What are some useful, undocumented Mathematica functions?



There have already been some questions about some undocumented functionality in Mathematica. Such as (please add to these lists!)

Also, other questions and answers that contained undocumented functions

Along with the "Undocumented (or scarcely documented) Features" segment of the What is in your Mathematica tool bag? question.

Szabolcs also maintains a list of Mathematica tricks which contains a list of "undocumented stuff".

So, what undocumented functions do you know and how do you use them? (Added useful information is maybe how you discovered the functions and any version dependence.)


Posted 2011-12-11T03:45:20.323

Reputation: 9 867

5I guess I have to state the obvious here and that is using undocumented functions/options can break with major or even minor mathematica upgrades. My advice is to stay away unless you like to experiment and have a lot of time on your hands. – Eric – 2012-01-27T15:45:54.753


J.M. mentioned Internal`FromPiecewise in (27254) if anyone feels like writing an answer about it.

– Mr.Wizard – 2015-04-13T17:07:43.923

@Kuba, Is it possible to edit this question to the style of community wiki? – xyz – 2015-05-07T08:23:38.390

1I don't know. No need for that on the other hand, this is a good question. – Kuba – 2015-05-07T08:25:58.730

@Kuba, Thanks, I use Mathematica in my study everyday and I like Mathematica and StackSxchange which give me knowledge and happy.:) – xyz – 2015-05-07T08:31:28.030

Indeed, this is a nice question -looking forward to the answers – dr.blochwave – 2015-05-07T09:03:45.013

Extremely shocked to see Internal\CompareNumeric` missing, but then I realised that your image is not the full list. Good question. – Oleksandr R. – 2015-05-10T04:53:44.317

@OleksandrR., Thanks, Image is just a part of the full list:) – xyz – 2015-05-10T06:38:36.853

I wonder what the purpose of Internal`Square[] is? – dr.blochwave – 2015-05-10T14:03:51.967

7I would add Internal`AbsSquare. So much time lost computing the square root in Abs or Norm just to undo it a moment later in quantum mechanical calculations. – The Vee – 2016-04-04T15:49:33.603

An archived version of the dead link: http://web.archive.org/web/20140823044625/http://web.ift.uib.no/~szhorvat/mmatricks.php

– Vladimir Reshetnikov – 2018-06-07T06:19:51.593

(155433) – Mr.Wizard – 2019-12-20T14:30:28.930



  • LongestCommonSequencePositions and LongestCommonSubsequencePositions Their use is analogous to LongestCommon(Sub)sequence but they return the position of the first match instead.

    Update: These are documented since 10.2.

  • ClipboardNotebook[] can be used to access the clipboard. NotebookGet@ClipboardNotebook[] will give a Notebook expression with the current contents of the clipboard. I use this for pre-processing data before it is pasted (e.g. in the table paste palette). I am not sure if this can be used for copying at all---I use the Front End's Copy function directly for that (through FrontEndTokenExecute)

    Update: Since version 8 we have some documented clipboard functions.

  • PolynomialForm[] allows changing the order in which polynomial terms are printed by setting the option TraditionalOrder -> True

    In[1]:= PolynomialForm[1+x+x^2, TraditionalOrder->True]
    Out[1]= x^2+x+1
  • POST request: In version 8 Import has experimental support for the POST HTTP request method. Example usage for uploading an image to imgur:

    Import["http://api.imgur.com/2/upload", "XML", 
           "RequestMethod" -> "POST", 
           "RequestParameters" -> {"key" -> apikey, "image" -> image}]

    (Of course you'll need to insert your API key and a properly encoded image, as shown in the answer I linked to above.)

  • Internal`Deflatten[] will reconstruct higher dimensional tensor from a flat list. Example:

    In[1]:= arr = {{1, 2}, {3, 4}}
    Out[1]= {{1, 2}, {3, 4}}
    In[2]:= flatArr = Flatten[arr]
    Out[2]= {1, 2, 3, 4}
    In[3]:= Internal`Deflatten[flatArr, Dimensions[arr]]
    Out[3]= {{1, 2}, {3, 4}}

    Warning: If the dimensions passed to it don't match the length of the flat array, this will crash the kernel!

