# Ordinary Differential Equations

This chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The library provides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with full access to any intermediate steps. A driver object can be used as a high level wrapper for easy use of low level functions.

These functions are declared in the header file gsl_odeiv2.h'. This is a new interface in version 1.15 and uses the prefix gsl_odeiv2 for all functions. It is recommended over the previous gsl_odeiv implementation defined in gsl_odeiv.h' The old interface has been retained under the original name for backwards compatibility.

## Defining the ODE System

The routines solve the general n-dimensional first-order system,

dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))


for i = 1, \dots, n. The stepping functions rely on the vector of derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j. A system of equations is defined using the gsl_odeiv2_system datatype.

Data Type: gsl_odeiv2_system
This data type defines a general ODE system with arbitrary parameters.

int (* function) (double t, const double y[], double dydt[], void * params)
This function should store the vector elements f_i(t,y,params) in the array dydt, for arguments (t,y) and parameters params. The function should return GSL_SUCCESS if the calculation was completed successfully. Any other return value indicates an error. A special return value GSL_EBADFUNC causes gsl_odeiv2 routines to immediately stop and return. The user must call an appropriate reset function (e.g. gsl_odeiv2_driver_reset or gsl_odeiv2_step_reset) before continuing. Use return values distinct from standard GSL error codes to distinguish your function as the source of the error.
int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);
This function should store the vector of derivative elements in the array dfdt and the Jacobian matrix J_{ij} in the array dfdy, regarded as a row-ordered matrix J(i,j) = dfdy[i * dimension + j] where dimension is the dimension of the system. Not all of the stepper algorithms of gsl_odeiv2 make use of the Jacobian matrix, so it may not be necessary to provide this function (the jacobian element of the struct can be replaced by a null pointer for those algorithms). The function should return GSL_SUCCESS if the calculation was completed successfully. Any other return value indicates an error. A special return value GSL_EBADFUNC causes gsl_odeiv2 routines to immediately stop and return. The user must call an appropriate reset function (e.g. gsl_odeiv2_driver_reset or gsl_odeiv2_step_reset) before continuing. Use return values distinct from standard GSL error codes to distinguish your function as the source of the error.
size_t dimension;
This is the dimension of the system of equations.
void * params
This is a pointer to the arbitrary parameters of the system.

## Stepping Functions

The lowest level components are the stepping functions which advance a solution from time t to t+h for a fixed step-size h and estimate the resulting local error.

Function: gsl_odeiv2_step * gsl_odeiv2_step_alloc (const gsl_odeiv2_step_type * T, size_t dim)
This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim dimensions. Please note that if you use a stepper method that requires access to a driver object, it is advisable to use a driver allocation method, which automatically allocates a stepper, too.

Function: int gsl_odeiv2_step_reset (gsl_odeiv2_step * s)
This function resets the stepping function s. It should be used whenever the next use of s will not be a continuation of a previous step.

Function: void gsl_odeiv2_step_free (gsl_odeiv2_step * s)
This function frees all the memory associated with the stepping function s.

Function: const char * gsl_odeiv2_step_name (const gsl_odeiv2_step * s)
This function returns a pointer to the name of the stepping function. For example,

printf ("step method is '%s'\n",
gsl_odeiv2_step_name (s));


would print something like step method is 'rkf45'.

Function: unsigned int gsl_odeiv2_step_order (const gsl_odeiv2_step * s)
This function returns the order of the stepping function on the previous step. The order can vary if the stepping function itself is adaptive.

Function: int gsl_odeiv2_step_set_driver (gsl_odeiv2_step * s, const gsl_odeiv2_driver * d)
This function sets a pointer of the driver object d for stepper s, to allow the stepper to access control (and evolve) object through the driver object. This is a requirement for some steppers, to get the desired error level for internal iteration of stepper. Allocation of a driver object calls this function automatically.

Function: int gsl_odeiv2_step_apply (gsl_odeiv2_step * s, double t, double h, double y[], double yerr[], const double dydt_in[], double dydt_out[], const gsl_odeiv2_system * sys)
This function applies the stepping function s to the system of equations defined by sys, using the step-size h to advance the system from time t and state y to time t+h. The new state of the system is stored in y on output, with an estimate of the absolute error in each component stored in yerr. If the argument dydt_in is not null it should point an array containing the derivatives for the system at time t on input. This is optional as the derivatives will be computed internally if they are not provided, but allows the reuse of existing derivative information. On output the new derivatives of the system at time t+h will be stored in dydt_out if it is not null.

