thevenin.solvers#

The solvers module provides interfaces for solving the systems of ordinary differential equations that govern equivalent circuit models. The solvers are taken from scikit-sundae, a wrapper for SUNDIALS. Specifically, both the IDA (for differential-algebraic equations) and CVODE (for ordinary differential equations) interfaces are included.

These solvers offer robust numerical methods with settings to adjust the accuracy, stability, and performance. While the implementations from from the scikit-sundae package, they are included here to centralize documentation and streamline solver configuration within the ECM framework.

Classes#

`CVODEResult`	Results class for CVODE solver.
`CVODESolver`	SUNDIALS CVODE solver.
`IDAResult`	Results class for IDA solver.
`IDASolver`	SUNDIALS IDA solver.

Package Contents#

class thevenin.solvers.CVODEResult(**kwargs)[source]#

Results class for CVODE solver.

Inherits from RichResult. The solution class groups output from CVODE into an object with the fields:

Parameters:

message (str) – Human-readable description of the status value.
success (bool) – True if the solver was successful (status >= 0). False otherwise.
status (int) – Reason for the algorithm termination. Negative values correspond to errors, and non-negative values to different successful criteria.
t (ndarray, shape(n,)) – Solution time(s). The dimension depends on the method. Stepwise solutions will only have 1 value whereas solutions across a full ‘tspan’ will have many.
y (ndarray, shape(n, m)) – State variable values at each solution time. Rows correspond to indices in ‘t’ and columns match indexing from ‘y0’.
i_events (ndarray, shape(k, num_events) or None) –
Provides an array for each detected event ‘k’ specifying indices for which event(s) occurred. i_events[k,i] != 0 if ‘events[i]’ occurred at ‘t_events[k]’. The sign of ‘i_events’ indicates the direction of zero-crossing:
- -1 indicates ‘events[i]’ was decreasing
- +1 indicates ‘events[i]’ was increasing
Output for ‘i_events’ will be None when either ‘eventsfn’ was None or if no events occurred during the solve.
t_events (ndarray, shape(k,) or None) – Times at which events occurred or None if ‘eventsfn’ was None or no events were triggered during the solve.
y_events (ndarray, shape(k, m) or None) – State variable values at each ‘t_events’ value or None. Rows and columns correspond to ‘t_events’ and ‘y0’ indexing, respectively.
nfev (int) – Number of times that ‘rhsfn’ was evaluated.
njev (int) – Number of times the Jacobian was evaluated, ‘jacfn’ or internal finite difference method.

Notes

Terminal events are appended to the end of ‘t’ and ‘y’. However, if an event was not terminal then it will only appear in ‘*_events’ outputs and not within the main output arrays.

‘nfev’ and ‘njev’ are cumulative for stepwise solution approaches. The values are reset each time ‘init_step’ is called.

class thevenin.solvers.CVODESolver(rhsfn, **options)[source]#

SUNDIALS CVODE solver.

This class wraps the C-based variable-coefficient ordinary differential equations (CVODE) solver from SUNDIALS.

Parameters:

