RandFunm

Solver

template<typename MatType, int Flags, typename RandomEngine>
class RandFunm : public mocca::internal::Solver<RandFunm<MatType, Flags, RandomEngine>>

A solver for calculating a matrix function \(f(\gamma \mathbf{A})\) using a random set of rows and columns of the matrix. Any matrix function is supported as long as it can be represented as a power series:

\[ f(\gamma \mathbf{A}) = \sum_{k = 0}^\infty{c_k (\gamma \mathbf{A})^k} \]
where \( c_k > 0\) are the coefficients associated with each term and \(\gamma \in \mathbb{R}\) is the scaling parameter. For example, \( c_k = 1 / k! \) when calculating the matrix exponential.

After initializing the solver, call the funm() method to calculate the desired matrix function. You can use either the built-in functions (mittag_leffler, expm and inverse) or provide a function object that receive the value \( k \) and returns the coefficient \( c_k \).

If you only need the diagonal of the matrix function, you call the funm_diag() method instead. Likewise, providing a vector to the funm() method will compute the action of the matrix function over this vector. Both routines are more efficient than computing the full matrix function and then extract its diagonal or multiplying by the target vector.

Template Parameters:

  • MatType - Matrix type, either a CSRMatrix or a Matrix

  • Flags - Additional flags for the solver. See MonteCarloFlags for more information.

  • RandomEngine - Type of the Random Number Generator

Public Functions

inline explicit RandFunm(RandFunmOptions param = RandFunmOptions())

Initialize a RandFunm solver with the parameters param.

Complexity: Constant

Parameters:

param – [in] parameters of the solver (see RandFunmOptions for all the available parameters)

inline explicit RandFunm(MatType &A, RandFunmOptions param = RandFunmOptions())

Initialize a RandFunm solver with the input matrix A and parameters param for further computing the matrix function \(f(\gamma \mathbf{A})\).

Complexity: Constant

Parameters:

  • A – [in] input matrix

  • param – [in] parameters of the solver (see RandFunmOptions for all the available parameters)

inline void set_param(RandFunmOptions param)

Set the parameters of the RandFunm solver.

Complexity: Constant

Parameters:

param – [in] parameters of the solver (see RandFunmOptions for all the available parameters)

inline index_t rows() const
Returns:

the number of rows of the underlying matrix

inline index_t cols() const
Returns:

the number of columns of the underlying matrix

inline void set_matrix(MatType &A)

Initialize the RandFunm solver with the matrix A for further computing the matrix function \(f(\gamma \mathbf{A})\).

Parameters:

A – [in] input matrix Complexity: Constant

template<typename Func>
inline evaluator<internal::MatFuncFull> funm(Func func, value_type gamma)

Calculates the matrix function \(f(\gamma \mathbf{A})\).

Parameters:

  • func – [in] expm, mittag_leffler(alpha), inverse or a functor that receives a value k and returns the coefficient \( c_k\)

  • result – [out] result matrix (dense)

  • gamma – [in] scale parameter

template<typename Func, internal::TypeLayout L>
inline evaluator<sparse_vec_ref, internal::MatFuncAction> funm(Func func, value_type gamma, SparseVector<value_type> &v)

Calculates the action of the matrix function \(f(\gamma \mathbf{A})\) over the vector \( \vec{v} \).

Parameters:

  • func – [in] expm, mittag_leffler(alpha), inverse or a functor that receives a value k and returns the coefficient \( c_k\)

  • v – [in] input vector (dense or sparse)

  • result – [out] result vector (dense)

  • gamma – [in] scale parameter

template<typename Func>
inline evaluator<internal::MatFuncDiag> funm_diag(Func func, value_type gamma)

Calculates the diagonal of the matrix function \(f(\gamma \mathbf{A})\).

Parameters:

  • func – [in] expm, mittag_leffler(alpha), inverse or a functor that receives a value k and returns the coefficient \( c_k\)

  • result – [out] diagonal of the matrix function stored as a dense vector

  • gamma – [in] scale parameter

template<typename Func, internal::TypeLayout L>
inline value_type funm_single(Func func, index_t idx, value_type gamma, Vector<value_type, L> &v)

Calculates a single entry of \(f(\gamma \mathbf{A})v\).

Parameters:

  • func – [in] expm, mittag_leffler(alpha), inverse or a functor that receives a value k and returns the coefficient \( c_k\)

  • idx – [in] index of the entry of the solution to be calculated

  • v – [in] input vector

  • gamma – [in] scale parameter

Returns:

the entry i of \(f(\gamma \mathbf{A})v\)

Options

struct RandFunmOptions

Options for the RandFunm solver.

Public Members

index_t num_samples = 1E6

The number of random samples.

int max_num_terms = 100

Maximum number of terms in the Neumann Series.

double weight_cutoff = 1E-3

Weight tolerance. The Neumann series will be truncated when the weight of the random walk is less than this value

uint64_t seed = PCG32_DEFAULT_SEED

Indicate the random seed for the random number generator (PCG64).

uint64_t stream = PCG32_DEFAULT_STREAM

Indicate the random stream of the random number generator (PCG64).