Iterative Cayley-Hamilton

header-only C++ implementation of the iterative Cayley-Hamilton method for the evaluation of matrix power series (and their differentials) of complex square matrices of arbitrary size. For a more detailed description of the method, please see arXiv:2404.07704.

Content

cayley_hamilton.h

cayley_hamilton class

implementation of the iterative Cayley-Hamilton method for computing matrix power series. Callable class has three overloads, computing

• only the matrix polynomial,
• the matrix polynomial and its derivative in a given direction,
• the matrix polynomial and all its derivatives.

has additional subroutines:

• get_r_k computes and returns the Cayley-Hamilton decomposition coefficents for the matrix polynomial and of its derivatives
• get_r_dr computes and returns the Cayley-Hamilton decomposition coefficients of the matrix polynomial and the derivatives of these coefficients
• ch_mult computes the Cayley-Hamilton decomposition coefficients of the product of two matrix polynomials with given CH decomposition coefficients_

chexp class

implementation of the iterative Cayley-Hamilton method with scaling and squaring for computing matrix exponentials. Callable class has three overloads, computing

• only the matrix exponential,
• the matrix exponential and its derivative in a given direction,
• the matrix exponential and all its derivatives.

has additional subroutines:

• get_r_k computes and returns the Cayley-Hamilton decomposition coefficents for the matrix exponential and of its derivatives
• get_r_dr computes and returns the Cayley-Hamilton decomposition coefficients of the matrix exponential and the derivatives of these coefficients

nvexp class

callable class, implementing the matrix exponentiation using naive Taylor series definition in combination with scaling and squaring.

See comments in the header file for more details.

sun_algebra.h

sun_algebra class

implementation of a sun_algebra class, which provides:

• hermitian su(n) basis as sparse_mat objects
• member functions to transform between su(n) vectors and su(n) matrices in either anti-hermition or hermitian rep.:
  • anti-hermitian: get_alg_mat_ah(invec,outmat) <--> proj_ah(inmat,outvec)
  • hermitiaon: get_alg_mat_h(invec,outmat) <--> proj_h(inmat,outvec)
• member function to take matrix log of SU(N) matrices and return corresponding su(n) vector: log_ah(inmat,outvec)
• member function to compute the SU(N) matrix corresponding to an su(n) vector: get_grp_mat(invec,outmat) (using the cayley_hamilton class)

test.cpp

CPP file to run some basic tests with the cayley_hamilton, chexp, and sun_algebra classes.