rhsfn (Callable) – Right-hand-side function with signature f(t, y, yp[, userdata]). If ‘rhsfn’ has return values, they are ignored. Instead of using returns, the solver interacts directly with the ‘yp’ array memory. For more info see the notes.
**options (dict, optional) – Keyword arguments to describe the solver options. A full list of names, types, descriptions, and defaults is given below.
userdata (object or None, optional) – Additional data object to supply to all user-defined callables. If ‘rhsfn’ takes in 4 arguments, including the optional ‘userdata’, then this option cannot be None (default). See notes for more info.
method ({'Adams', 'BDF'}, optional) – Specifies the linear multistep method. The suggested settings are to use ‘Adams’ for nonstiff problems and ‘BDF’ for stiff problems. The default is ‘BDF’.
first_step (float, optional) – Specifies the initial step size. The default is 0, which uses an estimated value internally determined by SUNDIALS.
min_step (float, optional) – Minimum allowable step size. The default is 0.
max_step (float, optional) – Maximum allowable step size. Use 0 (default) for unbounded steps.
rtol (float, optional) – Relative tolerance. For example, 1e-4 means errors are controlled to within 0.01%. It is recommended to not use values larger than 1e-3 nor smaller than 1e-15. The default is 1e-5.
atol (float or array_like[float], optional) – Absolute tolerance. Can be a scalar float to apply the same value for all state variables, or an array with a length matching ‘y’ to provide tolerances specific to each variable. The default is 1e-6.
linsolver ({'dense', 'band'}, optional) – Choice of linear solver. When using ‘band’, don’t forget to provide ‘lband’ and ‘uband’ values. The default is ‘dense’.
lband (int or None, optional) – Lower Jacobian bandwidth. Given an ODE system yp = f(t, y), the Jacobian is J = df_i/dy_j. ‘lband’ should be set to the max distance between the main diagonal and the non-zero elements below the diagonal. This option cannot be None (default) if ‘linsolver’ is ‘band’. Use zero if no values are below the main diagonal.
uband (int or None, optional) – Upper Jacobian bandwidth. See ‘lband’ for the Jacobian description. ‘uband’ should be set to the max distance between the main diagonal and the non-zero elements above the diagonal. This option cannot be None (default) if ‘linsolver’ is ‘band’. Use zero if no elements are above the main diagonal.
max_order (int, optional) – Specifies the maximum order for the linear multistep method. BDF and Adams allow values in ranges [1, 5] and [1, 12], respectively. The default if the method’s max, i.e., 5 for BDF and 12 for Adams.
max_num_steps (int, optional) – Specifies the maximum number of steps taken by the solver in each attempt to reach the next output time. The default is 500.
max_nonlin_iters (int, optional) – Specifies the maximum number of nonlinear solver iterations in one step. The default is 3.
max_conv_fails (int, optional) – Specifies the max number of nonlinear solver convergence failures in one step. The default is 10.
constraints_idx (array_like[int] or None, optional) – Specifies indices ‘i’ in the ‘y’ state variable array for which inequality constraints should be applied. Constraints types must be specified in ‘constraints_type’, see below. The default is None.
constraints_type (array_like[int] or None, optional) – If ‘constraints_idx’ is not None, then this option must include an array of equal length specifying the types of constraints to apply. Values should be in {-2, -1, 1, 2} which apply y[i] < 0, y[i] <= 0, y[i] >=0, and y[i] > 0, respectively. The default is None.
eventsfn (Callable or None, optional) –
Events function with signature g(t, y, events[, userdata]). Return values from this function are ignored. Instead, the solver directly interacts with the ‘events’ array. Each ‘events[i]’ should be an expression that triggers an event when equal to zero. If None (default), no events are tracked. See the notes for more info.

The ‘num_events’ option is required when ‘eventsfn’ is not None so memory can be allocated for the events array. The events function can also have the following attributes:

terminal: list[bool, int], optional
A list with length ‘num_events’ that tells how the solver how to respond to each event. If boolean, the solver will terminate when True and will simply record the event when False. If integer, termination occurs at the given number of occurrences. The default is [True]*num_events.

direction: list[int], optional
A list with length ‘num_events’ that tells the solver which event directions to track. Values must be in {-1, 0, 1}. Negative values will only trigger events when the slope is negative (i.e., ‘events[i]’ went from positive to negative). Alternatively, positive values track events with positive slope. If zero, either direction triggers the event. When not assigned, direction = [0]*num_events.

You can assign attributes like eventsfn.terminal = [True] to any function in Python, after it has been defined.
num_events (int, optional) – Number of events to track. Must be greater than zero if ‘eventsfn’ is not None. The default is 0.
jacfn (Callable or None, optional) – Jacobian function like J(t, y, yp, JJ[, userdata]). Fills the pre-allocated 2D matrix ‘JJ’ with values defined by the Jacobian JJ[i,j] = dyp_i/dy_j. An internal finite difference method is applied when None (default). As with other user-defined callables, return values from ‘jacfn’ are ignored. See notes for more info.

Notes

Return values from ‘rhsfn’, ‘eventsfn’, and ‘jacfn’ are ignored by the solver. Instead the solver directly reads from pre-allocated memory. The ‘yp’, ‘events’, and ‘JJ’ arrays from each user-defined callable should be filled within each respective function. When setting values across the entire array/matrix at once, don’t forget to use [:] to fill the existing array rather than overwriting it. For example, using yp[:] = f(t, y) is correct whereas yp = f(t, y) is not. Using this method of pre-allocated memory helps pass data between Python and the SUNDIALS C functions. It also keeps the solver fast, especially for large problems.

