### 26.1 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.

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