{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T07:48:20Z","timestamp":1761896900837,"version":"3.37.3"},"reference-count":41,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,10,8]],"date-time":"2018-10-08T00:00:00Z","timestamp":1538956800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Computational Intelligence and Soft Computing"],"published-print":{"date-parts":[[2018,10,8]]},"abstract":"<jats:p>Conjugate gradient is an iterative method that solves a linear system <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>A<\/mml:mi><mml:mi>x<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math> is a positive definite matrix. We present this new iterative method for solving linear interval systems <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mover accent=\"false\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mo>=<\/mml:mo><mml:mover accent=\"false\"><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math> is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mover accent=\"false\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mover accent=\"false\"><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math> at every step while the norm is sufficiently small. In addition, we present another iterative method that solves <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mover accent=\"false\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mo>=<\/mml:mo><mml:mover accent=\"false\"><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math> is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mover accent=\"false\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math> for linear interval systems, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mover accent=\"false\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mover accent=\"false\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><mml:mo>=<\/mml:mo><mml:mover accent=\"false\"><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:math>; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.<\/jats:p>","DOI":"10.1155\/2018\/2797038","type":"journal-article","created":{"date-parts":[[2018,10,8]],"date-time":"2018-10-08T19:34:58Z","timestamp":1539027298000},"page":"1-13","source":"Crossref","is-referenced-by-count":7,"title":["Two Iterative Methods for Solving Linear Interval Systems"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7903-9244","authenticated-orcid":true,"given":"Esmaeil","family":"Siahlooei","sequence":"first","affiliation":[{"name":"Department of Computer Science, Yazd University, Yazd, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3724-8689","authenticated-orcid":true,"given":"Seyed Abolfazl","family":"Shahzadeh Fazeli","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Yazd University, Yazd, Iran"},{"name":"Parallel Processing Laboratory, Yazd University, Yazd, Iran"}]}],"member":"311","reference":[{"year":"1996","key":"1"},{"year":"2010","key":"2"},{"year":"2010","key":"3"},{"year":"2008","key":"4"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1002\/wics.82"},{"first-page":"205","volume-title":"Uncertainty theory","year":"2007","key":"6"},{"year":"1983","key":"7"},{"first-page":"11","volume-title":"Fuzzy Dual Numbers","year":"2018","key":"8"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-016-0382-0"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/s40819-015-0113-z"},{"year":"1966","key":"11"},{"key":"12","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/0024-3795(69)90024-X","volume":"2","year":"1969","journal-title":"Linear Algebra and its Applications"},{"year":"1990","series-title":"Cambridge, UK","key":"13"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488505003710"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.22436\/jmcs.08.03.02"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.12989\/sem.2002.13.3.299"},{"first-page":"193","volume-title":"Limit state imprecise interval analysis in geotechnical engineering","year":"2015","key":"18"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-42056-1_12"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011470131086"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717716"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-017-2668-6"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1023\/A:1020505620702"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1137\/130914358"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.02.012"},{"issue":"13-16","key":"27","first-page":"607","volume":"5","year":"2011","journal-title":"Applied Mathematical Sciences"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1007\/s10287-017-0290-9"},{"issue":"6","key":"30","first-page":"15777","volume":"10","year":"2015","journal-title":"International Journal of Applied Engineering Research"},{"key":"31","first-page":"1","volume":"1000","year":"2018","journal-title":"Journal of Physics: Conference Series"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1137\/0730044"},{"issue":"7","key":"33","first-page":"514","volume":"1","year":"1995","journal-title":"J.UCS. 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