When ‘rhsfn’ (or ‘eventsfn’, or ‘jacfn’) require data outside of their normal arguments, you can supply ‘userdata’ as an option. When given, ‘userdata’ must appear in the function signatures for ALL of ‘rhsfn’, ‘eventsfn’ (when not None), and ‘jacfn’ (when not None), even if it is not used in all of these functions. Note that ‘userdata’ only takes up one argument position; however, ‘userdata’ can be any Python object. Therefore, to pass more than one extra argument you should pack all of the data into a single tuple, dict, dataclass, etc. and pass them all together as ‘userdata’. The data can be unpacked as needed within a function.

Examples

The following example solves the stiff Van der Pol equation, a classic ODE test problem. The same example is provided by MATLAB for comparison.

import numpy as np
import sksundae as sun
import matplotlib.pyplot as plt

def rhsfn(t, y, yp):
    yp[0] =  y[1]
    yp[1] = 1000.*(1. - y[0]**2)*y[1] - y[0]

solver = sun.cvode.CVODE(rhsfn)

tspan = np.array([0, 3000])
y0 = np.array([2, 0])

soln = solver.solve(tspan, y0)

plt.plot(soln.t, soln.y[:,0])
plt.show()

init_step(t0, y0)#

Initialize the solver.

This method is called automatically when using ‘solve’. However, it must be run manually, before the ‘step’ method, when solving with a step-by-step approach.

Parameters:

t0 (float) – Initial value of time.
y0 (array_like[float], shape(m,)) – State variable values at ‘t0’. The length should match the number of equations in ‘rhsfn’.

Returns:

CVODEResult – Custom output class for CVODE solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

MemoryError – Failed to allocate memory for the CVODE solver.
RuntimeError – A SUNDIALS function returned NULL or was unsuccessful.

solve(tspan, y0)#

Return the solution across ‘tspan’.

Parameters:

tspan (array_like[float], shape(n >= 2,)) – Solution time span. If len(tspan) == 2, the solution will be saved at internally chosen steps. When len(tspan) > 2, the solution saves the output at each specified time.
y0 (array_like[float], shape(m,)) – State variable values at ‘tspan[0]’. The length should match the number of equations in ‘rhsfn’.

Returns:

CVODEResult – Custom output class for CVODE solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

ValueError – ‘tspan’ must be strictly increasing or decreasing.
ValueError – ‘tspan’ length must be >= 2.

step(t, method='normal', tstop=None)#

Return the solution at time ‘t’.

Before calling the ‘step’ method, you must first initialize the solver by running ‘init_step’.

Parameters:

t (float) – Value of time.
method ({'normal', 'onestep'}, optional) – Solve method for the current step. When ‘normal’ (default), output is returned at time ‘t’. If ‘onestep’, output is returned after one internal step toward ‘t’. Both methods stop at events, if given, regardless of how ‘eventsfn.terminal’ was set.
tstop (float, optional) – Specifies a hard time constraint for which the solver should not pass, regardless of the ‘method’. The default is None.

Returns:

CVODEResult – Custom output class for CVODE solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

ValueError – ‘method’ value is invalid. Must be ‘normal’ or ‘onestep’.
ValueError – ‘init_step’ must be run prior to ‘step’.

Notes

In general, when solving step by step, times should all be provided in either increasing or decreasing order. The solver can output results at times taken in the opposite direction of integration if the requested time is within the last internal step interval; however, values outside this interval will raise errors. Rather than trying to mix forward and reverse directions, choose each sequential time step carefully so you get all of the values you need.

SUNDIALS provides a convenient graphic to help users understand how the step method and optional ‘tstop’ affect where the integrator stops. To read more, see their documentation here.

class thevenin.solvers.IDAResult(**kwargs)[source]#

Results class for IDA solver.

Inherits from RichResult. The solution class groups output from IDA into an object with the fields:

