{"id":210662,"created":"2025-01-19T01:11:52.110471+00:00","metadata":{"_oai":{"id":"oai:ipsj.ixsq.nii.ac.jp:00210662","sets":["581:10433:10437"]},"path":["10437"],"owner":"44499","recid":"210662","title":["xが小さい場合のクンマー関数U(a,b,x)の数値計算"],"pubdate":{"attribute_name":"公開日","attribute_value":"2021-04-15"},"_buckets":{"deposit":"0b589cc9-d7a2-4571-9758-88315f4fd769"},"_deposit":{"id":"210662","pid":{"type":"depid","value":"210662","revision_id":0},"owners":[44499],"status":"published","created_by":44499},"item_title":"xが小さい場合のクンマー関数U(a,b,x)の数値計算","author_link":["533864","533863","533865","533868","533867","533866"],"item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"xが小さい場合のクンマー関数U(a,b,x)の数値計算"},{"subitem_title":"Computation of Kummer Function U(a,b,x) for Small Argument x","subitem_title_language":"en"}]},"item_keyword":{"attribute_name":"キーワード","attribute_value_mlt":[{"subitem_subject":"[一般論文] クンマー関数，合流型超幾何関数，ガンマ関数の逆数のテイラー展開","subitem_subject_scheme":"Other"}]},"item_type_id":"2","publish_date":"2021-04-15","item_2_text_3":{"attribute_name":"著者所属","attribute_value_mlt":[{"subitem_text_value":"中部大学"},{"subitem_text_value":"中部大学"},{"subitem_text_value":"中部大学"}]},"item_2_text_4":{"attribute_name":"著者所属(英)","attribute_value_mlt":[{"subitem_text_value":"Chubu University","subitem_text_language":"en"},{"subitem_text_value":"Chubu University","subitem_text_language":"en"},{"subitem_text_value":"Chubu University","subitem_text_language":"en"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"jpn"}]},"publish_status":"0","weko_shared_id":-1,"item_file_price":{"attribute_name":"Billing file","attribute_type":"file","attribute_value_mlt":[{"url":{"url":"https://ipsj.ixsq.nii.ac.jp/record/210662/files/IPSJ-JNL6204010.pdf","label":"IPSJ-JNL6204010.pdf"},"date":[{"dateType":"Available","dateValue":"2023-04-15"}],"format":"application/pdf","billing":["billing_file"],"filename":"IPSJ-JNL6204010.pdf","filesize":[{"value":"525.2 kB"}],"mimetype":"application/pdf","priceinfo":[{"tax":["include_tax"],"price":"660","billingrole":"5"},{"tax":["include_tax"],"price":"330","billingrole":"6"},{"tax":["include_tax"],"price":"0","billingrole":"8"},{"tax":["include_tax"],"price":"0","billingrole":"44"}],"accessrole":"open_date","version_id":"9d67213f-d14b-45b1-b03a-a882c68a362d","displaytype":"detail","licensetype":"license_note","license_note":"Copyright (c) 2021 by the Information Processing Society of Japan"}]},"item_2_creator_5":{"attribute_name":"著者名","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"吉田, 年雄"}],"nameIdentifiers":[{}]},{"creatorNames":[{"creatorName":"奥居, 哲"}],"nameIdentifiers":[{}]},{"creatorNames":[{"creatorName":"足達, 義則"}],"nameIdentifiers":[{}]}]},"item_2_creator_6":{"attribute_name":"著者名(英)","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"Toshio, Yoshida","creatorNameLang":"en"}],"nameIdentifiers":[{}]},{"creatorNames":[{"creatorName":"Satoshi, Okui","creatorNameLang":"en"}],"nameIdentifiers":[{}]},{"creatorNames":[{"creatorName":"Yoshinori, Adachi","creatorNameLang":"en"}],"nameIdentifiers":[{}]}]},"item_2_source_id_9":{"attribute_name":"書誌レコードID","attribute_value_mlt":[{"subitem_source_identifier":"AN00116647","subitem_source_identifier_type":"NCID"}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourceuri":"http://purl.org/coar/resource_type/c_6501","resourcetype":"journal article"}]},"item_2_source_id_11":{"attribute_name":"ISSN","attribute_value_mlt":[{"subitem_source_identifier":"1882-7764","subitem_source_identifier_type":"ISSN"}]},"item_2_description_7":{"attribute_name":"論文抄録","attribute_value_mlt":[{"subitem_description":"$a>0$，$b\\ge 0$で，$x$（$>0$）が小さい場合のクンマー関数$U(a,b,x)$の新しい数値計算法を述べている．