The stepping function returns GSL_FAILURE if it is unable to compute the requested step. Also, if the user-supplied functions defined in the system sys return a status other than GSL_SUCCESS the step will be aborted. In that case, the elements of y will be restored to their pre-step values and the error code from the user-supplied function will be returned. Failure may be due to a singularity in the system or too large step-size h. In that case the step should be attempted again with a smaller step-size, e.g. h/2.

If the driver object is not appropriately set via gsl_odeiv2_step_set_driver for those steppers that need it, the stepping function returns GSL_EFAULT. If the user-supplied functions defined in the system sys returns GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user must call gsl_odeiv2_step_reset before calling this function again.

The following algorithms are available,

Step Type: gsl_odeiv2_step_rk2
Explicit embedded Runge-Kutta (2, 3) method.

Step Type: gsl_odeiv2_step_rk4
Explicit 4th order (classical) Runge-Kutta. Error estimation is carried out by the step doubling method. For more efficient estimate of the error, use the embedded methods described below.

Step Type: gsl_odeiv2_step_rkf45
Explicit embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.

Step Type: gsl_odeiv2_step_rkck
Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

Step Type: gsl_odeiv2_step_rk8pd
Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

Step Type: gsl_odeiv2_step_rk1imp
Implicit Gaussian first order Runge-Kutta. Also known as implicit Euler or backward Euler method. Error estimation is carried out by the step doubling method. This algorithm requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver.

Step Type: gsl_odeiv2_step_rk2imp
Implicit Gaussian second order Runge-Kutta. Also known as implicit mid-point rule. Error estimation is carried out by the step doubling method. This stepper requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver.

Step Type: gsl_odeiv2_step_rk4imp
Implicit Gaussian 4th order Runge-Kutta. Error estimation is carried out by the step doubling method. This algorithm requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver.

Step Type: gsl_odeiv2_step_bsimp
Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems. This stepper requires the Jacobian.

A variable-coefficient linear multistep Adams method in Nordsieck form. This stepper uses explicit Adams-Bashforth (predictor) and implicit Adams-Moulton (corrector) methods in P(EC)^m functional iteration mode. Method order varies dynamically between 1 and 12. This stepper requires the access to the driver object via gsl_odeiv2_step_set_driver.

Step Type: gsl_odeiv2_step_msbdf
A variable-coefficient linear multistep backward differentiation formula (BDF) method in Nordsieck form. This stepper uses the explicit BDF formula as predictor and implicit BDF formula as corrector. A modified Newton iteration method is used to solve the system of non-linear equations. Method order varies dynamically between 1 and 5. The method is generally suitable for stiff problems. This stepper requires the Jacobian and the access to the driver object via gsl_odeiv2_step_set_driver.

The control function examines the proposed change to the solution produced by a stepping function and attempts to determine the optimal step-size for a user-specified level of error.

Function: gsl_odeiv2_control * gsl_odeiv2_control_standard_new (double eps_abs, double eps_rel, double a_y, double a_dydt)
The standard control object is a four parameter heuristic based on absolute and relative errors eps_abs and eps_rel, and scaling factors a_y and a_dydt for the system state y(t) and derivatives y'(t) respectively.

The step-size adjustment procedure for this method begins by computing the desired error level D_i for each component,

D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)


and comparing it with the observed error E_i = |yerr_i|. If the observed error E exceeds the desired error level D by more than 10% for any component then the method reduces the step-size by an appropriate factor,

h_new = h_old * S * (E/D)^(-1/q)


where q is the consistency order of the method (e.g. q=4 for 4(5) embedded RK), and S is a safety factor of 0.9. The ratio E/D is taken to be the maximum of the ratios E_i/D_i.

If the observed error E is less than 50% of the desired error level D for the maximum ratio E_i/D_i then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level,

h_new = h_old * S * (E/D)^(-1/(q+1))


This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize, the overall scaling factor is limited to the range 1/5 to 5.

Function: gsl_odeiv2_control * gsl_odeiv2_control_y_new (double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the solution y_i(t). This is equivalent to the standard control object with a_y=1 and a_dydt=0.

Function: gsl_odeiv2_control * gsl_odeiv2_control_yp_new (double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the derivatives of the solution y'_i(t). This is equivalent to the standard control object with a_y=0 and a_dydt=1.

