[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83961-en":3,"doc-seo-83961-105":30,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83961,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","The mmatrix toolbox componentwise accurate algorithms for M-matrices with triplet representation","The mmatrix toolbox is a Matlab software package for componentwise accurate computations with M-matrices using left or right triplet representations. Its core is a Fortran Lapack-style implementation of unblocked, recursive, and blocked variants of the GTH algorithm, enabling accurate solution of linear systems, LU factorization, and inversion for M-matrix coefficients with nonnegative right-hand sides. Subtraction-free methods avoid cancellation, ensuring high componentwise accuracy for ill-conditioned problems. The toolbox also supports Schur complements, singular values, M-matrix square roots, and nonsymmetric algebraic Riccati equations, via an object-oriented Matlab interface.","arXiv :2607 .05437v 1 [ cs .MS] 3 Jul 2026  \nThe mmatrix toolbox: componentwise accurate algorithms for M-matrices with triplet representation ∗  \nBruno Iannazzo† Mehdi Najafi Kalyani‡ Federico Poloni‡  \nJuly 8, 2026  \nAbstract  \nWe introduce the mmatrix toolbox, a Matlab software package for componentwise accurate computations with M-matrices described through left or right triplet representations. The core of the toolbox is a Fortran implementation, in the Lapack style, of the unblocked, recursive, and blocked versions of the GTH algorithm and its applications for the accurate computation of the solution of linear systems with M-matrix coefficient and nonnegative right-hand side, the LU factorization of an M-matrix and its inverse. These algorithms avoid subtractive cancellation; this property ensures high componentwise accuracy even for ill-conditioned problems. The toolbox contains also accurate algorithms for related problems, such as computing the Schur complement, the singular values, the square root of an Mmatrix, and the solution of nonsymmetric algebraic Riccati equations associated with M-matrices. The Matlab interface is based on an object-oriented implementation allowing one to use standard Matlab operations on M-matrices with triplet representation.  \nKeywords M-matrix, mmatrix-toolbox, componentwise error  \nMSC Codes 65F30, 65Y15, 65F05, 15B48  \n1 Introduction  \nWe introduce the mmatrix-toolbox, a Matlab package designed for computations with M-matrices, available at [https://github.com/numpi/mmatrix](https://github.com/numpi/mmatrix).  \nAn M-matrix is a real matrix that can be written as A = sI − B, where B is componentwise nonnegative, I is the identity matrix and s ≥ ρ (B), where ρ (B) is the spectral radius of B. These matrices appear in numerous applications ranging from applied probability to network models and discretization of PDEs.  \nUnder mild assumptions an M-matrix possesses a triplet representation, that is made of three objects: the off-diagonal elements of A, a positive vector u and a nonnegative vector v such that  \n(right triplet) Au = v, (1a)  \n(left triplet) uTA = vT . (1b)  \n∗ This research has been supported by ICSC–Centro Nazionale di Ricerca in High Performance Computing, Big Data, and Quantum Computing funded by European Union–NextGenerationEU; by PRIN 2022ZK5ME7 CUP B53C24006410006 funded by Italian MUR; and by the project “AIDMIX” funded by the University of Perugia; BI and FP are members of INDAM (Istituto Nazionale di Alta Matematica) .  \n†Department of Mathematics and Computer Science, University of Perugia, Italy ([bruno.iannazzo@unipg.it](bruno.iannazzo@unipg.it)).  \n‡Department of Computer Science, University of Pisa, Italy ([mehdi.najafi@di.unipi.it](mehdi.najafi@di.unipi.it), [federico.poloni@unipi.it](federico.poloni@unipi.it)).  \nAlthough this representation is not unique, it allows the design of componentwise accurate algorithms for some typical problems associated with M-matrices such as the solution of a linear system, the inversion, and so on, through the so-called GTH-like (or just GTH) algorithm [17, 1], a subtraction-free variant of Gaussian elimination that exploits the M-matrix sign structure.  \nIn contrast, the customary linear system solvers included in linear algebra libraries such as Lapack do not guarantee componentwise accuracy. In particular, they typically deliver results within the relative norm accuracy predicted by the condition number, and this may lead to large relative errors in small entries.  \nThe triplet representation allows one to get full componentwise accuracy in the linear system solvers, when the right-hand side is a nonnegative vector. From these linear system solvers it is possible to design componentwise accurate algorithms for more complicated problems such as the square root of an M-matrix and the solution of matrix equations associated with M-matrices.  \nThe mmatrix toolbox provides a practical framework to work with M-matrices with triple","cbCaigMZuDk7oqQC","https://ap.wps.com/l/cbCaigMZuDk7oqQC","pdf",570671,3,1,27,"English","en",105,"# Introduction\n# Preliminaries\n## Triplet representations","[{\"question\":\"Which M-matrix problems are supported beyond linear system solving?\",\"answer\":\"The toolbox also addresses tasks such as LU factorization and inversion, Schur complement computation, singular values, square roots of M-matrices, and solutions of nonsymmetric algebraic Riccati equations.\"}]",1784191679,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"the-mmatrix-toolbox-componentwise-accurate-algorithms-for-m-matrices-with-triplet-representation","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/the-mmatrix-toolbox-componentwise-accurate-algorithms-for-m-matrices-with-triplet-representation/83961/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Which M-matrix problems are supported beyond linear system solving?","Question",{"text":75,"@type":76},"The toolbox also addresses tasks such as LU factorization and inversion, Schur complement computation, singular values, square roots of M-matrices, and solutions of nonsymmetric algebraic Riccati equations.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]