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It seems that interval arithmetic can be used to quantify floating point truncation error in computational calculations. Does anyone know if it is possible to use interval arithmetic in a finite difference scheme, to automatically quantify the error of the scheme?

For example, using interval arithmetic, given some complicated f(x), and given a finite precision interval which contains the infinite precision value of x, I can get an exact representation (in finite precision) of an interval which contains f(x). This will allow me to know the impact of the truncation/rounding errors that have occurred during the calculation.

Now, if f(N,x) is an N-point approximation to f(x), can I somehow use interval arithmetic to determine a good interval [f1, f2] which will contain f(x) when computed using the approximation f(N,x)?

In particular, I'm wondering if I can get an error bound of f(x) when it is the solution to a PDE which is solved using a finite difference scheme.

While I have seen this alluded to, I have not seen a good description or reference.

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Indeed, interval analysis not only accounts for rounding errors by turning all numbers into intervals but also brings into numerical analysis a number of interval-based methods based on calculating function enclosures. For example, the last chapter of [1] describes "various procedures for enclosing the solution to an ordinary differential equation (ODE)" by enclosing the integration of the known RHS function in the ODE.

A different problem is to approximate an unknown function given its values at N points. If the function is at least known to be in a certain smoothness class, the approximation method could provide the error (absolute difference) bounds.

[1] Tucker Warwick, "Validated Numerics: A Short Introduction to Rigorous Computations", 2011

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  • $\begingroup$ The Warwick book is an excellent reference..thank you! $\endgroup$ – David H Mar 24 '14 at 16:58

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