Does anybody know how useful LBFGS is for estimating the Hessian matrix in the case of many (>10 000) dimensions? When running scipy's implementation on a simple 100D quadratic form the algorithm does already seem to struggle. Are there any general results about special cases (i.e. a dominant diagonal) in which the approximated Hessian is reasonably trustworthy?
Finally, one immediate drawback in scipy's implementation to me seems that the initial estimate of the Hessian is the identity matrix which might lead to a slower convergence. Do you know how important this effect is, i.e. how would the algorithm be affected if I would have a good idea of what the diagonal elements would be?
Here are two sets of example plots for a rather diagonal dominant form, as well as for a case with strong off-diagonals. The first one shows the original covariance matrix and the latter one gives the approximated results using m=50 and m=500.
Code for running the experiment:
import numpy as np
from matplotlib import pyplot as plt
# Parameters
ndims = 100 # Dimensions for our problem
a = .2 # Relative importance of non-diagonal elements in covariance
m = 500 # Number of updates we allow in lbfgs
x0=1*np.random.rand(ndims) # Initial starting point for LBFGS
# Generate covariance matrix
A = np.matrix([np.random.randn(ndims) + np.random.randn(1)*a for i in range(ndims)])
A = A*np.transpose(A)
D_half = np.diag(np.diag(A)**(-0.5))
cov= D_half*A*D_half
invcov = np.linalg.inv(cov)
assert(np.all(np.linalg.eigvals(cov) > 0))
# Define quadratic form and its derivative
def gauss(x,invcov):
res = 0.5*x.T#invcov#x
return res[0,0]
def gaussgrad(x,invcov):
res = np.asarray(x.T#invcov)
return res[0]
# Put function in lambda shape
fgauss = lambda x: gauss(x,invcov=invcov)
fprimegauss = lambda x: gaussgrad(x,invcov=invcov)
# Run the lbfgs variant and retrieve the inverse Hessian approximation
x, f, d, s, y = fmin_l_bfgs_b(func=fgauss,x0=x0,fprime=fprimegauss,m=m,approx_grad=False)
invhess = LbfgsInvHess(s, y)
# Plot the results
plt.imshow(cov)
plt.colorbar()
plt.show()
plt.imshow(invhess.todense(),vmin=np.min(cov),vmax=np.max(cov))
plt.colorbar()
plt.show()
plt.imshow(invhess.todense()-cov)
plt.colorbar()
plt.show()
As scipy does not give the vectors from which the Hessian is reconstructed we need to call a marginally modified function (based on scipy.optimize.lbfgsb.py):
import numpy as np
from numpy import array, asarray, float64, zeros
from scipy.optimize import _lbfgsb
from scipy.optimize.optimize import (MemoizeJac, OptimizeResult,
_check_unknown_options, _prepare_scalar_function)
from scipy.optimize._constraints import old_bound_to_new
from scipy.sparse.linalg import LinearOperator
__all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct']
def fmin_l_bfgs_b(func, x0, fprime=None, args=(),
approx_grad=0,
bounds=None, m=10, factr=1e7, pgtol=1e-5,
epsilon=1e-8,
iprint=-1, maxfun=15000, maxiter=15000, disp=None,
callback=None, maxls=20):
"""
Minimize a function func using the L-BFGS-B algorithm.
Parameters
----------
func : callable f(x,*args)
Function to minimize.
x0 : ndarray
Initial guess.
fprime : callable fprime(x,*args), optional
The gradient of `func`. If None, then `func` returns the function
value and the gradient (``f, g = func(x, *args)``), unless
`approx_grad` is True in which case `func` returns only ``f``.
args : sequence, optional
Arguments to pass to `func` and `fprime`.
approx_grad : bool, optional
Whether to approximate the gradient numerically (in which case
`func` returns only the function value).
bounds : list, optional
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None or +-inf for one of ``min`` or
``max`` when there is no bound in that direction.
