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.
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.
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.
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.
This function creates a new control object which uses the same algorithm
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.
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.
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).
This function frees all the memory associated with the control function c.
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
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.
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'
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.
This function sets a pointer of the driver object d for control object c.