Numerical Analysis Questions & Answers

Question by Student 201327118 Proffesor, I dont understand that emax=2^8-1-1? Isnt emax=2^8-1=255? In this case IEEE single precision format In C, single precision is float float x; /*4Bytes finding Largest number*/ You teached emax=2^8-1-1=254 This is not a picture. i attached my note I drawed. image.jpeg	09.10.16

No attached pictures are allowed in the QNA thread except if they are drawings. The mathematics must be typeset within your question using LATEX, not using pictures.

Question by Student 201327118 Proffesor, I dont understand that emax=2^8-1-1? Isnt emax=2^8-1=255? In this case IEEE single precision format In C, single precision is float float x; /*4Bytes finding Largest number*/ You teached emax=2^8-1-1=254

You haven't read correctly my response to your question. To give you more time to think carefully about how to formulate a question using LATEX, you are now limited to one new question every 7 days.	09.11.16

Question by Student 201327103 professor, I don't know why $f_{min}=0.00....1$ in denormal number. I wonder why $f_{min}$ is different in normal($f_{min}=0.000...0$) and denormal($f_{min}=0.00....1$)	09.18.16

I don't understand the question fully. If I ask for the smallest positive number, then this can not be zero whether the number is normal or denormal because zero is not positive. I'll give you just 0.5 point bonus boost because you should put your question in more context so that I can understand better what you mean.	09.19.16

Question by Student 201029134 professor, When you explained ROUNDOFF ERROR, You solved $x=-g+\sqrt{g^{2}+1}$ with $\varepsilon_{mach} = 10^{-8}$ at Float But When you solved $x=\frac{1}{g+\sqrt{g^{2}+1}}$ you used $\varepsilon_{mach} = 10^{-7}$ at Float What is the difference between two $\varepsilon_{mach}$

When using float variables in C, $\epsilon_{\rm mach}$ should be set to $6\times 10^{-8}$. But I won't take away points if you use a slightly more conservative value of $10^{-7}$. The order of magnitude is what counts here. It was a good question, I'll give you 1.5 points bonus boost for it.

Question by Student 201327107 Professor, when you explained about float type machine precision you solved like this $$\epsilon_{machine}=\frac{(1*2^{-24})+1}{1}$$ But I don't know how does it derived. Could you explain how it derived	09.20.16

No, this should read: $$ \epsilon_{\rm mach}=\frac{1+2^{-24}-1}{1} $$ The first two terms on the numerator $1+2^{-24}$ correspond to the sum of the smallest number 1 and the largest possible round-off error $2^{-24}$. The last term on the numerator is the smallest number 1. I'll give you 0.5 point bonus boost.

Question by Student 201527145 Professor, I have a question about an assignment #1_Question #2_(d). Does the smallest possible number mean the "positive" smallest possible number? Or should I consider the "negative" smallest possible number?

Yes you are right: in Question A1Q2b and A1Q2d, we are seeking the smallest possible positive number. Thanks for pointing this out. I'll give you 2 points bonus boost.

Question by Student 201327107 Professor, I don't understand about float type denormal number. I learned that denormal number condition is $$e=00000000$$then exponent should be $0-127$? But you explained exponent is $-126$. How comes?	09.21.16

The exponent of the denormal number is the same as the smallest exponent (-126) but the difference is with the significant which is 0.f instead of 1.f. Thus, the maximum denormal number is just below the minimum positive normal number. If the exponent would be -127, then there would be a large gap between the smallest positive normal number and the largest denormal number. Not a good thing! I liked your question, I'll give you 2 points bonus boost.

Question by Student 201327102 Professor, I have a question. In double type, you taught us that $$p=e-g$$$$g=\frac{{e}_{max}-1}{2}$$ And you substitute $${2}^{11}-1(e={11111111111}_{base 2})$$ for $${e}_{max}$$ then in $${p}_{max}={e}_{max}-g$$ we should use different value in $${e}_{max}(for{11111111111}_{base 2}-1)$$ Although e=11111111111 is not possible(because it indicates infinite # or NaN), why we use e=11111111111 in former $${e}_{max}$$ 'after all'? What's the matter that we use e=11111111111-1 in former $${e}_{max}?$$

09.22.16

I understand what is confusing you. When determining $g$, $e_\max$ refers to the maximum possible positive exponent. But in other cases it refers to the maximum possible positive exponent minus one (because the maximum positive exponent is reserved). They should have been written with 2 symbols in class to avoid confusion. In your notes, rewrite $e_\max$ to $e_\max^\prime$ when determining $p_\max$, with $e_\max^\prime=e_\max-1$. Good point, I'll give you 2 points bonus.

Question by Student 201029134 professor, I think you should give some interval like a to b in The Question#1 on your assignment but you just give initial point $x_{0}$. and I think it is same in The Question#3 So, I wonder whether it is right or not.	10.02.16

For A2Q3, there is no initial interval. For A2Q1, I have made a change to the question formulation. I'll give you 1.5 points bonus boost for pointing this out — I would have given more if you had not made spelling mistakes.

Question by Student 201029134 professor, I am sorry but I actually ask the question above this about the Question#2 not the Question#3. I think secant method need two points like $x_{0}$ and $x_{1}$ for calculating.	10.03.16

The secant method only needs one $x$ for the initial guess (it functions similarly to the Newton method). But you also need to specify a small $\Delta x$ initially to calculate the derivatives. Just set it to a small value of your choice: this won't affect much the convergence history as long as it is not too small.

Question by Student 201527151 Professor, when we use function of sin(x), we include<math.h> library and also write a function code of double sin(double x){     double ret;     ret = sin(x);     return ret; } . However, the code still doesn't work, and when I compile, it says: undefined reference to 'sin'. Is there something else we should add to a code about sin then?	10.04.16

I don't understand. Are you trying to code a sin(x) function? If so, there is no need to code it because the function sin(x) is already defined in the math library.

Question by Student 201129143 Professor, I wrote $$LX'=B$$ "Back substitution, but starting from top" on note. But another book say that this process is Forward substitution. So I wonder that what is the different.	10.10.16

You can call it forward substitution if you wish, this is the correct term. In class, I mentioned that this is the same process as back substitution (but going from top to bottom rather than bottom to top) to make it clear that this is not a new type of process. I'll give you 1.5 point bonus boost for sharing this with the class.

Question by Student 201327116 Professor, Assignment #3 Q#1 (b) In C, We can solve by using the upper triangular. In Compile, The denominator is zero when there is the problem What ever it is that you can use when the denominator is zero?	10.13.16

Are you getting the same problem when solving it by hand? If not, then it means there is a bug in your C code. If yes, then outline the problem in another question below.

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