m : int, optional
The maximum number of variable metric corrections
used to define the limited memory matrix. (The limited memory BFGS
method does not store the full hessian but uses this many terms in an
approximation to it.)
factr : float, optional
The iteration stops when
``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
where ``eps`` is the machine precision, which is automatically
generated by the code. Typical values for `factr` are: 1e12 for
low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
high accuracy. See Notes for relationship to `ftol`, which is exposed
(instead of `factr`) by the `scipy.optimize.minimize` interface to
L-BFGS-B.
pgtol : float, optional
The iteration will stop when
``max{|proj g_i | i = 1, ..., n} <= pgtol``
where ``pg_i`` is the i-th component of the projected gradient.
epsilon : float, optional
Step size used when `approx_grad` is True, for numerically
calculating the gradient
iprint : int, optional
Controls the frequency of output. ``iprint < 0`` means no output;
``iprint = 0`` print only one line at the last iteration;
``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
``iprint = 99`` print details of every iteration except n-vectors;
``iprint = 100`` print also the changes of active set and final x;
``iprint > 100`` print details of every iteration including x and g.
disp : int, optional
If zero, then no output. If a positive number, then this over-rides
`iprint` (i.e., `iprint` gets the value of `disp`).
maxfun : int, optional
Maximum number of function evaluations.
maxiter : int, optional
Maximum number of iterations.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
maxls : int, optional
Maximum number of line search steps (per iteration). Default is 20.
Returns
-------
x : array_like
Estimated position of the minimum.
f : float
Value of `func` at the minimum.
d : dict
Information dictionary.
* d['warnflag'] is
- 0 if converged,
- 1 if too many function evaluations or too many iterations,
- 2 if stopped for another reason, given in d['task']
* d['grad'] is the gradient at the minimum (should be 0 ish)
* d['funcalls'] is the number of function calls made.
* d['nit'] is the number of iterations.
See also
--------
minimize: Interface to minimization algorithms for multivariate
functions. See the 'L-BFGS-B' `method` in particular. Note that the
`ftol` option is made available via that interface, while `factr` is
provided via this interface, where `factr` is the factor multiplying
the default machine floating-point precision to arrive at `ftol`:
``ftol = factr * numpy.finfo(float).eps``.
Notes
-----
License of L-BFGS-B (FORTRAN code):
The version included here (in fortran code) is 3.0
(released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd,
and Jorge Nocedal <nocedal#ece.nwu.edu>. It carries the following
condition for use:
This software is freely available, but we expect that all publications
describing work using this software, or all commercial products using it,
quote at least one of the references given below. This software is released
under the BSD License.
References
----------
* R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
Constrained Optimization, (1995), SIAM Journal on Scientific and
Statistical Computing, 16, 5, pp. 1190-1208.
* C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
FORTRAN routines for large scale bound constrained optimization (1997),
ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
* J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
FORTRAN routines for large scale bound constrained optimization (2011),
ACM Transactions on Mathematical Software, 38, 1.
"""
# handle fprime/approx_grad
if approx_grad:
fun = func
jac = None
elif fprime is None:
fun = MemoizeJac(func)
jac = fun.derivative
else:
fun = func
jac = fprime
# build options
if disp is None:
disp = iprint
opts = {'disp': disp,
'iprint': iprint,
'maxcor': m,
'ftol': factr * np.finfo(float).eps,
'gtol': pgtol,
'eps': epsilon,
'maxfun': maxfun,
'maxiter': maxiter,
'callback': callback,
'maxls': maxls}
res, s, y = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
**opts)
d = {'grad': res['jac'],
'task': res['message'],
'funcalls': res['nfev'],
'nit': res['nit'],
'warnflag': res['status']}
f = res['fun']
x = res['x']
return x, f, d, s, y
def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None,
disp=None, maxcor=10, ftol=2.2204460492503131e-09,
gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000,
iprint=-1, callback=None, maxls=20,
finite_diff_rel_step=None, **unknown_options):
"""
Minimize a scalar function of one or more variables using the L-BFGS-B
algorithm.
