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Taking into account the associated four\u2013term recurrence relation, this problem can be formulated as a generalized eigenvalue problem, involving a lower bidiagonal matrix and a 2\u2013banded lower Hessenberg matrix of order\n                    <jats:italic>n<\/jats:italic>\n                    . Unfortunately, the considered generalized eigenvalue problem is very ill\u2013conditioned, and classical balancing procedures do not improve it. Therefore, customary techniques for solving the generalized eigenvalue problem, like the\n                    <jats:italic>QZ<\/jats:italic>\n                    method, yield unreliable results. Here, we propose a novel balancing procedure that drastically reduces the ill\u2013conditioning of the eigenvalues of the involved matrix pencil. Moreover, we propose a fast and reliable algorithm, with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ \\mathcal {O}(n^2) $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    computational complexity and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ \\mathcal {O}(n) $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    memory, exploiting the structure of the considered matrix pencil.\n                  <\/jats:p>","DOI":"10.1007\/s11075-025-02021-z","type":"journal-article","created":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T03:52:32Z","timestamp":1738986752000},"page":"1507-1526","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On computing the zeros of Laguerre\u2013Sobolev polynomials"],"prefix":"10.1007","volume":"100","author":[{"given":"T.","family":"Laudadio","sequence":"first","affiliation":[]},{"given":"N.","family":"Mastronardi","sequence":"additional","affiliation":[]},{"given":"F. 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