    Update: Version 9.0 introduced the documented equivalent ArrayReshape.

  • Image capture start/stop IMAQ`StartCamera[] and IMAQ`StopCamera[] start and stop the webcam.

  • Undocumented interesting contexts to dig through: Internal`, Experimental`, Language`, NotebookTools` (similar to what the AuthorTools package offers), IMAQ` (IMage AQcuisition)

    There are lots of functions in these contexts, generally undocumented, but sometimes with self-explanatory names (e.g. Internal`RealValuedNumericQ seems obvious). Note that these functions might change in later versions. Some of the ones listed by ?Internal`* are even from old versions and no longer work in M- 8.

    Some functions from Language` are described here.

  • SystemOptions[] The functions to set and read these options are not undocumented, but the options themselves unfortunately are.

    • Experimental`SystemOptionsEditor[] In version 8 this gives a GUI for viewing/setting system options.

    • "TableCompileLength" (and other similar options from the "CompileOptions") section set the length of a Table above which it attempts to compile its argument.

      Example: SystemOptions["CompileOptions" -> "TableCompileLength"] will show that the default value is 250.

    • "SparseArrayOptions" -> {"TreatRepeatedEntries" -> 1}

      Setting this option to 1 will cause repeated entries to be summed up when creating a sparse array. See an example use and explanation here.

      In[1]:= Normal@SparseArray[{2 -> 1, 4 -> 1}]
      Out[1]= {0, 1, 0, 1}
      In[2]:= Normal@SparseArray[{2 -> 1, 4 -> 1, 2 -> 1}]
      Out[2]= {0, 1, 0, 1}
      In[3]:= SetSystemOptions["SparseArrayOptions" -> {"TreatRepeatedEntries" -> 1}]
      In[4]:= Normal@SparseArray[{2 -> 1, 4 -> 1, 2 -> 1}]
      Out[4]= {0, 2, 0, 1}

This MathGroup thread has some interesting information too.


Posted 2011-12-11T03:45:20.323

Reputation: 213 047

1I cannot for the life of me figure out what you mean by "The functions to set and read these options are not undocumented, but the options themselves unfortunately are." There's likely a typo which I can't figure out; would you kindly fix? Thanks. – QuantumDot – 2016-10-05T02:06:23.723

@QuantumDot "are not undocumented" means "are documented". So, it means we know how to set options, but unfortunately do not know what those options mean. – Vladimir Reshetnikov – 2018-06-08T17:19:36.880


One undocumented function I find useful is Precedence:

For example:

{#, Precedence@#} & /@ {Plus, Minus, Times, Power, Apply, Map, Factor,
    Prefix, Postfix, Infix} // TableForm


Plus    310.
Minus   480.
Times   400.
Power   590.
Apply   620.
Map     620.
Factor  670.
Prefix  640.
Postfix 70.
Infix   630.

Precedence is described in the lecture A New Mathematica Programming Style by Kris Carlson.


One from about a year ago, which was then considered 'under development', is TableView. I wonder what has happened to it?

For example:

Array[Subscript[a, ##] &, {4, 3}] // TableView


Example of a TableView

For the recent version 10.3,TableView cannot work normaly.

Example of TableView not working in version 10.3


Posted 2011-12-11T03:45:20.323

Reputation: 8 863

I was thinking of adding Precedence here. Luckily I noticed your answer. Appropriately enough, I learned of this from a comment you made to a question on stackoverflow. +1 – acl – 2012-01-29T01:23:53.120

5At a Wolfram Training Course in April '15 we were advised that TableView was being deprecated and not to use it as it caused frequent Kernel crashes. Its been deleted from the help, although curiously still gives a hit for ListPicker. – Gordon Coale – 2015-05-12T12:15:23.767

3As per a talk at WTC '18 I believe TableView might be coming back in 12.x. – Carl Lange – 2018-12-11T23:58:46.797


The following simulates Mathematica's behaviour after using it for more than 24 hrs.


