Numerical Recipes in C/C语言数值算法

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英文版。用C写的数值算法,有数学原理和代码实现。
Numerical Recipes in C.rar
  • Numerical Recipes in C
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/624a12b8fc37f87c243c2159/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624a12b8fc37f87c243c2159/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">2.7<span class="_"> </span>Sparse<span class="_ _0"> </span>Linear<span class="_ _0"> </span>Systems<span class="_ _1"> </span><span class="ff2 fs1">71</span></div><div class="t m1 x2 h3 y2 ff3 fs2 fc0 sc0 ls0 ws0">Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)</div><div class="t m1 x3 h3 y2 ff3 fs2 fc0 sc0 ls0 ws0">Copyright (C) 1988-1992 by Cambridge University Press.<span class="_"> </span>Programs Copyright (C) 1988-1992 by Numerical Recipes Software. </div><div class="t m1 x4 h3 y2 ff3 fs2 fc0 sc0 ls0 ws0">Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-</div><div class="t m1 x5 h3 y2 ff3 fs2 fc0 sc0 ls0 ws0">readable files (including this one) to any server<span class="_"> </span>computer, is strictly prohibited. To order Numerical Recipes books<span class="_ _2"> </span>or CDROMs, visit website</div><div class="t m1 x6 h3 y2 ff3 fs2 fc0 sc0 ls0 ws0">http://www.nr.com or call 1-800-872-7423 (North America only),<span class="_"> </span>or send email to directcustserv@cambridge.org (outside North America).</div><div class="t m0 x7 h4 y3 ff4 fs3 fc0 sc0 ls0 ws0">2.7<span class="_"> </span>Sparse<span class="_ _3"> </span>Linear<span class="_ _3"> </span>Systems</div><div class="t m0 x8 h2 y4 ff2 fs1 fc0 sc0 ls0 ws0">A<span class="_ _4"> </span>system<span class="_ _4"> </span>of<span class="_"> </span>linear<span class="_ _4"> </span>equations<span class="_"> </span>is<span class="_ _4"> </span>called<span class="_ _4"> </span><span class="ff5">sparse<span class="_"> </span></span>if<span class="_ _4"> </span>only<span class="_ _4"> </span>a<span class="_"> </span>relatively<span class="_"> </span>small<span class="_ _4"> </span>number</div><div class="t m0 x7 h2 y5 ff2 fs1 fc0 sc0 ls0 ws0">of<span class="_ _5"> </span>its<span class="_ _5"> </span>matrix<span class="_ _5"> </span>elements<span class="_ _5"> </span><span class="ff6">a</span></div><div class="t m0 x9 h5 y6 ff7 fs4 fc0 sc0 ls1 ws0">ij</div><div class="t m0 xa h2 y7 ff2 fs1 fc0 sc0 ls0 ws0">are<span class="_ _5"> </span>nonzero.<span class="_ _6"> </span>It<span class="_ _5"> </span>is<span class="_ _5"> </span>wasteful<span class="_ _5"> </span>to<span class="_ _7"> </span>use<span class="_ _5"> </span>general<span class="_ _5"> </span>methods<span class="_ _5"> </span>of</div><div class="t m0 x7 h2 y8 ff2 fs1 fc0 sc0 ls0 ws0">linear<span class="_ _4"> </span>algebra<span class="_"> </span>on<span class="_ _4"> </span>such<span class="_ _4"> </span>problems,<span class="_ _4"> </span>because<span class="_ _4"> </span>most<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _4"> </span><span class="ff6">O<span class="ff8">(</span>N</span></div><div class="t m0 xb h6 y9 ff9 fs4 fc0 sc0 ls0 ws0">3</div><div class="t m0 xc h2 ya ff8 fs1 fc0 sc0 ls0 ws0">)<span class="_ _4"> </span><span class="ff2">arithmetic<span class="_ _4"> </span>operations</span></div><div class="t m0 x7 h2 yb ff2 