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• Additional problem 1: Consider least squares approximation of a function $f(x) = e^{-x}$ on the continuous interval $[0, 1]$ as a linear combination $f(x) \approx \sum_{k=0}^{n} a_k g_k(x)$, where we choose $g_k(x) = x^k$. Construct the resulting normal equations in the form $Aa = b$ and solve them using CG. You should be able to derive analytic formulas for the entries $a_{ij}$ and $b_i$, but you should then construct and solve the system $Aa = b$ numerically. (To compute $b_i$ analytically, you will need to use integration by parts.) Plot your recovered approximation $\sum_{k=0}^{n} a_k g_k(x)$ on the interval $[0, 1]$ for $n = 5, 10$. Meanwhile, what do you notice about the convergence rate of CG as you take $n$ much larger? Support your observation with numerical evidence / convergence plots. What could be causing this behavior?

          • Additional problem 1: Consider least squares approximation of a function $f(x) = e^{-x}$ on the continuous interval $[0, 1]$ as a linear combination

$f(x) \approx \sum_{k=0}^{n} a_k g_k(x)$, 

where we choose $g_k(x) = x^k$. Construct the resulting normal equations in the form $Aa = b$ and solve them using CG. You should be able to derive analytic formulas for the entries $a_{ij}$ and $b_i$, but you should then construct and solve the system $Aa = b$ numerically. (To compute $b_i$ analytically, you will need to use integration by parts.) Plot your recovered approximation $\sum_{k=0}^{n} a_k g_k(x)$ on the interval $[0, 1]$ for $n = 5, 10$. Meanwhile, what do you notice about the convergence rate of CG as you take $n$ much larger? Support your observation with numerical evidence / convergence plots. What could be causing this behavior?

• Additional problem 1: Consider least squares approximation of a function f(x) = e^-x on the continuous interval [0, 1] as a linear combination

f(x) ≈∑k=0^n ak gk(x),

where we choose gk(x) = x^k. Construct the resulting normal equations in the form Aa = b and solve them using CG. You should be able to derive analytic formulas for the entries aij and bi, but you should then construct and solve the system Aa = b numerically. (To compute bi analytically, you will need to use integration by parts.) Plot your recovered approximation ∑k=0^n ak gk(x) on the interval [0, 1] for n = 5, 10. Meanwhile, what do you notice about the convergence rate of CG as you take n much larger? Support your observation with numerical evidence / convergence plots. What could be causing this behavior?

Added by Zachary G.

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