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

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4
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1answer
41 views

Polynomial GCD in the presence of floating-point errors

The crucial requirement for using root isolation methods based on Vincent's theorem is that the input polynomial does not have multiple zeros. One way to remove the multiple zeros is to use polynomial ...
0
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0answers
11 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 ...
0
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1answer
70 views

Gosper Formula for inv $\pi$, properties.

I need to understand very good how the properties of this formula $\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $ Taken from ...
5
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1answer
56 views

What are examples of cases where floating-point $aaaa\ne(aa)(aa)$?

As explained in answers to this question on SO, due to non-associativity of floating-point arithmetic repeated multiplication like $aaaa$ can't be optimized to $(aa)(aa)$. Of course, aside from just ...
1
vote
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 ...
0
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0answers
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 ...
2
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1answer
49 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
0
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2answers
77 views

Curve Fitting to Represent Any Data

I'm a programmer seeking to take a bunch of data and represent it as a curve. Specifically, I want to take several hundred/thousand (floating) points and represent those points to a specified level of ...
2
votes
1answer
73 views

IEEE-754 Format Conversion

Represent 11.0011 x 2^10 using the IEEE-754 standard for 32-bit floating point representation. 0-Sign Bit 10001010-Exponent 1001100000000000000000-Mantissa Is this answer is correct ? I am bit ...
11
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4answers
748 views

why is $2.2250738585072014\text{e}{-308}$ not a number? [closed]

In programming the min value of a float is: $$2.2250738585072014\text{e}{-308}$$ but when I type this into a calculator, it says Not a Number. what I am wondering ...
2
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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 ...
2
votes
1answer
75 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
0
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2answers
37 views

Simplifying a decimal number under modular arithmetic – $9.9 \pmod{13}$

Can you please help me simplify the relation $9.9 \pmod{13}$? It may seem like a stupid question (!) but your answers will help me very much. Thank you.
0
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1answer
16 views

approximation using floating point arithmetic

Let $x=2.14366$ and $y=2.14363$ and $d=x-y.$ If $d*$ is the value of d computed using $5-$digit decimal floating point arithmetic, find the relative error. For this question I know how to calculate ...
0
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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 ...
2
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1answer
36 views

Audio frequency increment yielding wrong results

I am writing something in the ChucK programming language, which is designed specifically for audio time functions (in this case, hertz). I'm having a really difficult time with a mathematical ...
2
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3answers
64 views

Why do I get a big relative error for my function? (Numerical Analisys - floating point)

When evaluating on the computer the following function: $$f(x)=\frac{x^2}{(\cos(\sin(x)))^2-1}$$ there is a big relative error for values $x\approx0$ (values very close to zero). I used the Taylor ...
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2answers
407 views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...
0
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1answer
19 views

Precision and Accuracy

How would I go about calculating the precision and accuracy of a given number? For example 0.05 has an accuracy of 2 and a precision of 3. 1 has an accuracy of 0 and a precision of 1. Is ...
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2answers
85 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 ...
0
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0answers
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 ...
0
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1answer
17 views

Storing a real matrix - fl notation

I have two matrix $A$ and $B$ and $fl(A), fl(B)$ denote the stored version them, respectively. Let $fl(AB)$ be the stored version of the product of $A$ and $B$. Is it true that $fl(AB) = fl(A) * ...
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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 ...
0
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1answer
437 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 ...
0
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1answer
458 views

How to convert from floating point binary to decimal in half precision(16 bits)?

I'm trying to convert a 16 bit precision binary number to decimal format however I am completely failing to do so. The binary I'm trying to convert is $0101011101010000$ My current method is: ...
6
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1answer
160 views

Explain why catastrophic cancellation happens

After my own research, the following picture emerges as the most frequently used example of catastrophic cancellation (It is indeed used in my class). Could anyone explain why the plot takes that ...
0
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0answers
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? ...
0
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1answer
51 views

How can I add the following 32-bit IEEE floating-point numbers?

How can I add the following two 32-bit IEEE floating-point numbers in binary? FEDCBA98(base 16) + 89ABCDEF(base 16) = a 33-bit binary number. How can this be possible?
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1answer
26 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?
1
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1answer
56 views

Gaps between successive floating point numbers

(all numbers discussed are in decimal) lets say we have a floating point data type that is like : m * 10 ^ e ...
0
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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 ...
0
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1answer
24 views

Selecting denominator for relative error margins

When looking at this page: http://floating-point-gui.de/errors/comparison/ there are values a, and b that are being compared ...
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0answers
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: ...
1
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0answers
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 ...
0
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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 ...
1
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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 ...
2
votes
4answers
129 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 ...
0
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0answers
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 ...
2
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2answers
106 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 - ...
0
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1answer
231 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
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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1answer
31 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 ...
2
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3answers
345 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 ...
6
votes
1answer
200 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
73 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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3answers
42 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
118 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
vote
1answer
70 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
1k 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 $2$.
0
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1answer
102 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 $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...