# Questions tagged [numerical-analysis]

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### How to determine the set of real numbers corresponding to a given floating point number?

Let's say we consider IEEE 754 double precision floating-point numbers, and we use RNTE - Round To Nearest, Ties to Even - rounding. I know that the RNTE rounding works this way: given two consecutive ...
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### How to represent zero as floating point number?

For the floating-point number, we have the form $\pm d_0.d_1d_2···d_{P-1}\times\beta^E$ $\pm$ --------------------------- sign $d_0.d_1d_2···d_{p-1}$ --------- significant $\beta$ --------------------...
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### How can I compute logarithm when comparison is undecidable?

In Haskell, I have the following datatypes that encodes arbitrary real numbers and arbitrary complex numbers, respectively: ...
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### Cancellation in C++

I am trying to figure out what the problem with the following expression in C++ is: y=std::log(std::cosh(x)); My first intention was that there might occure a ...
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### $fl(x)=x(1+\delta)$

The floating point representation of a real number $x$ in a machine is given by $fl(x)=x(1+\delta),\: |\delta| = \frac{|x^*-x|}{|x|} \le \epsilon$. But I do not find this equation very insightful. ...
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### Validity of Algorithm for Testing Two Floating Point Numbers

This question is related to the epsilon- (or delta- if you prefer) test for floating point equality. But my question is not how to do it. Instead I have a related algorithm for testing equality, and I ...
298 views

### fast and stable x * tanh(log1pexp(x)) computation

$$f(x) = x \tanh(\log(1 + e^x))$$ The function (mish activation) can be easily implemented using a stable log1pexp without any significant loss of precision. Unfortunately, this is computationally ...
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### Computing an Expression

I am writing code to evaluate the following expression: $$\frac{(a+b+c)!}{a! b! c!}$$ where $a$, $b$ and $c$ are on the range of $10$ to $500$. The result is going to be a floating point number. ...
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### Is order of matrix multiplication affecting numerical accuracy of the result?

I have to multiply three matrices of floats: A (100x8000), B (8000x27) and C (27x1). Is ...
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### How much can we trust mathematical software when working with large numbers, and how much memory it needs to work with these numbers?

For example, I want to evaluate the expression: $3^{3^{{3}^{3}}}$ so I used wolframalpha.com (it's free, and I don't own any software), which returned the scientific notation of the number above, ...
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### numerically stable log1pexp calculation

What are good approximations for computing log1pexp for single precision and double precision floating point numbers? Note: ...
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### Stable and fast computation of the squared euclidean distance matrix

Let's say I want to compute the matrix $M$ of the squared euclidean distances between each pair of vectors $(x, y)$ belonging to two sets $X$ and $Y$ respectively. The sets of vectors $X$ and $Y$ have ...
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### Proof that (x-y)(x+y) is more accurate than x²-y²

I was carrying on my reading of What Every Computer Scientist Should Know About Floating-Point Arithmetic but got stuck on the proof of Theorem 2 (page 34). At some point it says: \begin{align} (x \...
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### Proof that a guard digit bound the error of subtraction

I was reading What Every Computer Scientist Should Know About Floating-Point Arithmetic, which is extremely interesting. But I have some troubles understanding the proof of Theorem 9 (page 33). First ...
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### Rigorous error bounds for eigenvalue solvers

I computed the first four eigenvalues of a quite large ($2^{24}\times 2^{24}$) but very sparse matrix. I used pythons in-build function sparse.linalg.eigsh to compute them. I need a validation that ...
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### How to estimate floating-point precision of function?

Let's say I have a function that consists solely of floating-point operations where the last operation rounds the computed value to a predefined number of digits. And I feed this function with a range ...
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### predictor/corrector and gear integration for positive and negative time steps

I have a predictor method that takes differential equations for 214 biological species and integrates them in time using positive time steps. Right now it is able to successfully integrate the ...
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### Which implementation for the Maclaurin Series for the cosine function is better?

First, sorry if this post is off-topic. I consider it too analytic for stack overflow. In Numerical Analysis subject I must explain which one is better (has less error). The recursive ...
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### Avoiding overflows while computing $e^x$ by Taylor series

I'm coding a program to calculate the value of $e^x$ by using the Taylor expansion, that is: $$e^x =\sum_{k=0}^\infty \frac{x^k}{k!}$$ ...
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### Why is computation of this function numerically unstable? [closed]

Why is computing the function at certain ranges of h $$f(x) = \dfrac{\sqrt{-x+a} - \sqrt{2x+a}}{ 4a}$$ unstable? How would I rewrite this function so that it is stable?
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### Do formulas involving fewer repetitions of variables give higher numerical precision?

I'm having some trouble doing SICP exercise 2.15. Please note that this question is not closed related to Lisp. Instead, it's closely related to numerical analysis. Exercise 2.15. Eva Lu Ator, ...