Preface to the Third Edition
A new edition of a text presents not only an opportunity for corrections
and minor changes but also for adding new material. Thus we strived to
improve the presentation of Hermite interpolation and B-splines in Chap-
ter 2, and we added a new Section 2.4.6 on multi-resolution methods and
B-splines, using, in particular, low order B-splines for purposes of illustra-
tion. The intent is to draw attention to the role of B-splines in this area,
and to familiarize the reader with, at least, the principles of multi-resolution
methods, which are fundamental to modern applications in signal- and im-
The chapter on di
erential equations was enlarged, too: A new Section
7.2.18 describes solving di
erential equations in the presence of disconti-
nuities whose locations are not known at the outset. Such discontinuities
occur, for instance, in optimal control problems where the character of a
erential equation is a
ected by control function changes in response to
Many applications, such as parameter identification, lead to di
equations which depend on additional parameters. Users then would like
to know how sensitively the solution reacts to small changes in these pa-
Techniques for such sensitivity analyses are the subject of the
new Section 7.2.19.
Multiple shooting methods are among the most powerful for solving
boundary value problems for ordinary di
erential equations. We dedicated,
therefore, a new Section 7.3.8 to new advanced techniques in multiple shoot-
ing, which especially enhance the e
ciency of these methods when applied
to solve boundary value problems with discontinuities, which are typical
for optimal contol problems.
Among the many iterative methods for solving large sparse linear equa-
tions, Krylov space methods keep growing in importance.
treated these methods in Section 8.7 more systematically by adding new
subsections dealing with the GMRES method (Section 8.7.2), the biorthog-
onalization method of Lanczos and the (principles of the) QMR method
(Section 8.7.3), and the Bi-CG and Bi-CGSTAB algorithms (Section 8.7.4).
Correspondingly, the final Section 8.10 on the comparison of iterative meth-
ods was updated in order to incorporate the findings for all Krylov space
methods described before.
The authors are greatly indebted to the many who have contributed