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

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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 ...
2
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1answer
24 views

Entropy of floating number array

I am familiar with shanon's definition of entropy. $$ H(P) = - \sum_{i=1}^n p_i \cdot \log_2(\mathcal p_i) $$ I am today in the situation that I'd like to compute an entropy like function but for a ...
2
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4answers
49 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
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0answers
19 views

Different SVD results in Matlab

my question relates to calculating SVD in Matlab. I have been reading a lot and somehow I have jumbled up all the facts. It would be great if you experts could get me to the right track. My task is ...
2
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2answers
99 views

Floating point arithmetic: $(x-2)^9$

This is taken from Trefethen and Bau, 13.3. Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression: $$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - ...
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1answer
52 views

Converting double precision IEEE 754 hex to base 10 with repeating decimals

The number is 0x4001 8CCC CCCC CCCC. So far I have the stored exponent as 1000000000 which equals ...
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0answers
26 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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1answer
24 views

Finite binary representation of float number

How can I prove that float number $x$ has the finite binary representation if and only if it is written like that: $ x = m / 2^n $, where $m, n \in \mathbb N$. Should I consider something like 3 ...
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3answers
156 views

What is the maximum difference between two successive real numbers in the given floating point representation?

The following is a scheme for floating point number representation using 16 bits. Sign :- Bit 15 Exponent:-Bit 14-9 Mantissa :- Bit 8-0 Let $s, e,$ and $m$ be the numbers represented in binary in ...
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0answers
13 views

Activity on arrow network and block chart

I have to figure this out : ...
6
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1answer
132 views

How to calculate floating point numbers?

Here are two locations in small memory: 0110 | 1111 1110 1101 0011 0111 | 0000 0110 1101 1001 Interpret locations 6 and 7 as an IEEE floating point number. ...
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5answers
51 views

Not clear on what we mean with numbers with infinite digits

I am confused on a rather simplistic question. 1/3 = 0.333333333333 to infinity. So it has infinite digits. How is it possible to multiply such a number with another one and get a finite number? 6/3 = ...
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1answer
25 views

How to interpret fractional number of bits of precision

In double-precision floating-point format there're effective $53$ bits of mantissa stored. This lets us estimate maximum number of decimal digits of precision available: ...
2
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1answer
76 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 rounding rules, as accurately as possible. Please ...
1
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1answer
46 views

Converting 16bit float to Base10 and vice versa

Hi! I have some difficulties understanding how I'm supposed to calculate this 16bit float to base10. This is something that is coming up on a test so I would be pleased to learn how this is supposed ...
5
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1answer
840 views

What is the most significant digit?

What is the most significant digit of $$0.00234$$ I have a problem of figuring out where it is $0$ or $2$.
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0answers
37 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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1answer
75 views

Given the normalised floating point number system, calculate smallest possible value of y - x

First I present the problem and then my workings and thoughts: Given the normalized floating point number system $(\beta, t, L, U) = (10, 7, -6, 4)$ where $\beta$ is the base and $t$ is the ...
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3answers
91 views

Is this approach for testing orthogonality/parallelity of vectors wrong as I think?

In a math book four methods are written for testing parallelity/orthogonality of two vectors that are(notice $\vec v$ and $\vec w$ are approximations of vectors and we have x,y and z components of ...
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1answer
57 views

testing parallelity/perpendicularity of two 3D vectors with lengths close to zero using dot product

as u know dot product and vector norm(euclidean) definitions are as following: $ \vec v . \vec w = \left\lVert \vec v \right\rVert \times \left\lVert \vec w \right\rVert \times \cos\theta = (v_x ...
2
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1answer
39 views

Question about making a custom 'formula'

I am working on some program for myself, and I am stuck on one thing, which I am not sure how to make exactly, I do know how to calculate it manually on paper, but I don't know any kind of formula or ...
2
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1answer
66 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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2answers
58 views

Find point in 3D plane

I have four points in a 3D space, example: $$(0,0,1),\ (1,0,1),\ (1,0,2)\ \mbox{and}\ (0,0,2).$$ Then I have a 2D position on that square plane: $$x = 0.5,\ y = 0.5.$$ I need to find out the 3D ...
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2answers
55 views

binary and floating point representation

suppose that we have following binary digits $00011001.110 $,we can do following thing $00011001.110=1\cdot2^4+1\cdot2^3+1\cdot2^0+1\cdot2^{-1}+1\cdot2^{-2}=25.75$ then what does means? We then ...
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1answer
28 views

Comparison between float point numbers

How can I demostrate that given two numbers A and B, if A <= B, then fl(A) <= fl(B). fl(A) is the float point representation of A. Which is equal to A(1+x), and |x| <= the machine epsilon ...
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1answer
85 views

Prove that in any base the number of digits composing the repetitive mantissa of the reciprocal of a prime $p$ never exceeds $p-1$.