Parameters:

message (str) – Human-readable description of the status value.
success (bool) – True if the solver was successful (status >= 0). False otherwise.
status (int) – Reason for the algorithm termination. Negative values correspond to errors, and non-negative values to different successful criteria.
t (ndarray, shape(n,)) – Solution time(s). The dimension depends on the method. Stepwise solutions will only have 1 value whereas solutions across a full ‘tspan’ will have many.
y (ndarray, shape(n, m)) – State variable values at each solution time. Rows correspond to indices in ‘t’ and columns match indexing from ‘y0’.
yp (ndarray, shape(n, m)) – State variable time derivate values at each solution time. Row and column indexing matches ‘y’.
i_events (ndarray, shape(k, num_events) or None) –
Provides an array for each detected event ‘k’ specifying indices for which event(s) occurred. i_events[k,i] != 0 if ‘events[i]’ occurred at ‘t_events[k]’. The sign of ‘i_events’ indicates the direction of zero-crossing:
- -1 indicates ‘events[i]’ was decreasing
- +1 indicates ‘events[i]’ was increasing
Output for ‘i_events’ will be None when either ‘eventsfn’ was None or if no events occurred during the solve.
t_events (ndarray, shape(k,) or None) – Times at which events occurred or None if ‘eventsfn’ was None or no events were triggered during the solve.
y_events (ndarray, shape(k, m) or None) – State variable values at each ‘t_events’ value or None. Rows and columns correspond to ‘t_events’ and ‘y0’ indexing, respectively.
yp_events (ndarray, shape(k, m) or None) – State variable time derivative values at each ‘t_events’ value or None. Row and column indexing matches ‘y_events’.
nfev (int) – Number of times that ‘resfn’ was evaluated.
njev (int) – Number of times the Jacobian was evaluated, ‘jacfn’ or internal finite difference method.

Notes

Terminal events are appended to the end of ‘t’, ‘y’, and ‘yp’. However, if an event was not terminal then it will only appear in ‘*_events’ outputs and not within the main output arrays.

‘nfev’ and ‘njev’ are cumulative for stepwise solution approaches. The values are reset each time ‘init_step’ is called.

class thevenin.solvers.IDASolver(resfn, **options)[source]#

SUNDIALS IDA solver.

This class wraps the implicit differential algebraic (IDA) solver from SUNDIALS. It can be used to solve both ordinary differential equations (ODEs) and differiential agebraic equations (DAEs).

Parameters:

resfn (Callable) – Residual function with signature f(t, y, yp, res[, userdata]). If ‘resfn’ has return values, they are ignored. Instead of using returns, the solver interacts directly with the ‘res’ array memory. For more info see the notes.
**options (dict, optional) – Keyword arguments to describe the solver options. A full list of names, types, descriptions, and defaults is given below.
userdata (object or None, optional) – Additional data object to supply to all user-defined callables. If ‘resfn’ takes in 5 arguments, including the optional ‘userdata’, then this option cannot be None (default). See notes for more info.
calc_initcond ({'y0', 'yp0', None}, optional) – Specifies which initial condition, if any, to calculate prior to the first time step. The options ‘y0’ and ‘yp0’ will correct ‘y0’ or ‘yp0’ values at ‘t0’, respectively. When not None (default), the ‘calc_init_dt’ value should be used to specify the direction of integration.
calc_init_dt (float, optional) – Relative time step to take during the initial condition correction. Positive vs. negative values provide the direction of integration as forwards or backwards, respectively. The default is 0.01.
algebraic_idx (array_like[int] or None, optional) – Specifies indices ‘i’ in the ‘y[i]’ state variable array that are purely algebraic. This option should always be provided for DAEs; otherwise, the solver can be unstable. The default is None.
first_step (float, optional) – Specifies the initial step size. The default is 0, which uses an estimated value internally determined by SUNDIALS.
min_step (float, optional) – Minimum allowable step size. The default is 0.
max_step (float, optional) – Maximum allowable step size. Use 0 (default) for unbounded steps.
rtol (float, optional) – Relative tolerance. For example, 1e-4 means errors are controlled to within 0.01%. It is recommended to not use values larger than 1e-3 nor smaller than 1e-15. The default is 1e-5.
atol (float or array_like[float], optional) – Absolute tolerance. Can be a scalar float to apply the same value for all state variables, or an array with a length matching ‘y’ to provide tolerances specific to each variable. The default is 1e-6.
linsolver ({'dense', 'band'}, optional) – Choice of linear solver. When using ‘band’, don’t forget to provide ‘lband’ and ‘uband’ values. The default is ‘dense’.
lband (int or None, optional) – Lower Jacobian bandwidth. Given a DAE system 0 = F(t, y, yp), the Jacobian is J = dF_i/dy_j + c_j*dF_i/dyp_j where ‘c_j’ is determined internally based on both step size and order. ‘lband’ should be set to the max distance between the main diagonal and the non-zero elements below the diagonal. This option cannot be None (default) if ‘linsolver’ is ‘band’. Use zero if no values are below the main diagonal.
uband (int or None, optional) – Upper Jacobian bandwidth. See ‘lband’ for the Jacobian description. ‘uband’ should be set to the max distance between the main diagonal and the non-zero elements above the diagonal. This option cannot be None (default) if ‘linsolver’ is ‘band’. Use zero if no elements are above the main diagonal.
max_order (int, optional) – Specifies the maximum order for the linear multistep BDF method. The value must be in the range [1, 5]. The default is 5.
max_num_steps (int, optional) – Specifies the maximum number of steps taken by the solver in each attempt to reach the next output time. The default is 500.
max_nonlin_iters (int, optional) – Specifies the maximum number of nonlinear solver iterations in one step. The default is 4.
max_conv_fails (int, optional) – Specifies the max number of nonlinear solver convergence failures in one step. The default is 10.
constraints_idx (array_like[int] or None, optional) – Specifies indices ‘i’ in the ‘y’ state variable array for which inequality constraints should be applied. Constraints types must be specified in ‘constraints_type’, see below. The default is None.
constraints_type (array_like[int] or None, optional) – If ‘constraints_idx’ is not None, then this option must include an array of equal length specifying the types of constraints to apply. Values should be in {-2, -1, 1, 2} which apply y[i] < 0, y[i] <= 0, y[i] >=0, and y[i] > 0, respectively. The default is None.
eventsfn (Callable or None, optional) –
Events function with signature g(t, y, yp, events[, userdata]). Return values from this function are ignored. Instead, the solver directly interacts with the ‘events’ array. Each ‘events[i]’ should be an expression that triggers an event when equal to zero. If None (default), no events are tracked. See the notes for more info.