Function: gsl_odeiv2_control * gsl_odeiv2_control_scaled_new (double eps_abs, double eps_rel, double a_y, double a_dydt, const double scale_abs[], size_t dim)
This function creates a new control object which uses the same algorithm as gsl_odeiv2_control_standard_new but with an absolute error which is scaled for each component by the array scale_abs. The formula for D_i for this control object is,

D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)


where s_i is the i-th component of the array scale_abs. The same error control heuristic is used by the Matlab ODE suite.

Function: gsl_odeiv2_control * gsl_odeiv2_control_alloc (const gsl_odeiv2_control_type * T)
This function returns a pointer to a newly allocated instance of a control function of type T. This function is only needed for defining new types of control functions. For most purposes the standard control functions described above should be sufficient.

Function: int gsl_odeiv2_control_init (gsl_odeiv2_control * c, double eps_abs, double eps_rel, double a_y, double a_dydt)
This function initializes the control function c with the parameters eps_abs (absolute error), eps_rel (relative error), a_y (scaling factor for y) and a_dydt (scaling factor for derivatives).

Function: void gsl_odeiv2_control_free (gsl_odeiv2_control * c)
This function frees all the memory associated with the control function c.

Function: int gsl_odeiv2_control_hadjust (gsl_odeiv2_control * c, gsl_odeiv2_step * s, const double y[], const double yerr[], const double dydt[], double * h)
This function adjusts the step-size h using the control function c, and the current values of y, yerr and dydt. The stepping function step is also needed to determine the order of the method. If the error in the y-values yerr is found to be too large then the step-size h is reduced and the function returns GSL_ODEIV_HADJ_DEC. If the error is sufficiently small then h may be increased and GSL_ODEIV_HADJ_INC is returned. The function returns GSL_ODEIV_HADJ_NIL if the step-size is unchanged. The goal of the function is to estimate the largest step-size which satisfies the user-specified accuracy requirements for the current point.

Function: const char * gsl_odeiv2_control_name (const gsl_odeiv2_control * c)
This function returns a pointer to the name of the control function. For example,

printf ("control method is '%s'\n",
gsl_odeiv2_control_name (c));


would print something like control method is 'standard'

Function: int gsl_odeiv2_control_errlevel (gsl_odeiv2_control * c, const double y, const double dydt, const double h, const size_t ind, double * errlev)
This function calculates the desired error level of the ind-th component to errlev. It requires the value (y) and value of the derivative (dydt) of the component, and the current step size h.

Function: int gsl_odeiv2_control_set_driver (gsl_odeiv2_control * c, const gsl_odeiv2_driver * d)
This function sets a pointer of the driver object d for control object c.

## Evolution

The evolution function combines the results of a stepping function and control function to reliably advance the solution forward one step using an acceptable step-size.

Function: gsl_odeiv2_evolve * gsl_odeiv2_evolve_alloc (size_t dim)
This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.

Function: int gsl_odeiv2_evolve_apply (gsl_odeiv2_evolve * e, gsl_odeiv2_control * con, gsl_odeiv2_step * step, const gsl_odeiv2_system * sys, double * t, double t1, double * h, double y[])
This function advances the system (e, sys) from time t and position y using the stepping function step. The new time and position are stored in t and y on output.

The initial step-size is taken as h. The control function con is applied to check whether the local error estimated by the stepping function step using step-size h exceeds the required error tolerance. If the error is too high, the step is retried by calling step with a decreased step-size. This process is continued until an acceptable step-size is found. An estimate of the local error for the step can be obtained from the components of the array e->yerr[].

If the user-supplied functions defined in the system sys returns GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user must call gsl_odeiv2_step_reset and gsl_odeiv2_evolve_reset before calling this function again.

Otherwise, if the user-supplied functions defined in the system sys or the stepping function step return a status other than GSL_SUCCESS, the step is retried with a decreased step-size. If the step-size decreases below machine precision, a status of GSL_FAILURE is returned if the user functions returned GSL_SUCCESS. Otherwise the value returned by user function is returned. If no acceptable step can be made, t and y will be restored to their pre-step values and h contains the final attempted step-size.

If the step is successful the function returns a suggested step-size for the next step in h. The maximum time t1 is guaranteed not to be exceeded by the time-step. On the final time-step the value of t will be set to t1 exactly.

