The problem of the solution of a given se t of lin ear equations Ax = b on a high-speed digital computer has been studied intensively, and there are a large number of method s, more or less sati sfactory , for carrying out such a solution. Nevertheless occasions arise when existing methods are inadequate, either because the solutions are required exactly, or because the coefficie nt matrix A is "ill-conditioned." A notorious exam ple of the latter is furnished by the Hilbert matrices A = H" given by
{"title":"Solving equations exactly","authors":"M. Newman","doi":"10.6028/JRES.071B.023","DOIUrl":"https://doi.org/10.6028/JRES.071B.023","url":null,"abstract":"The problem of the solution of a given se t of lin ear equations Ax = b on a high-speed digital computer has been studied intensively, and there are a large number of method s, more or less sati sfactory , for carrying out such a solution. Nevertheless occasions arise when existing methods are inadequate, either because the solutions are required exactly, or because the coefficie nt matrix A is \"ill-conditioned.\" A notorious exam ple of the latter is furnished by the Hilbert matrices A = H\" given by","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"726 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132380510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper gives a theore m on combinations of segs in a finite, connec ted , undirected graph. The n the theore m is specialized to combinations of cut sets, giving a theore m first pro ven by Ma yeda. The paper contains an example showing that modifiers added to Ma yeda' s theore m by Yau , in the Journal of th e Franklin Institute, Janu ary 1962, yield a fal se theorem. Finally, the pa pe r discusses the prac· ticability of algorithms de veloped by Mayeda and Yau and based on Mayeda's theore m.
{"title":"Remarks on Cut-Sets","authors":"J. Grossman","doi":"10.6028/JRES.071B.025","DOIUrl":"https://doi.org/10.6028/JRES.071B.025","url":null,"abstract":"This paper gives a theore m on combinations of segs in a finite, connec ted , undirected graph. The n the theore m is specialized to combinations of cut sets, giving a theore m first pro ven by Ma yeda. The paper contains an example showing that modifiers added to Ma yeda' s theore m by Yau , in the Journal of th e Franklin Institute, Janu ary 1962, yield a fal se theorem. Finally, the pa pe r discusses the prac· ticability of algorithms de veloped by Mayeda and Yau and based on Mayeda's theore m.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"28 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120813286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Matrices of class J2","authors":"J. Maybee","doi":"10.6028/JRES.071B.029","DOIUrl":"https://doi.org/10.6028/JRES.071B.029","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121846535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A generalized matrix version of Rennie's inequality","authors":"J. Hearon","doi":"10.6028/JRES.071B.011","DOIUrl":"https://doi.org/10.6028/JRES.071B.011","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125551813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A ne w al~orilhm is ~ive n fur cumpulin ~ Ihe soluliun of a ny secu nd-orde r lin ea r diffe re nce e qua lion which is a ppli cable when s impl e rec urrence proce dures ca nnol be used becau se uf in sla bilil Y. Co mpare d wilh Ih e we ll -knuwn lV1ill e r a l ~o rilhm Ih e ne w m elhod has Ih e advanla~es of (i) a Ul umal ic ally deLermining the currec t number of rec urrence steps. (ii ) app lying to illllOlllo~en eou s differe nce eqlla ~ li uns, (iii ) e nab lin g mure powerful e rrur a na lyses lu be co ns lru c led. The me lhud is illuSlra led by num e rica l comp ulalion s, ineludin~ e rror ana lyses. uf A n~e rWe be r . S iruve, a nd Besse l fun c li uns, and Ih e s u luli un of a differe nli a l e qualiun in C he bys he v se ri es.
