Numerical Analysis Questions & Answers

Question by Student 201327107 Professor, I learned partial pivoting in order to prevent possible division by 0. As a result, for 4 rows and one column of A matrix is ​​0 in Assignment3 Q1, then should I have to use partial pivoting in C code and by hand?	10.16.16

No, you shouldn't be using pivoting here because the question asks to use Gaussian elimination. Double check your solution: if you think there is a mistake in the question formulation then outline your full solution below in LATEX until the problem occurs and I will check it.

Question by Student 201327107 In Assignment3 Question1 (b) $$int main()$${ $$ $$ $$for(row=0;row<N;row++)$${ $$for(row2=row+1;row2<N;row2++)$${ $$assert(A[row2]A[row]!=0.0);$$ $$fact=-A[row]A[row]/A[row2]A[row];$$ In code when $row=0$,$row2=3$, then $A[row2][row]$ is zero. So compile is stop. I think row4,col1 of matrix A that should not be the zero.	10.17.16

No, I asked that you post the solution by hand and show at which step there is a problem (using LATEX).

Question by Student 201327102 Professor, when you taught me about finding convergence of each method, you put $g(y)=\frac{1}{1+y}$ and expand $g(y)$ with TAYLOR SERIES at $y=0$ so you wrote $$g(y)=g(0)+(y-0)g'(y)+\frac{{y}^{2}}{2}g' '(y)+...$$ But according to the original form of TAYLOR SERIES is $$g(y)=g(0)+(y-0)g'(0)+\frac{{y}^{2}}{2}g' '(0)+...$$ Isn't there any wrong in your notation?	11.01.16

Yes, you are right, it should be: $$ g(y)=g(0)+(y-0)g'(0)+\frac{{y}^{2}}{2}g' '(0)+... $$ If I wrote otherwise on the board, then this is a mistake obviously so please change your notes in consequence. This is a good observation, I'll give you 2 points bonus boost.

Question by Student 201527110 Professor, I wonder 'curve fitting and interporation'assignment is due to 'thursday' or 15th. In the assignment page, you announced like Thursday 15th Nov. But 15th is tuesday as you know.	11.13.16

Oops, it's due on Thursday November 17th. Thanks for correcting this. I'll give you 2 points bonus boost.

Question by Student 201527110 Professor, I have a question during studing Cubic Spline boundary conditions. For define $f'_i (x)=α_L$ and $f'_i(x_{i+1})=α_R$, $α_L$ and $α_R$ in here, are user-specified constant. Is taht means it could be any arbitary number? Or do I have to define exact real numbers for that?	11.16.16

Well, user-specified constants are numbers that are specified by you, the user of the code. Of course, such could be any real number you wish to specify..

Question by Student 201527142 Professor, in assignment #6, I found you didn't define n, the number of data points, before writing function f. Is it necessary, isn't it?	11.23.16

Well, the number of data points $N$ can be obtained from the data shown in the tables..

Question by Student 201327102 Professor, I think I found wrong notation in Assignment 6# Question #2 reminder. In reminder, last row, you told interval of $i$ to $ 2\leq i\leq N$. But, in my note, you taught us that interval of $i$ is $ 2\leq i\leq N-1$ Isn't it necessary to revise that point?

In the reminder, last row, it is written $2\le i\le N-1$, not $2\le i\le N$..

Question by Student 201327107 Professor, I have question about Big O notation. Sometimes you write $$(b-a)O(\delta(x^2))$$ but sometimes you write just $$O(\delta(x^2))$$ except $(b-a)$. Do these two have same meaning?	11.25.16

I'm not sure what your question is.. The big O notation $O(\Delta x^2)$ means that the average truncation error leading term scales with $\Delta x^2$, that is all. Your question is not clear and is not well typeset either. I'll give you 0.5 point bonus boost only.

Question by Student 201029134 Professor, I got a grade and want to check my score. I'm sorry but When do I go to your office?	12.27.16

I'll be in Dec. 30th, Jan 2nd, and Jan 3rd from 9am till 6pm. Note that the grades can't be changed after January 3rd.

Question by Student 201542124 Professor, I have a question about the homework. In your homepage, there is a homework#1 Today I learned about IEEE in that class. So, When should we submit the homework? And, if so, should we upload the homework in your website or write in the paper and submit in person?	09.06.17

I'm not sure yet when Assign 1 will be due. When I decide on the date, I will let you know on my website, and you will receive an email. You should write the Assignment on paper and submit it at the beginning of the class.

Question by Student 201427565 professor, I don't understand the reason why I have to use $e =11111110$ instead of $e=11111111$ when i'm finding max P. and also for the min P, why is that $e=00000001$? not just the zero?	09.10.17

This was explained in class. You need to exclude the reserved exponents for the special cases.

Question by Student 201627148 Professer, in roundoff error, You wrote $$x={-2g+\sqrt{4g*g+4}}/2$$ $$sqrt(4*g*g+4)=2000.001+\sqrt{8000\epsilon_{mach}}$$ But i think you forgot to put $$\sqrt{8000\epsilon_{mach}}$$ to get x. Is it possible to erase because it is too small?	09.16.17

Let's do it again step by step. $$ {\rm sqrt}(4.0*g*g+4.0)=\sqrt{4(g\pm\epsilon_{\rm mach}g)^2+4\pm 4\epsilon_{\rm mach}} $$ $$ {\rm sqrt}(4.0*g*g+4.0)=\sqrt{4g^2(1\pm\epsilon_{\rm mach})^2+4 \pm 4\epsilon_{\rm mach}} $$ For $\epsilon_{\rm mach}\ll 1$: $$ {\rm sqrt}(4.0*g*g+4.0)\approx\sqrt{4g^2(1 \pm 2\epsilon_{\rm mach})+4\pm 4\epsilon_{\rm mach}} $$ Or $$ {\rm sqrt}(4.0*g*g+4.0)\approx\sqrt{4g^2+4 \pm 8g^2 \epsilon_{\rm mach}\pm 4\epsilon_{\rm mach}} $$ But for $8g^2 \gg 4$: $$ {\rm sqrt}(4.0*g*g+4.0)\approx\sqrt{4g^2+4 \pm 8g^2 \epsilon_{\rm mach}} $$ But for $8g^2\epsilon_{\rm mach}\ll 4g^2+4$ can show that $$ {\rm sqrt}(4.0*g*g+4.0)\approx\sqrt{4g^2+4} \pm \frac{1}{2} \frac{8g^2 \epsilon_{\rm mach}}{4g^2+4}\sqrt{4g^2+4} $$ If $4g^2\gg 4$: $$ {\rm sqrt}(4.0*g*g+4.0)\approx\sqrt{4g^2+4} \pm \epsilon_{\rm mach}\sqrt{4g^2+4} $$ Thus, for $\epsilon_{\rm mach}$ much smaller than 1 and $g$ much greater than 1, a very good approximation to the error associated with the sqrt of $4g^2+4$ is $\epsilon_{\rm mach}$ times the sqrt of $4g^2+4$ (I think that was what I wrote last class). If not, make a correction. Good question. I'll give you 2 points bonus boost.

Question by Student 201427564 Professor, when you calculate the smallest possible positive denormal number, why did you put $0.f_{min}$ instead of $1.f_{min}$ ?

Well because for the same exponent (the smallest possible one), 0.f will be just below 1.f. 1 point bonus boost.

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