Function: int gsl_odeiv2_evolve_apply_fixed_step (gsl_odeiv2_evolve * e, gsl_odeiv2_control * con, gsl_odeiv2_step * step, const gsl_odeiv2_system * sys, double * t, const double h, double y[])
This function advances the ODE-system (e, sys, con) from time t and position y using the stepping function step by a specified step size h. If the local error estimated by the stepping function exceeds the desired error level, the step is not taken and the function returns GSL_FAILURE. Otherwise the value returned by user function is returned.

Function: int gsl_odeiv2_evolve_reset (gsl_odeiv2_evolve * e)
This function resets the evolution function e. It should be used whenever the next use of e will not be a continuation of a previous step.

Function: void gsl_odeiv2_evolve_free (gsl_odeiv2_evolve * e)
This function frees all the memory associated with the evolution function e.

Function: int gsl_odeiv2_evolve_set_driver (gsl_odeiv2_evolve * e, const gsl_odeiv2_driver * d)
This function sets a pointer of the driver object d for evolve object e.

If a system has discontinuous changes in the derivatives at known points, it is advisable to evolve the system between each discontinuity in sequence. For example, if a step-change in an external driving force occurs at times t_a, t_b and t_c then evolution should be carried out over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1) separately and not directly over the range (t_0,t_1).

## Driver

The driver object is a high level wrapper that combines the evolution, control and stepper objects for easy use.

Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_y_new (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const double hstart, const double epsabs, const double epsrel)
Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_yp_new (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const double hstart, const double epsabs, const double epsrel)
Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_standard_new (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const double hstart, const double epsabs, const double epsrel, const double a_y, const double a_dydt)
Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_scaled_new (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const double hstart, const double epsabs, const double epsrel, const double a_y, const double a_dydt, const double scale_abs[])
These functions return a pointer to a newly allocated instance of a driver object. The functions automatically allocate and initialise the evolve, control and stepper objects for ODE system sys using stepper type T. The initial step size is given in hstart. The rest of the arguments follow the syntax and semantics of the control functions with same name (gsl_odeiv2_control_*_new).

Function: int gsl_odeiv2_driver_set_hmin (gsl_odeiv2_driver * d, const double hmin)
The function sets a minimum for allowed step size hmin for driver d. Default value is 0.

Function: int gsl_odeiv2_driver_set_hmax (gsl_odeiv2_driver * d, const double hmax)
The function sets a maximum for allowed step size hmax for driver d. Default value is GSL_DBL_MAX.

Function: int gsl_odeiv2_driver_set_nmax (gsl_odeiv2_driver * d, const unsigned long int nmax)
The function sets a maximum for allowed number of steps nmax for driver d. Default value of 0 sets no limit for steps.

Function: int gsl_odeiv2_driver_apply (gsl_odeiv2_driver * d, double * t, const double t1, double y[])
This function evolves the driver system d from t to t1. Initially vector y should contain the values of dependent variables at point t. If the function is unable to complete the calculation, an error code from gsl_odeiv2_evolve_apply is returned, and t and y contain the values from last successful step.

If maximum number of steps is reached, a value of GSL_EMAXITER is returned. If the step size drops below minimum value, the function returns with GSL_ENOPROG. If the user-supplied functions defined in the system sys returns GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user must call gsl_odeiv2_driver_reset before calling this function again.

Function: int gsl_odeiv2_driver_apply_fixed_step (gsl_odeiv2_driver * d, double * t, const double h, const unsigned long int n, double y[])
This function evolves the driver system d from t with n steps of size h. If the function is unable to complete the calculation, an error code from gsl_odeiv2_evolve_apply_fixed_step is returned, and t and y contain the values from last successful step.

Function: int gsl_odeiv2_driver_reset (gsl_odeiv2_driver * d)
This function resets the evolution and stepper objects.

Function: int gsl_odeiv2_driver_reset_hstart (gsl_odeiv2_driver * d, const double hstart)
The routine resets the evolution and stepper objects and sets new initial step size to hstart. This function can be used e.g. to change the direction of integration.

Function: int gsl_odeiv2_driver_free (gsl_odeiv2_driver * d)
This function frees the driver object, and the related evolution, stepper and control objects.