ne w al ~ orilhm是~我n毛皮cumpulin ~国内soluliun纽约secu nd-orde林ea r而再保险指标e作为狮子ppli电缆当s impl e rec urrence过程接受的ca nnol因为佛罗里达大学在猩猩使用sla bilil y试管d wilh Ih e我们会-knuwn lV1ill e r l ~ o rilhm Ih e ne w m elhod Ih e advanla ~ es(我)一个Ul umal ic盟友deLermining currec t rec urrence步骤的数量。(2)应用程序可以在不同的应用程序中使用不同的应用程序;(3)应用程序可以使用更强大的应用程序;(3)应用程序可以在不同的应用程序中使用不同的应用程序。这一领域的研究主要集中在多个领域，包括数据分析和数据分析。我们是r。我和贝斯都很喜欢我的作品，而我的作品中有一种不同的风格，我的作品中有一种独特的品质，我的作品中有一种独特的品质。
{"title":"Numerical solution of second-order linear difference equations","authors":"F. Olver","doi":"10.6028/JRES.071B.018","DOIUrl":"https://doi.org/10.6028/JRES.071B.018","url":null,"abstract":"A ne w al~orilhm is ~ive n fur cumpulin ~ Ihe soluliun of a ny secu nd-orde r lin ea r diffe re nce e qua lion which is a ppli cable when s impl e rec urrence proce dures ca nnol be used becau se uf in sla bilil Y. Co mpare d wilh Ih e we ll -knuwn lV1ill e r a l ~o rilhm Ih e ne w m elhod has Ih e advanla~es of (i) a Ul umal ic ally deLermining the currec t number of rec urrence steps. (ii ) app lying to illllOlllo~en eou s differe nce eqlla ~ li uns, (iii ) e nab lin g mure powerful e rrur a na lyses lu be co ns lru c led. The me lhud is illuSlra led by num e rica l comp ulalion s, ineludin~ e rror ana lyses. uf A n~e rWe be r . S iruve, a nd Besse l fun c li uns, and Ih e s u luli un of a differe nli a l e qualiun in C he bys he v se ri es.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123651449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the hypothesis (H) that X admits a me tric p (yielding th e co rrect topo logy) s uch that I is a p-co ntrac tion. In a prev ious paper [1] 1 we gave co nditi ons on X s uch that f, if for some me tri c it sati sfied (1.1) locally at all points of X, would in fac t obey (H). No te that s uc h result s have a hypothes is whi c h is metric in nature. The prese nt paper is concern ed with topological conditions for (H), co nditi ons whose state me nts do not refer to any particular me tri c or me tri cs on X. In view of the Banach Contraction Theore m [2], two conditions which naturally s uggest the mselves are that , for some ~EX,
考虑假设(H) X允许一个矩阵p(产生正确的拓扑)，使得I是一个p-co -简并。在之前的一篇论文[1]1中，我们给出了关于X s的几个条件，使得f，如果对于某些metric，它在X的所有点局部满足(1.1)，那么它实际上会服从(H)。不，这个结果有一个假设，即H本质上是度量的。本文研究(H)的拓扑条件，这些条件的状态不指向任何特定的me - tri c或x上的me - tri c。根据巴拿赫收缩定理[2]，有两个条件自然表明，对于某些~EX，
{"title":"A Converse to Banach/s Contraction Theorem","authors":"P. R. Meyers","doi":"10.6028/JRES.071B.014","DOIUrl":"https://doi.org/10.6028/JRES.071B.014","url":null,"abstract":"Consider the hypothesis (H) that X admits a me tric p (yielding th e co rrect topo logy) s uch that I is a p-co ntrac tion. In a prev ious paper [1] 1 we gave co nditi ons on X s uch that f, if for some me tri c it sati sfied (1.1) locally at all points of X, would in fac t obey (H). No te that s uc h result s have a hypothes is whi c h is metric in nature. The prese nt paper is concern ed with topological conditions for (H), co nditi ons whose state me nts do not refer to any particular me tri c or me tri cs on X. In view of the Banach Contraction Theore m [2], two conditions which naturally s uggest the mselves are that , for some ~EX,","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126562420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that if A is a bounded linear operator with closed range on a Hilbe rt space then A can be fac tored as A = UH, with U a partial isometry and H nonnegative and self adjoint. For the finite dimensional case a s tri ctly matrix-theoretic derivation is given based on the concept of a ge neralized inverse. Certain properti es of the factors are give n as well as conditions under whic h H or both U and H are uniquely de termined by A. A pivotal ite m in the derivation is the representation of a square partial isometry a s the produc t of a unitary matrix and a n orthogonal projection. Thi s representa tion is new, of some int e rest in itse lf and greatl y s impli fies the de rivations.
