Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case
Identifieur interne : 000F70 ( Hal/Corpus ); précédent : 000F69; suivant : 000F71Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case
Auteurs : Y.-F. S. Pétermann ; Jean-Luc RémySource :
English descriptors
Abstract
1. Let $s$ be a real number. We prove that, if $s\ge1/2$, $s\not=1$ and $s$ can be written with $D_s$ bits in base 2, then in order to compute $\zeta(s)$ in any relative precision $P\ge11$, that is, in order to compute a $P-$bit number $\zeta_P(s)$such that $|\zeta_P(s)-\zeta(s)|$ is certified to be smaller than the number $ulp(\zeta_P(s))$ represented by a ``1'' at the $P-$th (and last) significant bit-place of $|\zeta_P(s)|$, it is sufficient to perform all the computations (i.e. additions, subtractions, multiplications, divisions, and computation of $k^s$ for integers $k\ge2$) with an internal precision $$ D= \max\left(D_s,P+\max\left(14, \left\lceil\frac3\logP2\log2+2.71\right\rceil\right)\right) $$ (all this contributing an error less than $ulp(\zeta_P(s)/2$), and then to round to the nearest $P-$bit number. For instance if the wanted precision is $P=1000$ (and if $s$ has no more than 1018 significant bits), then an internal precision $D=1018$ is sufficient. 2. Let $s=\sigma+it$ be a complex non real number. Assume $\sigma\ge1/2$ and $t>0$. First we address the problem of exploiting an error relative to modulus in order to estimate the relative errors of each of the real and imaginary parts of the computed $\zeta(s)^*$. Determining regions of the complex plane where these parts cannot vanish could help.Then we establish an easily computable upper bound for a crucial quantity in the error analysis (for the error relative to modulus), subject to the truth of an open conjecture of Brent on the size of the error committed while computing the Bernoulli numbers; we note that the upper bound one can obtain without this conjecture can become so large that even for certain ``reasonable'' value of $s$ it is of no practical use.
Url:
Links to Exploration step
Hal:inria-00070174Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case</title>
<author><name sortKey="Petermann, Y F S" sort="Petermann, Y F S" uniqKey="Petermann Y" first="Y.-F. S." last="Pétermann">Y.-F. S. Pétermann</name>
</author>
<author><name sortKey="Remy, Jean Luc" sort="Remy, Jean Luc" uniqKey="Remy J" first="Jean-Luc" last="Rémy">Jean-Luc Rémy</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-2364" status="OLD"><idno type="RNSR">200221401H</idno>
<orgName>Solving problems through algebraic computation and efficient software</orgName>
<orgName type="acronym">SPACES</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/spaces</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">HAL</idno>
<idno type="RBID">Hal:inria-00070174</idno>
<idno type="halId">inria-00070174</idno>
<idno type="halUri">https://hal.inria.fr/inria-00070174</idno>
<idno type="url">https://hal.inria.fr/inria-00070174</idno>
<date when="2006">2006</date>
<idno type="wicri:Area/Hal/Corpus">000F70</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case</title>
<author><name sortKey="Petermann, Y F S" sort="Petermann, Y F S" uniqKey="Petermann Y" first="Y.-F. S." last="Pétermann">Y.-F. S. Pétermann</name>
</author>
<author><name sortKey="Remy, Jean Luc" sort="Remy, Jean Luc" uniqKey="Remy J" first="Jean-Luc" last="Rémy">Jean-Luc Rémy</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-2364" status="OLD"><idno type="RNSR">200221401H</idno>
<orgName>Solving problems through algebraic computation and efficient software</orgName>
<orgName type="acronym">SPACES</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/spaces</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
</analytic>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>ARBITRARY PRECISION</term>
<term>CERTIFIED PRECISION</term>
<term>ERROR ANALYSIS</term>