## Examples

The following program solves the second-order nonlinear Van der Pol oscillator equation,

u"(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0


This can be converted into a first order system suitable for use with the routines described in this chapter by introducing a separate variable for the velocity, v = u'(t),

u' = v
v' = -u + \mu v (1-u^2)


The program begins by defining functions for these derivatives and their Jacobian. The main function uses driver level functions to solve the problem. The program evolves the solution from (u, v) = (1, 0) at t=0 to t=100. The step-size h is automatically adjusted by the controller to maintain an absolute accuracy of 10^{-6} in the function values (u, v). The loop in the example prints the solution at the points t_i = 1, 2, \dots, 100.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv2.h>

int
func (double t, const double y[], double f[],
void *params)
{
double mu = *(double *)params;
f[0] = y[1];
f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
return GSL_SUCCESS;
}

int
jac (double t, const double y[], double *dfdy,
double dfdt[], void *params)
{
double mu = *(double *)params;
gsl_matrix_view dfdy_mat
= gsl_matrix_view_array (dfdy, 2, 2);
gsl_matrix * m = &dfdy_mat.matrix;
gsl_matrix_set (m, 0, 0, 0.0);
gsl_matrix_set (m, 0, 1, 1.0);
gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
dfdt[0] = 0.0;
dfdt[1] = 0.0;
return GSL_SUCCESS;
}

int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};

gsl_odeiv2_driver * d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
1e-6, 1e-6, 0.0);
int i;
double t = 0.0, t1 = 100.0;
double y[2] = { 1.0, 0.0 };

for (i = 1; i <= 100; i++)
{
double ti = i * t1 / 100.0;
int status = gsl_odeiv2_driver_apply (d, &t, ti, y);

if (status != GSL_SUCCESS)
{
printf ("error, return value=%d\n", status);
break;
}

printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}

gsl_odeiv2_driver_free (d);
return 0;
}


The user can work with the lower level functions directly, as in the following example. In this case an intermediate result is printed after each successful step instead of equidistant time points.

int
main (void)
{
const gsl_odeiv2_step_type * T
= gsl_odeiv2_step_rk8pd;

gsl_odeiv2_step * s
= gsl_odeiv2_step_alloc (T, 2);
gsl_odeiv2_control * c
= gsl_odeiv2_control_y_new (1e-6, 0.0);
gsl_odeiv2_evolve * e
= gsl_odeiv2_evolve_alloc (2);

double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};

double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };

while (t < t1)
{
int status = gsl_odeiv2_evolve_apply (e, c, s,
&sys,
&t, t1,
&h, y);

if (status != GSL_SUCCESS)
break;

printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}

gsl_odeiv2_evolve_free (e);
gsl_odeiv2_control_free (c);
gsl_odeiv2_step_free (s);
return 0;
}


For functions with multiple parameters, the appropriate information can be passed in through the params argument in gsl_odeiv2_system definition (mu in this example) by using a pointer to a struct.

It is also possible to work with a non-adaptive integrator, using only the stepping function itself, gsl_odeiv2_driver_apply_fixed_step or gsl_odeiv2_evolve_apply_fixed_step. The following program uses the driver level function, with fourth-order Runge-Kutta stepping function with a fixed stepsize of 0.001.

int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = { func, jac, 2, &mu };

gsl_odeiv2_driver *d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
1e-3, 1e-8, 1e-8);

double t = 0.0;
double y[2] = { 1.0, 0.0 };
int i, s;

for (i = 0; i < 100; i++)
{
s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);

if (s != GSL_SUCCESS)
{
printf ("error: driver returned %d\n", s);
break;
}

printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}

gsl_odeiv2_driver_free (d);
return s;
}


• Ascher, U.M., Petzold, L.R., Computer Methods for Ordinary Differential and Differential-Algebraic Equations, SIAM, Philadelphia, 1998.
• Hairer, E., Norsett, S. P., Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, 1993.
• Hairer, E., Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996.

Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,

• Abramowitz & Stegun (eds.), Handbook of Mathematical Functions, Section 25.5.

The implicit Bulirsch-Stoer algorithm bsimp is described in the following paper,

• G. Bader and P. Deuflhard, "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations.", Numer. Math. 41, 373--398, 1983.

The Adams and BDF multistep methods msadams and msbdf are based on the following articles,

• G. D. Byrne and A. C. Hindmarsh, "A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations.", ACM Trans. Math. Software, 1, 71--96, 1975.
• P. N. Brown, G. D. Byrne and A. C. Hindmarsh, "VODE: A Variable-coefficient ODE Solver.", SIAM J. Sci. Stat. Comput. 10, 1038--1051, 1989.
• A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers.", ACM Trans. Math. Software 31, 363--396, 2005.