The ‘num_events’ option is required when ‘eventsfn’ is not None so memory can be allocated for the events array. The events function can also have the following attributes:

terminal: list[bool, int], optional
A list with length ‘num_events’ that tells how the solver how to respond to each event. If boolean, the solver will terminate when True and will simply record the event when False. If integer, termination occurs at the given number of occurrences. The default is [True]*num_events.

direction: list[int], optional
A list with length ‘num_events’ that tells the solver which event directions to track. Values must be in {-1, 0, 1}. Negative values will only trigger events when the slope is negative (i.e., ‘events[i]’ went from positive to negative). Alternatively, positive values track events with positive slope. If zero, either direction triggers the event. When not assigned, direction = [0]*num_events.

You can assign attributes like eventsfn.terminal = [True] to any function in Python, after it has been defined.
num_events (int, optional) – Number of events to track. Must be greater than zero if ‘eventsfn’ is not None. The default is 0.
jacfn (Callable or None, optional) – Jacobian function like J(t, y, yp, res, cj, JJ[, userdata]). The function should fill the pre-allocated 2D matrix ‘JJ’ with the values defined by JJ[i,j] = dres_i/dy_j + cj*dres_i/dyp_j. An internal finite difference method is applied when None (default). As with other user-defined callables, return values from ‘jacfn’ are ignored. See notes for more info.

Notes

Return values from ‘resfn’, ‘eventsfn’, and ‘jacfn’ are ignored by the solver. Instead the solver directly reads from pre-allocated memory. The ‘res’, ‘events’, and ‘JJ’ arrays from each user-defined callable should be filled within each respective function. When setting values across the entire array/matrix at once, don’t forget to use [:] to fill the existing array rather than overwriting it. For example, using res[:] = F(t, y, yp) is correct whereas res = F(t, y, yp) is not. Using this method of pre-allocated memory helps pass data between Python and the SUNDIALS C functions. It also keeps the solver fast, especially for large problems.

When ‘resfn’ (or ‘eventsfn’, or ‘jacfn’) require data outside of their normal arguments, you can supply ‘userdata’ as an option. When given, ‘userdata’ must appear in the function signatures for ALL of ‘resfn’, ‘eventsfn’ (when not None), and ‘jacfn’ (when not None), even if it is not used in all of these functions. Note that ‘userdata’ only takes up one argument position; however, ‘userdata’ can be any Python object. Therefore, to pass more than one extra argument you should pack all of the data into a single tuple, dict, dataclass, etc. and pass them all together as ‘userdata’. The data can be unpacked as needed within a function.