Options
-------
disp : None or int
If `disp is None` (the default), then the supplied version of `iprint`
is used. If `disp is not None`, then it overrides the supplied version
of `iprint` with the behaviour you outlined.
maxcor : int
The maximum number of variable metric corrections used to
define the limited memory matrix. (The limited memory BFGS
method does not store the full hessian but uses this many terms
in an approximation to it.)
ftol : float
The iteration stops when ``(f^k -
f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``.
gtol : float
The iteration will stop when ``max{|proj g_i | i = 1, ..., n}
<= gtol`` where ``pg_i`` is the i-th component of the
projected gradient.
eps : float or ndarray
If `jac is None` the absolute step size used for numerical
approximation of the jacobian via forward differences.
maxfun : int
Maximum number of function evaluations.
maxiter : int
Maximum number of iterations.
iprint : int, optional
Controls the frequency of output. ``iprint < 0`` means no output;
``iprint = 0`` print only one line at the last iteration;
``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
``iprint = 99`` print details of every iteration except n-vectors;
``iprint = 100`` print also the changes of active set and final x;
``iprint > 100`` print details of every iteration including x and g.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
maxls : int, optional
Maximum number of line search steps (per iteration). Default is 20.
finite_diff_rel_step : None or array_like, optional
If `jac in ['2-point', '3-point', 'cs']` the relative step size to
use for numerical approximation of the jacobian. The absolute step
size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``,
possibly adjusted to fit into the bounds. For ``method='3-point'``
the sign of `h` is ignored. If None (default) then step is selected
automatically.
Notes
-----
The option `ftol` is exposed via the `scipy.optimize.minimize` interface,
but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The
relationship between the two is ``ftol = factr * numpy.finfo(float).eps``.
I.e., `factr` multiplies the default machine floating-point precision to
arrive at `ftol`.
"""
#_check_unknown_options(unknown_options)
m = maxcor
pgtol = gtol
factr = ftol / np.finfo(float).eps
x0 = asarray(x0).ravel()
n, = x0.shape
if bounds is None:
bounds = [(None, None)] * n
if len(bounds) != n:
raise ValueError('length of x0 != length of bounds')
# unbounded variables must use None, not +-inf, for optimizer to work properly
bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in bounds]
# LBFGSB is sent 'old-style' bounds, 'new-style' bounds are required by
# approx_derivative and ScalarFunction
new_bounds = old_bound_to_new(bounds)
# check bounds
if (new_bounds[0] > new_bounds[1]).any():
raise ValueError("LBFGSB - one of the lower bounds is greater than an upper bound.")
# initial vector must lie within the bounds. Otherwise ScalarFunction and
# approx_derivative will cause problems
x0 = np.clip(x0, new_bounds[0], new_bounds[1])
if disp is not None:
if disp == 0:
iprint = -1
else:
iprint = disp
sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps,
bounds=new_bounds,
finite_diff_rel_step=finite_diff_rel_step)
func_and_grad = sf.fun_and_grad
fortran_int = _lbfgsb.types.intvar.dtype
nbd = zeros(n, fortran_int)
low_bnd = zeros(n, float64)
upper_bnd = zeros(n, float64)
bounds_map = {(None, None): 0,
(1, None): 1,
(1, 1): 2,
(None, 1): 3}
for i in range(0, n):
l, u = bounds[i]
if l is not None:
low_bnd[i] = l
l = 1
if u is not None:
upper_bnd[i] = u
u = 1
nbd[i] = bounds_map[l, u]
if not maxls > 0:
raise ValueError('maxls must be positive.')