{"title":"Polar Factorization of a Matrix","authors":"J. Hearon","doi":"10.6028/JRES.071B.012","DOIUrl":"https://doi.org/10.6028/JRES.071B.012","url":null,"abstract":"It is known that if A is a bounded linear operator with closed range on a Hilbe rt space then A can be fac tored as A = UH, with U a partial isometry and H nonnegative and self adjoint. For the finite dimensional case a s tri ctly matrix-theoretic derivation is given based on the concept of a ge neralized inverse. Certain properti es of the factors are give n as well as conditions under whic h H or both U and H are uniquely de termined by A. A pivotal ite m in the derivation is the representation of a square partial isometry a s the produc t of a unitary matrix and a n orthogonal projection. Thi s representa tion is new, of some int e rest in itse lf and greatl y s impli fies the de rivations.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"12 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116126093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this note , which is expository, is to present proofs of two fundamental theorems on sets of commuting matrices. The theorems are classical, but existing proofs tend to be unnecessarily complicated and furthermore are diffi c ult to find in the literature. For these reasons the following simple proofs (along the lines set down by Frobenius and Schur in their original memoirs on group r epresentations) are of interest. In fac t the basic tools are the concept of irreducibility and a simplified form of Schur's lemma, both from the theory of group representations. In addition some special information concerning normal matrices will be required , which we summarize briefly below. (Complete proofs may be found in MacDuffee's book.)' For simplicity, all the matrices considered below are over the complex field. A matrix A is normal if it commutes with A *, the conjugate transpose of A. A matrix V is unitary if V* = VI. Unitary matrices are themselves normal matrices, the unitary matrices form a group, and V* A V is normal if and only if A is normal. It is known that A is normal if and only if A = V* DV, where V is unitary and D diagonal.
这篇说明性的笔记的目的是给出关于交换矩阵集合的两个基本定理的证明。这些定理是经典的，但现有的证明往往是不必要的复杂，而且很难在文献中找到。由于这些原因，下面的简单证明(沿着Frobenius和Schur在他们关于群表示的原始回忆录中所提出的思路)是有趣的。事实上，基本工具是不可约的概念和舒尔引理的简化形式，两者都来自群表示理论。此外，还需要一些关于正态矩阵的特殊信息，我们将在下面简要地总结一下。(完整的证明可以在MacDuffee的书中找到)为了简单起见，下面考虑的所有矩阵都在复域上。矩阵A是正规的，如果它与A *， A的共轭转置交换，矩阵V是酉矩阵，如果V* = VI。酉矩阵本身是正规矩阵，这些酉矩阵组成一个群，并且V* A V是正规的当且仅当A是正规的。已知A是正规的当且仅当A = V* DV，其中V是酉的，D是对角的。
{"title":"Two classical theorems on commuting matrices","authors":"M. Newman","doi":"10.6028/JRES.071B.013","DOIUrl":"https://doi.org/10.6028/JRES.071B.013","url":null,"abstract":"The purpose of this note , which is expository, is to present proofs of two fundamental theorems on sets of commuting matrices. The theorems are classical, but existing proofs tend to be unnecessarily complicated and furthermore are diffi c ult to find in the literature. For these reasons the following simple proofs (along the lines set down by Frobenius and Schur in their original memoirs on group r epresentations) are of interest. In fac t the basic tools are the concept of irreducibility and a simplified form of Schur's lemma, both from the theory of group representations. In addition some special information concerning normal matrices will be required , which we summarize briefly below. (Complete proofs may be found in MacDuffee's book.)' For simplicity, all the matrices considered below are over the complex field. A matrix A is normal if it commutes with A *, the conjugate transpose of A. A matrix V is unitary if V* = VI. Unitary matrices are themselves normal matrices, the unitary matrices form a group, and V* A V is normal if and only if A is normal. It is known that A is normal if and only if A = V* DV, where V is unitary and D diagonal.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125142428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of s tudying the growth of the error is most important for the numerical so lution of. differential equations. In thi s paper the Wilrs crite rion is generaJized to be applied for sys tems of diffe re ntial eq uations. A general th eore m is inves tigated and regions of s tability have to be determined. The use of an electronic computer is more essential for s uch region s to be c haracterized. These regions of s tability have the property that , the error introduced at any stage te nds to decay. The regions of stabi lity for partic ular numerical methods are exp licitl y determined.