Examples

The following example solves the Robertson problem, which is a classic test problem for programs that solve stiff ODEs. A full description of the problem is provided by MATLAB. While initializing the solver, algebraic_idx=[2] specifies y[2] is purely algebraic, and calc_initcond='yp0' tells the solver to determine the values for ‘yp0’ at ‘tspan[0]’ before starting to integrate. That is why ‘yp0’ can be initialized as an array of zeros even though plugging in ‘y0’ to the residuals expressions actually gives yp0 = [-0.04, 0.04, 0]. The initialization is checked against the correct answer after solving.

import numpy as np
import sksundae as sun
import matplotlib.pyplot as plt

def resfn(t, y, yp, res):
    res[0] = yp[0] + 0.04*y[0] - 1e4*y[1]*y[2]
    res[1] = yp[1] - 0.04*y[0] + 1e4*y[1]*y[2] + 3e7*y[1]**2
    res[2] = y[0] + y[1] + y[2] - 1.0

solver = sun.ida.IDA(resfn, algebraic_idx=[2], calc_initcond='yp0')

tspan = np.hstack([0, 4*np.logspace(-6, 6)])
y0 = np.array([1, 0, 0])
yp0 = np.zeros_like(y0)

soln = solver.solve(tspan, y0, yp0)
assert np.allclose(soln.yp[0], [-0.04, 0.04, 0], rtol=1e-3)

soln.y[:, 1] *= 1e4  # scale y[1] so it is visible in the figure
plt.semilogx(soln.t, soln.y)
plt.show()

init_step(t0, y0, yp0)#

Initialize the solver.

This method is called automatically when using ‘solve’. However, it must be run manually, before the ‘step’ method, when solving with a step-by-step approach.

Parameters:

t0 (float) – Initial value of time.
y0 (array_like[float], shape(m,)) – State variable values at ‘t0’. The length must match that of ‘yp0’ and the number of residual equations in ‘resfn’.
yp0 (array_like[float], shape(m,)) – Time derivatives for the ‘y0’ array, evaluated at ‘t0’. The length and indexing should be consistent with ‘y0’.

Returns:

IDAResult – Custom output class for IDA solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

MemoryError – Failed to allocate memory for the IDA solver.
RuntimeError – A SUNDIALS function returned NULL or was unsuccessful.
ValueError – ‘y0’ and ‘yp0’ must be the same length.

solve(tspan, y0, yp0)#

Return the solution across ‘tspan’.

Parameters:

tspan (array_like[float], shape(n >= 2,)) – Solution time span. If len(tspan) == 2, the solution will be saved at internally chosen steps. When len(tspan) > 2, the solution saves the output at each specified time.
y0 (array_like[float], shape(m,)) – State variable values at ‘tspan[0]’. The length must match that of ‘yp0’ and the number of residual equations in ‘resfn’.
yp0 (array_like[float], shape(m,)) – Time derivatives for the ‘y0’ array, evaluated at ‘tspan[0]’. The length and indexing should be consistent with ‘y0’.

Returns:

IDAResult – Custom output class for IDA solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

ValueError – ‘tspan’ must be strictly increasing or decreasing.
ValueError – ‘tspan’ length must be >= 2.

step(t, method='normal', tstop=None)#

Return the solution at time ‘t’.

Before calling the ‘step’ method, you must first initialize the solver by running ‘init_step’.

Parameters:

t (float) – Value of time.
method ({'normal', 'onestep'}, optional) – Solve method for the current step. When ‘normal’ (default), output is returned at time ‘t’. If ‘onestep’, output is returned after one internal step toward ‘t’. Both methods stop at events, if given, regardless of how ‘eventsfn.terminal’ was set.
tstop (float, optional) – Specifies a hard time constraint for which the solver should not pass, regardless of the ‘method’. The default is None.

Returns:

IDAResult – Custom output class for IDA solutions. Includes pretty-printing consistent with scipy outputs. See the class definition for more information.

Raises:

ValueError – ‘method’ value is invalid. Must be ‘normal’ or ‘onestep’.
ValueError – ‘init_step’ must be run prior to ‘step’.

Notes

In general, when solving step by step, times should all be provided in either increasing or decreasing order. The solver can output results at times taken in the opposite direction of integration if the requested time is within the last internal step interval; however, values outside this interval will raise errors. Rather than trying to mix forward and reverse directions, choose each sequential time step carefully so you get all of the values you need.

SUNDIALS provides a convenient graphic to help users understand how the step method and optional ‘tstop’ affect where the integrator stops. To read more, see their documentation here.