x = array(x0, float64)
f = array(0.0, float64)
g = zeros((n,), float64)
wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64)
iwa = zeros(3*n, fortran_int)
task = zeros(1, 'S60')
csave = zeros(1, 'S60')
lsave = zeros(4, fortran_int)
isave = zeros(44, fortran_int)
dsave = zeros(29, float64)
task[:] = 'START'
n_iterations = 0
while 1:
# x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \
_lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr,
pgtol, wa, iwa, task, iprint, csave, lsave,
isave, dsave, maxls)
task_str = task.tobytes()
if task_str.startswith(b'FG'):
# The minimization routine wants f and g at the current x.
# Note that interruptions due to maxfun are postponed
# until the completion of the current minimization iteration.
# Overwrite f and g:
f, g = func_and_grad(x)
elif task_str.startswith(b'NEW_X'):
# new iteration
n_iterations += 1
if callback is not None:
callback(np.copy(x))
if n_iterations >= maxiter:
task[:] = 'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT'
elif sf.nfev > maxfun:
task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS '
'EXCEEDS LIMIT')
else:
break
task_str = task.tobytes().strip(b'\x00').strip()
if task_str.startswith(b'CONV'):
warnflag = 0
elif sf.nfev > maxfun or n_iterations >= maxiter:
warnflag = 1
else:
warnflag = 2
# These two portions of the workspace are described in the mainlb
# subroutine in lbfgsb.f. See line 363.
s = wa[0: m*n].reshape(m, n)
y = wa[m*n: 2*m*n].reshape(m, n)
print(x.shape)
# See lbfgsb.f line 160 for this portion of the workspace.
# isave(31) = the total number of BFGS updates prior the current iteration;
n_bfgs_updates = isave[30]
n_corrs = min(n_bfgs_updates, maxcor)
inv_hess = LbfgsInvHess(s[:n_corrs], y[:n_corrs])
task_str = task_str.decode()
return OptimizeResult(fun=f, jac=g, nfev=sf.nfev,
njev=sf.ngev,
nit=n_iterations, status=warnflag, message=task_str,
x=x, success=(warnflag == 0), hess_inv=inv_hess), s[:n_corrs], y[:n_corrs]
class LbfgsInvHess(LinearOperator):
"""Linear operator for the L-BFGS approximate inverse Hessian.
This operator computes the product of a vector with the approximate inverse
of the Hessian of the objective function, using the L-BFGS limited
memory approximation to the inverse Hessian, accumulated during the
optimization.
Objects of this class implement the ``scipy.sparse.linalg.LinearOperator``
interface.
Parameters
----------
sk : array_like, shape=(n_corr, n)
Array of `n_corr` most recent updates to the solution vector.
(See [1]).
yk : array_like, shape=(n_corr, n)
Array of `n_corr` most recent updates to the gradient. (See [1]).
References
----------
.. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited
storage." Mathematics of computation 35.151 (1980): 773-782.
"""
def __init__(self, sk, yk):
"""Construct the operator."""
if sk.shape != yk.shape or sk.ndim != 2:
raise ValueError('sk and yk must have matching shape, (n_corrs, n)')
n_corrs, n = sk.shape
super().__init__(dtype=np.float64, shape=(n, n))
self.sk = sk
self.yk = yk
self.n_corrs = n_corrs
self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
def _matvec(self, x):
"""Efficient matrix-vector multiply with the BFGS matrices.
This calculation is described in Section (4) of [1].
Parameters
----------
x : ndarray
An array with shape (n,) or (n,1).
Returns
-------
y : ndarray
The matrix-vector product
"""
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
q = np.array(x, dtype=self.dtype, copy=True)
if q.ndim == 2 and q.shape[1] == 1:
q = q.reshape(-1)
alpha = np.empty(n_corrs)
for i in range(n_corrs-1, -1, -1):
alpha[i] = rho[i] * np.dot(s[i], q)
q = q - alpha[i]*y[i]
r = q
for i in range(n_corrs):
beta = rho[i] * np.dot(y[i], r)
r = r + s[i] * (alpha[i] - beta)
return r
def todense(self):
"""Return a dense array representation of this operator.