fs1 fc0 sc0 ls0 ws0">dev<span class="_ _8"></span>oted<span class="_ _0"> </span>to<span class="_ _0"> </span>solving<span class="_ _0"> </span>the<span class="_ _0"> </span>set<span class="_"> </span>of<span class="_ _0"> </span>equations<span class="_ _0"> </span>or<span class="_ _0"> </span>in<span class="_ _8"></span>verting<span class="_ _9"> </span>the<span class="_"> </span>matrix<span class="_ _9"> </span>in<span class="_ _8"></span>volve<span class="_ _9"> </span>zero<span class="_ _0"> </span>operands.</div><div class="t m0 x7 h2 yc ff2 fs1 fc0 sc0 ls0 ws0">Furthermore,<span class="_ _5"> </span>you<span class="_ _5"> </span>might<span class="_ _5"> </span>wish<span class="_ _5"> </span>to<span class="_ _5"> </span>work<span class="_ _5"> </span>problems<span class="_ _a"> </span>so<span class="_ _5"> </span>large<span class="_ _5"> </span>as<span class="_ _5"> </span>to<span class="_ _5"> </span>tax<span class="_ _5"> </span>your<span class="_ _5"> </span>a<span class="_ _8"></span>vailable</div><div class="t m0 x7 h2 yd ff2 fs1 fc0 sc0 ls0 ws0">memory<span class="_ _5"> </span>space,<span class="_ _5"> </span>and<span class="_ _5"> </span>it<span class="_ _7"> </span>is<span class="_ _5"> </span>wasteful<span class="_ _5"> </span>to<span class="_ _7"> </span>reserve<span class="_ _a"> </span>storage<span class="_ _5"> </span>for<span class="_ _5"> </span>unfruitful<span class="_ _5"> </span>zero<span class="_ _5"> </span>elements.</div><div class="t m0 x7 h2 ye ff2 fs1 fc0 sc0 ls0 ws0">Note<span class="_ _a"> </span>that<span class="_ _5"> </span>there<span class="_ _5"> </span>are<span class="_ _a"> </span>two<span class="_ _5"> </span>distinct<span class="_ _a"> </span>(and<span class="_ _5"> </span>not<span class="_ _a"> </span>always<span class="_ _5"> </span>compatible)<span class="_ _4"> </span>goals<span class="_ _5"> </span>for<span class="_ _a"> </span>any<span class="_ _5"> </span>sparse</div><div class="t m0 x7 h2 yf ff2 fs1 fc0 sc0 ls0 ws0">matrix<span class="_ _a"> </span>method:<span class="_ _b"> </span>saving<span class="_ _a"> </span>time<span class="_ _5"> </span>and/or<span class="_ _a"> </span>saving<span class="_ _5"> </span>space.</div><div class="t m0 x8 h2 y10 ff2 fs1 fc0 sc0 ls0 ws0">W<span class="_ _c"></span>e<span class="_ _d"> </span>hav<span class="_ _8"></span>e<span class="_ _d"> </span>already<span class="_ _7"> </span>considered<span class="_ _d"> </span>one<span class="_ _7"> </span>archetypal<span class="_ _d"> </span>sparse<span class="_ _d"> </span>form<span class="_ _7"> </span>in<span class="_ _d"> </span><span class="ffa">&#167;</span>2.4,<span class="_ _3"> </span>the<span class="_ _e"> </span>band</div><div class="t m0 x7 h2 y11 ff2 fs1 fc0 sc0 ls0 ws0">diagonal<span class="_ _4"> </span>matrix.<span class="_ _f"> </span>In<span class="_ _a"> </span>the<span class="_ _a"> </span>tridiagonal<span class="_ _a"> </span>case,<span class="_ _5"> </span>e.g.,<span class="_ _a"> </span>we<span class="_ _5"> </span>saw<span class="_ _4"> </span>that<span class="_ _5"> </span>it<span class="_ _a"> </span>was<span class="_ _a"> </span>possible<span class="_ _a"> </span>to<span class="_ _5"> </span>sa<span class="_ _8"></span>ve</div><div class="t m0 x7 h2 y12 ff2 fs1 fc0 sc0 ls0 ws0">both<span