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

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How to identify whether a fractional part of a number contains more that 2 digits.

EX. I want to accept numbers which have maximum of 2 digits after decimal points. i, e, 10.23 should be allowed and 10.233 should not be allowed. What mathematical operations can be done to ...
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2answers
86 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
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2answers
141 views

Accurate computation of $\exp(a x^2) Q(x)$ for big values of $x$?

I was wondering how one can accurately compute the value of $\exp(a x^2) Q(b x)$ for large values of $$x \left(Q(x) \triangleq \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-\frac{u^2}{2}} du \right)?.$$ ...
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2answers
33 views

Floating Point Calculation

Take the polynomial $x^2+(-4*10^3)x+2)$. On the floating point system, b=10, m=4, e=4, if I wanted to find the roots using the quadratic formula what would be the values of the roots? I got 3.999 as ...
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2answers
79 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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1answer
99 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. ...
2
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1answer
87 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
35 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
66 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 ...
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1answer
58 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 ...
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1answer
78 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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1answer
439 views

Convert hexadecimal numbers to floating-point format using single-precision IEEE 754 format

I need help with converting hexadecimal numbers to floating-point format using single-precision IEEE 754 format. Hex numbers such as 312A. how do I go about converting it? I know how to convert from ...
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1answer
27 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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1answer
80 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
103 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
96 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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1answer
180 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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1answer
61 views

Multiplication of floating numbers to a modulus

As we all know, the integers follow the following identity : $$(A\cdot B\cdot C) \bmod M = ((A\cdot B) \bmod M\cdot C) \bmod M$$ But it does not work for real numbers having fractional part. For ...
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1answer
56 views

Accurate computation of KL divergence between binary RVs

I was wondering how one can compute the KL Divergence between two binary distributions (say, with parameters $p$ and $q$ and assume $p < \frac12$ and $q < \frac12$ for simplicity) accurately. ...
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0answers
53 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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0answers
40 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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178 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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0answers
29 views

What is the short name for a set of all floating-point numbers?

We can name the set of all real numbers $\mathbb{R}$, and the set of all integers $\mathbb{Z}$. Is there a commonly accepted short name for the set of all IEEE 754 floating point numbers? I understand ...
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25 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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33 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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85 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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121 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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0answers
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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0answers
12 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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57 views

Is is possible to define a sign convention for eigenvectors calculated with a small uncertainty?

I'm working with a numerical method that involves the diagonalization of a real, symmetric $n \times n$ matrix $H$. Now obviously the sign of the (normalized) eigenvectors $\phi_i$ is not well ...
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38 views

Binary to Decimal Floating Point Number

I seem to be having difficulties trying to figure this out: I have a Binary 0101011101010000 and I'm trying to compute the decimal floating point number, in the IEEE-754 format Can somebody help? ...
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16 views

Number Systems and Floating Point Arethmetic

I am confused about the term of finite precision number, is that the same with the floating point number? Moreover, I have a number system: ...
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89 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 ...
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0answers
38 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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30 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 ...