Returns
-------
arr : ndarray, shape=(n, n)
An array with the same shape and containing
the same data represented by this `LinearOperator`.
"""
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
I = np.eye(*self.shape, dtype=self.dtype)
Hk = I
for i in range(n_corrs):
A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i]
A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] *
s[i][np.newaxis, :])
return Hk
Edit: Typo in code.
I have a bit of a simple question regarding solving a differential equation I'm hoping someone can help me answer. I have a differential equation for a simple neuron model, with varying input (i.e. integrate and fire model of a neuron, with a "square" input):
def impulse_func(t, V):
if 100 <= t <= 200:
return 0.25
return 0
def passive_membrane(t, V, El, R, tau, I_app):
dVdt = (-(V - El) + R*I_app(t, V))/tau
return dVdt
t = np.arange(0, 400, 0.1)
V0 = np.array([-60.])
V = solve_ivp(fun = passive_membrane,
t_span = [0, 400],
t_eval = t,
y0=V0,
args = (-60., 100., 10., impulse_func),
dense_output=True,
rtol = 1e-8, atol = 1e-8)
In this case, the values of El, R, and tau will be remaining constant. This works just as expected, with the plot being My problem is that I now need to create a "spike" in the solution which is dependent on the solution.
Specifically, if the solution passes my "threshold" value of -37, for a single unit of time I would like to set the solution value to 60 and then return to my "resting" which is -65 in the next unit of time. I've tried modifying the impulse_func and the passive_membrane functions, but it just led to an exploding solution. I'm able to do this when I write my own solver, but I'm lost on how to implement it with solve_ivp, which I'd like to use since it has much more functionality than my own poorly optimized RK2 method and because I want to be able to pass variable thresholds, times, etc.
THIS PART IS JUST BACKGROUND IF YOU NEED IT
I am developing a numerical solver for the Second-Order Kuramoto Model. The functions I use to find the derivatives of theta and omega are given below.
# n-dimensional change in omega
def d_theta(omega):
return omega
# n-dimensional change in omega
def d_omega(K,A,P,alpha,mask,n):
def layer1(theta,omega):
T = theta[:,None] - theta
A[mask] = K[mask] * np.sin(T[mask])
return - alpha*omega + P - A.sum(1)
return layer1
These equations return vectors.
QUESTION 1
I know how to use odeint for two dimensions, (y,t). for my research I want to use a built-in Python function that works for higher dimensions.
QUESTION 2
I do not necessarily want to stop after a predetermined amount of time. I have other stopping conditions in the code below that will indicate whether the system of equations converges to the steady state. How do I incorporate these into a built-in Python solver?
WHAT I CURRENTLY HAVE
This is the code I am currently using to solve the system. I just implemented RK4 with constant time stepping in a loop.