class="_ _e"> </span>time<span class="_ _e"> </span>(order<span class="_ _e"> </span><span class="ff6">N<span class="_ _b"> </span></span>instead<span class="_ _7"> </span>of<span class="_ _d"> </span><span class="ff6">N</span></div><div class="t m0 xd h6 y13 ff9 fs4 fc0 sc0 ls0 ws0">3</div><div class="t m0 xe h2 y14 ff2 fs1 fc0 sc0 ls0 ws0">)<span class="_ _e"> </span>and<span class="_ _e"> </span>space<span class="_ _e"> </span>(order<span class="_ _e"> </span><span class="ff6">N<span class="_ _b"> </span></span>instead<span class="_ _e"> </span>of<span class="_ _e"> </span><span class="ff6">N</span></div><div class="t m0 xf h6 y13 ff9 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x10 h2 y14 ff2 fs1 fc0 sc0 ls0 ws0">).<span class="_ _10"> </span>The</div><div class="t m0 x7 h2 y15 ff2 fs1 fc0 sc0 ls0 ws0">method<span class="_ _a"> </span>of<span class="_ _a"> </span>solution<span class="_ _a"> </span>was<span class="_ _5"> </span>not<span class="_ _a"> </span>different<span class="_ _4"> </span>in<span class="_ _5"> </span>principle<span class="_ _4"> </span>from<span class="_ _5"> </span>the<span class="_ _a"> </span>general<span class="_ _a"> </span>method<span class="_ _a"> </span>of<span class="_ _a"> </span><span class="ff6 ls2">LU</span></div><div class="t m0 x7 h2 y16 ff2 fs1 fc0 sc0 ls0 ws0">decomposition;<span class="_ _2"> </span>it<span class="_ _9"> </span>was<span class="_ _9"> </span>just<span class="_ _9"> </span>applied<span class="_ _2"></span>clev<span class="_ _8"></span>erly<span class="_ _8"></span>,<span class="_ _2"> </span>and<span class="_ _9"> </span>with<span class="_ _9"> </span>due<span class="_ _2"> </span>attention<span class="_ _2"> </span>to<span class="_ _9"> </span>the<span class="_ _9"> </span>bookkeeping</div><div class="t m0 x7 h2 y17 ff2 fs1 fc0 sc0 ls0 ws0">of<span class="_ _9"> </span>zero<span class="_ _9"> </span>elements.<span class="_ _5"> </span>Man<span class="_ _8"></span>y<span class="_ _0"> </span>practical<span class="_ _9"> </span>schemes<span class="_ _0"> </span>for<span class="_ _9"> </span>dealing<span class="_ _9"> </span>with<span class="_ _9"> </span>sparse<span class="_"> </span>problems<span class="_ _2"> </span>have<span class="_ _2"> </span>this</div><div class="t m0 x7 h2 y18 ff2 fs1 fc0 sc0 ls0 ws0">same<span class="_ _0"> </span>character<span class="_ _c"></span>.<span class="_ _5"> </span>The<span class="_ _8"></span>y<span class="_"> </span>are<span class="_ _9"> </span>fundamentally<span class="_ _9"> </span>decomposition<span class="_ _9"> </span>schemes,<span class="_"> </span>or<span class="_ _9"> </span>else<span class="_"> </span>elimination</div><div class="t m0 x7 h2 y19 ff2 fs1 fc0 sc0 ls0 ws0">schemes<span class="_ _0"> </span>akin<span class="_ _9"> </span>to<span class="_"> </span>Gauss-Jordan,<span class="_ _2"> </span>but<span class="_ _9"> </span>carefully<span class="_ _0"> </span>optimized<span class="_ _9"> </span>so<span class="_"> </span>as<span class="_ _9"> </span>to<span class="_ _0"> </span>minimize<span class="_ _9"> </span>the<span class="_"> </span>number</div><div class="t m0 x7 h2 y1a ff2 fs1 fc0 sc0 ls0 ws0">of<span class="_ _4"> </span>so-called<span class="_ _4"> </span><span class="ff5">&#64257;ll-ins</span>,<span class="_ _a"> </span>initially<span class="_ _4"> </span>zero<span class="_ _4"> </span>elements<span class="_ _4"> </span>which<span class="_ _a"> </span>must<span class="_ _4"> </span>become<span class="_ _4"> </span>nonzero<span class="_ _4"> </span>during<span class="_ _4"> </span>the</div><div class="t m0 x7 h2 y1b ff2 fs1 fc0 sc0 ls0 ws0">solution<span class="_ _a"> </span>process,<span class="_ _a"> </span>and<span class="_ _a"> </span>for<span class="_ _a"> </span>which<span class="_ _a"> </span>storage<span class="_ _a"> </span>must<span class="_ _a"> </span>be<span class="_ _a"> </span>reserved.</div><div class="t m0 x8 h2 y1c ff2 fs1 fc0 sc0 ls0 ws0">Direct<span class="_ _7"> </span>methods<span class="_ _7"> </span>for<span class="_ _e"> </span>solving<span class="_ _7"> </span>sparse<span class="_ _7"> </span>equations,<span class="_ _e"> </span>then,<span class="_ _d"> </span>depend<span class="_ _7"> </span>crucially<span class="_ _7"> </span>on<span class="_ _7"> </span>the</div><div class="t m0 x7 h2 y1d ff2 fs1 fc0 sc0 ls0 ws0">precise<span class="_ _4"> </span>pattern<span class="_ _4"> </span>of<span class="_ _a"> </span>sparsity<span class="_ _4"> </span>of<span class="_ _a"> </span>the<span class="_ _4"> </span>matrix.<span class="_ _11"> </span>Patterns<span class="_ _4"> </span>that<span class="_ _a"> </span>occur<span class="_ _4"> </span>frequently<span class="_ _c"></span>,<span class="_ _4"> </span>or<span class="_ _a"> </span>that<span class="_ _4"> </span>are</div><div class="t m0 x7 h2 y1e ff2 fs1 fc0 sc0 ls0 ws0">useful<span class="_ _4"> </span>as<span class="_ _4"> </span>way-stations<span class="_"> </span>in<span class="_ _4"> </span>the<span class="_ _4"> </span>reduction<span class="_"> </span>of<span class="_ _4"> </span>more<span class="_ _4"> </span>general<span class="_"> </span>forms,<span class="_ _4"> </span>already<span class="_ _4"> </span>ha<span class="_ _8"></span>ve<span class="_ _4"> </span>special</div><div class="t m0 x7 h2 y1f ff2 fs1 fc0 sc0 ls0 ws0">names<span class="_"> </span>and<span class="_"> </span>special<span class="_ _4"> </span>methods<span class="_"> </span>of<span class="_"> </span>solution.<span class="_ _7"> </span>W<span class="_ _c"></span>e<span class="_ _4"> </span>do<span class="_ _4"> </span>not<span class="_"> </span>hav<span class="_ _8"></span>e<span class="_"> </span>space<span class="_ _4"> </span>here<span class="_"> </span>for<span class="_"> </span>any<span class="_"> </span>detailed</div><div class="t m0 x7 h2 y20 ff2 fs1 fc0 sc0 ls0 ws0">revie<span class="_ _8"></span>w<span class="_"> </span>of<span class="_"> </span>these.<span class="_ _5"> </span>References<span class="_"> </span>listed<span class="_"> </span>at<span class="_ _4"> </span>the<span class="_"> </span>end<span class="_"> </span>of<span class="_"> </span>this<span class="_"> </span>section<span class="_"> </span>will<span class="_ _4"> </span>furnish<span class="_"> </span>you<span class="_"> </span>with<span class="_"> </span>an</div><div class="t m0 x7 h2 y21 ff2 fs1 fc0 sc0 ls0 ws0">&#8220;in&#8221;<span class="_ _a"> </span>to<span class="_ _a"> </span>the<span class="_ _a"> </span>specialized<span class="_ _a"> </span>literature,<span class="_ _a"> </span>and<span class="_ _a"> </span>the<span