I was trying to find bases where the reciprocals of primes have a short repetitive mantissa. Here is what I found: http://imagizer.imageshack.us/a/img835/7738/c7gb.png The bases are on the left. The ...
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1answer
44 views

Using rgb triplets in “scalar” math

So I'm reading a computer graphics research paper and I'm confused as to how to interpret certain formulas that are being used. In the paper, a value σ is defined as an RGB triplet. Later on, that ...
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2answers
52 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
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1answer
85 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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0answers
48 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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0answers
28 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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0answers
22 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 ...
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1answer
65 views

Max value - Mantissa calculation clarification?

I've been reading this article about floating point representation floating point representation is : Where mantissa is : All understood. But they also say : How did they get to ...
2
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1answer
27 views

How many bits of difference in a relative error?

I would like to know if there is a formula or any other way to find out how many bits of difference between two values given the relative error. For instance: $$\epsilon_{\text{rel}} = \frac{V - ...
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1answer
50 views

Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
2
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1answer
116 views

A general method to efficiently calculate the floor of an element of $\mathbb{Q}[\sqrt{2}]$

I had to decide whether to post this question here on the Mathematics Stack Exchange or on Stack Overflow, but I decided that the question was essentially a mathematical one despite being inspired by ...
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1answer
100 views

Converting base2 scientific notation to base10 scientific notation

I would now like to know how to convert a value in base2 scientific notation (is that the correct terminology?), say 1.93 * 2 ^ 88, into the form of A * 10 ^ B. I want to do this without expressing ...
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0answers
71 views

What is the result of the calculation: $8.375 - 0.375-0.875$?

Given the base $\beta=2$ (binary) and $t=4$ digits to represent the number in computer with hidden bit representation and symmetric rounding, what is the result of the calculation in floating point ...
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1answer
29 views

Range of number systems [closed]

I don't understand how to get the fraction part. This is what I came up with for the integer part. A) For 12 bit unsigned = 0 to 4095 B) For 12 bit signed = -2048 to 2047 C) For 12 bit in 2's ...
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1answer
58 views

What makes a floating point number finite?

We started studying about the binary floating point system, and talked mainly about the limitations when trying to represent fractions on computer systems, and how mostly what we get when manipulating ...
2
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2answers
38 views

Representing $-2.5$ as a floating point number

I am trying to understand the following: For example the number $-2.5 = -1*1.25*2^1$ is strored as: $S = 1$, Exponent $= 1+127 = 128$, Mantissa = $0.25 $ This got me. How do you connect all ...
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0answers
49 views

Biased form with hidden bit

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 remaining ...
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1answer
95 views

Check if a point is on a plane? (Minimize the use of multiplications and divisions)

In $\mathbb R3$, given a plane $\mathcal P$ defined by three 3D points points $v_0, v_1, v_2$, I want to check if another point $p$ belongs to that plane, while avoiding the use of multiplications and ...
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1answer
71 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
114 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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1answer
73 views

IEEE-754 64-bit representation of any number between -1 and +1 [closed]

In 64-bit IEEE 754 format, a 64-bit number is represented using 1 bit for the sign bit, 11 bits for the exponent width, and 52 bits for the fraction precision. How many bits is used to represent ...
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1answer
30 views

Problem in deriving distribution of the product of two floating point numbers

I'm studying the Hamming's article "On the Distribution of Numbers", but i'm confused in the way he obtain the (cumulative) distribution of the product of two floating point numbers. Maybe it's only ...
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2answers
52 views

How to compute the output of floating numbers? [closed]

Is it possible to "prove" that the following program outputs "1.73472e-17" ...
2
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1answer
75 views

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 $c$ ...
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1answer
80 views

How to find the ratio of two numbers written in scientific notation, such as $ 2.69 \cdot 10^{-8} $ and $ 2.23 \cdot 10^{-7} $?

The numbers are forces and he wants us to put them in a ratio in order to compare them, but I'm really bad at ratios so. $ 2.69 \cdot 10^{-8} $ Newtons is obviously smaller than $ 2.23 \cdot 10^{-7} ...