{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:42:06Z","timestamp":1776793326262,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,1,6]],"date-time":"2012-01-06T00:00:00Z","timestamp":1325808000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In various applications, it is necessary to differentiate a matrix functional\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"w left-parenthesis bold upper A left-parenthesis bold x right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>w<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">A<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">x<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">w(\\textbf {A}(\\textbf {x}))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper A left-parenthesis bold x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">A<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">x<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {A}(\\textbf {x})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a matrix depending on a parameter vector\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold x\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mtext mathvariant=\"bold\">x<\/mml:mtext>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {x}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Usually, the functional itself can be readily computed from a triangular factorization of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper A left-parenthesis bold x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">A<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">x<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {A}(\\textbf {x})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This paper develops several methods that also use the triangular factorization to efficiently evaluate the first and second derivatives of the functional. Both the full and sparse matrix situations are considered. There are similarities between these methods and algorithmic differentiation. However, the methodology developed here is explicit, leading to new algorithms. It is shown how the methods apply to several applications where the functional is a log determinant, including spline smoothing, covariance selection and restricted maximum likelihood.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02451-8","type":"journal-article","created":{"date-parts":[[2011,1,6]],"date-time":"2011-01-06T13:42:37Z","timestamp":1294321357000},"page":"1585-1600","source":"Crossref","is-referenced-by-count":4,"title":["Differentiation of matrix functionals using triangular factorization"],"prefix":"10.1090","volume":"80","author":[{"given":"F.","family":"de Hoog","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Anderssen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Lukas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,1,6]]},"reference":[{"issue":"1-3","key":"1","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/S0024-3795(97)10009-X","article-title":"Monte Carlo estimates of the log determinant of large sparse matrices","volume":"289","author":"Barry, Ronald Paul","year":"1999","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"R. 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