Mathematical questions concerning floating point numbers, a finite approximation of the real numbers used in computing.

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Quadratic Root Equation Error

Suppose a machine with the floating-point system $\beta = 10$, $p = 8$, $L = -50$, and $U = 50$ is used to calculate the roots of a quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ ...
2
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1answer
146 views

Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
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1answer
47 views

The arctangent is a strange floating point number

I have 2 players in a game (Cod 4). I read X, Y, Z and store them in: ...
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1answer
58 views

error bound in function approximation algorithm

Suppose we have the set of floating point number with "m" bit mantissa and "e" bits for exponent. Suppose more over we want to approximate a function "f". From the theory we know that usually a ...
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1answer
265 views

Why is this number the smallest positive normalised binary value?

In the AQA A2 Computing textbook (Bond and Langfield, 2009), they say that this number is the smallest positive normalised value, given a 10 bit mantissa and a 6 bit exponent: ...
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34 views

Need an Ambiguous example

Suppose I have a floating-point number with $m$ where $(m > 0)$ digits after the decimal point. Now if I want to round it up to $d$ where $(0 ≤ d < m)$ digits after the decimal point, sometimes ...
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66 views

Finding the mantissa from binary with floating point numbers?

Here is the example problem slide I am working with: I understand how to get the exponent, its just 2+128=130-127=3 I understand the first bit is the sign bit for positive or negative. I just ...
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1answer
13 views

Computer arithmetic - size of error.

I'm actually reading What Every Computer Scientist Should Know About Floating-Points Arithmetics and I have a little problem with understaing one thing. Here's a quote: Write $(b \otimes b) ...
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1answer
64 views

Floating point rounding

Having trouble proving these two statements are true if we assume no overflow occurs and all rounding modes(round down, round up, round to zero, round to nearest) are valid. 1) If x is non zero ...
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1answer
32 views

Stability of (floating point) computed variance

Homework Question from Accuracy and Stability of Numerical Algorithms, 2nd Edition, by Nicholas J. Higham, page 33: So every time we store an number and do a operation, we introduce an error bounded ...
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1answer
26 views

standard notation to handle representation of a real number on a computer

Is there a standard notation to handle the effective representation of some real number $x$ on a finite machine ? I have in mind some kind of braces, but I am not sure it is appropriate. Let me try to ...
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1answer
216 views

Floating point number,Mantissa,Exponent

In this computer, numbers are stored in 12-bits. We will also assume that for a floating point (real) number, 6 bits of these bits are reserved for the mantissa (or significand) with 2^(k-1)-1 as the ...
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41 views

How would you convert the following 32-bit IEEE floating-point to decimal form?

I have got -1.101 1100 1011 1010 1001 1000 * 2^(9) How can I convert this to decimal form?
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94 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...
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1answer
189 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
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134 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
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381 views

Data Representation Question

A computer stores a number of $16$ bits word using floating-point arrangement. Given that the first bit is reserved for the sign and followed by $6$ bits for the exponent using biased form. The ...
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0answers
56 views

How quickly can one compare exp(m/n) to a given rational?

For positive integers $\hspace{.06 in}m_{\hspace{.02 in}0}\hspace{.02 in},n_0\hspace{.02 in},m_1,n_1\:$, $\;$ how difficult is it to decide whether $$\exp\left(\hspace{-0.03 in}\frac{m_{\hspace{.02 ...
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34 views

How are Floating Point approximations done by integer operations? (Source Wikipedia)

Please help me understand the mathematics involved in Wikipedia page of Floating point, section of Piecewise Linear approximation to exponential and logarithm. Following is the link Piecewise linear ...
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63 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
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202 views

Check for Ill Conditioned matrix

How can I efficiently check if a tridiagonal system with 1's in diagonal is ill-conditioned or not ? The common way is to get the ratio of largest and smallest singular values and see if its greater ...
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9 views

Calculating range and eps-machine of floating-point system

Suppose I have a 5-bit floating point system with a 3 bit exponent with radix $\beta = 2$. What is the range and $\epsilon_{machine}$ of this system? I know that I can write numbers as: $$sign ...
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26 views

How do we tell arithemetic addition or subtraction from floating point numbers?

Source: C++ for Engineers and Scientists, Gary J. Bronson Source: Programmable Logic Controllers: The Complete Guide to the Technology by Clarence T. Jones In the table $12345.67_{10}$ = ...
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26 views

If $1/2 ≤ p/q ≤ 2$ , then $p-q$ is representable exactly on the computer

I've found the following affirmation in an article. I've been thinking about it but I don't know the way to prove it: It is not hard to prove that if $p$ and $q$ are two of a computer’s ...
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17 views

How to bound the input parameters to a chaotic function to obtain exact result in a finite precision setting?

While I was reading the paper entitled (http://dx.doi.org/10.1109/ISCAS.2003.1204947) Kocarev, Ljupco, and Zarko Tasev. "Public-key encryption based on Chebyshev maps." Circuits and Systems, ...
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49 views

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation?

