Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes
The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gaus...
Saved in:
| Published in: | IEEE transactions on information theory Vol. 46; no. 4; pp. 1495 - 1509 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
IEEE
01.07.2000
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects: | |
| ISSN: | 0018-9448, 1557-9654 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss-Markov random processes (GMrp), this result provides a closed-form factored expression for the covariance matrix of a first-order GMrp. This factored form leads to the interpretation of a first-order GMrp as the product of three independent processes: a forward independent-increments process, a backward independent-increments process, and a variance-stationary process. We explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy. We then consider the issue of approximating general nonstationary Gaussian processes by Gauss-Markov processes under two optimality criteria: the Kullback-Leibler distance and maximum entropy. The problem reduces to approximating general covariances by covariance matrices whose inverses are banded. Our inversion result is an efficient algorithmic solution to this problem. We evaluate the information loss between the original process and its Gauss-Markov approximation. |
|---|---|
| AbstractList | The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss-Markov random processes (GMrp), this result provides a closed-form factored expression for the covariance matrix of a first-order GMrp. This factored form leads to the interpretation of a first-order GMrp as the product of three independent processes: a forward independent-increments process, a backward independent-increments process, and a variance-stationary process. We explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy. We then consider the issue of approximating general nonstationary Gaussian processes by Gauss-Markov processes under two optimality criteria: the Kullback-Leibler distance and maximum entropy. The problem reduces to approximating general covariances by covariance matrices whose inverses are banded. Our inversion result is an efficient algorithmic solution to this problem. We evaluate the information loss between the original process and its Gauss-Markov approximation. The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss-Markov random processes (GMrp), this result provides a closed-form factored expression for the covariance matrix of a first-order GMrp. This factored form leads to the interpretation of a first-order GMrp as the product of three independent processes: a forward independent-increments process, a backward independent-increments process, and a variance-stationary process. We explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy. We then consider the issue of approximating general nonstationary Gaussian processes by Gauss-Markov processes under two optimality criteria: the Kullback-Leibler distance and maximum entropy. The problem reduces to approximating general covariances by covariance matrices whose inverses are banded. Our inversion result is an efficient algorithmic solution to this problem. We evaluate the information loss between the original process and its Gauss-Markov approximation |
