**Sektion 6**

Freitag, 22.09.2000, 17.00–17.20 Uhr, WIL C 129

#### Uwe Schnabel, TU Dresden, Institut für Numerische Mathematik

#### Gerd Pönisch, TU Dresden, Institut
für Numerische Mathematik

A point (x^{*}, ^{*}, ^{*}) is called a winged cusp point of the nonlinear system F (x, , ) = 0 , F : ^{n} × ^{1} × ^{3} ^{n},
if

and if the Ljapunov-Schmidt reduced function has the normal form g(, ) = ±^{3} ± ^{2}, g : ^{1} × ^{1} ^{1}. A
minimally extended system F (x, , ) = 0, f(x, , ) = 0 is proposed for defining winged cusp points,
where f : ^{n} × ^{1} × ^{3} ^{4} are 4 scalar functions characterizing the winged cusp. The functions
f_{i} : ^{n} × ^{1} × ^{3} ^{1}, i = 1, ..., 4, depend on the partial derivatives _{
}g, _{}g, _{}^{2}g, and _{}_{}g. For
the description of the f_{i} an evaluable type of Ljapunov-Schmidt reduced function is introduced. The
regular solution (x^{*}, ^{*}, ^{*}) of the (n + 4)-dimensional extended system delivers the desired winged cusp
point (x^{*}, ^{*}) as first part. For numerically solving these systems, a two-stage Newton-type method are
proposed. Computational differentiation is applied for computing the several partial derivatives needed. A
numerical example is given.

The talk is based on the papers:

[1] Schnabel, U., Pönisch, G., Janovský, V. Reduced functions characterizing singular points
and their relations. Preprint IOKOMO–04–1999, TU Dresden, 1999.

[2] Schnabel, U., Walther, A. Berechnung singulärer Punkte: Newtonverfahren und
Automatische Differentiation. Preprint IOKOMO–05–1999, TU Dresden, 1999.