Numerical Analysis Questions & Answers

Question by Student 201727153 Professor, I have a question about #6(b) in Assignment 3. Since every row needs work to be done at least 1 division, the range of n, isn't it from 1 to N?	10.30.18

I'm not sure what you mean and I can't find 6b. Please formulate your question better.

Question by Student 201727153 Sorry, I wrote it wrong. Problem #4(b) in assignment 3, it is written that the range of n is from 1 to N-1. However, we have to do the work to all the rows at least 1 division. So, is the range of n from 1 to N?

I'm still not sure what you mean. You need to explain this better. What do you mean by 1 division? division of what by what? You need to be more clear. ​ Keep in mind that if such a question is asked in the exam, I will give almost all the points for the quality of the explanation, not for the answer itself. How well you explain your steps and the quality of the logic is what matters the most.

Question by Student 201427128 Professor, I have a question about #3(b) in Assignment 4. Consider, $|\epsilon_{n+1}|=k|\epsilon_n|^p$ → $|x_{n+1}-\eta|=k|x_n-\eta|^p$. to get the order of convergence $p$, I have to know $\eta$ and $k$. how can I get the $\eta$ and $k$? or Is there another way to get order of convergence? I need help.	11.11.18

In this case, you have to find the order of convergence using the results obtained in (a). That is, don't derive an analytical expression for $p$, but find $p$ from the results. Hint: $p$ may change as the solution progresses towards the root. You can find the root (needed to evaluate the error) through the results as well. Good question: 2 points bonus.

Question by Jaehyuk Professor, I have question on A4Q4(b). As far as I know, the elements of Jacobians are the partial derivatives of the functions. The secant method, however, does not have partial derivatives in its terms. Then, is there any particular ways to derive partial derivatives from the secant method which that can be the elements of the Jacobians?	11.12.18

Well, solving a system of equations with the secant method is not so different to solving a single equation. When solving a single equation, the derivative $f^\prime$ is replaced by a first-order approximation. Do the same here and use very small $\Delta x$ and $\Delta y$ for the first iteration. 1 point bonus.

Question by Jaehyuk Professor, I have question on A#4Q3-(b). According to the definition of the order of convergence, we can derive p from $\mid\epsilon_{n+1}\mid=k\times\mid\epsilon_n\mid^{p}$. Here starts my problem. That is, for $x_1$, the root is -0.0025, the 1st guess is 0, the 1st iteration yields 5, and the 2nd iteration yields 4.91244. Assigning these values into the formula above, this ends up as follows;$\mid5-(-0.025)\mid=k\times\mid0-(-0.025)\mid^p$ and $\mid4.91244-(-0.025)\mid=k\times\mid5-(-0.025)\mid^p$. This yields $k=4.96393$ and $p=-0.003315$ which does not match with the answer. I am wondering which part of the process is wrong.

11.16.18

You're not doing anything wrong. If you would answer this in the exam, you would get full points. But, in general, we do not know what the exact root is. Thus, it is necessary to approximate it as the solution at the next iteration. If using the solution at the next iteration (i.e. 4.573738E+00) as the root, you'll get the answer listed. I made this more clear within the question formulation. 2 points bonus.

11.17.18

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?

11.19.18

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.	11.20.18

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.

11.21.18

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} $A^{-1} = 1/det(A)\begin{bmatrix} a_{22} & -a_{12} \\-a_{21} & a_{11} \end{bmatrix}$ $det(A)=a_{11}a_{22}-a_{12}a_{21}$ $W_{det(A)}=3$ ("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.

12.08.18

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.

Question by Student 201427128 professor, I have question in A7#3. In this case, N is even(=50). In simpson method, Interval $I_{i}$ need 3 data point. ex) $I_{1}$ mean interval between point ($i_{1},i_{2},i_{3}$) so when I use simpson method in this problem, $I_{49}$(between $i_{49},i_{50}$) is left. how can I solve it? Maybe I think the remaining Interval $I_{49}$ can be solved by trapezoidal rule. Is it the right way?

12.09.18

That's for you to find out. Check if you obtain reasonable convergence rates using your approach when $N$ is small (i.e. as you would expect for the Simpson rule) and if so, it means it's fine.

Question by Jaehyuk Proffessor, I have a question about 4th order Runge Kutta method. I made a assumetion that $f(\phi,t)=f(t)$. In this case, $$ k_{1}=\bigtriangleup t×f(t), k_{2}=k_{3}=\bigtriangleup t×f(t+\frac{\bigtriangleup t}{2}), k_{4}=\bigtriangleup t×f(t+\bigtriangleup t) $$. Substitute them to 4th order Runge Kutta, $$ \phi_{n+1}=\phi_{n}+\bigtriangleup t×(f(t)+4×f(t+\frac{\bigtriangleup t}{2})+f(t+\bigtriangleup t))/6 $$, which is same as the Simpson's Rule. Am I on the right track to relate 4th order Runge Kutta to Simpon's Rule?

12.10.18

Hm, didn't I mention this in class..? It's a good observation, but not a question.

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