qr_algorithm Namespace Reference

Functions to compute QR decomposition of any rectangular matrix. More...

Functions
template<typename T >
std::ostream &	operator<< (std::ostream &out, std::valarray< std::valarray< T > > const &v)
template<typename T >
std::ostream &	operator<< (std::ostream &out, std::valarray< T > const &v)
template<typename T >
double	vector_dot (const std::valarray< T > &a, const std::valarray< T > &b)
template<typename T >
double	vector_mag (const std::valarray< T > &a)
template<typename T >
std::valarray< T >	vector_proj (const std::valarray< T > &a, const std::valarray< T > &b)
template<typename T >
void	qr_decompose (const std::valarray< std::valarray< T > > &A, std::valarray< std::valarray< T > > *Q, std::valarray< std::valarray< T > > *R)
std::valarray< double >	eigen_values (std::valarray< std::valarray< double > > *A, bool print_intermediates=false)

Detailed Description

Functions to compute QR decomposition of any rectangular matrix.

Function Documentation

eigen_values()

std::valarray< double > qr_algorithm::eigen_values	(	std::valarray< std::valarray< double > > *	A,
		bool	print_intermediates = false )

Compute eigen values using iterative shifted QR decomposition algorithm as follows:

1. Use last diagonal element of A as eigen value approximation \(c\)
2. Shift diagonals of matrix \(A' = A - cI\)
3. Decompose matrix \(A'=QR\)
4. Compute next approximation \(A'_1 = RQ \)
5. Shift diagonals back \(A_1 = A'_1 + cI\)
6. Termination condition check: last element below diagonal is almost 0
   1. If not 0, go back to step 1 with the new approximation \(A_1\)
   2. If 0, continue to step 7
7. Save last known \(c\) as the eigen value.
8. Are all eigen values found?
   1. If not, remove last row and column of \(A_1\) and go back to step 1.
   2. If yes, stop.

Note

The matrix \(A\) gets modified

Parameters

[in,out]	A	matrix to compute eigen values for
[in]	print_intermediates	(optional) whether to print intermediate A, Q and R matrices (default = false)

                                                                     {
    int rows = A->size();
    int columns = rows;
    int counter = 0, num_eigs = rows - 1;
    double last_eig = 0;
 
    std::valarray<std::valarray<double>> Q(rows);
    std::valarray<std::valarray<double>> R(columns);
 
    /* number of eigen values = matrix size */
    std::valarray<double> eigen_vals(rows);
    for (int i = 0; i < rows; i++) {
        Q[i] = std::valarray<double>(columns);
        R[i] = std::valarray<double>(columns);
    }
 
    /* continue till all eigen values are found */
    while (num_eigs > 0) {
        /* iterate with QR decomposition */
        while (std::abs(A[0][num_eigs][num_eigs - 1]) >
               std::numeric_limits<double>::epsilon()) {
            // initial approximation = last diagonal element
            last_eig = A[0][num_eigs][num_eigs];
            for (int i = 0; i < rows; i++) {
                A[0][i][i] -= last_eig; /* A - cI */
            }
 
            qr_decompose(*A, &Q, &R);
 
            if (print_intermediates) {
                std::cout << *A << "\n";
                std::cout << Q << "\n";
                std::cout << R << "\n";
                printf("-------------------- %d ---------------------\n",
                       ++counter);
            }
 
            // new approximation A' = R * Q
            mat_mul(R, Q, A);
 
            for (int i = 0; i < rows; i++) {
                A[0][i][i] += last_eig; /* A + cI */
            }
        }
 
        /* store the converged eigen value */
        eigen_vals[num_eigs] = last_eig;
        // A[0][num_eigs][num_eigs];
        if (print_intermediates) {
            std::cout << "========================\n";
            std::cout << "Eigen value: " << last_eig << ",\n";
            std::cout << "========================\n";
        }
 
        num_eigs--;
        rows--;
        columns--;
    }
    eigen_vals[0] = A[0][0][0];
 
    if (print_intermediates) {
        std::cout << Q << "\n";
        std::cout << R << "\n";
    }
 
    return eigen_vals;
}

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operator<<() [1/2]

template<typename T >

std::ostream & qr_algorithm::operator<<	(	std::ostream &	out,
		std::valarray< std::valarray< T > > const &	v )

