Algorithms

Automatic Differentiation (auto_diff Module)

Algorithm	Namespace
Backward-mode automatic differentiation [15]	num_collect::auto_diff::backward
Forward-mode automatic differentiation [15]	num_collect::auto_diff::forward

Optimization (opt Module)

Algorithm	Class
Adaptive diagonal curves (ADC) [22]	num_collect::opt::adaptive_diagonal_curves
Backtracking line search	num_collect::opt::backtracking_line_searcher
Conjugate gradient method	num_collect::opt::conjugate_gradient_optimizer
Dividing rectangles (DIRECT) method	num_collect::opt::dividing_rectangles
Downhill simplex method [17]	num_collect::opt::downhill_simplex
Downhill simplex method with simulated annealing [17]	num_collect::opt::annealing_downhill_simplex
Gaussian process optimization [2, 23]	num_collect::opt::gaussian_process_optimizer
Quasi-Newton method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula	num_collect::opt::bfgs_optimizer
Quasi-Newton method with Davidon-Fletcher-Powell (DFP) formula	num_collect::opt::dfp_optimizer

Regularization (regularization Module)

Computation of Regularized Solutions

Algorithm	Class
Fast iterative shrinkage-thresholding algorithm (FISTA) [1]	num_collect::regularization::fista
Tikhonov regularization [3, 10]	num_collect::regularization::tikhonov, num_collect::regularization::full_gen_tikhonov

Selection of Regularization Parameters

Algorithm	Class
Generalized cross validation (GCV)	num_collect::regularization::explicit_gcv, num_collect::regularization::implicit_gcv
L-curve	num_collect::regularization::explicit_l_curve

RBF Interpolation (rbf Module)

Interpolator Using RBF Interpolation

Type of Length Parameter	Type of RBF	Use of Polynomial Terms	Alias for Class
Local	Global	No	num_collect::rbf::local_rbf_interpolator
Global	Global	No	num_collect::rbf::global_rbf_interpolator, num_collect::rbf::gaussian_process_interpolator
Local	Compact	No	num_collect::rbf::local_csrbf_interpolator
Global	Global	Yes	num_collect::rbf::global_rbf_polynomial_interpolator

Selection of Length Parameters

Algorithm	Class / Function
Maximum likelihood estimator (MLE) for length parameters [21]	optimize_length_parameter_scale member functions in interpolators.

Selection of Points

Algorithm	Class / Function
Halton nodes [5]	num_collect::rbf::generate_halton_nodes

RBFs

RBF	Class
Gaussian	num_collect::rbf::rbfs::gaussian_rbf
Inverse multi-quadric	num_collect::rbf::rbfs::inverse_multi_quadric_rbf
Inverse quadratic	num_collect::rbf::rbfs::inverse_quadratic_rbf
Multi-quadric	num_collect::rbf::rbfs::multi_quadric_rbf
Sech	num_collect::rbf::rbfs::sech_rbf
Wendland’s Compactly Supported RBF (CSRBF) [24]	num_collect::rbf::rbfs::wendland_csrbf

Root-Finding Algorithms (roots Module)

Algorithm	Class
Newton-Raphson method	num_collect::roots::newton_raphson

Special functions (functions Module)

Function	Class
Gamma function	num_collect::functions::gamma
Legendre function	num_collect::functions::legendre

Numerical Integration (integration Module)

Algorithm	Range	Class
Double Exponential (DE) rule	\((a,b)\)	num_collect::integration::de_finite_integrator
Double Exponential (DE) rule	\((-\infty, \infty)\)	num_collect::integration::de_infinite_integrator
Double Exponential (DE) rule	\((0, \infty)\)	num_collect::integration::de_semi_infinite_integrator
Gauss-Legendre formula	\((a,b)\)	num_collect::integration::gauss_legendre_integrator
Gauss-Legendre-Kronrod formula [16]	\((a,b)\)	num_collect::integration::gauss_legendre_kronrod_integrator
TANH rule	\((a,b)\)	num_collect::integration::tanh_finite_integrator

ODE Solvers (ode Module)

Formulas of Runge-Kutta method

Formula	Class
ARK4(3)6L[2]SA-ERK formula [13]	num_collect::ode::runge_kutta::ark43_erk_formula
ARK4(3)6L[2]SA-ESDIRK formula [13]	num_collect::ode::runge_kutta::ark43_esdirk_formula
ARK5(4)8L[2]SA-ESDIRK formula [13]	num_collect::ode::runge_kutta::ark54_esdirk_formula
DOPRI5 formula [9]	num_collect::ode::runge_kutta::dopri5_formula
ESDIRK45c formula [11]	num_collect::ode::runge_kutta::esdirk45_formula
Runge-Kutta 4 (RK4)	num_collect::ode::runge_kutta::rk4_formula
Runge-Kutta-Fehlberg 45 (RKF45)	num_collect::ode::runge_kutta::rkf45_formula
4th order SDIRK [9]	num_collect::ode::runge_kutta::sdirk4_formula
Tanaka Formula 1	num_collect::ode::runge_kutta::tanaka1_formula
Tanaka Formula 2	num_collect::ode::runge_kutta::tanaka2_formula

Formulas of Rosenbrock method

Formula	Class
RODASP formula	num_collect::ode::rosenbrock::rodasp_formula
RODASPR formula [19]	num_collect::ode::rosenbrock::rodaspr_formula
ROS3w formula [19]	num_collect::ode::rosenbrock::ros3w_formula
ROS34PW3 formula [19]	num_collect::ode::rosenbrock::ros34pw3_formula

Average vector field (AVF) method [18]

Order	Class
2	num_collect::ode::avf::avf2_formula
3	num_collect::ode::avf::avf3_formula
4	num_collect::ode::avf::avf4_formula

Symplectic formulas

Formula	Class
Leap-frog formula	num_collect::ode::symplectic::leap_frog_formula
Fourth-order symplectic integration formula in [4]	num_collect::ode::symplectic::symplectic_forest4_formula

Other formulas

Formula	Class
Implicit Euler formula	num_collect::ode::runge_kutta::implicit_euler_formula

Controllers of step sizes

Algorithm	Class
Basic algorithm in [8]	num_collect::ode::basic_step_size_controller
PI controller [7]	num_collect::ode::pi_step_size_controller

Solvers of Linear Equations (linear Module)

Solvers of Linear Equations

Algorithm	Class
Algebraic multigrid (AMG) method [20]	num_collect::linear::algebraic_multigrid_solver
Gauss-Seidel iteration [6]	num_collect::linear::gauss_seidel_iterative_solver
Symmetric successive over-relaxation [6]	num_collect::linear::symmetric_successive_over_relaxation, num_collect::linear::parallel_symmetric_successive_over_relaxation

Ordering Algorithms

Algorithm	Class
Cuthill-McKee (CM) ordering [6, 14]	num_collect::linear::cuthill_mckee_ordering
Reverse Cuthill-McKee (RCM) ordering [6, 14]	num_collect::linear::reverse_cuthill_mckee_ordering

Common Algorithms (util Module)

Algorithm	Class
Kahan summation [12]	num_collect::util::kahan_adder