class="_ _a"> </span>following<span class="_ _4"> </span>list<span class="_ _a"> </span>of<span class="_ _a"> </span>buzz<span class="_ _4"> </span>words<span class="_ _a"> </span>(and<span class="_ _a"> </span>Figure</div><div class="t m0 x7 h2 y22 ff2 fs1 fc0 sc0 ls0 ws0">2.7.1)<span class="_ _4"> </span>will<span class="_ _a"> </span>at<span class="_ _a"> </span>least<span class="_ _a"> </span>let<span class="_ _5"> </span>you<span class="_ _4"> </span>hold<span class="_ _a"> </span>your<span class="_ _4"> </span>own<span class="_ _4"> </span>at<span class="_ _a"> </span>cocktail<span class="_ _a"> </span>parties:</div><div class="t m0 x11 h2 y23 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">tridiagonal</span></div><div class="t m0 x11 h2 y24 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">band<span class="_ _a"> </span>diagonal<span class="_ _a"> </span>(or<span class="_ _a"> </span>banded)<span class="_ _a"> </span>with<span class="_ _5"> </span>bandwidth<span class="_ _4"> </span><span class="ff6">M</span></span></div><div class="t m0 x11 h2 y25 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">band<span class="_ _e"> </span>triangular</span></div><div class="t m0 x11 h2 y26 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">block<span class="_ _e"> </span>diagonal</span></div><div class="t m0 x11 h2 y27 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">block<span class="_ _e"> </span>tridiagonal</span></div><div class="t m0 x11 h2 y28 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">block<span class="_ _e"> </span>triangular</span></div><div class="t m0 x11 h2 y29 ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">cyclic<span class="_ _e"> </span>banded</span></div><div class="t m0 x11 h2 y2a ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">singly<span class="_ _5"> </span>(or<span class="_ _a"> </span>doubly)<span class="_ _a"> </span>bordered<span class="_ _a"> </span>block<span class="_ _a"> </span>diagonal</span></div><div class="t m0 x11 h2 y2b ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">singly<span class="_ _5"> </span>(or<span class="_ _a"> </span>doubly)<span class="_ _a"> </span>bordered<span class="_ _4"> </span>block<span class="_ _5"> </span>triangular</span></div><div class="t m0 x11 h2 y2c ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">singly<span class="_ _5"> </span>(or<span class="_ _a"> </span>doubly)<span class="_ _a"> </span>bordered<span class="_ _a"> </span>band<span class="_ _a"> </span>diagonal</span></div><div class="t m0 x11 h2 y2d ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">singly<span class="_ _5"> </span>(or<span class="_ _a"> </span>doubly)<span class="_ _a"> </span>bordered<span class="_ _a"> </span>band<span class="_ _a"> </span>triangular</span></div><div class="t m0 x11 h2 y2e ffa fs1 fc0 sc0 ls0 ws0">&#8226;<span class="_ _d"> </span><span class="ff2">other<span class="_ _d"> </span>(!)