Works as advertised! :D

rm -rf

Posted 2011-12-11T03:45:20.323

Reputation: 85 395

3That make my day! :D – ybeltukov – 2014-01-19T22:33:16.563

14OMG! I thought you were kidding... – Silvia – 2014-06-19T19:14:59.360

I wonder how it was discovered, can't find it in Names["*`*"] – navigaid – 2017-01-13T03:34:52.273

Oh wow ok I thought I was just crazy (I am but still) – CA Trevillian – 2019-06-11T03:39:50.560


Thinking about a recent answer made me wonder exactly which functions in Mathematica use Assumptions. You can find the list of System` functions that use that Option by running

Reap[Do[Quiet[If[Options[Symbol[i], Assumptions]=!={}, Sow[i], Options::optnf]], 
  {i, DeleteCases[Names["System`*"], _?(StringMatchQ[#, "$"~~__] &)]}]][[2, 1]]

which (can be more elegantly written using list comprehension and) returns (in version 8)

{"ContinuedFractionK", "Convolve", "DifferenceDelta", "DifferenceRootReduce", "DifferentialRootReduce", "DirichletTransform", "DiscreteConvolve", "DiscreteRatio", "DiscreteShift", "Expectation", "ExpectedValue", "ExponentialGeneratingFunction", "FinancialBond", "FourierCoefficient", "FourierCosCoefficient", "FourierCosSeries", "FourierCosTransform", "FourierSequenceTransform", "FourierSeries", "FourierSinCoefficient", "FourierSinSeries", "FourierSinTransform", "FourierTransform", "FourierTrigSeries", "FullSimplify", "FunctionExpand", "GeneratingFunction", "Integrate", "InverseFourierCosTransform", "InverseFourierSequenceTransform", "InverseFourierSinTransform", "InverseFourierTransform", "InverseZTransform", "LaplaceTransform", "Limit", "PiecewiseExpand", "PossibleZeroQ", "PowerExpand", "Probability", "ProbabilityDistribution", "Product", "Refine", "Residue", "Series", "SeriesCoefficient", "Simplify", "Sum", "SumConvergence", "TimeValue", "ToRadicals", "TransformedDistribution", "ZTransform"}

You can similarly look for functions that take assumptions that are not in the System` context and the main ones you find are in Names["Developer`*Simplify*"] which are (adding "Developer`" to the context path)

{"BesselSimplify", "FibonacciSimplify", "GammaSimplify", 
 "HolonomicSimplify", "PolyGammaSimplify", "PolyLogSimplify", 
 "PseudoFunctionsSimplify", "ZetaSimplify"}

These are all specialized simplification routines that are not called by Simplify but are called by FullSimplify. However, sometimes FullSimplify can take too long on large expressions and I can imagine calling these specialized routines would be useful. Here's a simple usage example

In[49]:= FunctionsWolfram[""] /. Equal -> Subtract // Simplify
         % // Developer`PolyLogSimplify

Out[49]= -Pi^2/6 + Log[1 - z] Log[z] + PolyLog[2, 1 - z] + PolyLog[2, z]

Out[50]= 0

(The FunctionsWolfram code is described here)

Another interesting assumption related context I noticed was Assumptions`. Once again, appending "Assumptions`" to the $ContextPath, Names["Assumptions`*"] returns the functions

{"AAlgebraicQ", "AAssumedIneqQ", "AAssumedQ", "ABooleanQ", 
"AComplexQ", "AEvaluate", "AEvenQ", "AImpossibleIneqQ", "AInfSup", 
"AIntegerQ", "AllAssumptions", "AMathIneqs", "AMod", "ANegative", 
"ANonNegative", "ANonPositive", "AOddQ", "APositive", "APrimeQ", 
"ARationalQ", "ARealIfDefinedQ", "ARealQ", "ASign", "AssumedFalse", 
"AUnequalQ", "AWeakSign", "ImpliesQ"}

These contain assumption aware versions of some standard system functions, e.g.