| Author | Kavcic, A. Moura, J.M.F. |
| Author_xml | – sequence: 1 givenname: A. surname: Kavcic fullname: Kavcic, A. organization: Div. of Eng. & Appl. Sci., Harvard Univ., Cambridge, MA, USA – sequence: 2 givenname: J.M.F. surname: Moura fullname: Moura, J.M.F. |
| BookMark | eNp9kU1vGyEQQFHlSrXTHHrtadVDoxw2ZhbYhd6iKB-VbOWSnNF4PbSk68WFtaP01wfHVg9RlBMM8-YNMBM26kNPjH0BfgbAzRT0mVGykfoDG4NSTWlqJUdszDno0kipP7FJSg85lAqqMbNzHKJvKRWPfvhdLLBf0rLw_ZZiovTjsPOhL7D7FWJmVqnIUOGwHXL8D4ddMrjiGjcplXOMf8K2WMeQndnwmX102CU6PqxH7P7q8u7ippzdXv-8OJ-VrRD1UCIHECScrHUDRguHgNA4UMQXy0YIrJb5XqikI0S1qF0-BnCGU-0WjRLiiJ3svbnz3w2lwa58aqnrsKewSdaArCuloc7k93fJSsuq5mKn_PYKfAib2OdXWDDKVLwSMkOne6iNIaVIzq6jX2F8ssDtbiIWtN1PJLPTV2zrh5f_GyL67s2Kr_sKT0T_zYfkM57DmFQ |
| CODEN | IETTAW |
| CitedBy_id | crossref_primary_10_1109_TGRS_2019_2946752 crossref_primary_10_1016_j_cviu_2006_12_004 crossref_primary_10_1515_eqc_2010_002 crossref_primary_10_1109_TCOMM_2016_2536664 crossref_primary_10_1109_TSP_2013_2256900 crossref_primary_10_1002_nla_425 crossref_primary_10_1109_TIT_2003_819324 crossref_primary_10_1007_s11424_022_0015_1 crossref_primary_10_1007_s13398_019_00707_3 crossref_primary_10_1016_j_acha_2023_101579 crossref_primary_10_1109_TIT_2008_2009855 crossref_primary_10_1016_j_aml_2014_02_015 crossref_primary_10_4028_www_scientific_net_AMM_590_795 crossref_primary_10_1109_LGRS_2010_2089494 crossref_primary_10_1109_TSP_2004_840709 crossref_primary_10_1016_j_laa_2003_07_014 crossref_primary_10_1109_ACCESS_2018_2868266 crossref_primary_10_1109_TSP_2008_927480 crossref_primary_10_3390_fractalfract7060468 crossref_primary_10_1049_wss2_12074 crossref_primary_10_1109_TSP_2015_2502556 crossref_primary_10_1002_mma_3416 crossref_primary_10_1109_TCOMM_2005_861651 crossref_primary_10_1109_JSTSP_2017_2731599 crossref_primary_10_1109_JCN_2013_000068 crossref_primary_10_1587_transfun_E100_A_1037 crossref_primary_10_1002_nla_455 crossref_primary_10_1109_TSP_2009_2037350 crossref_primary_10_1016_j_camwa_2009_07_069 crossref_primary_10_1109_TSP_2006_890915 crossref_primary_10_1080_03081087_2021_1884180 crossref_primary_10_1016_j_jmva_2016_04_001 crossref_primary_10_1109_49_920180 crossref_primary_10_1007_s10596_017_9667_7 crossref_primary_10_1016_j_laa_2024_04_035 |
| Cites_doi | 10.1109/TIT.1972.1054786 10.1109/18.119691 10.1109/78.709549 10.1109/18.817531 10.1109/18.30972 10.1002/0471200611 10.1016/0024-3795(84)90207-6 10.1016/0005-1098(94)00069-U 10.1109/83.799887 |
| ContentType | Journal Article |
| Copyright | Copyright Institute of Electrical and Electronics Engineers, Inc. (IEEE) Jul 2000 |
| Copyright_xml | – notice: Copyright Institute of Electrical and Electronics Engineers, Inc. (IEEE) Jul 2000 |
| DBID | RIA RIE AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D F28 FR3 |
| DOI | 10.1109/18.954748 |
| DatabaseName | IEEE All-Society Periodicals Package (ASPP) 1998-Present IEEE Electronic Library (IEL) CrossRef Computer and Information Systems Abstracts Electronics & Communications Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional ANTE: Abstracts in New Technology & Engineering Engineering Research Database |
| DatabaseTitle | CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional Engineering Research Database ANTE: Abstracts in New Technology & Engineering |
