RTOL_—_ATOL_—_MAXSTEP_ODE_error

RTOL — ATOL — MAXSTEP ODE error controls

The default values for ATOL and RTOL are 0.000001, while the default value of MAXSTEP is 50000.

Matrix Exponent solver [3]

For the matrix exponent solver, only an absolute tolerance (ATOL) applies. If ATOL is set to zero, it is reset internally to the square root of the machine precision (approximately 1.e-8).

LSODA/LSODE solver [4]

The estimated local error for each component, Y(i), is controlled to be less (in magnitude) than RTOL*abs(Y(i)) + ATOL. Thus, the local error test passes if, in each component, either the absolute error is less than ATOL or the relative error is less than RTOL. If RTOL is set to 0.0, then only ATOL is applied by the solver (pure absolute error control). Currently, the use of pure relative error control (that is, ATOL is set to 0.0) is not permitted for the LSODE/LSODA solver and can lead to errors. Note that both tolerances are local controls on the current integration interval, and the global error may exceed these values.

For LSODE and LSODA, MAXSTEP is the maximum number of steps allowed during one call to the solver and ‘MAXSTEP = 0’ means that, at most, 5,000 steps will be used.

DVERK (non-stiff) solver [7]

For the DVERK solver, only relative tolerance (RTOL) applies and serves as a global tolerance term (that is, the ATOL value is ignored): the solver attempts to control a norm of the local error in such a way that the global error is proportional to RTOL. Note that a global tolerance cannot be set to zero.

For this solver, MAXSTEP represents the maximum number of function evaluations allowed during one call to the solver, ‘MAXSTEP = 0’ means that there is no limit in the number of function evaluations.

DOPRI5 (non-stiff) solver [8]

The code keeps the local error of Y(i) below RTOL*abs(Y(i)) + ATOL. Thus, the local error test passes if, in each component, either the absolute error is less than ATOL or the relative error is less than RTOL. If RTOL is set to 0.0, then only ATOL is applied by the solver (pure absolute error control). Similarly, if ATOL is set to 0.0, only RTOL is applied by the solver.

For DOPRI5, MAXSTEP denotes the maximum number of steps allowed during one call to the solver and ‘MAXSTEP = 0’ means that, at most, 100,000 steps will be used.