# Tagged Questions

For questions regarding the underlying implementation of built-in functions.

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### How was Mathematica built?

I've been wondering this for a while now: How was Mathematica built? Is it written in Haskell? C++? I'd love to get a taste of the wizardry behind this incredible software.
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### Is there a method to look inside at the internal code of Built-In functions? [duplicate]

How can I look at the code of built-in functions? Is there something like this: LookInsideFunction[SubsetQ] RETURNS ...
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### What algorithm do the Compress and Uncompress functions use?

Compress and Uncompress are used internally by Mathematica to compress things like 3D data in notebook files - data that I'd like to read independently. Does anyone have any idea if the compression ...
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### How does Plus work on machine precision Real arguments?

I thought Kahan's summation method would make a nice example for students to use to think about round-off error [W. Kahan, Pracniques: Further Remarks on Reducing Truncation Errors, Commun. ACM 8  (...
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### Why this upvalue doesn't escape from Sum?

Probably a hard question, but I decide to cry out loud :). This is actually another problem I encountered when answering this question. Consider the following transform rule stored as an upvalue: <...
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### BoxWhiskerChart Notched: how is the notch calculated?

I am aware of one convention for the uncertainty in a median: it is 1.58 * IQR / Sqrt[n], where IQR is the inter-quartile range and n is the number of data points. This was proposed by McGill, Tukey, ...
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### What is the underlying algorithm to simplify sums of reciprocals of polynomials?

Flipping through Wolfram's blog entry on Leibniz, W noted Huygens' interview test for the young Leibniz, namely to determine: $$\sum_{n\ge2} \frac{1}{{n \choose 2}}$$ It's one thing to do this by ...
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### What are the algorithm details of FindRoot?

The Help page of FindRoot says: "by default, FindRoot uses Newton's method (Newton-Raphson) to solve a nonlinear system". But I ...
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### How to show steps of Integrate[Exp[-(I*(s*t))]/(Sqrt[2*Pi]*Sqrt[s]), {s, 0, Infinity}]? [duplicate]

I am trying to use Wolfram Alpha in Mathematica 10.2 to show me the steps for calculating the integral in the title, i.e I have written the following command: show steps of ...
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### How does Matematica compute the CDF of the binomial distribution?

I want to know, how Mathematica actually implements the cumulative distribution function of the binomial distribution. Which algorithm does Mathematica use to compute ...
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### Citation of the algorithms used in FindGraphCommunities

For my research work I need to put a comparison of different community detection algorithms. Instead of implementing all those algorithms once again, I would like to use the methods already provided ...
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### Does LinearSolve know that I gave it a Sparse lower triangular matrix?

Let $\mathbf L$ be a nonsingular numeric (non-symbolic) large, lower triangular sparse matrix. Let $\mathbf b$ be an appropriately sized vector. I want to solve $\mathbf L\mathbf x=\mathbf b$. ...
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### What values of $\delta$ and $\eta$ are being used in LatticeReduce?

The Lenstra–Lenstra–Lovász algorithm has a parameter $\delta$ with $1/4 < \delta < 1$, where roughly speaking the closer $\delta$ is to 1 the longer it takes, but the better basis reduction you ...
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### What type of solver does Mathematica use in LinearSolve

I have a question regarding linear equation solvers. For a specific 9-diagonal matrix, every method I tried in C++ (Gaus, GCC, BICGTAB) didn't work (even though they worked for other matrices). But in ...
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### Any ideas on how GeneralMiniMaxApproximation is implemented?

GeneralMiniMaxApproximation is used to construct minimax approximations of parametrically defined functions. I am curious about how ...
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### Default algorithm in NonlinearModelFit

Does anyone know the default algorithm used in NonlinearModelFit? The documentation gives several possible algorithms, but I can't figure out what it uses if you do not specify the algorithm. Thanks.
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### How to determine BLAS/LAPACK implementation used internally for numerical matrix operations?

Is there a command which reveals which implementation of BLAS and LAPACK are used in Mathematica's matrix operations such as Eigensystem? I asked a related question ...
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### What's the algorithm behind markov classifier in Classify function for text classification?

https://www.wolfram.com/mathematica/new-in-10/highly-automated-machine-learning/find-which-author-wrote-a-text.html The above url show the use of Classify ...
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### How does Accumulate work?

Accumulate can be used to compute the partial sums of a list. The partial sums can also be computed using a For loop but this ...
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### The difference between “SymbolicProcessing” -> 0 and restricting the function definition to numeric values only

The Documentation tells us that there are two ways to disable symbolic processing of the integrand by the NIntegrate function when it is known that it just slows ...
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### SetSystemOptions is obsolete? What is next？ [closed]

When running the following code on Mathematica 10.2 SetSystemOptions[ "CompileOptions" -> "CompileReportExternal" -> True]; I received the message: ...
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### What methods does Maximize use like $\mathtt{Maximize}\left[\left\{3 x^2+2 \sqrt{2} x y,x^4+y^4=1\right\},\{x,y\}\right]$

I used Maximize for the maximum value of $3 x^2+2 \sqrt{2} x y$ when $x^4+y^4=1$ , $x>0,y>0$. What methods does Maximize ...
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### Apart may use Padé method: what's that?

