## Selecting Absolute Tolerance and Relative Tolerance for Simulation

SimBiology® uses `AbsoluteTolerance` and `RelativeTolerance` to control the accuracy of integration during simulation. Specifically, `AbsoluteTolerance` is used to control the largest allowable absolute error at any step during simulation. It controls the error when a solution is small. Intuitively, when the solution approaches 0, `AbsoluteTolerance` is the threshold below which you do not worry about the accuracy of the solution since it is effectively 0. `RelativeTolerance` controls the relative error of a single step of the integrator. Intuitively, it controls the number of significant digits in a solution, except when it is smaller than the absolute tolerance, and $-{\mathrm{log}}_{10}\left(RelativeTolerance\right)$ is the number of correct digits.

### Algorithm

At each simulation step `i`, the solver estimates the local error `e` in the state `j` of the simulation. The solver reduces the size of time step `i` until the error of the state satisfies:

$|e\left(i,j\right)|\le \mathrm{max}\left(RelativeTolerance\ast |y\left(i,j\right)|,AbsoluteTolerance\left(i,j\right)\right)$

Thus at state values of larger magnitude, the accuracy is determined by `RelativeTolerance`. As the state values approach zero, the accuracy is controlled by `AbsoluteTolerance`.

The correct choice of values for `RelativeTolerance` and `AbsoluteTolerance` varies depending on the problem. The default values may work for first trials of the simulation. As you adjust the tolerances, consider that there are trade-offs between speed and accuracy:

• If the simulation takes too long, you can increase (or loosen) the values of `RelativeTolerance` and `AbsoluteTolerance` at the cost of some accuracy.

• If the results seem inaccurate, you can decrease (or tighten) the relative tolerance values by dividing with 10N, where N is a real positive number. But this tends to slow down the solver.

• If the magnitude of the state values is high, you can decrease the relative tolerance to get more accurate results.

### Absolute Tolerance Scaling

How SimBiology uses `AbsoluteTolerance` to determine the error depends on whether the `AbsoluteToleranceScaling` property is enabled. By default, `AbsoluteToleranceScaling` is enabled which means each state has its own absolute tolerance that may increase over the course of simulation:

$AbsoluteTolerance\left(i,j\right)=CSAbsTol*Scale\left(i,j\right)$

`CSAbsTol` is the `AbsoluteTolerance` property defined in `SolverOptions` of the active configuration set object.

For a state that has a nonzero initial value, the scale is the maximum magnitude over the state, as seen over the simulation thus far:

`$Scale\left(i,j\right)=\mathrm{max}\left(|y\left(1:i,j\right)|\right)$`

For a state that has an initial value of zero, the scale is estimated as the state value after taking a trial step of size `AbsoluteToleranceStepSize` using the Euler method. Let us call this value `ye(j)`. Then:

$Scale\left(i,j\right)=\mathrm{max}\left(|\left[ye\left(j\right);y\left(2:i,j\right)\right]|\right)$

If an initial state is zero and has no dynamic at time = 0, then:

$AbsoluteTolerance\left(i,j\right)=CSAbsTol$

Doses, events, and initial assignment rules at simulation time = 0 are not considered when calculating absolute tolerance scaling.

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