# Tagged Questions

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

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### Graph diameter algorithm

Does anyone know what the complexity of Mathematica's GraphDiameter algorithm is? I have a family of very large (and very regular and very sparse) graphs, and if Mathematica uses something like the ...
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ImageAdjust does an excellent job of auto-leveling an image. Although there doesn't appear to be an explicit way to get the contrast, brightness and gamma values ...
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### Rationalize error

The docs state that "Rationalize[x,dx] yields the rational number with smallest denominator that lies within dx of x." However, testing this out it appears to be false. ...
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### What is the Chebyshev method for InterpolationFunction?

With InterpolationOrder -> All, some methods of NDSolve returns interpolations with ...
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### What does the GraphLayout -> “SpectralEmbedding” option do (mean) when displaying a graph?

I have some relational data represented as a graph. As such some of the relational properties can be recognized. When I use the GraphLayout -> "SpectralEmbedding" option a very nice and sharply ...
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### How is Integrate Implemented? [duplicate]

At the end of this 90 minute lecture by Stephen Wolfram on the history of the development of Mathematica, Wolfram takes questions from the audience. One of those questions asked how the ...
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### Why patterns names aren't scoped by default?

In a recent question [1] OP asked how to make Module scope working inside Set, so how to get: ...
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### How does ListInterpolation work?

ListInterpolation takes an array of data and interpolates between the entries. Let us interpolate between the elements of the identity matrix. For reference, this ...
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### Mathematica's morphological thinning algorithm

Could anyone cite some papers that describe Mathematica's morphological thinning algorithm?
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### Where can I find documentation on the methods/algorithm used in FindGraphPartition in Mathematica 10?

I am working on a graph theoretical project where it would be helpful to know the name of the algorithm used in FindGraphPartition. Question: Where can I find ...
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### NIntegrate evaluates its 1st argument while it has the attribute HoldAll?

NIntegrate owns the attribute HoldAll: Attributes@NIntegrate (* {HoldAll, Protected} *) ...
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### Why NIntegrate is badly-behaved on $J_{\frac{9}{2}}(x)$ by default?

A friend of mine showed me this example: Plot[BesselJ[9/2, x], {x, 0, 1}, PlotLabel -> Style["The integrand seems to be simple", 14]] ...
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### What is the default method used by NSolve

I have a system of 6 polynomial equations of the form aijk xi xj + bik xi = ck (k=1,2,...,6; summation convention on i & j) I am using NSolve[{eq1,eq2,...,eq6},{x1,x2,...,x6},Reals] What ...
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### Classifying text and number features

I am new to the ML algorithms and I am checking the Classify function of Mathematica. I understand the math for the random forest algorithm, but I have a question ...
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### What is the default SignificanceTest setting for FindClusters with Optimize method?

According to the documentation, the default value of FindCluster's Method option is Optimize....
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### Can parallelization speed up FindClusters?

I have a clustering task that would take FindClusters 2-3 months1 to complete. Can this calculation be sped up with parallelization? (FWIW, the data to be ...
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### Are the triangles in DelaunayMesh oriented ccw?

MeshCells[R, 2] returns the triangles from R=DelaunayMesh[pts], where pts is a list of 2D ...
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### min and max in Ticks determination

I wonder, how Mathematica determines the values that it sends to custom tick functions, ie. the minimum and the maximum on the axes? Here is an example: ...
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### How does Mathematica compute the determinant of a matrix? [closed]

I have been trying to write efficient code for calculating the matrix determinant for some time now. I noticed last night that Mathematica is able to compute the determinant of a $200 \times 200$ ...
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### FrobeniusSolve: how does it work?

Can someone suggest any reference to read? I would like to understand how the algorithm works.
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### Where do DistributeDefinitions and ParallelNeeds put their definitions for newly-launched kernels to find

I'm sorry to keep banging on the definition distribution bell, but here is another (hopefully well-focused enough) question on that vein. If one is trying to distribute the definitions in a package ...
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### Accuracy of PrimeQ function

Using PrimeQ in Mathematica 10 on integers up to $2\cdot 10^{5717}$ the function appears to work. The Documentation for Mathematica 5 says that ...
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### Multiply integrand with -1, and the precision changes?

"After multiplying the integrand of NIntegrate with -1, the Precision of the output will ...
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### Meaning of the character sequence after an Entity's name

Mathematica can interpret an entity when you type Ctrl+=, like this: Then we'll get an entity We can get the InputForm You will get a canonical name ...
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### The theory/algorithm knowledge behind the built-in Discretize* and Boundary*

The ComputationalGeometry package was built-in in V10 distribution. In classical computational geometry field, the following questions were investigated. Convex ...
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### Show how Mathematica defines a function

Is there a way to show how Mathematica defines a function, such as In: Something[Sqrt], Out: Sqrt[x] -> x^(1/2) As far as I understand it the command ...
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### What kind of preprocessing does Mathematica do when I use EdgeDetect on an image?

I am doing my own implementation of Canny and Sobel in another language (not in Mathematica). I was testing the EdgeDetect method (Sobel and Canny), and my own, and ...
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### Algorithm used by Fit[]

Does anyone know what algorithm Mathematica's basic Fit[] function uses? Is it proprietary? I'm trying to do some pretty basic, linear, polynomial data fitting (...
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### Is the Method -> Automatic option for Classify not deterministic?

I just prepared an (well known) example in machine learning, the weather dependent determination of whether to play golf or not, as shown in this picture (of a corresponding Dataset) Now I set up ...
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### What CPU instruction set extensions does Mathematica support?

Most CPUs in desktops and laptops nowadays are backwards compatible with the x86-64 instruction sets. But newer CPUs also have additional instruction set extensions like AVX 2, FMA3, and BMI/BMI2 ...
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### What is the encoding used when exporting images as MOV with “VideoEncoding” -> “MPEG-4 Video”?

I have a few rasterised images generated in Mathematica, and I want to output them in a video file, ideally using H.264 encoding. I found the following command. ...
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### The implementation of FunctionRange

How exactly does the FunctionRange work? What kind of algorithm does it use? I looked it up in the manual but it doesn't provide any references or even a name of ...
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### How many clusters does DominantColors return?

Is there any documentation on how it automatically chooses the number of colors - does it use X-means or some other heuristic to determine the number of clusters it automatically returns?
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### What governs images resizing in the FrontEnd environment

What is responsible for the middle Row images' size? And why single image row is different? ...
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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 Mathematica calculate $\sin(\pi/5)$?

Consider: Sin[Pi/5] which returns: $$\sqrt{\frac58-\frac{\sqrt5}{8}}$$ Does anyone know how to figure out the trigonometric identities used by Mathematica to ...
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### What is the default ColorFunction for MatrixPlot?

I think the the default ColorFunction for ArrayPlot is ColorNegate[RGBColor[{#, #, #}]] &...
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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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### 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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### How can I access the Criterion function of KI method in FindThreshold for a dataset?

I want to find a threshold for some data set, using Kittler-Illingworth minimum error thresholding method (KI method). To do so, I use the Mathematica built-in function ...
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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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### 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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### 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'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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### 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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### SetSystemOptions is obsolete? What is next？ [closed]

When running the following code on Mathematica 10.2 SetSystemOptions[ "CompileOptions" -> "CompileReportExternal" -> True]; I received the message: ...
### 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 ...