++) {

    z1.mult(qv[i], Q);
    Q = z1;
    
  }
  
  R.mult(Q, mat);
  Q.transpose();
}
 
double in[][3] = {
  { 12, -51,   4},
  {  6, 167, -68},
  { -4,  24, -41},
  { -1,   1,   0},
  {  2,   0,   3},
};
 
int main()
{
  Matrix A(in); 
  Matrix Q, R;

  matrix_show(A,"A");  

  
  householder(A, R, Q);
  
  matrix_show(Q,"Q");
  matrix_show(R,"R");
 
  
  Matrix A_check;
  A_check.mult(Q, R);

  
  double l2 = matrix_compare(A,A_check);

  
  matrix_show(A_check, l2 < 1e-12 ? "A == Q * R ? yes" : "A == Q * R ? no");
 
  return EXIT_SUCCESS;
}






Python: 

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


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))


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))