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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 82490, 1715]*) (*NotebookOutlinePosition[ 83357, 1745]*) (* CellTagsIndexPosition[ 83313, 1741]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Ortogonaalsed pol\[UDoubleDot]noomid", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["Legendre pol\[UDoubleDot]noom", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["Defineerime funtsiooni Legendre'i pol\[UDoubleDot]noomide leidmiseks: \ ", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(L[n_, x_] := ExpandNumerator[ Simplify[\[PartialD]\_{x, n}\((x\^2 - 1)\)\^n\/\(2\^n\ \(n!\)\)]]\)], \ "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Siin D[f,{x,n}] leiab f n-nda tuletise x j\[ADoubleDot]rgi. // Simplify lihtsustab avalise ja // ExpandNumerator avab lugejas sulud.\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Lpoly1 = L[1, x]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Lpoly2 = L[2, x]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ M\[ADoubleDot]rkus. Mathematical on ka omal olemas funktsioon Legendre pol\ \[UDoubleDot]noomide leidmiseks. See on LegendreP[n,x].\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Joonestame Legendre pol\[UDoubleDot]noomide graafikud.", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Plot[{Lpoly1, Lpoly2}, {x, \(-1.5\), 1.5}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-1.5\), 1.5}, GridLines \[Rule] Automatic]; \)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell[CellGroupData[{ Cell["Tsheb\[OTilde]shevi pol\[UDoubleDot]noomid", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Tsheb\[OTilde]shevi pol\[UDoubleDot]noomi leidmisel v\[OTilde]iks l\ \[ADoubleDot]htuda valemist T[n_,x_] := Cos[n ArcCos[x]], kuid sel juhul tekivad hiljem raskused trigonomeetriliste avaliste \ lihtsustamisel. Seet\[OTilde]ttu on parem kasutada rekurrentset seost. Anname ette T0(x), \ T1(x) ja eeskirja, kuidas kahe eelneva pol\[UDoubleDot]noomi abil arvutada j\[ADoubleDot]rgmine \ pol\[UDoubleDot]noom.\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[{ \(T[0, x_] := 1; \), "\n", \(T[1, x_] := x; \), "\n", \(T[n_, x_] := Expand[2\ x\ T[n - 1, x] - T[n - 2, x]]\)}], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Tpoly2 = T[2, x]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Tpoly3 = T[3, x]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ M\[ADoubleDot]rkus. Mathematica funktsioon Tsheb\[OTilde]shevi pol\ \[UDoubleDot]noomide leidmiseks on ChebyshevT[n,x].\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Joonestame Tsheb\[OTilde]shevi pol\[UDoubleDot]noomide graafikud.", \ "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Plot[{T[1, x], Tpoly2, Tpoly3}, {x, \(-1.5\), 1.5}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-1.5\), 1.5}, GridLines \[Rule] Automatic]; \)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Numbriline diferentseerimine", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Numbrilisel diferentseerimisel leitakse k\[OTilde]igepealt andmetele \ interpolatsioonipol\[UDoubleDot]noom. Teoorias l\[ADoubleDot]htutakse sageli Newtoni interpolatsioonipol\ \[UDoubleDot]noomist. Meie siin kasutame Lagrange interpolatsioonipol\[UDoubleDot]noomi, sest selle leidmiseks vajalik \ programm oli meil mitme muutuja funktsiooni interpoleerimisel juba tarvitusel.\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(LagrangePoly[xx_List, ff_List, x_Symbol] := Module[{n = Length[xx]}, Expand[\[Sum]\+\(i = 1\)\%n\((\[Product]\+\(k = 1\)\%\(n - 1\)\(x - \ \(Drop[xx, {i}]\)\[LeftDoubleBracket]k\[RightDoubleBracket]\)\/\(xx\ \[LeftDoubleBracket]i\[RightDoubleBracket] - \(Drop[xx, {i}]\)\ \[LeftDoubleBracket]k\[RightDoubleBracket]\))\)\ ff\[LeftDoubleBracket] i\[RightDoubleBracket]]]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["L\[ADoubleDot]hteandmed on kujul {{x1,y1}, {x2,y2}, ...}", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[{ \(s\[OTilde]lmed = {{1, 2}, {1.5, 3.8}, {2, 4.2}, {2.8, 3.1}}; \), "\n", \(xx = s\[OTilde]lmed /. \[InvisibleSpace]{x_, y_} \[Rule] x; \), "\n", \(yy = s\[OTilde]lmed /. \[InvisibleSpace]{x_, y_} \[Rule] y; \)}], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Leiame interpolatsioonipol\[UDoubleDot]noomi:", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Lpoly = LagrangePoly[xx, yy, x]\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Leiame selle pol\[UDoubleDot]noomi tuletise:", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(dL = \[PartialD]\_x Lpoly\)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Seega, tuletise v\[ADoubleDot]\[ADoubleDot]rtus n\[ADoubleDot]iteks \ punktis x0 = 1.7 on:", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[{ \(x0 = 1.7; \), "\n", \(k = \[PartialD]\_x Lpoly /. \[InvisibleSpace]x \[Rule] x0\)}], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Graafiline interpretatsioon.", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Teatavasti v\[OTilde]ib funktsiooni tuletist vaadelda selle funktsiooni \ graafiku puutuja t\[OTilde]usuna. Kanname joonisele intrpolatsioonis\[OTilde]lmedeks olevad punktid, \ interpolatsioonipol\[UDoubleDot]noomi, punkti (x0,y0) ja selles punktis v\[OTilde]etud puutuja.\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[{ \(y0 = Lpoly /. \[InvisibleSpace]x \[Rule] x0; \), "\n", \(puutuja = \((x - x0)\)\ k + y0\)}], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Puutuja v\[OTilde]rrandi koostamisel l\[ADoubleDot]htume sirge \ v\[OTilde]rrandist y = (x-x0)k + y0\ \>", "Info", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Plot[{Lpoly, puutuja}, {x, 0, 3}, Epilog \[Rule] {PointSize[0.03], Point /@ s\[OTilde]lmed, Hue[0], Point[{x0, y0}]}]; \)], "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ \[CapitalUDoubleDot]lesanne: on antud punktid {{1,-0.2},{2,0.3},{3,1},{4,2.5},{5,2},{6,1.3}}. 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