Hermitian

Kernel Polynomial Method

The kernel polynomial method is an algorithm to obtain an approximation for the spectral density of a Hermitian matrix. This algorithm combines expansion in polynomials of Chebyshev [1], the stochastic trace [2] and a kernel smothing techinique in order to obtain the approximation for the spectral density

Applications

  • Hamiltonian matrices associated with quantum mechanics
  • Laplacian matrix associated with a graph
  • Magnetic Laplacian associated with directed graphs
  • etc

References

[1] Wang, L.W., 1994. Calculating the density of states and optical-absorption spectra of large quantum systems by the plane-wave moments method. Physical Review B, 49(15), p.10154.

[2] Hutchinson, M.F., 1990. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications in Statistics-Simulation and Computation, 19(2), pp.433-450.

emate.hermitian.kpm.pykpm(H, num_moments=10, num_vecs=10, extra_points=2, precision=32, lmin=None, lmax=None, epsilon=0.01, device='/gpu:0', swap_memory_while=False)

Kernel Polynomial Method using a Jackson’s kernel.

Parameters:
  • H (scipy sparse matrix) – The Hermitian matrix
  • num_moments (int) –
  • num_vecs (int) – Number of random vectors in oder to aproximate the trace
  • extra_points (int) –
  • precision (int) – Single or double precision
  • limin (float, optional) – The smallest eigenvalue
  • lmax (float) – The highest eigenvalue
  • epsilon (float) – Used to rescale the matrix eigenvalues into the interval [-1, 1]
  • device (str) – ‘/gpu:ID’ or ‘/cpu:ID’
Returns:

  • ek (array of floats) – An array with num_moments + extra_points approximated “eigenvalues”
  • rho (array of floats) – An array containing the densities of each “eigenvalue”
emate.hermitian.kpm.cupykpm(H, num_moments=10, num_vecs=10, extra_points=12, precision=32, lmin=None, lmax=None, epsilon=0.01)[source]

Kernel Polynomial Method using a Jackson’s kernel. CUPY version

Parameters:
  • H (scipy CSR sparse matrix) – The Hermitian matrix
  • num_moments (int) –
  • num_vecs (int) – Number of random vectors in oder to aproximate the trace
  • extra_points (int) –
  • precision (int) – Single or double precision
  • limin (float, optional) – The smallest eigenvalue
  • lmax (float) – The highest eigenvalue
  • epsilon (float) – Used to rescale the matrix eigenvalues into the interval [-1, 1]
Returns:

  • ek (array of floats) – An array with num_moments + extra_points approximated “eigenvalues”
  • rho (array of floats) – An array containing the densities of each “eigenvalue”
emate.hermitian.kpm.tfkpm(H, num_moments=10, num_vecs=10, extra_points=2, precision=32, lmin=None, lmax=None, epsilon=0.01, device='/gpu:0', swap_memory_while=False)[source]

Kernel Polynomial Method using a Jackson’s kernel.

Parameters:
  • H (scipy sparse matrix) – The Hermitian matrix
  • num_moments (int) –
  • num_vecs (int) – Number of random vectors in oder to aproximate the trace
  • extra_points (int) –
  • precision (int) – Single or double precision
  • limin (float, optional) – The smallest eigenvalue
  • lmax (float) – The highest eigenvalue
  • epsilon (float) – Used to rescale the matrix eigenvalues into the interval [-1, 1]
  • device (str) – ‘/gpu:ID’ or ‘/cpu:ID’
Returns:

  • ek (array of floats) – An array with num_moments + extra_points approximated “eigenvalues”
  • rho (array of floats) – An array containing the densities of each “eigenvalue”