John Ernsthausen identified and implemented examples for the numerical differential-algebraic
equation (DAE) integration methods developed in Rheinbold’s package *DAEPAK*. As Werner C. Rheinbold’s
and Patrick J. Rabier’s graduate student, John assisted in the documentation and debugging
of the *FORTRAN77* subroutines in *DAEPAK*. The experience required a deep understanding of the
numerical differential geometry package *MANPAK*, a collection of numerical algorithms implementing
computational differential geometry for submanifolds of .

### Computational differential geometry

A subset of is a manifold whenever a chart exists near each point in , roughly speaking. For example, a road map near your current location on Earth is a chart from flat into the surface of Earth, a 2-dimensional submanifold of .

More precisely, whenever each point in a subset of
can be diffeomorphically mapped into a (non-empty) subset of in
, with , and the
smoothness of the diffeomorphism is , then
is a -dimensional
submanifold of .
The diffeomorphism is a *local parameterization*, and the inverse of a local parameterization is a *chart*.

Differential geometry enables calculus on a (nonempty) submanifold of . Given any point , there is a local parameterization near . All local parameterizations near are an equivalent class, and the particular local parameterization is a representative from it. Locally near , a DAE can be reduced to an ordinary differential equations on a manifold. In this point of view, the path constructed by projecting any solution to the DAE through the local parameterization is a solution to the equivalent ordinary differential equation on a manifold, and the unique solution of that local ordinary differential equation on a manifold lifted through the representative local parameterization must be the unique solution of the DAE.

Ordinary differential equations describe dynamics in . Ordinary differential equations on a manifold describe dynamics restricted to the manifold. The solutions to ordinary differential equations on a manifold must live in the manifold.

Differential geometry is discussed in the language of submersions, immersions, and
diffeomorphisms for the purpose of constructing a coordinate subspace, local parameterization, and tangent space
near each point in the submanifold embedded into the ambient space .
These concepts are discussed in the *MANPAK* article [R1996] and
the Rabier and Rheinboldt book [RR2002].

While all these topics are required to understand the mathematics behind computational differential geometry,
a deep understanding of these topics is not required to apply the results.
Local parameterizations and their derivatives for general submanifolds of
are constructed in the software package *MANPAK*. *DAEPAK* is built on the software package *MANPAK*.

### LITERATURE

- R1996
- Rheinboldt W.C.: MANPAK: A set of algorithms for computations on implicitly defined manifolds. Computers & Mathematics with Applications, 32(12), 15-28 (1996). [link].
- RR2002
- Rabier P.J. and Rheinboldt W.C.:
Theoretical and numerical analysis of differential-algebraic equations.
In: P.G. Ciarlet, J.L. Lions (eds.) Solution of Equations in (Part 4),
Techniques of Scientific Computing (Part 4),
*Handbook of Numerical Analysis*, vol 8, pp. 183-540. Elsevier, Amsterdam (2002) [link].