In[22]:= Assuming[Element[x, Integers], {IntegerQ[x], AIntegerQ[x]}]
         Assuming[x > 0, {Positive[x], APositive[x]}]

Out[22]= {False, True}

Out[23]= {Positive[x], True}


Posted 2011-12-11T03:45:20.323

Reputation: 9 867


Sequential With

From Daniel Lichtblau's comment there is a new undocumented syntax for With introduced sometime after version 10.1 that allows:

With[{a = 0}, {a = a + 1}, {a = a + 1}, a]

Delayed With, Block, and Module

These scoping constructs support the use of := in definitions which allows one to handle unevaluated expressions.

With[{x := 2 + 2}, Trace[x]]
Block[{x := 2 + 2}, Trace[x]]
Module[{x := 2 + 2}, Trace[x]]
{2 + 2, 4}

{x, 2 + 2, 4}

{x$6064, 2 + 2, 4}


  • I find this most useful in With since it inserts definitions into held expressions.
    I commonly use this for in-place modification of DownValues:

     a[1] = Pi;
     a[2] = E;
     With[{dv := DownValues[a]},  dv = dv /. {1 -> 3};]
      a[3]   (*  out=  π  *)
  • Kuba finds it very useful for writing readable controllers definitions.


    asso = <|"nested" -> <|"key" -> <|
        "spec" -> <|"a" -> 1, "b" ->  0|>
      a := asso["nested", "key", "spec", "a"],
      b := asso["nested", "key", "spec", "b"]
        Slider[Dynamic[b, {Automatic, (a = b^2) &}]],

The earliest Stack Exchange usage of this that I can find is a post by Szabolcs.

I implemented a similar syntax in my listWith function which is itself an extension of With.


Posted 2011-12-11T03:45:20.323

Reputation: 259 163

Is just the example unfortunately chosen? It seems to me that it works the same without With. DownValues[a] = DownValues[a] /. {1 -> 3}; a[3] – BoLe – 2016-08-03T11:55:15.423

@Kuba I think I usually make Dynamic part of the definition in that case? Does that fail in your typical application? edit: Actually I remember that I use both, e.g. (88458). Would you care to edit this answer to include your example?

– Mr.Wizard – 2016-08-03T12:06:13.750

@Kuba Thanks for the example! – Mr.Wizard – 2016-08-03T12:10:49.530

1As an added variant, I often need to create random option values, and delayed With is perfect for this, e.g. AxesLabel -> With[{x := RandomChoice[{None, Automatic, "North"}]}, RandomChoice[{x, {x, x}}]]. – rcollyer – 2016-08-30T14:52:02.333

@rcollyer Now I am curious; why would you need random option values? – Mr.Wizard – 2016-08-31T06:54:04.267

4Testing. It's combinatorically impossible to cover all combinations, and you hope they're independent, we all know that isn't strictly true. So, for some tests, I'll generate random sets of options and see what blows up. Often it is the only way to get coverage. – rcollyer – 2016-08-31T12:52:02.283



Since version 11.2, this function is now buried in a deeper context: NumericalMath`NSequenceLimit[].

I quite like SequenceLimit[] myself; it is a function that numerically estimates the limit of a sequence by applying the Shanks transformation (as embodied in Wynn's $\varepsilon$ algorithm). The method is a particularly nice generalization of the probably more well-known Aitken $\delta^2$ transformation for accelerating the convergence of a sequence. Another way of looking at it is that if one applies the Shanks transformation to a sequence whose terms correspond to partial sums of a power series, the transformation gives the results corresponding to the diagonal Padé approximants formed from the partial sums.

Enough preamble, and let's see an example. Consider the sequence of iterates to the cosine's fixed point:

seq = NestList[Cos, N[1, 30], 20];

and let's generate the number for comparison purposes:

dottie = x /. FindRoot[x == Cos[x], {x, 3/4}, WorkingPrecision -> 40]



% - dottie



% - dottie

It can be seen here that applying the Shanks transformation to the sequence of iterates gave a result which had more good digits than any of the iterates themselves. This is the power of the function SequenceLimit[].