How does Apart work? The page tutorial/SomeNotesOnInternalImplementation#7441 says, "Apart ...
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### How does Mathematica resolve symbolic systems of inequalities? [closed]

Before getting into my question, let me provide some context: I recently started using Mathematica to automatically perform stability analyses that I previously did by hand. A major part of the ...
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### Mathematica Lattice Reduce Command

I'm going through a very old copy of "Mathematica: A System for Doing Mathematics by Computer" for self practice. I'm on chapter 3, and ran into the LatticeReduce ...
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### What are the hidden specifications for FindRoot

The Help page of FindRoot says: "by default, FindRoot uses Newton's method (Newton-Raphson) to solve a nonlinear system" (or a nonlinear equation I suppose)...
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### How to find matrix exponential using RootSum? [duplicate]

From what I've been able to deduce, Mathematica uses a root sum scheme to determine the exponential of a matrix. Can someone please explain to me the theory behind this? I've searched, but have had no ...
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### FindMinimum [] methods [closed]

Which algorithm is used when we do not use the option Method of FindMinimum[] function. FindMinimum[f,{x,x0}]
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### How to interpret the FullForm of a SparseArray?

SparseArrays are atomic objects, but they do have a FullForm which reveals information about them. What is the meaning of the ...
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### How does Mathematica calculate LaguerreL

About the function LaguerreL[n,a,x], the helping documents in Mathematica only say that this function satisfies equation $xy^{\prime\prime}+(a+1-x)y^\prime+ny=0$. ...
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### Functionality of FindMaximumFlow

I know that the superseeded function NetworkFlow (needs Combinatorica) uses Goldberg-Tarjan's Push-Relabel-Algorithm but i cant find any information on Mathematica's built-in-function (...
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### What Algorithm Does Mathematica use to Calculate Betweenness Centrality for Unweighted Graphs?

I recently asked this question on the Mathematics SE. Does anyone know, as a follow up, what algorithm Mathematica uses to calculate the betweenness centrality vector (of $N$ elements) for an ...
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### What am I missing in this highly oscillatory integral?

I want to numerically integrate this equation (in python without calling Mathematica): $\int_0^\infty {\rm d}k f(k) J_v(r k) J_n(s k)$ where $f(k)$ is a non-smooth function, $J_v$ are the Bessel ...
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### References on functions FindGraphPartition and FindGraphCommunties

Does somebody have some interesting references on the FindGraphPartition and FindGraphCommunties[.., Method -> "Spectral"] ...
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### what is The Formula behind the Kolmogorov-Smirnov test statistics?

I calculate the Kolmogrov-samirnov test statistic for the following data: ...
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### The default bandwidth of the SmoothKernelDistribution function

I am researching the Silverman rule of thumb for bandwidth selection. Below is a sample of my code, from which it seems that BW=63 is close to the default parameter given by Mathematica (2 images are ...
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### Documentation for the Bounded option of SmoothKernelDistribution

I have found the Bounded option useful in the SmoothKernelDistribution function, but I can't find any documentation for exactly ...
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### How does LocationTest choose its “AutomaticTest”?

Is there a way of determining how LocationTest chooses its "AutomaticTest"? Sometimes it's clear from the ...
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### How does Plot work?

I considered the following function: sin[x_] := Module[{}, Print["x=", x]; Sin[x] ] in Mathematica. Next, I tried to plot it using: ...
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### What algorithms does NSolve use?

I've been using NSolve to find roots and now I've been wondering what underlying algorithm Mathematica uses. It's probabaly similiar to ...
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### Specific Mathematica algorithms, for example LU Decomposition

Where I can find detailed information on algorithms used by Mathematica, especially for numerical methods. The Manual doesn't seem to iclude specifics in most cases. For example I get a different LU ...
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### What function generates random numbers in a compiled function?

I am writing a simulation for 2D random walk. I wrote several version of code, and found that the speed is like this (fastest to slowest): C++, Mathematica compiled procedural code, Mathematica ...
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### How to know the internal algorithms of functions like Predict or Classify?

The document centre seems only explain how to use these functions, but just in an very brief way. I know Mathmatica is not open source, so we can not expect to see the code of the function, but are ...
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### Algorithms behind implemented Mathematica functions [duplicate]

When I use Mathematica for analysis and I write a publication, readers often ask me: "What is the algorithm behind your calculation?". I answer that I did it with Mathematica and if they are ...
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### How to find out which method Mathematica selected?

There are commands like NonlinearModelFit[] or NDSolve[] that have the option Method it ...
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### How does Mathematica estimate error on fit parameters?

When performing a fit using NonlinearModelFit (or any of Mathematica fitting functions, but I am particularly interested in nonlinear models), Mathematica makes a ...
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### How does Mma compute Confidence Intervals?

I am trying to understand how Mma computes the Confidence Intervals after a NonlinearModelFit. Consider the following example: ...
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### Custom Methods for NMinimize/NonLinearModelFit

Is there a way to write custom methods for NonLinearModelFit or NMinimize? To give background, I currently have a batch of code ...