Reports and Investigations written in English

Algorithms and Programs

Generation of LaTeX Picture Environments (GaPFilL Method)



Algorithms and Programs

The "Sums of Powers Algorithm" (SPA) is the last one of six new central algorithms published in the book "Algorithmische Lineare Algebra" (Algorithmic Linear Algebra, Editors: Vieweg Verlag Wiesbaden, 1997, written in German). Here, a draft (Pdf file, 217 KB) was available for twelve years. In 2012, this paper was reworked on demand for a commemorative issue of a journal. It appeared in November 2012: "An Efficient Reliable Algorithm for the Approximation of All Polynomial Roots Based on the Method of D. Bernoulli", Mathematics and Informatics, 1, Dedicated to 75th Anniversary of Anatolii Alekseevich Karatsuba, Современные проблемы математики, 16, 52-65 (2012), Steklov Math. Inst., RAS, Moscow. In April 2013, a slightly corrected version with the same title was published in the Proceedings of the Steklov Institute of Mathematics, 2013, Vol. 280, Suppl 2, pp. S43-S55. The first stage of this algorithm is visualized by a Maple X program SPAview.mws (Maple X worksheet as zip file, 12 KB). An extensive documentation "Visualization of the First Stage of the SPA" (Pdf file, 312 KB) with an example gives some insight, even if Maple is not available.

The rootfinding program SPA (Maple-V5 text file, 32 KB), which belongs to the above mentioned algorithm and which was written in 2000, fulfills the well-known criteria of reliability and efficiency. Many of the underlying facts have been developed and tested by  D. GUNSTHÖVEL  who planned to write a dissertation in this field.

To open some files in this website, a password is needed which can be obtained by playing the Up-and-Down Game against the computer. This game and its explanation is contained in the Safe program (Java archive, 18 KB) which can be downloaded and then used independendly of a browser. To open it the first time, one has to right click (ctrl-click on Macintosh computers). Afterwards, it can be opened as usual. If the program belongs to the user directory, the command line opens it with: java -jar SafeE.jar.

An article on a first sufficient criterion of convergence for Laguerre's method of rootfinding was rejected by four journals specialized in this area. It is now available here with the title "Convergence and visualization of Laguerre’s rootfinding algorithm" (Pdf file, 1.2 MB).

As an application of the criterion, convergence properties of Laguerre's method can be visualized with the Cython program Laguerre.pyx (text file, 8 KB) in the CAS Sage. This program is commented in the Report on the Visualization of Laguerre's Method (Pdf file, 7.7 MB) which also contains the new result and 21 figures. It is complemented by Two Pdf-Figures for the Visualization of Laguerre's Method (Pdf file, 144 KB). Some of the figures were generated with the following modified Cython programs: Laguerre0.pyx (text file, 7 KB, simplified input and output), Laguerre1.pyx (text file, 6 KB, Laguerre iteration with the factor 1 in the denominator), Laguerre2.pyx (text file, 4 KB, computation of an optional number of iterations for a single starting value) and Laguerre3.pyx (text file, 7 KB, output of the figures as aethetically pleasing graphics without white or black elements).

Generation of LaTeX Picture Environments (GaPFilL Method)

"GaPFilL" (Graphics as PostScript Filtered for LaTeX) is a new method for the generation of LaTeX picture environments in a four step way:

i) Construction of the desired figure with a drawing program;

ii) Export as a PostScript file;

iii) Application of a Perl filter program to the PostScript text;

iv) Transfer of the resulting code into a LaTeX document.

The following two prototypes of filter programs are written with MacPerl 5.6 for the Macintosh version of Cabri GéomètreTMII and for the PostScript driver 'Virtual Printer'. The first program requires the package ebezier. The  second filter in addition supplies the package pict2e :

CABebez.pl (text file, 28 KB),

CABpict.pl (text file, 32 KB).

Since April 2006 these two Perl programs and an English documentation entitled "How to Generate LaTeX Picture Environments Using the GaPFilL Method" (Pdf-Programm, 607 KB) are available in the Internet archive "Comprehensive TeX Archive Network" (CTAN, http://dante.ctan.org/CTAN).

A new Perl filter program OOopict.pl (Text-File, 26 KB) is added for the powerful free drawing software OpenOffice.org 3 Draw which has its own PostScript generator not depending on the operating systems supplied by OpenOffice.org, namely Linux, Mac OS, Unix and Windows. The English documentation contained in the CTAN is entitled "How to Generate LaTeX Picture Environments Using OpenOffice.org 3 Draw" (Pdf file, 149 KB).

The complete actual folder of the CTAN can be downloaded as zip archive gapfill.zip (526 KB).

This Perl flter program can also be used with the free drawing programs Apache OpenOffice Draw and LibreOffice Draw which belong to the office suits succeeding OpenOffice.org.


The report "Elementary Analysis' and 'Algorithmic Linear Algebra" (Pdf file, 140 KB) was written for the ICMI (International Commission on Mathematical Instruction) Study 'On the Teaching and Learning of Mathematics at University Level' in November 1998. For reasons which we do not know the editors did not publish the text. It contains short descriptions of the most important projects carried out in the Heinrich-Behnke-Seminar für Didaktik der Mathematik.

A paper entitled Didactical Mathematics was submitted in September 2015 for the ICME-13 (International Congress on Mathematics Education 2016 in Hamburg). It contains short descriptions of all five hypertext books of the Pentatope Project. Since the accepted contribution can't be realized for health reasons, the paper is available here (Pdf file, 181 KB). In the references you find the links for downloading the books from this website.


A student who was enthusiastic about the above-mentioned book "Algorithmische Lineare Algebra" has voluntarily translated 180 of its 405 pages. Here is a scanned version of his script "Algorithmic Linear Algebra" (Pdf file, 1.5 MB).