random_spd_matrix.h

// Description: Random symmetric positive-definite matrix

#ifndef PASTELMATH_RANDOM_SPD_MATRIX_H
#define PASTELMATH_RANDOM_SPD_MATRIX_H

#include "pastel/math/matrix/random_matrix/random_orthogonal_matrix.h"
#include "pastel/sys/random/random_uniform.h"

#include <armadillo>
#include <algorithm>
#include <vector>

namespace Pastel
{

    //! Generates a random spd (n x n)-matrix S with given det(S).
    /*!
   Preconditions:
   n >= 0
   determinant > 0
   */
    template <typename Real>
    Matrix<Real> randomSymmetricPositiveDefinite(
        integer n,
        const NoDeduction<Real>& determinant)
    {
        ENSURE_OP(n, >=, 0);

        // Generate a random partition of the
        // interval [0, -ln(d)] to n subintervals
        // represented by (n + 1) partition points.

        Real xDelta = -std::log(determinant);

        std::vector<Real> partitionSet;
        partitionSet.reserve(n + 1);
        partitionSet.push_back(0);
        partitionSet.push_back(xDelta);

        for (integer i = 0;i < n - 1;++i)
        {

            partitionSet.push_back(random<Real>() * xDelta);
        }

        std::sort(partitionSet.begin(), partitionSet.end());

        // Generate a random rotation matrix.
        arma::Mat<Real> result = randomOrthogonal<Real>(n);

        // Multiply the columns of the rotation matrix
        // with square root of the diagonal element of D.
        for (integer j = 0;j < n;++j)
        {
            Real b = partitionSet[j + 1] - partitionSet[j];
            for (integer i = 0;i < n;++i)
            {

                result(i, j) *= std::exp(-b / 2);
            }
        }

        result *= transpose(result);

        return result;
    }

    //! Generates a random spd (n x n)-matrix S with given det(S) and cond(S).
    /*!
   Preconditions:
   n >= 0
   determinant > 0
   condition >= 1
   */
    template <typename Real>
    arma::Mat<Real> randomSymmetricPositiveDefinite(
        integer n,
        const NoDeduction<Real>& determinant,
        const NoDeduction<Real>& condition)
    {
        ENSURE_OP(n, >=, 0);
        ENSURE_OP(condition, >=, 1);
        ENSURE_OP(determinant, >, 0);

        Real a =
            ((n - 1) * std::log(condition) - 
            std::log(determinant)) / n;

        Real b =
            a - std::log(condition);

        // Generate a random rotation matrix.
        arma::Mat<Real> result = randomRotation<Real>(n);

        // Multiply the columns of the rotation matrix
        // with square roots of the diagonal elements of D.
        for (integer i = 0;i < n;++i)
        {
            result(i, 0) *= std::exp(-a / 2);
        }

        for (integer j = 1;j < n;++j)
        {
            for (integer i = 0;i < n;++i)
            {
                result(i, j) *= std::exp(-b / 2);
            }
        }

        result *= transpose(result);

        return result;
    }

}

#endif