<term>RIEMANN ZETA-FUNCTION</term>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">1. Let $s$ be a real number. We prove that, if $s\ge1/2$, $s\not=1$ and $s$ can be written with $D_s$ bits in base 2, then in order to compute $\zeta(s)$ in any relative precision $P\ge11$, that is, in order to compute a $P-$bit number $\zeta_P(s)$such that $|\zeta_P(s)-\zeta(s)|$ is certified to be smaller than the number $ulp(\zeta_P(s))$ represented by a ``1'' at the $P-$th (and last) significant bit-place of $|\zeta_P(s)|$, it is sufficient to perform all the computations (i.e. additions, subtractions, multiplications, divisions, and computation of $k^s$ for integers $k\ge2$) with an internal precision $$ D= \max\left(D_s,P+\max\left(14, \left\lceil\frac3\logP2\log2+2.71\right\rceil\right)\right) $$ (all this contributing an error less than $ulp(\zeta_P(s)/2$), and then to round to the nearest $P-$bit number. For instance if the wanted precision is $P=1000$ (and if $s$ has no more than 1018 significant bits), then an internal precision $D=1018$ is sufficient. 2. Let $s=\sigma+it$ be a complex non real number. Assume $\sigma\ge1/2$ and $t>0$. First we address the problem of exploiting an error relative to modulus in order to estimate the relative errors of each of the real and imaginary parts of the computed $\zeta(s)^*$. Determining regions of the complex plane where these parts cannot vanish could help.Then we establish an easily computable upper bound for a crucial quantity in the error analysis (for the error relative to modulus), subject to the truth of an open conjecture of Brent on the size of the error committed while computing the Bernoulli numbers; we note that the upper bound one can obtain without this conjecture can become so large that even for certain ``reasonable'' value of $s$ it is of no practical use.</div>
</front>
</TEI>
<hal api="V3"><titleStmt><title xml:lang="en">Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case</title>
<author role="aut"><persName><forename type="first">Y.-F.S.</forename>
<surname>Pétermann</surname>
</persName>
<email></email>
<idno type="halauthor">97818</idno>
</author>
<author role="aut"><persName><forename type="first">Jean-Luc</forename>
<surname>Rémy</surname>
</persName>
<email></email>
<idno type="halauthor">61771</idno>
<affiliation ref="#struct-2364"></affiliation>
</author>
<editor role="depositor"><persName><forename>Rapport De Recherche</forename>
<surname>Inria</surname>
</persName>
<email>rrrt-editeurs@inria.fr</email>
</editor>
</titleStmt>
<editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2006-05-19 19:19:38</date>
<date type="whenWritten">2006-02</date>
<date type="whenModified">2016-05-19 01:09:23</date>
<date type="whenReleased">2006-05-31 14:24:23</date>
<date type="whenProduced">2006</date>
<date type="whenEndEmbargoed">2006-05-19</date>
<ref type="file" target="https://hal.inria.fr/inria-00070174/document"><date notBefore="2006-05-19"></date>
</ref>
<ref type="file" n="1" target="https://hal.inria.fr/inria-00070174/file/RR-5852.pdf"><date notBefore="2006-05-19"></date>
</ref>
</edition>
<respStmt><resp>contributor</resp>
<name key="111068"><persName><forename>Rapport De Recherche</forename>
<surname>Inria</surname>
</persName>
<email>rrrt-editeurs@inria.fr</email>
</name>
</respStmt>
</editionStmt>
<publicationStmt><distributor>CCSD</distributor>
<idno type="halId">inria-00070174</idno>
<idno type="halUri">https://hal.inria.fr/inria-00070174</idno>
<idno type="halBibtex">petermann:inria-00070174</idno>
<idno type="halRefHtml">[Research Report] RR-5852, INRIA. 2006, pp.31</idno>
<idno type="halRef">[Research Report] RR-5852, INRIA. 2006, pp.31</idno>
</publicationStmt>
<seriesStmt><idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
<idno type="stamp" n="INRIA-RRRT">Rapports de recherche et Technique de l'Inria</idno>
<idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
<idno type="stamp" n="INPL">Institut National Polytechnique de Lorraine</idno>
