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TheAlgorithms/C++: numerical_methods/qr_decomposition.cpp File Reference

Program to compute the QR decomposition of a given matrix. More...

Go to the source code of this file.

Program to compute the QR decomposition of a given matrix.

Definition in file qr_decomposition.cpp.

main function

Definition at line 23 of file qr_decomposition.cpp.

23 {

24 unsigned int ROWS, COLUMNS;

25

26 std::cout << "Enter the number of rows and columns: ";

27 std::cin >> ROWS >> COLUMNS;

28

29 std::cout << "Enter matrix elements row-wise:\n";

30

31 std::valarray<std::valarray<double>> A(ROWS);

32 std::valarray<std::valarray<double>> Q(ROWS);

33 std::valarray<std::valarray<double>> R(COLUMNS);

34 for (int i = 0; i < std::max(ROWS, COLUMNS); i++) {

35 if (i < ROWS) {

36 A[i] = std::valarray<double>(COLUMNS);

37 Q[i] = std::valarray<double>(COLUMNS);

38 }

39 if (i < COLUMNS) {

40 R[i] = std::valarray<double>(COLUMNS);

41 }

42 }

43

44 for (int i = 0; i < ROWS; i++)

45 for (int j = 0; j < COLUMNS; j++) std::cin >> A[i][j];

46

47 std::cout << A << "\n";

48

49 clock_t t1 = clock();

51 double dtime = static_cast<double>(clock() - t1) / CLOCKS_PER_SEC;

52

53 std::cout << Q << "\n";

54 std::cout << R << "\n";

55 std::cout << "Time taken to compute: " << dtime << " sec\n ";

56

57 return 0;

58}

void qr_decompose(const std::valarray< std::valarray< T > > &A, std::valarray< std::valarray< T > > *Q, std::valarray< std::valarray< T > > *R)

template<typename T>

operator to print a matrix

Definition at line 33 of file qr_decompose.h.

34 {

35 const int width = 12;

36 const char separator = ' ';

37

38 out.precision(4);

39 for (size_t row = 0; row < v.size(); row++) {

40 for (size_t col = 0; col < v[row].size(); col++)

41 out << std::right << std::setw(width) << std::setfill(separator)

42 << v[row][col];

43 out << std::endl;

44 }

45

46 return out;

47}

template<typename T>

Decompose matrix \(A\) using Gram-Schmidt process.

\begin{eqnarray*} \text{given that}\quad A &=& *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\ \text{where}\quad\mathbf{a}_i &=& \left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column vectors)}\\ \text{then}\quad\mathbf{u}_i &=& \mathbf{a}_i *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\ \mathbf{e}_i &=&\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\ Q &=& \begin{bmatrix}\mathbf{e}_0 & \mathbf{e}_1 & \mathbf{e}_2 & \dots & \mathbf{e}_{N-1}\end{bmatrix}\\ R &=& \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots \\ 0 & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\ 0 & 0 & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}\\ \end{eqnarray*}

Parameters

A input matrix to decompose Q output decomposed matrix R output decomposed matrix

Definition at line 146 of file qr_decompose.h.

150 {

151 std::size_t ROWS = A.size();

152 std::size_t COLUMNS = A[0].size();

153 std::valarray<T> col_vector(ROWS);

154 std::valarray<T> col_vector2(ROWS);

155 std::valarray<T> tmp_vector(ROWS);

156

157 for (int i = 0; i < COLUMNS; i++) {

158

159 int j;

160 R[0][i] = 0.;

161

162

163#ifdef _OPENMP

164

165#pragma omp for

166#endif

167 for (j = 0; j < ROWS; j++) {

168 tmp_vector[j] = A[j][i];

169 col_vector[j] = A[j][i];

170 }

171 for (j = 0; j < i; j++) {

(

k = 0;

< ROWS;

++) {

col_vector2[

] = Q[0][

][j];

174 }

col_vector2 =

(col_vector, col_vector2);

176 tmp_vector -= col_vector2;

177 }

178

180

181#ifdef _OPENMP

182

183#pragma omp for

184#endif

185 for (j = 0; j < ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;

186

187

188#ifdef _OPENMP

189

190#pragma omp for

191#endif

192 for (int kk = 0; kk < ROWS; kk++) {

193 col_vector[kk] = Q[0][kk][i];

194 }

195

196#ifdef _OPENMP

197

198#pragma omp for

199#endif

(

k = i;

< COLUMNS;

++) {

201 for (int kk = 0; kk < ROWS; kk++) {

col_vector2[kk] = A[kk][

];

203 }

R[0][i][

] = (col_vector * col_vector2).

();

205 }

206 }

207}

double k(double x)

Another test function.

T sum(const std::vector< std::valarray< T > > &A)

std::valarray< T > vector_proj(const std::valarray< T > &a, const std::valarray< T > &b)

double vector_mag(const std::valarray< T > &a)

double mag(const std::array< double, 3 > &vec)

Calculates the magnitude of the mathematical vector from it's direction ratios.

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