, 590 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 524 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 460 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 296 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 247 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 196 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 151 min read
Beware, this three-dimensional graphic needs some time to load. You can rotate the graphic around any axis. This is in continuation of: Stability Regions for BDF and Tendler's Formulas Stability Regions for Tischer's Formulas Stability Regions for eTendler Formulas Below is the output…
, 58 min read
In Searching for Tendler-like formulas we developed new cyclic linear multistep formulas for the numerical solution of ordinary differential equations. These cyclic formulas are similar to Tendler's formulas. Order by order they improve on Widlund wedge angle and Widlund distance. In Die…
, 23 min read
1. Einleitung 1.1 Problemstellung und gesellschaftliche Relevanz 1.2 Zielsetzung und Aufbau der Arbeit 1.3 Methodisches Vorgehen 1.4 Thematische Abgrenzungen 2. Theoretischer Rahmen und Grundannahmen 2.1 Begriff und Zielsetzung des Einsatzes von KI in der Verwaltung 2.2. Digitale Infrastrukturen…
, 71 min read
1. Dahlquist's test equation 2. Logarithmic error in double precision 3. Logarithmic error in single precision and other machines 4. Scripts to generate results 1. Dahlquist's test equation We tested the BDF, Tendler's formulas, new Tendler-like formulas, and Tischer's formulas on the classical…
, 12 min read
1. Baseline 2. Specific new formulas 3. Searching across grids 4. Various new formulas 5. stabregion2.c 1. Baseline The formulas from Tendler from 1973 are our baseline. Clearly, we want to improve them. So here we summarize their characteristics: p is the order l is the cycle length α is the…
, 2 min read
Problem statement: Loop over a variable number of loops. I.e., we want to search a parameter space and therefore want to loop over multiple loops. The number of loops is variable. The initial approach goes like this: for (p1=1; p1<=10; ++p1) for (p2=1; p2<=10; ++p2) for (p3=1;…
, 8 min read
1. Tischer's formulas All cyclic linear multistep methods were designed to only have root at 1, and all other parasitic roots to be zero. See Tischer, Peter E. and Sacks-Davis, Ron: “A New Class of Cyclic Multistep Formulae for Stiff Systems”. 2. Donelson & Hansen formulas See Donelson III,…
, 8 min read
The parasitic roots $\lambda_i$ of a multistep method are the roots, which are not 1. In below table the root 1 is indexed with 0. The roots are for the matrix polynomial $$ \rho(\lambda) = A_m \lambda^m + A_{m-1} \lambda^{m-1} + \cdots + A_1 \lambda + A_0 $$ for the multistep method $$ …
, 62 min read
1. Zusammenhang zwischen Picard und Newton Iteration 2. Newton Iteration im Programm TENDLER 3. Die Picard Iteration im Programm TENDLER 4. Der Konvergenztest im Programm TENDLER 5. Verhinderung langsamer Konvergenz in TENDLER 6. Der Hindmarsh-Test im Programm TENDLER 7. Der Konvergenztest im…
, 26 min read
1. Einbau neuer zyklischer Formeln in das Programm TENDLER 2. Alternativen zum Prädiktor 3. Verwebung von Prädiktor und Korrektor 4. Speicherung des Prädiktors im Programm TENDLER Donald A. Calahan (1972), S.236, schreibt hierzu: Without prediction $\ldots$ typically 3–5 corrector…
, 13 min read
1. Definition der Differentialgleichung 2. Lösung der Differentialgleichung 3. Extras Das Programm TENDLER dient zur näherungsweisen Lösung von gewöhnlichen Differentialgleichungssystemen erster Ordnung und zwar von Anfangswertproblemen. Differentialgleichungen höherer Ordnung lassen sich…
, 25 min read
1. Die Sprache C 2. Objektorientierte Programmierung 3. Evolution von Differentialgleichungslösern 4. Statische und dynamische Organisation des Programmes TENDLER 5. Programmrealisierungen von Mehrschrittformeln 6. Speicherplatzaufwand bei Mehrschrittformeln In the late 1960's we witnessed a…
, 21 min read
1. Strategien und Rechnergenauigkeit 2. STINT vers. GEAR und EPISODE 3. STINT versus LSODE 4. Rationalisierte Schrittweiten und dessen Einfluß Es gibt eine Fülle von Differentialgleichungslösern, sowohl für steife, als auch für nicht-steife Gleichungen. Allerdings sind nicht alle Löser auch…
, 22 min read
1. Pädiktor-Korrektor-Verfahren und lineare Differenzengleichungen 2. Newton Iteration und Prädiktor-Korrektor-Verfahren 3. Gegenüberstellung von $\beta_\kappa/\alpha_\kappa$ für 3 Verfahren 4. Anzahl der Newton Iterationen und Konsistenzordnung Implizite lineare Mehrschrittverfahren…