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+ StyleBox[ + PaneBox[ + TagBox[ + GridBox[{{ + TagBox[ + GridBox[{{ + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + AbsoluteThickness[1.6], + GrayLevel[0]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + AbsoluteThickness[1.6], + GrayLevel[0]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + AbsoluteThickness[1.6], + RGBColor[1, 0, 0]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + AbsoluteThickness[1.6], + RGBColor[1, 0, 0]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}}, + GridBoxAlignment -> { + "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, + AutoDelete -> False, + GridBoxDividers -> { + "Columns" -> {{False}}, "Rows" -> {{False}}}, + GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, + GridBoxSpacings -> { + "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, + GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, + AutoDelete -> False, + GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], + "Grid"], Alignment -> Left, AppearanceElements -> None, + ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> + "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { + FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> + False], TraditionalForm]& ), + Editable->True, + InterpretationFunction:>(RowBox[{"LineLegend", "[", + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",", + InterpretationBox[ + ButtonBox[ + TooltipBox[ + GraphicsBox[{{ + GrayLevel[0], + RectangleBox[{0, 0}]}, { + GrayLevel[0], + RectangleBox[{1, -1}]}, { + GrayLevel[0], + RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> + True, FrameStyle -> GrayLevel[0.], FrameTicks -> None, + PlotRangePadding -> None, ImageSize -> + Dynamic[{ + Automatic, 1.35 CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification]}]], "GrayLevel[0]"], + Appearance -> None, BaseStyle -> {}, BaselinePosition -> + Baseline, DefaultBaseStyle -> {}, ButtonFunction :> + With[{Typeset`box$ = EvaluationBox[]}, + If[ + Not[ + AbsoluteCurrentValue["Deployed"]], + SelectionMove[Typeset`box$, All, Expression]; + FrontEnd`Private`$ColorSelectorInitialAlpha = 1; + FrontEnd`Private`$ColorSelectorInitialColor = + GrayLevel[0]; + FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; + MathLink`CallFrontEnd[ + FrontEnd`AttachCell[Typeset`box$, + FrontEndResource["GrayLevelColorValueSelector"], { + 0, {Left, Bottom}}, {Left, Top}, + "ClosingActions" -> { + "SelectionDeparture", "ParentChanged", + "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> + Automatic, Method -> "Preemptive"], + GrayLevel[0], Editable -> False, Selectable -> False]}], + "]"}], ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",", + InterpretationBox[ + ButtonBox[ + TooltipBox[ + GraphicsBox[{{ + GrayLevel[0], + RectangleBox[{0, 0}]}, { + GrayLevel[0], + RectangleBox[{1, -1}]}, { + RGBColor[1, 0, 0], + RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> + True, FrameStyle -> RGBColor[0.6666666666666666, 0., 0.], + FrameTicks -> None, PlotRangePadding -> None, ImageSize -> + Dynamic[{ + Automatic, 1.35 CurrentValue["FontCapHeight"]/ + AbsoluteCurrentValue[Magnification]}]], + "RGBColor[1, 0, 0]"], Appearance -> None, BaseStyle -> {}, + BaselinePosition -> Baseline, DefaultBaseStyle -> {}, + ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, + If[ + Not[ + AbsoluteCurrentValue["Deployed"]], + SelectionMove[Typeset`box$, All, Expression]; + FrontEnd`Private`$ColorSelectorInitialAlpha = 1; + FrontEnd`Private`$ColorSelectorInitialColor = + RGBColor[1, 0, 0]; + FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; + MathLink`CallFrontEnd[ + FrontEnd`AttachCell[Typeset`box$, + FrontEndResource["RGBColorValueSelector"], { + 0, {Left, Bottom}}, {Left, Top}, + "ClosingActions" -> { + "SelectionDeparture", "ParentChanged", + "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> + Automatic, Method -> "Preemptive"], + RGBColor[1, 0, 0], Editable -> False, Selectable -> + False]}], "]"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{#, ",", #2}], "}"}], ",", + RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", + RowBox[{"LabelStyle", "\[Rule]", + RowBox[{"{", "}"}]}], ",", + RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )], + Scaled[{0.75, 0.75}], ImageScaled[{0.5, 0.5}], + BaseStyle->{FontSize -> Larger}, + FormatType->StandardForm]}, + AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], + Axes->{True, True}, + AxesLabel->{None, None}, + AxesOrigin->{0, 0}, + DisplayFunction->Identity, + Frame->{{False, False}, {False, False}}, + FrameLabel->{{None, None}, {None, None}}, + FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, + GridLines->{None, None}, + GridLinesStyle->Directive[ + GrayLevel[0.5, 0.4]], + Method->{ + "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> + AbsolutePointSize[6], "ScalingFunctions" -> None}, + PlotRange->{{-1, 1}, {0., 0.9999994079562567}}, + PlotRangeClipping->True, + PlotRangePadding->{{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, + Ticks->{Automatic, Automatic}], + InterpretTemplate[Legended[ + 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a/Lectures_my/NumMet/Lecture3/lecture3.tex +++ b/Lectures_my/NumMet/Lecture3/lecture3.tex @@ -81,16 +81,16 @@ we have $R[P] \equiv \max_{a \leq x \leq b} |f (x) - P(x)| < \varepsilon$, [\dots] \end{block} Using interpolation, we can construct a polynomial $P_n(x)$ of degree $n$ - that approximates $f$ up to an approximation error of $R_n(x)$: + that approximates $f$ up to an approximation error of $R_n$: \begin{equation*} - R_n(x) \equiv \max_{a \leq x \leq b} \left|f(x) - P_n(x)\right|\,. + R_n \equiv \max_{a \leq x \leq b} \left|f(x) - P_n(x)\right|\,. \end{equation*} \begin{alert}{Conclusion} Weierstrass's theorem says that there is \emph{at least one} polynomial for each choice of the residual error $\varepsilon$. It does not say that either \begin{itemize} - \item $P_n$ is in the set of polynomial for a given $\varepsilon$, or + \item $P_n$ is \emph{in} the set of polynomial for a given $\varepsilon$, or \item $R_n \to 0$ if $n \to \infty$! \end{itemize} \begin{center}{\color{red} Increasing $n$ might be harmful!}\end{center} @@ -99,9 +99,9 @@ \begin{frame}{When Interpolation fails} So far, you have been confronted with examples in which interpolation -works nicely. Let us now discuss an example, which behaves pathologically.\\[\medskipamount] +works nicely. Let us now discuss an example, which behaves pathologically.\\ -\begin{overlayarea}{\textwidth}{6cm} +\begin{overlayarea}{\textwidth}{7cm} \begin{columns} \begin{column}[T]{.5\textwidth} Consider the classical example by Runge: @@ -115,6 +115,7 @@ \item<2-> {\color{blue}the interpolating polynomial to degree $4$} \item<3-> {\color{red}the interpolating polynomial to degree $6$} \end{itemize} +\only<2->{for equidistant interpolation points on $[-1, +1]$.} \end{column} \begin{column}[T]{.5\textwidth} \begin{tikzpicture} @@ -138,17 +139,26 @@ } \end{axis} \end{tikzpicture} + \begin{center}\only<4->{$R_n \to \infty$ as $n \to \infty$}\end{center} \end{column} \end{columns} - \begin{center}\only<4->{$R_n \to \infty$ as $n \to \infty$}\end{center} \end{overlayarea} \end{frame} \begin{frame}{Pieceswise linear/cubic/... interpolations} - Piecewise interpolation of the target function $f$ can be achieved - with relatively small degrees of freedom for each of the interpolating polynomials. + \begin{columns} + \begin{column}{.6\textwidth} + This pathological behaviour can be avoided by piecewise interpolation with a small value of + the polynomial degree $n$. Most popular are linear ($n=1$) or cubic ($n=3$) piecewise interpolations - (``splines''), which do not suffer from the problem that was just described. + (``splines''), which do not suffer from the problem that was just described.\\ \medskip + \end{column} + \begin{column}{.4\textwidth} + \includegraphics[width=.9\textwidth]{runge.pdf} + \end{column} + \end{columns} + \medskip + The concrete approaches have been discussed last lecture by Marcin. \end{frame} \begin{frame}{Interpolation in more than $D=1$ dimensions} @@ -311,14 +321,55 @@ \end{itemize} \end{frame} -\begin{frame}{Algorithm for Bilinear Interpolation} +\begin{frame}[shrink]{Algorithm for Bilinear Interpolation} + \begin{columns} + \begin{column}[T]{.8\textwidth} \begin{enumerate} - \item Create a rectilinear grid in the $(x,y)$ plane - \item For each rectangle, map the rectangle to the unit square - \item Evaluate the function $f$ on the four - corners of the (mapped) unit square $P_{0,0}$, $P_{0, 1}$, $P_{1, 1}$, $P_{1, 0}$. - \item Make an ansatz: $f(x, y) = a_{0,0} + a_{1,0} x + a_{0,1} y + a_{1,1} x y$. - \item Solve the interpolation equation + \item<1-> Create a rectilinear grid in the $(x,y)$ plane + \item<2-> For each rectangle, map the rectangle to the unit square + \item<3-> Evaluate the function $f$ on the four + corners of the (mapped) unit square ${\color{purple}Q_{0,0}}$, + ${\color{red}Q_{1, 0}}$, ${\color{orange}Q_{1, 1}}$, ${\color{Gold}Q_{0, 1}}$. + \item<4-> Make an ansatz: $P(x, y) = a_{0,0} + a_{1,0} x + a_{0,1} y + a_{1,1} x y$. + \end{enumerate} + \end{column} + \begin{column}[T]{.2\textwidth} + \resizebox{\textwidth}{!