</span></div><div class="t m0 x7 h2 y2f ff2 fs1 fc0 sc0 ls0 ws0">Y<span class="_ _c"></span>ou<span class="_ _5"> </span>should<span class="_ _5"> </span>also<span class="_ _7"> </span>be<span class="_ _5"> </span>aware<span class="_ _5"> </span>of<span class="_ _7"> </span>some<span class="_ _5"> </span>of<span class="_ _7"> </span>the<span class="_ _7"> </span>special<span class="_ _5"> </span>sparse<span class="_ _7"> </span>forms<span class="_ _7"> </span>that<span class="_ _5"> </span>occur<span class="_ _7"> </span>in<span class="_ _5"> </span>the</div><div class="t m0 x7 h2 y30 ff2 fs1 fc0 sc0 ls0 ws0">solution<span class="_ _0"> </span>of<span class="_ _0"> </span>partial<span class="_ _0"> </span>differential<span class="_ _9"> </span>equations<span class="_ _9"> </span>in<span class="_"> </span>two<span class="_ _9"> </span>or<span class="_"> </span>more<span class="_ _0"> </span>dimensions.<span class="_ _a"> </span>See<span class="_"> </span>Chapter<span class="_ _9"> </span>19.</div><div class="t m0 x8 h2 y31 ff2 fs1 fc0 sc0 ls0 ws0">If<span class="_"> </span>your<span class="_ _4"> </span>particular<span class="_"> </span>pattern<span class="_"> </span>of<span class="_"> </span>sparsity<span class="_ _4"> </span>is<span class="_ _4"> </span>not<span class="_"> </span>a<span class="_ _4"> </span>simple<span class="_ _4"> </span>one,<span class="_"> </span>then<span class="_"> </span>you<span class="_ _4"> </span>may<span class="_"> </span>wish<span class="_ _4"> </span>to</div><div class="t m0 x7 h2 y32 ff2 fs1 fc0 sc0 ls0 ws0">try<span class="_"> </span>an<span class="_ _9"> </span><span class="ff5">analyze/factorize/operate<span class="_ _0"> </span></span>package,<span class="_ _9"> </span>which<span class="_"> </span>automates<span class="_ _0"> </span>the<span class="_"> </span>procedure<span class="_ _2"> </span>of<span class="_"> </span>&#64257;guring</div><div class="t m0 x7 h2 y33 ff2 fs1 fc0 sc0 ls0 ws0">out<span class="_ _4"> </span>ho<span class="_ _8"></span>w<span class="_ _4"> </span>&#64257;ll-ins<span class="_ _4"> </span>are<span class="_ _4"> </span>to<span class="_ _4"> </span>be<span class="_ _4"> </span>minimized.<span class="_ _e"> </span>The<span class="_ _4"> </span><span class="ff5">analyze<span class="_ _4"> </span></span>stage<span class="_ _4"> </span>is<span class="_ _4"> </span>done<span class="_ _4"> </span>once<span class="_"> </span>only<span class="_ _4"> </span>for<span class="_ _4"> </span>each</div><div class="t m0 x7 h2 y34 ff2 fs1 fc0 sc0 ls0 ws0">pattern<span class="_ _4"> </span>of<span class="_ _4"> </span>sparsity<span class="_ _c"></span>.<span class="_ _3"> </span>The<span class="_ _4"> </span><span class="ff5">factorize<span class="_ _4"> </span></span>stage<span class="_ _a"> </span>is<span class="_ _4"> </span>done<span class="_ _4"> </span>once<span class="_ _4"> </span>for<span class="_ _4"> </span>each<span class="_ _4"> </span>particular<span class="_ _4"> </span>matrix<span class="_ _4"> </span>that</div><div class="t m0 x7 h2 y35 ff2 fs1 fc0 sc0 ls0 ws0">&#64257;ts<span class="_ _5"> </span>the<span class="_ _5"> </span>pattern.<span class="_ _12"> </span>The<span class="_ _5"> </span><span class="ff5">operate<span class="_ _a"> </span></span>stage<span class="_ _5"> </span>is<span class="_ _5"> </span>performed<span class="_ _a"> </span>once<span class="_ _5"> </span>for<span class="_ _a"> </span>each<span class="_ _5"> </span>right-hand<span class="_ _a"> </span>side<span class="_ _5"> </span>to</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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