document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1046540>Question 1046540</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The symmetrix positive definite matrix \r<BR>\n" );
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document.write( "A = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix+%28+3%2C+3%2C+16%2C+-8%2C+-4%2C+-8%2C+29%2C+12%2C+-4%2C+12%2C+41%29%29 ALIGN=MIDDLE  ALT=\"%28matrix+%28+3%2C+3%2C+16%2C+-8%2C+-4%2C+-8%2C+29%2C+12%2C+-4%2C+12%2C+41%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "can be written as the product of a lower triangular matrix \r<BR>\n" );
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document.write( "L= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix+%28+3%2C+3%2C+l%5B11%5D%2C+0%2C+0%2C+l%5B21%5D%2C+l%5B22%5D%2C+0%2C+l%5B31%5D%2C+l%5B32%5D%2C+l%5B33%5D%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and it's transpose <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5ET ALIGN=MIDDLE  ALT=\"L%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" >, that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=LL%5ET ALIGN=MIDDLE  ALT=\"A=LL%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Find L and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5ET ALIGN=MIDDLE  ALT=\"L%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661999 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52814)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=661999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "For some useful info related to this subject see this Wikipedia article\r<BR>\n" );
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document.write( "<A  rel=nofollow HREF=https://en.wikipedia.org/wiki/Cholesky_decomposition>https://en.wikipedia.org/wiki/Cholesky_decomposition</A>\r<BR>\n" );
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document.write( "https://en.wikipedia.org/wiki/Cholesky_decomposition\r<BR>\n" );
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document.write( "Somewhere in the beginning of 80-ies I was in need to have my own computer program (subroutine) based on this algorithm \r\n" );
document.write( "(Cholesky LU-decomposition) to use it in the more wide finite element code for solving systems of linear equations.\r\n" );
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document.write( "I was lucky: I found very good description of the algorithm in the book by Wilkinson and Reinsch \"Handbook for Automatic Computations\". \r\n" );
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document.write( "I learned this algorithm, wrote the subroutine, wrote the entire finite-element code and used it during some years, \r\n" );
document.write( "making my research computer simulations.\r\n" );
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document.write( "It worked successfully in solving matrix equations of the size 1000 - 3000 - 5000 in one (in each of the two) matrix dimensions.\r\n" );
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document.write( "For more grandiose matrices the algorithm lost its effectiveness, and other methods were required.  \r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Some reminiscences . . . \r<BR>\n" );
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document.write( "Surely, I know the algorithm, but its presentation requires a lot of writing.\r<BR>\n" );
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document.write( "This is why I refer you to that article.\r<BR>\n" );
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document.write( "See also in the Internet with keywords \"Cholesky decomposition\",  \"LU-decomposition\" . . . \r<BR>\n" );
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document.write( "Also, good sources of information are the books\r<BR>\n" );
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document.write( "   &nbsp;&nbsp;&nbsp;&nbsp;- \"Numerical recipies, vol. I\"   &nbsp;&nbsp;&nbsp;&nbsp;(classic)<BR>\n" );
document.write( "   &nbsp;&nbsp;&nbsp;&nbsp;- \"LINPACK user's guide\" (1979) by Dongarra and others . . . &nbsp;&nbsp;&nbsp;&nbsp;(classic too)\r<BR>\n" );
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document.write( "As well as any authoritative contemporary guide/textbook on Matrix Computations.\r<BR>\n" );
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document.write( "In nowadays, you can even find an online matrix calculator in the Internet making LU-decomposition for free ! ! !\r<BR>\n" );
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document.write( "I just found one such a calculator for you under the link \r<BR>\n" );
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document.write( "<A  rel=nofollow HREF=http://comnuan.com/cmnn0100d/>http://comnuan.com/cmnn0100d/</A>\r<BR>\n" );
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document.write( "http://comnuan.com/cmnn0100d/\r<BR>\n" );
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