As with any powerful tool, however, some care is needed in its use. Consider for instance this example:

seq = N[Accumulate[((-1)^Range[0, 30]) Range[0,30]!], 30];

We have generated here a rather violently divergent sequence of partial sums $\sum\limits_k (-1)^k k!$. One would rightly be wary of trying to derive results from a sequence like this, but SequenceLimit[] manages to do something, even if it does spit out a warning:

   SequenceLimit::seqlim: The general form of the sequence could not be determined,
   and the result may be incorrect. >>

% - (-E ExpIntegralEi[-1])

and in fact the result can be justified through analytic continuation. However, that the algorithm can give unexpected results for divergent sequences is something to be mindful and careful of.

J. M.'s ennui

Posted 2011-12-11T03:45:20.323

Reputation: 115 520

Interesting, it has a usage message, but is not in the documentation - it's also in the System` context, which is rare for undocumented functions. Do you think it forms the backend for NSum[..., Method -> "WynnEpsilon"]? – Simon – 2012-01-28T03:38:51.067

1Yes, it's the method behind the "WynnEpsilon" options of NSum[]/NProduct[]. – J. M.'s ennui – 2012-01-28T03:47:29.620


This answer of mine http://mathematica.stackexchange.com/questions/95126/limit-of-partial-sums-involving-inverse-squares/95166#95166 has implementations of the Shanks transformation and Richardson extrapolation. I really wanted to include Richardson extrapolation in NSum, but got involved in NIntegrate ... By the way, the original author of NSum (Jerry Keiper) implemented his own modifications of the Wynn Epsilon algorithm.

– Anton Antonov – 2015-10-07T11:21:21.137

@Anton, it does seem unfortunate that Keiper did not mention his modifications, as you say, of the Wynn algorithm. In any event: I note that your implementation of Shanks does not use the "rhombus rule", and your implementation of Richardson extrapolation does not exploit the recurrence's "triangular" nature. You might consider looking at this...

– J. M.'s ennui – 2015-10-07T11:32:20.983

It is worth mentioning that despite its N prefix the function works on symbolic sequences as well, e.g. NumericalMath`NSequenceLimit[{a0, a1, a2, a3, a4, a5}]. – Vladimir Reshetnikov – 2018-06-07T06:13:25.677

@Vladimir, I suppose that would correspond to symbolically applying the Shanks transformation on the list of expressions given to it. – J. M.'s ennui – 2018-09-24T11:06:00.657

I gave an improved version of NumericalMath`NSequenceLimit, in both Mathematica and C++ versions, along with symbolic and numeric examples here: https://mathematica.stackexchange.com/a/178537/14489

– Chris Chiasson – 2018-09-25T13:48:59.643



The default value for the overhang parameter k (3rd argument) in ListCorrelate is None.

Internal` Predicates

Internal`LinearQ[expr, var] yields True if expr is a polynonial of exactly order one in var, and yields False otherwise.

Internal`RationalFunctionQ[expr,var] returns True if expr is a rational function of the symbol var, and returns False otherwise. Internal`RationalFunctionQ[expr,{var1, var2,...}] checks that expr is rational in each of the var_i. (ref)

Internal`RealValuedNumberQ[expr] yields True if expr is a real-valued number, and False otherwise.

Internal`RealValuedNumericQ[expr] yields True if expr is a real-valued numeric quantity, and False otherwise.

Internal`DependsOnQ[expr, form] yields True if a subexpression in expr matches form (excluding heads) and mathematically depends on form, and yields False otherwise. Takes a third argument (True/False, but behavior seems to be independent of choice) but seems to include heads also (ref)

Internal`EPolyQ[expr,var] yields True if expr is a polynomial in var and is in expanded form with respect to var. (New in 11.2)

Internal`ExceptionFreeQ[expr] yields True if expr evaluates to something that contains Infinity, DirectedInfinity, or Indeterminate, and yields False otherwise.

