least square approximations

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

Re: least square approximations

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**alphabeta89** Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

More context would help with this, like is the data known and you want the actual coefficients, or do you want something like the regression equations?

CB

Re: least square approximations

Quote:

Originally Posted by

**CaptainBlack** More context would help with this, like is the data known and you want the actual coefficients, or do you want something like the regression equations?

CB

Sorry! I realised I left out some details!

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Define $\displaystyle f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$

.

Define $\displaystyle f(t_{j})$ by $\displaystyle x_{j}$ for $\displaystyle j=0,...,99.$Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

Re: least square approximations

Quote:

Originally Posted by

**alphabeta89** Sorry! I realised I left out some details!

Let $\displaystyle t_{j}=j/100, a_{j}=j,b_{j}=-j$ for $\displaystyle j=0,...,99$. Define

$\displaystyle f(t)= \sum_{k=0}^{99}(a_{k}\textup{cos}(2\pi{kt})+b_{k}{ sin}(2\pi{kt}))$

.

Define $\displaystyle f(t_{j})$ by $\displaystyle x_{j}$ for $\displaystyle j=0,...,99.$Determine the values of $\displaystyle c_{l},d_{m}$ for $\displaystyle l=0,...,5,m=1,...,4$, so that

$\displaystyle P(t)=c_{0}+\sum_{k=1}^{4}(c_{k}\textup{cos}(2\pi{k t})+d_{k}\textup{sin}(2\pi{kt}))+c_{5}\textup{cos} (10\pi{t})$

is the least squares approximation to the data points $\displaystyle (t_j,x_j)$ for $\displaystyle j=0,...,99$.

This is now a numerical problem, consider using the Excel solver to find the best fit.

(Fourier theory suggests that $\displaystyle c_0\approx 0$ and $\displaystyle c_i\approx 100$ for $\displaystyle i=1..5$ and $\displaystyle d_i\approx 100$ for $\displaystyle i=1 .. 4$ )

CB