From ea73cb66aa910579df478c91578b488c80aeaea9 Mon Sep 17 00:00:00 2001 From: "Claudio Maggioni (maggicl)" Date: Sat, 16 May 2020 13:58:14 +0200 Subject: [PATCH] hw4: done 1,2,3,4 --- hw4/hw4.pdf | Bin 98567 -> 104291 bytes hw4/hw4.tex | 99 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 99 insertions(+) diff --git a/hw4/hw4.pdf b/hw4/hw4.pdf index 6c6585c614762cfd9840545462296850c28aecc1..5fc10f86aad8816827daaf035f6089664201dcfe 100644 GIT binary patch delta 51547 zcmZs?Q*fY7)UF*)Y)@?4wr$(C@x+?gwk8wXoOoi}ww?U%SG&IY5B5G=)n~n`?(VBs zT^$kpW54D{9TzBQMk6<uKskZhnf18XskM1s z!w~QWS%(}Rq}0&%rbRW`G&q25MpW3=LA^B{@ZF(ReNB*-PX3cUWCBUE_v_&oQyB6; zYRE4ZgOmasKug{5cBC|HHFzl`^K&fBPp|q@(C5j$xk9kYF*_f~27GcD1=$U^8JAj- z(#idS`{y@-g7;^X4VF?n#b%#qP{09 z1>5)Pab=GLZ+vE-v8llv8iPlnr^9o*W9<;h_kY3y#F@zrJq5xTg^CSoazR8EnGIzI z&-vjwr-`)oss(;FFfF_c}MQ2uHPTbEi9>Q&H&B)Ux499z->)Ko{=O1HDXM${F+9whbJ?# zn>e}B6u;1k7!lJH#1;}q&kP1w2pgU2Y_?^5%PwLx?2E#?&Bc&U*@k9l^OJJ7*U 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a/hw4/hw4.tex b/hw4/hw4.tex index 48eb38f..be039b7 100644 --- a/hw4/hw4.tex +++ b/hw4/hw4.tex @@ -75,4 +75,103 @@ $$a_2 = f[0,1,3] = \frac{2-(-3)}{3-0} = \frac{5}{3}$$ $$p(x) = \left(\frac{5}{3}(x-1)-3\right) x + 0 = \frac{5}{3}x^2 - \frac{14}{3}x$$ The interpolating polynomials are indeed equal. + +\section*{Question 4} +\subsection*{Point a)} + +The node coordinates to which fix the quadratic spline are $(1/2, y_0), (1, y_1),(3/2, y_2),(2, y_3)$. + +Then, we can start formulating the equations for the linear system: + +$$y_0 = s\left(\frac{1}{2}\right) = a_{-1}B_2\left(\frac{1}{2} + 1\right) + a_{0}B_2\left(\frac{1}{2}\right) + a_{1}B_2\left(\frac{1}{2} - 1\right) + $$$$ ++ a_{2}B_2\left(\frac{1}{2} - 2\right) ++ a_{-1}B_2\left(\frac{1}{2} - 3\right) ++ a_{0}B_2\left(\frac{1}{2} - 4\right) ++ a_{1}B_2\left(\frac{1}{2} - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot \frac{1}{2} + a_{1} \cdot \frac{1}{2} + a_{2} \cdot 0 + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{a_0 + a_1}{2}$$ + +$$y_1 = s\left(1\right) = a_{-1}B_2\left(1 + 1\right) + a_{0}B_2\left(1\right) + a_{1}B_2\left(1 - 1\right) + $$$$ ++ a_{2}B_2\left(1 - 2\right) ++ a_{-1}B_2\left(1 - 3\right) ++ a_{0}B_2\left(1 - 4\right) ++ a_{1}B_2\left(1 - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot \left(\frac{1}{8}\right) + a_{1} \cdot \frac{3}{4} + a_{2} \cdot \left(\frac{1}{8}\right) + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = +\frac{1}{8} a_{0} + \frac{3}{4} a_{1} + \frac{1}{8} a_{2}$$ + +$$y_2 = s\left(\frac{3}{2}\right) = a_{-1}B_2\left(\frac{3}{2} + 1\right) + a_{0}B_2\left(\frac{3}{2}\right) + a_{1}B_2\left(\frac{3}{2} - 