| DatabaseTitleList | Computer and Information Systems Abstracts Technology Research Database |
| Database_xml | – sequence: 1 dbid: RIE name: IEEE Electronic Library (IEL) url: https://ieeexplore.ieee.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Computer Science |
| EISSN | 1557-9654 |
| EndPage | 1509 |
| ExternalDocumentID | 57255649 10_1109_18_954748 954748 |
| GroupedDBID | -~X .DC 0R~ 29I 3EH 4.4 5GY 5VS 6IK 97E AAJGR AARMG AASAJ AAWTH ABAZT ABFSI ABQJQ ABVLG ACGFO ACGFS ACGOD ACIWK AENEX AETEA AETIX AGQYO AGSQL AHBIQ AI. AIBXA AKJIK AKQYR ALLEH ALMA_UNASSIGNED_HOLDINGS ASUFR ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ CS3 DU5 E.L EBS EJD F5P HZ~ H~9 IAAWW IBMZZ ICLAB IDIHD IFIPE IFJZH IPLJI JAVBF LAI M43 MS~ O9- OCL P2P PQQKQ RIA RIE RNS RXW TAE TN5 VH1 VJK AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D RIG F28 FR3 |
| ID | FETCH-LOGICAL-c336t-a0113e3f46871983fa1a17f15e0bd733a2ddeda54feaa5b6f0bd11f90e6fb7533 |
| IEDL.DBID | RIE |
| ISICitedReferencesCount | 65 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000088206200022&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0018-9448 |
| IngestDate | Sun Sep 28 04:30:47 EDT 2025 Sun Sep 28 08:21:28 EDT 2025 Fri Jul 25 04:22:01 EDT 2025 Tue Nov 18 22:24:49 EST 2025 Sat Nov 29 02:54:40 EST 2025 Tue Aug 26 21:00:27 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Language | English |
| License | https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c336t-a0113e3f46871983fa1a17f15e0bd733a2ddeda54feaa5b6f0bd11f90e6fb7533 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 |
| PQID | 195920234 |
| PQPubID | 23500 |
| PageCount | 15 |
| ParticipantIDs | proquest_journals_195920234 crossref_primary_10_1109_18_954748 crossref_citationtrail_10_1109_18_954748 proquest_miscellaneous_28426033 proquest_miscellaneous_914625816 ieee_primary_954748 |
| PublicationCentury | 2000 |
| PublicationDate | 2000-07-01 |
| PublicationDateYYYYMMDD | 2000-07-01 |
| PublicationDate_xml | – month: 07 year: 2000 text: 2000-07-01 day: 01 |
| PublicationDecade | 2000 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | IEEE transactions on information theory |
| PublicationTitleAbbrev | TIT |
| PublicationYear | 2000 |
| Publisher | IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Publisher_xml | – name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| References | golub (ref2) 1983 lev-ari (ref11) 1984 ref12 jain (ref5) 1968 ref10 van trees (ref13) 1968; i ref1 ref8 ref7 ref9 ref4 ref3 ref6 rivlin (ref15) 1974 scharf (ref14) 1991 |
| References_xml | – ident: ref4 doi: 10.1109/TIT.1972.1054786 – year: 1983 ident: ref2 publication-title: Matrix Computations Baltimore MD – year: 1991 ident: ref14 publication-title: Statistical Signal Processing Detection Estimation and Time Series Analysis – volume: i year: 1968 ident: ref13 publication-title: Detection Estimation and Modulation Theory – ident: ref3 doi: 10.1109/18.119691 – year: 1984 ident: ref11 article-title: Autoregressive models for nonstationary discrete time processes publication-title: IEEE Int Symp on Information Theory – ident: ref1 doi: 10.1109/78.709549 – year: 1968 ident: ref5 publication-title: Fundamentals of Digital Image Processing – ident: ref6 doi: 10.1109/18.817531 – ident: ref12 doi: 10.1109/18.30972 – ident: ref9 doi: 10.1002/0471200611 – ident: ref10 doi: 10.1016/0024-3795(84)90207-6 – ident: ref8 doi: 10.1016/0005-1098(94)00069-U – year: 1974 ident: ref15 publication-title: The Chebyshev Polynomials – ident: ref7 doi: 10.1109/83.799887 |