}{ + \begin{tikzpicture} + \draw[black,thick] (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0); + \draw[black,thick] (1,0) -- (2,0) -- (2,1) -- (1,1) -- (1,0); + \draw[black,thick] (0,1) -- (1,1) -- (1,2) -- (0,2) -- (0,1); + \draw[black,thick] (1,1) -- (2,1) -- (2,2) -- (1,2) -- (1,1); + \only<2->{ + \fill[gray,thick] (0,1) -- (1,1) -- (1,2) -- (0,2) -- (0,1); + } + \only<3->{ + \fill[purple] (0,1) circle [radius=2pt]; + \fill[red] (1,1) circle [radius=2pt]; + \fill[orange] (1,2) circle [radius=2pt]; + \fill[Gold] (0,2) circle [radius=2pt]; + } + \end{tikzpicture} + \bigskip + } + \only<2->{ + \resizebox{\textwidth}{!}{ + \begin{tikzpicture} + \draw[black,thick] (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0); + \only<3->{ + \fill[purple] (0,0) circle [radius=2pt]; + \fill[red] (1,0) circle [radius=2pt]; + \fill[orange] (1,1) circle [radius=2pt]; + \fill[Gold] (0,1) circle [radius=2pt]; + } + \end{tikzpicture} + } + } + \end{column} + \end{columns} + \begin{enumerate} + \setcounter{enumi}{4} + \item<5-> Solve the interpolation equation \begin{equation*} \left[\begin{matrix} 1 & 0 & 0 & 0\\ @@ -335,13 +386,15 @@ \end{matrix}\right] = \left[\begin{matrix} - f(P_{0, 0})\\ - f(P_{1, 0})\\ - f(P_{0, 1})\\ - f(P_{1, 1}) + f(Q_{0, 0})\\ + f(Q_{1, 0})\\ + f(Q_{0, 1})\\ + f(Q_{1, 1}) \end{matrix}\right] \end{equation*} - \item Map the unit square back to the $(x,y)$ plane. + We only need to invert the Vandermonde matrix once! + \item<6-> Map the result back from the unit square back to the grid piece in $(x,y)$ plane. + \item<7-> Jump back to \#2. \end{enumerate} \end{frame} @@ -635,8 +688,12 @@ \begin{columns} \begin{column}[T]{.5\textwidth} - The problem here is the starting point, and the existence of a bump between the starting - point $h_1 = 1/2$ and the point of interest ($h=0$). + The problem here are + \begin{itemize} + \item the choice of the starting point, + \item the existence of a bump between the starting + point $h_1 = 1/2$ and the point of interest ($h=0$). + \end{itemize} \end{column} \begin{column}[T]{.5\textwidth} \begin{tikzpicture} @@ -658,6 +715,67 @@ \end{columns} \end{frame} +\begin{frame}{Speeding up extrapolations} + Let's assume you have carried out an extrapolation to some point $x_0$ + for several different values of the degree $n$: + \begin{align*} + P_n(x) & :\,\text{interpolating polynomial} & + p_n & \equiv P_n(x_0)\,. + \end{align*} + The $p_n$ are a sequence that (hopefully) converges to the value you seek.\\ + + For definiteness, let's use the previous example of $f(x) = \cos(x)$. The first + few elements of the sequence $p_n$ were: + \begin{align*} + p_0 & = 0.877583 & + p_1 & = 1.06024 & + p_2 & = 1.00056 \\ + p_3 & = 0.99996 & + p_4 & = 1.00000 + \end{align*} + + Assuming that any sequence $p_n$ is convergent, it would be nice to have a way + to accelerate the sequence without any additional (potentially costly!) evaluations + of the function $f(x)$! +\end{frame} + +\begin{frame}{Delta-Square rule by Aitkens} + Define a new sequence $q_n$ as follows: + \begin{equation*} + q_n \equiv p_n - \frac{\left[\Delta(p)_{n}\right]^2}{\Delta^2(p)_{n}}\,, + \end{equation*} + where + \begin{align*} + \Delta(p)_k & = p_{k + 1} - p_k\,, & + \Delta^2(p)_k & = \Delta(p)_{k + 1} - \Delta(p)_k\,. + \end{align*} + This could be written simpler, but in order to avoid numerical instabilities (floating point numbers!) + the version written here is preferable. +\end{frame} + +\begin{frame}{Accelerating our example} + \begin{tikzpicture} + \begin{axis}[% + ymin=0.99990, + ymax=1.00010, + xlabel=$n$ + ] + \addplot[black, mark=*, only marks] coordinates { + (3, 0.99996) + (4, 1.00000) + (5, 1.00000) + }; + \only<2->{ + \addplot[blue, mark=*, only marks] coordinates { + (3, 0.999954) + (4, 0.999998) + }; + } + \addplot[red, mark=none, domain=0:6] { 1 }; + \end{axis} + \end{tikzpicture} +\end{frame} + \backupbegin \begin{frame}\frametitle{Backup}