Computing Jacobi's $\theta$ in quasi-linear time
Identifieur interne : 000235 ( Main/Exploration ); précédent : 000234; suivant : 000236Computing Jacobi's $\theta$ in quasi-linear time
Auteurs : Hugo Labrande [France]Source :
English descriptors
Abstract
Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $\theta$.
Url:
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Hal, to step Corpus: 001771
- to stream Hal, to step Curation: 001771
- to stream Hal, to step Checkpoint: 000207
- to stream Main, to step Merge: 000235
- to stream Main, to step Curation: 000235
Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Computing Jacobi's $\theta$ in quasi-linear time</title>
<author><name sortKey="Labrande, Hugo" sort="Labrande, Hugo" uniqKey="Labrande H" first="Hugo" last="Labrande">Hugo Labrande</name>
<affiliation wicri:level="1"><hal:affiliation type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno>
<idno type="IdUnivLorraine">[UL]100--</idno>
<orgName>Université de Lorraine</orgName>
<orgName type="acronym">UL</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-lorraine.fr/</ref>
</desc>
</hal:affiliation>
<country>France</country>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">HAL</idno>
<idno type="RBID">Hal:hal-01227699</idno>
<idno type="halId">hal-01227699</idno>
<idno type="halUri">https://hal.inria.fr/hal-01227699</idno>
<idno type="url">https://hal.inria.fr/hal-01227699</idno>
<date when="2015-11-13">2015-11-13</date>
<idno type="wicri:Area/Hal/Corpus">001771</idno>
<idno type="wicri:Area/Hal/Curation">001771</idno>
<idno type="wicri:Area/Hal/Checkpoint">000207</idno>
<idno type="wicri:explorRef" wicri:stream="Hal" wicri:step="Checkpoint">000207</idno>
<idno type="wicri:Area/Main/Merge">000235</idno>
<idno type="wicri:Area/Main/Curation">000235</idno>
<idno type="wicri:Area/Main/Exploration">000235</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">Computing Jacobi's $\theta$ in quasi-linear time</title>
<author><name sortKey="Labrande, Hugo" sort="Labrande, Hugo" uniqKey="Labrande H" first="Hugo" last="Labrande">Hugo Labrande</name>
<affiliation wicri:level="1"><hal:affiliation type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno>
<idno type="IdUnivLorraine">[UL]100--</idno>
<orgName>Université de Lorraine</orgName>
<orgName type="acronym">UL</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-lorraine.fr/</ref>
</desc>
</hal:affiliation>
<country>France</country>
</affiliation>
</author>
</analytic>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>Arithmetico-geometric mean</term>
<term>Number theory</term>
<term>Quasi-linear time complexity</term>
<term>Theta functions</term>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $\theta$.</div>
</front>
</TEI>
<affiliations><list><country><li>France</li>
</country>
</list>
<tree><country name="France"><noRegion><name sortKey="Labrande, Hugo" sort="Labrande, Hugo" uniqKey="Labrande H" first="Hugo" last="Labrande">Hugo Labrande</name>
</noRegion>
</country>
</tree>
</affiliations>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000235 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 000235 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Lorraine |area= InforLorV4 |flux= Main |étape= Exploration |type= RBID |clé= Hal:hal-01227699 |texte= Computing Jacobi's $\theta$ in quasi-linear time }}
This area was generated with Dilib version V0.6.33. |