# This function randomly samples initial values in the domain and returns whether the solution converged
# Inputs:
# f change in theta (d_theta)
# g change in omega (d_omega)
# tol when step size is lower than tolerance, the solution is said to converge
# h size of the time step
# max_iter maximum number of steps Runge-Kutta will perform before giving up
# max_laps maximum number of laps the solution can do before giving up
# fixed_t vector of fixed points of theta
# fixed_o vector of fixed points of omega
# n number of dimensions
# theta initial theta vector
# omega initial omega vector
# Outputs:
# converges true if it nodes restabilizes, false otherwise
def kuramoto_rk4_wss(f,g,tol_ss,tol_step,h,max_iter,max_laps,fixed_o,fixed_t,n):
def layer1(theta,omega):
lap = np.zeros(n, dtype = int)
converges = False
i = 0
tau = 2 * np.pi
while(i < max_iter): # perform RK4 with constant time step
p_omega = omega
p_theta = theta
T1 = h*f(omega)
O1 = h*g(theta,omega)
T2 = h*f(omega + O1/2)
O2 = h*g(theta + T1/2,omega + O1/2)
T3 = h*f(omega + O2/2)
O3 = h*g(theta + T2/2,omega + O2/2)
T4 = h*f(omega + O3)
O4 = h*g(theta + T3,omega + O3)
theta = theta + (T1 + 2*T2 + 2*T3 + T4)/6 # take theta time step
mask2 = np.array(np.where(np.logical_or(theta > tau, theta < 0))) # find which nodes left [0, 2pi]
lap[mask2] = lap[mask2] + 1 # increment the mask
theta[mask2] = np.mod(theta[mask2], tau) # take the modulus
omega = omega + (O1 + 2*O2 + 2*O3 + O4)/6
if(max_laps in lap): # if any generator rotates this many times it probably won't converge
break
elif(np.any(omega > 12)): # if any of the generators is rotating this fast, it probably won't converge
break
elif(np.linalg.norm(omega) < tol_ss and # assert the nodes are sufficiently close to the equilibrium
np.linalg.norm(omega - p_omega) < tol_step and # assert change in omega is small
np.linalg.norm(theta - p_theta) < tol_step): # assert change in theta is small
converges = True
break
i = i + 1
return converges
return layer1
Thanks for your help!
You can wrap your existing functions into a function accepted by odeint (option tfirst=True) and solve_ivp as
def odesys(t,u):
theta,omega = u[:n],u[n:]; # or = u.reshape(2,-1);
return [ *f(omega), *g(theta,omega) ]; # or np.concatenate([f(omega), g(theta,omega)])
u0 = [*theta0, *omega0]
t = linspan(t0, tf, timesteps+1);
u = odeint(odesys, u0, t, tfirst=True);
#or
res = solve_ivp(odesys, [t0,tf], u0, t_eval=t)
The scipy methods pass numpy arrays and convert the return value into same, so that you do not have to care in the ODE function. The variant in comments is using explicit numpy functions.
While solve_ivp does have event handling, using it for a systematic collection of events is rather cumbersome. It would be easier to advance some fixed step, do the normalization and termination detection, and then repeat this.
If you want to later increase efficiency somewhat, use directly the stepper classes behind solve_ivp.
I am looking for a way to set a fixed step size for solving my initial value problem by Runge-Kutta method in Python. Accordingly, how I can tell the scipy.integrate.RK45 to keep a constant update (step size) for its integration procedure?
Thank you very much.
Scipy.integrate is usually used with changeable step method by controlling the TOL(one step error) while integrating numerically. The TOL is usually computed by checking with another numerical method. For example RK45 uses the 5th order Runge-Kutta to check the TOL of the 4th order Runge-Kutta method to determine the integrating step.
Hence if you must integrate ODEs with fixed step, just turn off the TOL check by setting atol, rtol with a rather large constant. For example, like the form:
solve_ivp(your function, t_span=[0, 10], y0=..., method="RK45", max_step=0.01, atol = 1, rtol = 1)
The TOL check is set to be so large that the integrating step would be the max_step you choose.
It is quite easy to code the Butcher tableau for the Dormand-Prince RK45 method.
