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July 23rd, 2018, 03:13 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3  Need some insights for approximating curves
Data Point: (3, 1), (1, 0), (0, 5), (2, 0), (4, 1) Using the following basis to approxmiate the above points.. The five basis: $\displaystyle 1, \sin(x \frac{\pi}{2}), \sin(x \frac{\pi}{4}), \cos(x \frac{\pi}{2}), \cos(x \frac{\pi}{6}) $ $\displaystyle \begin{bmatrix} 1 & 1 & \frac{1}{\sqrt{2}} & 0 & 0 \\ 1 & 1 & \frac{1}{\sqrt{2}} & 0 & \frac{\sqrt{3}}{2} \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0.5 \\ 1 & 0 & 0 & 1 & 0.5 \end{bmatrix} \begin{bmatrix} t_4 \\ t_3 \\ t_2 \\ t_1 \\ t_0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 5 \\ 0 \\ 1 \end{bmatrix} $ Would anyone please share any insights as to how the above big matrix was formed? Last edited by skipjack; July 24th, 2018 at 12:54 AM. 
July 23rd, 2018, 06:52 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,427 Thanks: 1314 
let that matrix be $A$ let $b$ be a vector of the 5 basis functions as a function of $x$ let $d = (d_1,d_2)$ be the vector of ordered pairs $A_{i,j} = b_j(d_{i,1})$ 
July 24th, 2018, 03:25 PM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
How the basis were chosen? I sensed there were hidden rules for choosing these 5 basis. I did some experiments with the following basis of my choosing. $\displaystyle 1, \sin(x \frac{\pi}{2}), \cos(x \frac{\pi}{4}), \sin(x \frac{\pi}{6}), \cos(x \frac{\pi}{8}) $ It did not yield the correct result of t0, t1, t2, t3 and t4. Last edited by skipjack; July 24th, 2018 at 05:35 PM. 
July 24th, 2018, 03:32 PM  #4 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
I think it is based on Fourier series....

July 24th, 2018, 04:11 PM  #5 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
Here is the formula: $\displaystyle y(x) = a_0 + a_1 \sin(x \frac{\pi}{2}) + a_2 \sin(x \frac{\pi}{4}) + a_k \sin(x \frac{\pi}{2k}) + b_1 \cos(x \frac{\pi}{2}) + b_2 \cos(x \frac{\pi}{4}) +\, ... + b_{k1} \cos( x \frac{\pi}{2(k  1)}) $ Last edited by skipjack; July 24th, 2018 at 05:34 PM. 
July 25th, 2018, 05:14 AM  #6 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
But.. what term should I pick for the matrix of nxn sizze??

July 25th, 2018, 03:57 PM  #7 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3  Would anyone able to tell me what formula is that? I thought it was fourier but it wasn't....

July 25th, 2018, 05:14 PM  #8 
Senior Member Joined: Sep 2015 From: USA Posts: 2,427 Thanks: 1314  

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approximating, curves, insights 
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