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation? Let $x_T=0.2349$, $x_A=0.2299$, then absolute error = $|x_T-x_A|=0.0050=\frac{1}{2}\times ...
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28 views

Error accumulation

Assume that $a_j\leq u$ and $p_j\in\{-1,+1\}$ for $j=1,2,\ldots,n$ and that $nu<1$ where $u:=2^{-t-1}$. Show that following is true: $$\prod^{n}_{j=1}(1+a_j)^{p_j}=1+\theta_n$$ where ...
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81 views

Rounding up a binary number

I just converted the fraction $\frac{3}{5}$ to the following floating point binary number: $(1.001100110011001100110011\cdots)_{2}\times 2^{-1}$. Now, I am trying to find the two nearest machine ...
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52 views

smallest poisitive integer does not belong in floating point system F

A floating-point number system F is a subset of real numbers whose elements have the form: $x=\pm(m/\beta^t)\beta^e$ base (beta), precision (t), and exponent range (emin and emax). The mantissa m ...
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41 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
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139 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
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47 views

Displaying $2 \cdot 10^5$ as $200000$

How can I in Maple write out the number $2 \cdot 10^5$, so it displays the full number with all digits: $200000$? By typing increasing zeros I can test Maples treshold, and I get: $20$, $200$, ...
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33 views

How do I round this binary number to the nearest even

I have this binary representation of 0.1: 0.00011001100110011001100110011001100110011001100110011001100110 I need to round it ...
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34 views

Automatic curve fitting to find order of an algorithm?

I'm a newbies in mathematics. I'm looking for an automatic best curve fitting function to find the order of an algorithm. I would like to know if it does exists a math library function that would ...
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15 views

Whether assigning of single precision IEEE754 float to double is reversible?

Within scope of IEEE754 standard let's assign single precision variable s to double precision variable d and then assign d to single precision variable s'. Whether this operation is ...
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20 views

What should be the precision to uniquely recover the binary double float number from the decimal one?

Assume we store binary float number in IEEE 754 format to decimal format. Then we recover decimal to binary back. With regard to single floats - according to this article When a binary IEEE single ...
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15 views

Is there any difference between fixed point and decimal point?

Source: Introduction to Computers' 1999 Ed.1999 Edition Fixed point number 774.3675 is just a decimal number with a decimal point to show a fractional part 3675/10000. I see no difference in the ...
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16 views

Given a base, a mantissa size and a range for the exponent, what's the largest number?

I was introduced to floating-point numbers week ago or so, and although I think I understood the basics, I'm still not sure how to apply them. A floating-point number is usually represented with the ...
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14 views

Simple computations on decimal floating point arithmetic

Can anyone please check my working on the computations of $2*0.3-0.6$ and $3*0.3-0.9$ on decimal floating point? My working: $2*0.3-0.6=(2.0*10^0*3.0*10^{-1})-6.0*10^{-1}=6.0*10^{-1}-6.0*10^{-1}=0$ ...
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25 views

What are 2*0.3-0.6 and 3*0.3-0.9 in decimal floating point numbers?

Investigate the values $a=2*0.3-0.6$ and $b=3*0.3-0.9$. What do we expect to get on a computer using decimal floating point numbers? My reason and doubt: It doesn't really mention how many degree ...
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20 views

Floating point binary to half precision floating point

I am trying to convert to $16$ bit half precision floating point however I ran into a possible error and am unsure if a negative exponent is ok. I am trying to convert $0010011100010000$ I separate ...
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10 views

What line-polygon clipping algorithm can I use to ensure that the resultant endpoints are always within the polygon?

I have a 2D plane, partitioned into n-sided, convex polygons. I'm using WRF's PNPOLY algorithm for polygon inclusion to ensure that a point belongs inside one and only one polygon. Is there an ...
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119 views

floating point subtraction for binary numbers

Consider that I want to do a binary operation on the following floating point numbers: 0.35-0.62 I can reach the end but I can not figure out how the sign bit is ...
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16 views

Fixed point exponent faithfull rounding? does it make any sense?

i've a stupid question that just come up in my mind. Let's say we have a floating point number $x = (-1)^{s_x} 2^{e_x} 1.m_x$, let's say for some reason we need to divide the exponent by $2$, maybe ...
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33 views

Longest string whose $\log$ probability can be represented

The problem starts out with a binary string of length $n$, so the probability of any random string in the set is $\cfrac{1}{2^n}$. The smallest positive floating point value that the system can ...
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26 views

Computing relative roundoff error of a correctly rounded binary number

This is related to a question that was asked and answered a moment ago. I need to answer the following: If $\displaystyle \frac{3}{5}$ is correctly rounded to the binary number ...
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44 views

How to Adequately Implement Phase I of Two-Phase Simplex Algorithm on a Computer with Floating Point Error

I'm currently trying to write some code that implements Phase I of the two-phase Simplex Algorithm described here: http://www.statslab.cam.ac.uk/~ff271/teaching/opt/notes/notes8.pdf In order to test ...
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5 views

Prove that a condition for a product fitting in a range holds

Given the sets $$L = \{x \in \mathbb Z : -2^{63} \le x < 2^{63} \}$$ $$F = \{m \cdot 2^e \in \mathbb R : m \in \mathbb Z, -2^{24} + 1 \le m \le 2^{24} - 1, e \in \mathbb Z\}$$ and ...
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44 views

Heuristics for sequence convergence

Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ...
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52 views

Identity for fractional summation

I would like to know if there's an identity to represent the following summation $\sum_{i=0}^{n}\frac{x_i}{y_i}$ Where x and y are non integer values. The result of this is being calculated using ...