0
1/5 | 1/5
3/10 | 3/40 9/40
4/5 | 44/45 −56/15 32/9
8/9 | 19372/6561 −25360/2187 64448/6561 −212/729
1 | 9017/3168 −355/33 46732/5247 49/176 −5103/18656
1 | 35/384 0 500/1113 125/192 −2187/6784 11/84
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| 35/384 0 500/1113 125/192 −2187/6784 11/84 0
| 5179/57600 0 7571/16695 393/640 −92097/339200 187/2100 1/40
first in a function for a single step
import numpy as np
def DoPri45Step(f,t,x,h):
k1 = f(t,x)
k2 = f(t + 1./5*h, x + h*(1./5*k1) )
k3 = f(t + 3./10*h, x + h*(3./40*k1 + 9./40*k2) )
k4 = f(t + 4./5*h, x + h*(44./45*k1 - 56./15*k2 + 32./9*k3) )
k5 = f(t + 8./9*h, x + h*(19372./6561*k1 - 25360./2187*k2 + 64448./6561*k3 - 212./729*k4) )
k6 = f(t + h, x + h*(9017./3168*k1 - 355./33*k2 + 46732./5247*k3 + 49./176*k4 - 5103./18656*k5) )
v5 = 35./384*k1 + 500./1113*k3 + 125./192*k4 - 2187./6784*k5 + 11./84*k6
k7 = f(t + h, x + h*v5)
v4 = 5179./57600*k1 + 7571./16695*k3 + 393./640*k4 - 92097./339200*k5 + 187./2100*k6 + 1./40*k7;
return v4,v5
and then in a standard fixed-step loop
def DoPri45integrate(f, t, x0):
N = len(t)
x = [x0]
for k in range(N-1):
v4, v5 = DoPri45Step(f,t[k],x[k],t[k+1]-t[k])
x.append(x[k] + (t[k+1]-t[k])*v5)
return np.array(x)
Then test it for some toy example with known exact solution y(t)=sin(t)
def mms_ode(t,y): return np.array([ y[1], sin(sin(t))-sin(t)-sin(y[0]) ])
mms_x0 = [0.0, 1.0]
and plot the error scaled by h^5
for h in [0.2, 0.1, 0.08, 0.05, 0.01][::-1]:
t = np.arange(0,20,h);
y = DoPri45integrate(mms_ode,t,mms_x0)
plt.plot(t, (y[:,0]-np.sin(t))/h**5, 'o', ms=3, label = "h=%.4f"%h);
plt.grid(); plt.legend(); plt.show()
to get the confirmation that this is indeed an order 5 method, as the graphs of the error coefficients come close together.
By looking at the implementation of the step, you'll find that the best you can do is to control the initial step size (within the bounds set by the minimum and maximum step size) by setting the attribute h_abs prior to calling RK45.step:
In [27]: rk = RK45(lambda t, y: t, 0, [0], 1e6)
In [28]: rk.h_abs = 30
In [29]: rk.step()
In [30]: rk.step_size
Out[30]: 30.0
If you are interested in data-wise fix step size, then I highly recommend you to use the scipy.integrate.solve_ivp function and its t_eval argument.
This function wraps up all of the scipy.integrate ode solvers in one function, thus you have to choose the method by giving value to its method argument. Fortunately, the default method is the RK45, so you don't have to bother with that.
What is more interesting for you is the t_eval argument, where you have to give a flat array. The function samples the solution curve at t_eval values and only returns these points. So if you want a uniform sampling by the step size then just give the t_eval argument the following: numpy.linspace(t0, tf, samplingResolution), where t0 is the start and tf is the end of the simulation.
Thusly you can have uniform sampling without having to resort fix step size that causes instability for some ODEs.
You've said you want a fixed-time step behaviour, not just a fixed evluation time step. Therefore, you have to "hack" your way through that if you not want to reimplement the solver yourself. Just set the integration tolerances atol and rtol to 1e90, and max_step and first_step to the value dt of the time step you want to use. This way the estimated integration error will always be very small, thus tricking the solver into not shrinking the time step dynamically.
However, only use this trick with EXPLICIT algortithms (RK23,RK45,DOP853) !
The implicit algorithms from "solve_ivp" (Radau, BDF, maybe LSODA as well) adjust the precision of the nonlinear Newton solver according to atol and rtol, therefore you might end up having a solution which does not make any sense...
I suggest to write your own rk4 fixed step program in py. There are many internet examples to help. That guarantees that you know precisely how each value is being computed. Furthermore, there will normally be no 0/0 calculations and if so they will be easy to trace and prompt another look at the ode's being solved.