Step size of ode45 and reltol abstol

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The stepsize of ode45 is not constant, but chosen such that the error tolerances you prescribe are met. Thus mean step size can only be controlled indirectly by strengthening (-> smaller stepsize) or weakening (-> larger stepsize) the error tolerances RelTol and AbsTol.

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RelTol Is the relative accuracy tolerance , Used to control the correct number of digits in the calculated answer .AbsTol Is the absolute error tolerance , Used to control the difference between the calculated answer and the actual solution . In each step , Solution component i The error in e Will satisfy

|e(i)| ≤ max(RelTol*abs(y(i)),AbsTol(i))

Roughly speaking , This means that you want RelTol Correct the number of digits in all solution components （ But less than the threshold AbsTol(i) Except for the solution component of ）. When the weight y(i) Very hour , Even if you are not interested in this weight , You must still specify a small enough AbsTol(i) value , In order to obtain y(i) Some correct digits in , So as to accurately calculate the components you are more interested in .

At each step, the solver estimates the local error e in the ith component of the solution. This error must be less than or equal to the acceptable error, which is a function of the specified relative tolerance, RelTol, and the specified absolute tolerance, AbsTol.
|e(i)| ≤ max(RelTol*abs(y(i)),AbsTol(i))

For relative accuracy, adjust RelTol. For the absolute error tolerance, the scaling of the solution components is important: if |y| is somewhat smaller than AbsTol, the solver is not constrained to obtain any correct digits in y. You might have to solve a problem more than once to discover the scale of solution components.

Roughly speaking, this means that you want RelTol correct digits in all solution components except those smaller than thresholds AbsTol(i). Even if you are not interested in a component y(i) when it is small, you may have to specify AbsTol(i) small enough to get some correct digits in y(i) so that you can accurately compute more interesting components.