1\right) + $$$$ ++ a_{2}B_2\left(\frac{3}{2} - 2\right) ++ a_{-1}B_2\left(\frac{3}{2} - 3\right) ++ a_{0}B_2\left(\frac{3}{2} - 4\right) ++ a_{1}B_2\left(\frac{3}{2} - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot \frac{1}{2} + a_{2} \cdot \frac{1}{2} + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{a_1 + a_2}{2}$$ + +$$y_3 = s\left(2\right) = a_{-1}B_2\left(2 + 1\right) + a_{0}B_2\left(2\right) + a_{1}B_2\left(2 - 1\right) + $$$$ ++ a_{2}B_2\left(2 - 2\right) ++ a_{-1}B_2\left(2 - 3\right) ++ a_{0}B_2\left(2 - 4\right) ++ a_{1}B_2\left(2 - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot \left(\frac{1}{8}\right) + a_{2} \cdot \frac{3}{4} + a_{-1} \cdot \left(\frac{1}{8}\right) + a_{0} \cdot 0 + a_{1} \cdot 0 = +\frac{1}{8} a_{1} + \frac{3}{4} a_{2} + \frac{1}{8} a_{-1}$$ + +The linear system in matrix form is: + +\[\frac{1}{8} \cdot +\begin{bmatrix} +4&4&0&0\\ +1&6&1&0\\ +0&4&4&0\\ +0&1&6&1\\ +\end{bmatrix} +\begin{bmatrix} +a_0\\a_1\\a_2\\a_{-1}\\ +\end{bmatrix} += +\begin{bmatrix} +y_0\\y_1\\y_2\\y_{3}\\ +\end{bmatrix}\] + +The determinant of that matrix is 0 so the matrix is singular. + +\subsection*{Point b)} + +The node coordinates to which fix the quadratic spline are $(0, y_0), (1, y_1),(2, y_2),(3, y_3)$. + +Then, we can start formulating the equations for the linear system: +$$y_0 = s(0) = a_{-1}B_2\left(1\right) + a_{0}B_2\left(0\right) + a_{1}B_2\left(- 1\right) + $$$$ ++ a_{2}B_2\left(- 2\right) ++ a_{-1}B_2\left(- 3\right) ++ a_{0}B_2\left(- 4\right) ++ a_{1}B_2\left(- 5\right)=$$$$ +a_{-1} \cdot \frac{1}{8} + a_{0} \cdot \frac{3}{4} + a_{1} \cdot \frac{1}{8} + a_{2} \cdot 0 + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = +\frac{1}{8} a_{-1} + \frac{3}{4} a_{0} + \frac{1}{8} a_{1}$$ + +$$y_1 = s\left(1\right) = \frac{1}{8} a_{0} + \frac{3}{4} a_{1} + \frac{1}{8} a_{2}$$ + +$$y_2 = s\left(2\right) = \frac{1}{8} a_{1} + \frac{3}{4} a_{2} + \frac{1}{8} a_{-1}$$ + +$$y_3 = s\left(3\right) = a_{-1}B_2\left(3 + 1\right) + a_{0}B_2\left(3\right) + a_{1}B_2\left(3 - 1\right) + $$$$ ++ a_{2}B_2\left(3 - 2\right) ++ a_{-1}B_2\left(3 - 3\right) ++ a_{0}B_2\left(3 - 4\right) ++ a_{1}B_2\left(3 - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 + a_{2} \cdot \frac{1}{8} + a_{-1} \cdot \frac{3}{4} + a_{0} \cdot \frac{1}{8} + a_{1} \cdot 0 = +\frac{1}{8} a_{2} + \frac{3}{4} a_{-1} + \frac{1}{8} a_{0}$$ + +The linear system in matrix form is: + +\[\frac{1}{8} \cdot +\begin{bmatrix} +6&1&0&1\\ +1&6&1&0\\ +0&1&6&1\\ +1&0&1&6\\ +\end{bmatrix} +\begin{bmatrix} +a_0\\a_1\\a_2\\a_{-1}\\ +\end{bmatrix} += +\begin{bmatrix} +y_0\\y_1\\y_2\\y_{3}\\ +\end{bmatrix}\] + \end{document}