brainpy.integrators.ode.CashKarp

class brainpy.integrators.ode.CashKarp(f, var_type=None, dt=None, name=None, adaptive=None, tol=None, show_code=False, state_delays=None, neutral_delays=None)

The Cash–Karp method for ODEs.

The Cash–Karp method was proposed by Professor Jeff R. Cash from Imperial College London and Alan H. Karp from IBM Scientific Center. it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms.

It has the characteristics of:

• method stage = 6

• method order = 4

• Butcher Tables:

\[\begin{split}\begin{array}{l|lllll} 0 & & & & & & \\ 1 / 5 & 1 / 5 & & & & & \\ 3 / 10 & 3 / 40 & 9 / 40 & & & \\ 3 / 5 & 3 / 10 & -9 / 10 & 6 / 5 & & \\ 1 & -11 / 54 & 5 / 2 & -70 / 27 & 35 / 27 & & \\ 7 / 8 & 1631 / 55296 & 175 / 512 & 575 / 13824 & 44275 / 110592 & 253 / 4096 & \\ \hline & 37 / 378 & 0 & 250 / 621 & 125 / 594 & 0 & 512 / 1771 \\ & 2825 / 27648 & 0 & 18575 / 48384 & 13525 / 55296 & 277 / 14336 & 1 / 4 \end{array}\end{split}\]

References

[1]

https://en.wikipedia.org/wiki/Cash%E2%80%93Karp_method

[2]

J. R. Cash, A. H. Karp. “A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides”, ACM Transactions on Mathematical Software 16: 201-222, 1990. doi:10.1145/79505.79507

__init__(f, var_type=None, dt=None, name=None, adaptive=None, tol=None, show_code=False, state_delays=None, neutral_delays=None)

Methods

__init__(f[, var_type, dt, name, adaptive, ...])
build()
cpu()	Move all variable into the CPU device.
cuda()	Move all variables into the GPU device.
load_state_dict(state_dict[, warn, compatible])	Copy parameters and buffers from state_dict into this module and its descendants.
load_states(filename[, verbose])	Load the model states.
nodes([method, level, include_self])	Collect all children nodes.
register_implicit_nodes(*nodes[, node_cls])
register_implicit_vars(*variables[, var_cls])
save_states(filename[, variables])	Save the model states.
set_integral(f)	Set the integral function.
state_dict()	Returns a dictionary containing a whole state of the module.
to(device)	Moves all variables into the given device.
tpu()	Move all variables into the TPU device.
train_vars([method, level, include_self])	The shortcut for retrieving all trainable variables.
tree_flatten()	Flattens the object as a PyTree.
tree_unflatten(aux, dynamic_values)	Unflatten the data to construct an object of this class.
unique_name([name, type_])	Get the unique name for this object.
vars([method, level, include_self, ...])	Collect all variables in this node and the children nodes.

Attributes

A
B1
B2
C
arguments	All arguments when calling the numer integrator of the differential equation.
dt	The numerical integration precision.
integral	The integral function.
name	Name of the model.
neutral_delays	neutral delays.
parameters	The parameters defined in the differential equation.
state_delays	State delays.
variables	The variables defined in the differential equation.