| SSID | ssj0014512 |
| Score | 1.9242272 |
| Snippet | The paper considers the inversion of full matrices whose inverses are L-banded. We derive a nested inversion algorithm for such matrices. Applied to a... |
| SourceID | proquest crossref ieee |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 1495 |
| SubjectTerms | Algorithms Approximation Factorization Inversions Mathematical analysis Matrices Matrix inversion Matrix methods Maximum entropy |
| Title | Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes |
| URI | https://ieeexplore.ieee.org/document/954748 https://www.proquest.com/docview/195920234 https://www.proquest.com/docview/28426033 https://www.proquest.com/docview/914625816 |
| Volume | 46 |
| WOSCitedRecordID | wos000088206200022&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVIEE databaseName: IEEE Electronic Library (IEL) customDbUrl: eissn: 1557-9654 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0014512 issn: 0018-9448 databaseCode: RIE dateStart: 19630101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1NT9wwEB0V1EM5FLql6kKhVsWBSyCOndjmViGglyIOVOIWOc4YIUGC1rv8fvy1K6rSAzfLmciRJ-PxeDzvARwIWnfWUFn4vUJfcGlM0fmoo7BCV4xj4LzikWxCXF7Kmxt1lXG2Yy0MIsbLZ3gUmjGX349mEY7KjlXNBZdrsCZEk0q1VgkDXtMEDO6HVT7kyCBCtFTHVB6lF_9yPZFL5Z8FOHqV8803fc8WfMybR_IzafsTvMNhAptLYgaS7XQCGy9QBj9D-zvC8KMj4dCVdOHYuCd3Q7iQge4kt7x-iL6_HWde5sERL0QSF08u1CSjJRd64VwRynvGJ_KYSgzQbcOf87Pr019F5lUoDGPNvNDephkyyxsfLSnJrKaaCktrLLteMKYrv-b1uuYWta67xvpuSq0qsbGdD2_YF1gfxgG_AilZZ0rTYaAF5LaSutGMom4qUwljlJrC4XLKW5NBxwP3xX0bg49StVS2aRqn8GMl-piQNl4TmgQ1rASWvbtLNbbZBF0bUHMCNzyfwvfVU287ISGiBxwXrvWu2YdzjE2B_EdCeUdS1ZI2O68OvAsfUml-uL77DdbnswXuwXvzNL9zs_34hz4De1rnNA |
| linkProvider | IEEE |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1Nb9QwEB1BQQIOLSxUbAvUqjhwSRvHTmL3hhCliHbVQ5F6sxxnjCq1SbXe7e_HX7uiohy4Wc5EjjwZj8fjeQ_gY0vrzhoqCr9X6AsujCk6H3UUttUV4xg4r3gkm2hnM3F5Kc8zznashUHEePkMD0Iz5vL70SzDUdmhrHnLxWN4UnNelalYa50y4DVN0OB-YOmDjgwjREt5SMVBevWe84lsKn8twdGvHG_91xe9hM28fSSfk75fwSMcJrC1omYg2VIn8OIPnMHXoM4iED86Eo5dSRcOjntyNYQrGeiOcstriOjrX-Pcy9w44oVIYuPJpZpktOSbXjpXhAKf8Y7cpiIDdG_g5_HXiy8nRWZWKAxjzaLQ3qoZMssbHy9JwaymmraW1lh2fcuYrvyq1-uaW9S67hrruym1ssTGdj7AYduwMYwDvgVSss6UpsNADMhtJXSjGUXdVKZqjZFyCp9WU65Mhh0P7BfXKoYfpVRUqDSNU9hfi94mrI2HhCZBDWuBVe_uSo0qG6FTATcnsMPzKeytn3rrCSkRPeC4dMo7Zx_QMTYF8g8J6V1JVQva7Dw48B48O7k4O1Wn32c_duF5KtQPl3nfwcZivsT38NTcLa7c_EP8W38DoLzqew |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Matrices+with+banded+inverses%3A+inversion+algorithms+and+factorization+of+Gauss-Markov+processes&rft.jtitle=IEEE+transactions+on+information+theory&rft.au=Kavcic%2C+A.&rft.au=Moura%2C+J.M.F.&rft.date=2000-07-01&rft.pub=IEEE&rft.issn=0018-9448&rft.volume=46&rft.issue=4&rft.spage=1495&rft.epage=1509&rft_id=info:doi/10.1109%2F18.954748&rft.externalDocID=954748 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0018-9448&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0018-9448&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0018-9448&client=summon |