The ore_algebra package provides an implementation of Ore algebras for Sage. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing; desingularization; solvers for polynomials, ...

# Dr. Fredrik Johansson

### Research Area

Computation of special functions, arbitrary-precision arithmetic## Software

## Publications

### 2014

### Ore Polynomials in Sage

#### Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson

In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho, Martin Weimann (ed.), Lecture Notes in Computer Science , pp. ?-?. 2014. tba. [pdf] [ps]@

author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

**inproceedings**{RISC4944,author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

### Fast and rigorous computation of special functions to high precision

#### F. Johansson

RISC. PhD Thesis. 2014. [pdf]@

author = {F. Johansson},

title = {{Fast and rigorous computation of special functions to high precision}},

language = {english},

year = {2014},

translation = {0},

school = {RISC},

length = {0}

}

**phdthesis**{RISC4972,author = {F. Johansson},

title = {{Fast and rigorous computation of special functions to high precision}},

language = {english},

year = {2014},

translation = {0},

school = {RISC},

length = {0}

}

### Using functional equations to enumerate 1324-avoiding permutations

#### Fredrik Johansson, Brian Nakamura

Advances in Applied Mathematics 56(0), pp. 20 - 34. 2014. ISSN 0196-8858. [url]@

author = {Fredrik Johansson and Brian Nakamura},

title = {{Using functional equations to enumerate 1324-avoiding permutations}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {56},

number = {0},

pages = {20 -- 34},

isbn_issn = {ISSN 0196-8858},

year = {2014},

refereed = {yes},

keywords = {Enumeration algorithm},

length = {15},

url = {http://www.sciencedirect.com/science/article/pii/S0196885814000256}

}

**article**{RISC4987,author = {Fredrik Johansson and Brian Nakamura},

title = {{Using functional equations to enumerate 1324-avoiding permutations}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {56},

number = {0},

pages = {20 -- 34},

isbn_issn = {ISSN 0196-8858},

year = {2014},

refereed = {yes},

keywords = {Enumeration algorithm},

length = {15},

url = {http://www.sciencedirect.com/science/article/pii/S0196885814000256}

}

### 2013

### Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

#### Fredrik Johansson, Manuel Kauers, Marc Mezzarobba

In: Proceedings of ISSAC'13, Manuel Kauers (ed.), pp. 211-218. 2013. isbn 978-1-4503-2059-7/13/06. [pdf] [ps]@

author = {Fredrik Johansson and Manuel Kauers and Marc Mezzarobba},

title = {{Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions. },

pages = {211--218},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4707,author = {Fredrik Johansson and Manuel Kauers and Marc Mezzarobba},

title = {{Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions. },

pages = {211--218},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

### 2012

### Efficient implementation of the Hardy-Ramanujan-Rademacher formula

#### Fredrik Johansson

LMS Journal of Computation and Mathematics(15), pp. 341-359. 2012. ISSN 1461-1570. [url] [pdf]@

author = {Fredrik Johansson},

title = {{Efficient implementation of the Hardy-Ramanujan-Rademacher formula}},

language = {english},

abstract = {We describe how the Hardy–Ramanujan–Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy–Ramanujan–Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver’s tabulation of 76 065 congruences. Supplementary materials are available with this article.},

journal = {LMS Journal of Computation and Mathematics},

number = {15},

pages = {341--359},

isbn_issn = {ISSN 1461-1570},

year = {2012},

refereed = {yes},

length = {19},

url = {http://journals.cambridge.org/action/displayAbstract?aid=8710297}

}

**article**{RISC4597,author = {Fredrik Johansson},

title = {{Efficient implementation of the Hardy-Ramanujan-Rademacher formula}},

language = {english},

abstract = {We describe how the Hardy–Ramanujan–Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy–Ramanujan–Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver’s tabulation of 76 065 congruences. Supplementary materials are available with this article.},

journal = {LMS Journal of Computation and Mathematics},

number = {15},

pages = {341--359},

isbn_issn = {ISSN 1461-1570},

year = {2012},

refereed = {yes},

length = {19},

url = {http://journals.cambridge.org/action/displayAbstract?aid=8710297}

}