<idno type="stamp" n="LORIA2">Publications du LORIA</idno>
<idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno>
<idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno>
<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno>
</seriesStmt>
<notesStmt><note type="audience" n="0">Not set</note>
<note type="report" n="6">Research Report</note>
</notesStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case</title>
<author role="aut"><persName><forename type="first">Y.-F.S.</forename>
<surname>Pétermann</surname>
</persName>
<idno type="halAuthorId">97818</idno>
</author>
<author role="aut"><persName><forename type="first">Jean-Luc</forename>
<surname>Rémy</surname>
</persName>
<idno type="halAuthorId">61771</idno>
<affiliation ref="#struct-2364"></affiliation>
</author>
</analytic>
<monogr><idno type="reportNumber">RR-5852</idno>
<imprint><biblScope unit="pp">31</biblScope>
<date type="datePub">2006</date>
</imprint>
<authority type="institution">INRIA</authority>
</monogr>
</biblStruct>
</sourceDesc>
<profileDesc><langUsage><language ident="en">English</language>
</langUsage>
<textClass><keywords scheme="author"><term xml:lang="en">CERTIFIED PRECISION</term>
<term xml:lang="en">RIEMANN ZETA-FUNCTION</term>
<term xml:lang="en">ERROR ANALYSIS</term>
<term xml:lang="en">ARBITRARY PRECISION</term>
</keywords>
<classCode scheme="halDomain" n="info.info-oh">Computer Science [cs]/Other [cs.OH]</classCode>
<classCode scheme="halTypology" n="REPORT">Reports</classCode>
</textClass>
<abstract xml:lang="en">1. Let $s$ be a real number. We prove that, if $s\ge1/2$, $s\not=1$ and $s$ can be written with $D_s$ bits in base 2, then in order to compute $\zeta(s)$ in any relative precision $P\ge11$, that is, in order to compute a $P-$bit number $\zeta_P(s)$such that $|\zeta_P(s)-\zeta(s)|$ is certified to be smaller than the number $ulp(\zeta_P(s))$ represented by a ``1'' at the $P-$th (and last) significant bit-place of $|\zeta_P(s)|$, it is sufficient to perform all the computations (i.e. additions, subtractions, multiplications, divisions, and computation of $k^s$ for integers $k\ge2$) with an internal precision $$ D= \max\left(D_s,P+\max\left(14, \left\lceil\frac3\logP2\log2+2.71\right\rceil\right)\right) $$ (all this contributing an error less than $ulp(\zeta_P(s)/2$), and then to round to the nearest $P-$bit number. For instance if the wanted precision is $P=1000$ (and if $s$ has no more than 1018 significant bits), then an internal precision $D=1018$ is sufficient. 2. Let $s=\sigma+it$ be a complex non real number. Assume $\sigma\ge1/2$ and $t>0$. First we address the problem of exploiting an error relative to modulus in order to estimate the relative errors of each of the real and imaginary parts of the computed $\zeta(s)^*$. Determining regions of the complex plane where these parts cannot vanish could help.Then we establish an easily computable upper bound for a crucial quantity in the error analysis (for the error relative to modulus), subject to the truth of an open conjecture of Brent on the size of the error committed while computing the Bernoulli numbers; we note that the upper bound one can obtain without this conjecture can become so large that even for certain ``reasonable'' value of $s$ it is of no practical use.</abstract>
</profileDesc>
</hal>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Hal/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000F70 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Hal/Corpus/biblio.hfd -nk 000F70 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Lorraine |area= InforLorV4 |flux= Hal |étape= Corpus |type= RBID |clé= Hal:inria-00070174 |texte= Arbitrary Precision Error Analysis for computing $\zeta(s)$ with the Cohen-Olivier algorithm: Complete description of the real case and preliminary report on the general case }}
![]() | This area was generated with Dilib version V0.6.33. | ![]() |