{"title":"Criterion for the stability of numerical integration methods for the solution of systems of differential equations","authors":"A. Karim","doi":"10.6028/JRES.071B.016","DOIUrl":"https://doi.org/10.6028/JRES.071B.016","url":null,"abstract":"The problem of s tudying the growth of the error is most important for the numerical so lution of. differential equations. In thi s paper the Wilrs crite rion is generaJized to be applied for sys tems of diffe re ntial eq uations. A general th eore m is inves tigated and regions of s tability have to be determined. The use of an electronic computer is more essential for s uch region s to be c haracterized. These regions of s tability have the property that , the error introduced at any stage te nds to decay. The regions of stabi lity for partic ular numerical methods are exp licitl y determined.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130140082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Expressions are derived for the indefinite integrals, J,!( r ) Co(ar )dr Irf(r ) q (ar )elr I ,!(r) C,,(ar ) Co(f3r)elr a "" f3 y,!(r) Co(ar ) Z,,(Ar )dr where Co(ar) are zero order Bessel fun c tions, ZoO...,.) are zero orde r modified Bessel fun ction s and f( r ) is a polynomial in r.In general , the expressions given for the integrals are given in te rms of prescribed fun c tions of the Bessel fun c tions, and th e coefficients of these func tions are determined from a finite series, the te rm s of which are found from recurre nce relationships that involve only the polynomial fIr) _ Coeffic ients of the term s of the fin ite series are given in tabular form for up to an eleventh degree polynomial.
为不定积分，J，!(r)有限公司(ar) Irf博士(r) q (ar) elr我! C (r), (ar)有限公司(f3r) elr”“f3 y ! (r)有限公司(ar) Z (ar)博士有限公司(ar)是零阶贝塞尔有趣C,动物园…,)为零奥德r修正贝塞尔有趣引发反应和f (r)是r.In一般的多项式,给出的表达式规定积分给出了te rms的有趣设计C的贝塞尔有趣C,并这些函数的系数确定从有限级数,其中的项s是从只涉及多项式的递归关系中得到的。对于最高为11次多项式的有限级数，项s的系数以表格形式给出。
{"title":"Indefinite integrals involving Bessel functions","authors":"B. Peavy","doi":"10.6028/JRES.071B.019","DOIUrl":"https://doi.org/10.6028/JRES.071B.019","url":null,"abstract":"Expressions are derived for the indefinite integrals, J,!( r ) Co(ar )dr Irf(r ) q (ar )elr I ,!(r) C,,(ar ) Co(f3r)elr a \"\" f3 y,!(r) Co(ar ) Z,,(Ar )dr where Co(ar) are zero order Bessel fun c tions, ZoO...,.) are zero orde r modified Bessel fun ction s and f( r ) is a polynomial in r.In general , the expressions given for the integrals are given in te rms of prescribed fun c tions of the Bessel fun c tions, and th e coefficients of these func tions are determined from a finite series, the te rm s of which are found from recurre nce relationships that involve only the polynomial fIr) _ Coeffic ients of the term s of the fin ite series are given in tabular form for up to an eleventh degree polynomial.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116719256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：