Numerical Analysis Questions & Answers  
Question by Jaehyuk
Professor, I have question about the number of arithmetic operation of Lagrage Polynomial. For example, with 4 data points, $p_{3}(x)=\frac{(x-x_{2})(x-x_{3})(x-x_{4})}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}+\frac{(x-x_{1})(x-x_{3})(x-x_{4})}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}+\frac{(x-x_{1})(x-x_{2})(x-x_{4})}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}+\frac{(x-x_{1})(x-x_{2})(x-x_{3})}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}$. Then change the order of deniminator as follows; $p_{3}(x)=-\frac{(x-x_{2})(x-x_{3})(x-x_{4})}{(x_{2}-x_{1})(x_{3}-x_{1})(x_{4}-x_{1})}+\frac{(x-x_{1})(x-x_{3})(x-x_{4})}{(x_{2}-x_{1})(x_{3}-x_{2})(x_{4}-x_{2})}-\frac{(x-x_{1})(x-x_{2})(x-x_{4})}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{4}-x_{3})}+\frac{(x-x_{1})(x-x_{2})(x-x_{3})}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}$. By doing so, I can reduce the number of subtraction of denominator; $(x_{4}-x_{3}), (x_{4}-x_{2}), (x_{4}-x_{1}), (x_{3}-x_{2}), (x_{3}-x_{1}), (x_{2}-x_{1})$. This can reduce the number of arithmetic operation to find and save the value of denominator from 20(3subtraction and 2multiplication for 4 terms$((3+2)\times4)$) to 14(2multiplication for 4terms and 6 subtration$(2\times{4}+6)$). Am I on the right track to reduce the number of the operation?
I don't understand well your question. Why are there only 3 terms within $p_3(x)$ and not 4? Also, the rest of the question doesn't make much sense to me. You need to explain this better. 0.5 point bonus for the effort. Make sure to use the PREVIEW button and check if the question looks as intended. Also, use \$\$ and not \$ for long math expressions.
Question by Jaehyuk
Professor, I have question about A#6 Reminder. In the last row which includes $b_{i+1}$, the range of i is $2\leqq{i}\leqq(N-1)$. As far as I know, however,if i goes up to (N-1), then this would create $b_{N}$. This means there are N equations not (N-1). I am confusing why there are N equations for cubic splines.
Good question. We need to find $b_N$ because $a_{N-1}$ and $c_{N-1}$ depend on $b_N$. So, we only need to find $N-1$ intervals, but we need to find $N$ bs. 2 points bonus.
Question by Student 201527105
Professor, i have a question about Piecewise linear interpolation. I wonder if Piecewise liner interpolation's function has always straight grape line in each intervals. This interpolation is simple to use, but it seems to be bad when viewed from the side of derivative. In each intervals, derivatives may be discontinuous. And i also think that this is a little bad method to estimate arbitrary values in the given intervals. What is the big difference of this method when compared to other interpolations.
Hm, I'm not sure what confuses you. Answers to your questions were given in class.
Question by Student 201427113
Professor, When you explain Analytical matrix immersion, Work time about Inverse matrix for 2X2
A= \begin{bmatrix} a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix} $A^{-1} = 1/det(A)\begin{bmatrix} a_{22} & -a_{12} \\-a_{21} & a_{11} \end{bmatrix}$

$W_{det(A)}=3$ ("2" multi. and "1" sub.)
$W_{A^{-1}}=W_{det(A)}+ 4Div. = 3+4 = 7$
But When Change marix \begin{bmatrix} a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix} to \begin{bmatrix} a_{22} & -a_{12} \\-a_{21} & a_{11} \end{bmatrix} Why Work time is not apllied in matrix change.
There's little work involved when moving numbers around in memory compared with additions or multiplications. Hence why it's not counted. 1 point bonus.
Question by Student 201527105
professor, i have a question about Assignment7-Q#4. To calculate the error, the actual value of intergal should be known. According to the trapezoidal rule and the simpson rule, i obtained appoximate values. But i don't know the actual value of given integral. How can i get it? / And i also wonder if the error always decreases as N increases. (Intuitively, i think it is true.)
You can find the exact value by using a very large $N$. So, just double $N$ until the value of the integral doesn't change significantly anymore and use this as the exact value. 2 points bonus.
Question by Student 201427128
professor, I have question in operation counting about gaussian elimination with partial pivoting. for example, In the process of gaussian elim, if \[ {m}=\begin{bmatrix} 1&0&5&5 \\ 0&0&6&6 \\ 0&1&7&7 \end{bmatrix} \] I have to do partial pivoting $P_{23}$ \[ {P_{23}m}= \begin{bmatrix} 1&0&0\\ 0&0&1\\ 0&1&0\\ \end{bmatrix} \begin{bmatrix} 1&0&5&5 \\ 0&0&6&6 \\ 0&1&7&7 \end{bmatrix} =\begin{bmatrix} 1&0&5&5 \\ 0&1&7&7 \\ 0&0&6&6 \end{bmatrix} \] In this course, Operation counts of $P_{23}=3*(3*4)=36?$, Is what I'm doing right? or Is there another way to calculate?
No, there is no work done when doing the pivoting operation because there is no addition or multiplication. This was mentioned above for a similar question.
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