operator to print a matrix

                                                               {
    const int width = 12;
    const char separator = ' ';
 
    out.precision(4);
    for (size_t row = 0; row < v.size(); row++) {
        for (size_t col = 0; col < v[row].size(); col++)
            out << std::right << std::setw(width) << std::setfill(separator)
                << v[row][col];
        out << std::endl;
    }
 
    return out;
}

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operator<<() [2/2]

template<typename T >

std::ostream & qr_algorithm::operator<<	(	std::ostream &	out,
		std::valarray< T > const &	v )

operator to print a vector

                                                                 {
    const int width = 10;
    const char separator = ' ';
 
    out.precision(4);
    for (size_t row = 0; row < v.size(); row++) {
        out << std::right << std::setw(width) << std::setfill(separator)
            << v[row];
    }
 
    return out;
}

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qr_decompose()

template<typename T >

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

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

  {
    std::size_t ROWS = A.size();        // number of rows of A
    std::size_t COLUMNS = A[0].size();  // number of columns of A
    std::valarray<T> col_vector(ROWS);
    std::valarray<T> col_vector2(ROWS);
    std::valarray<T> tmp_vector(ROWS);
 
    for (int i = 0; i < COLUMNS; i++) {
        /* for each column => R is a square matrix of NxN */
        int j;
        R[0][i] = 0.; /* make R upper triangular */
 
        /* get corresponding Q vector */
#ifdef _OPENMP
// parallelize on threads
#pragma omp for
#endif
        for (j = 0; j < ROWS; j++) {
            tmp_vector[j] = A[j][i]; /* accumulator for uk */
            col_vector[j] = A[j][i];
        }
        for (j = 0; j < i; j++) {
            for (int k = 0; k < ROWS; k++) {
                col_vector2[k] = Q[0][k][j];
            }
            col_vector2 = vector_proj(col_vector, col_vector2);
            tmp_vector -= col_vector2;
        }
 
        double mag = vector_mag(tmp_vector);
 
#ifdef _OPENMP
// parallelize on threads
#pragma omp for
#endif
        for (j = 0; j < ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;
 
            /* compute upper triangular values of R */
#ifdef _OPENMP
// parallelize on threads
#pragma omp for
#endif
        for (int kk = 0; kk < ROWS; kk++) {
            col_vector[kk] = Q[0][kk][i];
        }
 
#ifdef _OPENMP
// parallelize on threads
#pragma omp for
#endif
        for (int k = i; k < COLUMNS; k++) {
            for (int kk = 0; kk < ROWS; kk++) {
                col_vector2[kk] = A[kk][k];
            }
            R[0][i][k] = (col_vector * col_vector2).sum();
        }
    }
}

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vector_dot()

template<typename T >

double qr_algorithm::vector_dot ( const std::valarray< T > & a, const std::valarray< T > & b )

inline

Compute dot product of two vectors of equal lengths

If \(\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\) and \(\vec{b}=\left[b_0,b_1,b_1,...,b_L\right]\) then \(\vec{a}\cdot\vec{b}=\displaystyle\sum_{i=0}^L a_i\times b_i\)

Returns

\(\vec{a}\cdot\vec{b}\)

                                                                           {
    return (a * b).sum();
    // could also use following
    // return std::inner_product(std::begin(a), std::end(a), std::begin(b),
    // 0.f);
}

vector_mag()

template<typename T >

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

inline

Compute magnitude of vector.

If \(\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\) then \(\left|\vec{a}\right|=\sqrt{\displaystyle\sum_{i=0}^L a_i^2}\)

Returns

\(\left|\vec{a}\right|\)

                                                  {
    double dot = vector_dot(a, a);
    return std::sqrt(dot);
}

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vector_proj()

template<typename T >

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

Compute projection of vector \(\vec{a}\) on \(\vec{b}\) defined as

\[\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^2}\vec{b}\]

Returns

NULL if error, otherwise pointer to output

check for division by zero using machine epsilon

                                                      {
    double num = vector_dot(a, b);
    double deno = vector_dot(b, b);
 
    /*! check for division by zero using machine epsilon */
    if (deno <= std::numeric_limits<double>::epsilon()) {
        std::cerr << "[" << __func__ << "] Possible division by zero\n";
        return a;  // return vector a back
    }
 
    double scalar = num / deno;
 
    return b * scalar;
}

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