Questions tagged [floating-point]
Mathematical questions concerning floating point numbers, a finite approximation of the real numbers used in computing.
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Expression of sum in floating point system
This is a question of an exam on Numerical Analysis I had:
Consider the floating point system of base $2$, maximum number of decimals $53$, maximum exponent $1025$ and minimum exponent $-1022$. That ...
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Evaluating $a(b + c)$ more accurately with FMA
I'm using machine-precision floating-point arithmetic, and every so often it happens that I need to evaluate an expression of the form $a(b + c)$. I found that the accuracy can be improved using FMA (...
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Numerically stable evaluation of factored univariate real polynomial
Suppose we have a real univariate factored polynomial, meaning we have its factors: an arbitrary number of polynomials of degree less than or equal to two. To simplify things, if necessary, let's ...
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Bias in Single Precision Floating numbers
I had a doubt regarding Single Precision Floating point numbers. It is about the bias number which can be derived from exponent part of this representation of numbers.
On searching up on google, most ...
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How to compute this "smooth max operator"?
I was seeking for an alternate way to activate each neuron of a neural network non-linearly. Eventually, I came up with the following binary operation:
$$
x \lor y = \log (\exp x + \exp y)
$$
With $-\...
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How to Multiply 2 arrays with unique non-integers to prodice an array with unique results?
Is there an Algortihm/formulae to multiply two arrays (1D & 2D) of unique numbers such that the resultant array contains unique results.
Would one have to create the 2 initial arrays in a certain ...
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What is the set of all numbers that can be represented with a floating-point format?
Computers use single- (or, for more precise calculations, double-) precision floating-point formats to represent a subset of real numbers. While a decent chunk of real numbers can be stored with these ...
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Method for finding the largest positive difference between two pairs of IEEE 754 double precision floating point numbers and fixed-point numbers
I have two pairs of IEEE 754 double precision (64-bit) floating-point numbers and unsigned fixed-point numbers, and I'm trying to find which pair has the greatest difference.
The fixed-point numbers ...
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Is converting between roots and coefficients of a polynomial numerically stable?
Assume we're on a computer using 32-bit floats (or something similar), and I'm converting back and forth between the $n$ coefficients of a polynomial and the corresponding $n$ roots of the polynomial. ...
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storing decimal number into computer with finite mantissa
I am learning about numerical methods and the following link caught my attention:
https://www.iro.umontreal.ca/~mignotte/IFT2425/Disasters.html
So from what I understand 0.1 is not exactly ...
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Proof of loss of orthogonality in Gram-Schmidt
I am stuck at understanding about how to derive the following proofs related to error bounds which are given in the following slides. Can anyone please explain to me how these are derived?
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fl(A) where A is a square matrix
We defined $fl(x)$ to be the function $fl:\mathbb{R} \rightarrow \mathbb R_b (t, s)$ (i.e., takes reals and outputs the float). What does $fl(A)$ mean when $A \in \mathbb R ^{n \times n} $? I assume ...
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trouble understanding floating point representation
I had a quiz last week on floating point representation. After he graded the quiz, he walked us through each step so that we could see what we did wrong. I took notes so that I could study his ...
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Determine The Base of The Venusian Numeration System [closed]
this question is from Thomas Koshy's book called "Discrete Mathematics With Applications":
Any idea how to do this question? I can tell that the base of the system is at least 3 (since we ...
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Fast computation of $x^{1/p}$, where $x\in\mathbb{R}^+$ and $p=2^{n}$, where $n\in\mathbb{N}$ with bit shifts?
There is plenty of literature regarding the legendary Fast inverse square root routine from Quake, but can we do something similar to compute $x^{1/p}$ as given in the title?
Given that $p$ is a power ...
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