Overview

We now move on to MATLAB's capabilities for evaluating integrals and solving ordinary and partial differential equations.

Most of the functions discussed in this chapter support mixed absolute/relative error tests, with tolerances AbsTol and RelTol, respectively. This means that they test whether an estimate err of some measure of the error in the vector x is small enough by testing whether, for all i,

  err(i) <= max(AbsTol,RelTol*abs(x(i)))

If AbsTol is zero this is a pure relative error test and if RelTol is zero it is a pure absolute error test. Since we cannot expect to obtain an answer with more correct significant digits than the 16 or so to which MATLAB works, RelTol should be no smaller than about eps; and since x = 0 is a possibility we should also take AbsTol = 0. A rough way of interpreting the mixed error test above is that err(i) is acceptably small if x(i) has as many correct digits as specified by RelTol or is smaller than AbsTol in absolute value. AbsTol can be a vector of absolute tolerances, in which case the test is

  err(i) <= max(AbsTol(i),RelTol*abs(x(i)))