**8:45 AM / Wednesday, February 14, 1996**
## Invited Presentation 5

Numerical Integration with Verification

Automatic differentiation shows its real strength in connection with interval arithmetic, in use already for several decades. Powerful programming environments (PASCAL-XSC, ACRITH-XSC, C-XSC) have been developed that allow easy use of both tools. Arithmetic for automatic differentiation and generation of Taylor coefficients including interval data are provided as part of the XSC-tools. Derivatives, Taylor-coefficients, gradients, Jacobian and Hessian matrices or enclosures of these can directly be computed out of the expression by a simple type change of the operands.
Verified integration of definite integrals (even with singularities) or differential equations are typical applications. Safe enclosures of the remainder term of integration formulas can be used to control the step size and to compute close bounds of the solution.

**Ulrich Kulisch**

Technical University of Karlsruhe, Germany

MEM, 12/29/95