# Questions tagged [numerical-analysis]

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### Algorithm to calculate lower incomplete gamma function

I'm trying to understand the implementation of the algoritm here. Please see GammaLowerRegularized(a, x) function. I understand 1st part of the function for ...
• 131
1 vote
29 views

### What does it mean unambiguously that a number is value 0 up to numerical precision?

I was reading that a quantity $x$ is $0$ upt to numerical precision. What does this statement formally mean -- especially in the context of numerical methods or real computers. I looked up in google ...
• 2,910
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### (Numerical Analysis) What is the largest double float represented for the gamma function and $n!$

Consider that \begin{align} \Gamma(n+1) = n! \end{align} for any integers. I then got the following two questions: What is the largest value of $n$ for which $Γ(n+1)$ and $n!$ can be exactly ...
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### Sparse Recovery and Feature Selection

This may be a silly question but are sparse recovery and feature selection problems fundamentally the same? Namely, say we have a set $N$ of features and $S \subset N$ which are the "important&...
86 views

### Precise algorithm for finding higher order derivatives

I'm trying to make an algorithm that finds the first 10 or so terms of a function's Taylor series, which requires finding the nth derivative of the function for the nth term. It's easy to implement ...
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1 vote
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### Finding points of local maximum error in Remez algorithm

So the Remez algorithm is an algorithm for finding optimal polynomial approximations or at the very least for converging towards them. To find an approximation of $f$ by an $N$th degree polynomial the ...
• 3,738
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### Why aren't Grobner bases more common?

I have only heard of / seen Grobner bases in passing, but my understanding is that they are much like vector bases but for polynomials instead of linear systems. I know polynomials are used in CAD and ...
• 125
1 vote
234 views

### What is the machine epsilon and number of mantissa bits for TI-83?

I am trying to determine how many bits the TI-83 Plus uses to store floating point numbers. I am using the algorithm for approximating the machine epsilon given in "Numerical Mathematics and ...
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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 ...
2k views

### 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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1 vote
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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 ...
1 vote
94 views

### $fl(x)=x(1+\delta)$

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. ...
• 124
1 vote
62 views

### 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 ...
• 111
404 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 ...
• 151
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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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287 views

### numerically stable log1pexp calculation

What are good approximations for computing log1pexp for single precision and double precision floating point numbers? Note: ...
• 265
347 views

### 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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556 views

### 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 ...
• 423
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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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