$U(a,b,x)$は$U(a,b,x)=(\\pi/\\sin \\pi b)\\{M(a,b,x)/(\\Gamma(a+1-b)\\Gamma(b))-x^{1-b}M(a+1-b,2-b,x)/(\\Gamma(a)\\Gamma(2-b))\\}$で定義される．上式の$\\{\\ \\}$の前半部と後半部の減算では，$b$が整数に近くないときには，桁落ちは起こらないので，そのまま計算すればよいが，$b$が整数に近いときには，桁落ちが起こり，工夫が必要である．たとえば，$0\\le b\\le 0.5$の場合には，上式の$\\{\\ \\}$の部分は$b/(\\Gamma(1+b)\\Gamma(a+1-b))$+$\\sum_{k=1}^{\\infty}\\{A_{k}(a,b)-B_{k}(a,b)\\}x^{k}$の形で表わすことができる．ここで，$A_{k}(a,b)$は2つの項の差で表され，特に$A_{1}(a,b)=a/(\\Gamma(1+b)\\Gamma(a+1-b))-1/(\\Gamma(2-b)\\Gamma(a))$である．この式は，$b$が0に近いときには桁落ちが生ずるが，ガンマ関数の逆数$1/\\Gamma(1+b)$，$1/\\Gamma(2-b)$，$1/\\Gamma(a+1-b)$の$b=0$周りのテイラー展開を用いれば，桁落ちを回避することができる．そのため，本論文では，プログラム実行時に，$1/\\Gamma(1+b)$や$1/\\Gamma(2-b)$，さらに，変数$a$を含む$1/\\Gamma(a+1-b)$の場合でも，必要な次数までの微分係数を能率的に求められる方法を提案している．$A_{k}(a,b)\\ (k\\ge 2)$は$A_{k-1}(a,b)$から漸化式を用いて計算できる．このようにして，桁落ちを回避することにより$U(a,b,x)$を求めている．","subitem_description_type":"Other"}]},"item_2_description_8":{"attribute_name":"論文抄録(英)","attribute_value_mlt":[{"subitem_description":"Numerical method for computation of Kummer functions $U(a,b,x)$ with small argument $x$ ($>0$) $a>0$, $b\\ge 0$）is proposed. The function $U(a,b,x)$ is defined as $U(a,b,x)=(\\pi/\\sin \\pi b)\\{M(a,b,x)/(\\Gamma(a+1-b)\\Gamma(b))-x^{1-b}M(a+1-b,2-b,x)/(\\Gamma(a)\\Gamma(2-b))\\}$. When $b$ is not near to an integer, the cancellation of significant digits does not occur by subtraction of these two parts and so we can compute the above expression as it is. However, when $b$ is near to an integer, such cancellation occurs. For example, in case of $0\\le b\\le 0.5$, the part $\\{\\ \\}$ of the above equation can be expressed as $b/(\\Gamma(b+1)\\Gamma(a+1-b))$+$\\sum_{k=1}^{\\infty}\\{A_{k}(a,b)-B_{k}(a,b)\\}x^{k}$, where $A_{k}(a,b)$ is a subtraction of two terms, especially $A_{1}(a,b)=a/(\\Gamma(1+b)\\Gamma(a+1-b))-1/(\\Gamma(2-b)\\Gamma(a))$. When $b$ is near to 0, the cancellation of significant digits occurs in this equation. By using Taylor series about $b=0$ for reciprocal gamma functions, $1/\\Gamma(1+b)$, $1/\\Gamma(2-b)$ and $1/\\Gamma(a+1-b)$, such cancellation can be removed. In this paper, we describe an efficient method for the runtime computation of derivatives as much as necessary in case of $1/\\Gamma(1+b)$, $1/\\Gamma(2-b)$ and $1/\\Gamma(a+1-b)$ which contains a variable $a$. By the recurrence, $A_{k}(a,b)\\ (k\\ge 2)$ is computed from $A_{k-1}(a,b)$. Removing the cancellation of significant digits in this way, we can compute $U(a,b,x)$.","subitem_description_type":"Other"}]},"item_2_biblio_info_10":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicPageEnd":"1064","bibliographic_titles":[{"bibliographic_title":"情報処理学会論文誌"}],"bibliographicPageStart":"1056","bibliographicIssueDates":{"bibliographicIssueDate":"2021-04-15","bibliographicIssueDateType":"Issued"},"bibliographicIssueNumber":"4","bibliographicVolumeNumber":"62"}]},"relation_version_is_last":true,"item_2_identifier_registration":{"attribute_name":"ID登録","attribute_value_mlt":[{"subitem_identifier_reg_text":"10.20729/00210557","subitem_identifier_reg_type":"JaLC"}]},"weko_creator_id":"44499"},"updated":"2025-01-19T17:58:22.808445+00:00","links":{}}