This page gives mathematical details behind the double exponential transformation technique for numerical integration. Software for implementing this method is available in the article Fast Numerical Integration.

The trapezoid rule is a crude method of numerical integration often presented in calculus classes. The error in approximating integrals by the trapezoid rule is typically too large for the method to be practical in application. However, in some special cases, this crude method is astonishingly accurate. In particular, the trapezoid rule works remarkably well for integrating analytic functions over the real line that approach zero rapidly in the tails (i.e.* f*(*x*) goes to zero like exp(-*c* exp |*x*|) as |*x*| → ∞). The double exponential integration method takes advantage the excellent performance in this case by transforming other integrals to have this form.

## Transformation from finite interval to real line

There have been several transformations studied for transforming a function over the finite open interval (a, b) into a function over the real line (-∞, ∞). We use the transformation

*x* = tanh( π sinh(*t*)/2 )

in the software referenced above to map *x* values in (-1, 1) into t values in (-∞, ∞). The functions hyperbolic tangent and hyperbolic sine functions are not evaluated at run time; they enter the software via pre-calculated node and weight values.

For information on other transformations from finite intervals to the real line in the double exponential integration method, see [1]. Also, [2] has more information about the specific transformation *x* = tanh( π sinh(*t*)/2 ).

## Error estimates

The error in the double exponential integration method is on the order of

exp( – *c* *N*/log(*N*) )

where *N* is the number of integration points. This is *optimal* in the sense that no integration method achieves better accuracy for the same number of function evaluations for integrands belonging to the Hardy space *H ^{p}* for

*p*> 1.

Note that while elementary numerical integration schemes have error estimates that decrease like a *polynomial* in *N*, i.e. *N*^{–k}, the method presented here has error estimates that decrease *exponentially*.

## References

[1] Masatake Mori and Masaaki Sugihara. The double-exponential transformation in numerical analysis. Journal of Computational and Applied Mathematics, 127 (2001), 287–296.

[2] Mayinur Muhammad and Masatake Mori. Double exponential formulas for numerical indefinite integration. Journal of Computational and Applied Mathematics, 161 (2003) 431–448.

[3] Masatake Mori. Quadrature formulas obtained by variable transformation and the DE-rule, Journal of Computational and Applied Mathematics 12 & 13 (1985) 119–130.

[4] Hidetosi Takahasi and Masatake Mori. Double Exponential Formulas for Numerical Integration. Publ. RIMS, Kyoto University 9 (1974) 721–741.