Internal`FundamentalDiscriminantQ[expr] yields True if expr is a fundamental discriminant Integer with the exception of 1, and False otherwise.

Internal`GregorianLeapYearQ[expr] yields True if the expr is an integer that corresponds to a leap year of the Gregorian Canlendar, and False otherwise.

Internal`LiterallyOccurringQ[expr, form] yields True if a subexpression in expr explicitly matches form, ignoring any Attributes of heads that might ordinarily influence the pattern matcher, and yields False otherwise.

Internal`LiterallyAbsentQ[expr, form] yields True if no subexpression in expr matches form, and yields False otherwise.

Internal`TestIntegerQ[number, form] yields {number, True} if number is an Integer, and {number, False} otherwise.

Internal`WouldBeNumericQ[expr, {var_1, var_2, ...}] yields True if expr would become a numeric quantity if the var_i were all numeric quantities, and False otherwise.

Internal`PatternFreeQ[expr] yields True if expr does not contain any of {Alternatives, Blank, BlankNullSequence, BlankSequence, Except, Longest, Optional, OptionsPattern, OrderlessPatternSequence, PatternSequence, Repeated, RepeatedNull, Shortest}, and False otherwise.

Internal`PatternPresentQ[expr] yields True if expr contains any of {Alternatives, Blank, BlankNullSequence, BlankSequence, Except, Longest, Optional, OptionsPattern, OrderlessPatternSequence, PatternSequence, Repeated, RepeatedNull, Shortest}, and False otherwise.

Internal`PolynomialFunctionQ[expr, var] yields True if expr is a polynomial in var, and yields False otherwise. InternalPolynomialFunctionQ[expr, {var1, var2,...}] yieldsTrueif expr is a polynomial in all var_i, and yieldsFalse` otherwise. (more info)

Internal`RadicalMemberQ[rad, {poly1, poly2, ...}] tests whether rad belongs to the radical of the polynomial ideal generated by poly1, poly2,... (ref)

Internal`SyntacticNegativeQ[expr] gives True if expr has a minus sign vague... needs to be clarified (application)

Other Internal`'s

Internal`BinomialPrimePowerDecomposition[n,m] gives a Internal`FactoredNumber object containing the list of prime factors of the binomial coefficient (n,m) together with their exponents.

Internal`ConditionalValueBody[inputHead,{test1, test2, ...},{{msgName1,arg1A,arg1B,...},{msgName2,arg2A,arg2B,...},...},body] evaluates each of the test_i in turn and if any yields False, immediately generates the corresponding inputHead::msgName_i Message with arguments arg_iA, arg_iB,... and returns Fail; otherwise evaluates body.

Internal`CompareNumeric[prec, a, b] returns -1, 0, or 1 according to whether a is less, equal, or greater than b when compared at the precision of a or b (whichever is less) minus prec decimal digits of "tolerance". It is the fundamental operation underlying Less, Equal, Greater, LessEqual etc. for finite-precision numeric types. (ref) (more info)

Internal`DiracGammaMatrix[n, "Metric" -> {list of +/-1}, "Basis" -> ("Dirac"/"Chiral")] returns the nth Dirac Gamma matrix.

Internal`ListMin[matrix] returns matrix with rows that has all elements greater or equal to elements of another removed. (ref)

Internal`Metric is an option to Internal`DiracGammaMatrix.

Internal`JoinOrFail[list1, list2] returns the list formed by appending list2 to the end of list1.

Internal`PerfectPower[integer] gives the list of integers {n,p} such that integer is n^p.

Internal`RiccatiSolve[{a, b}, {q, r}] solves the continuous time algebraic Riccati equation. (this is a documented System function)

Internal`DiscreteRiccatiSolve[{a, b}, {q, r}] solves the discrete time algebraic Riccati equation. (this is a documented System function)

Internal`MakePolynomial[expr] returns the form of expr in terms of new variables, such that expr is polynomial in the new variables. (ref)

Internal`ToEquivalenceClass[expr, toll] replaces the floating point numbers in expr with their equivalence class representatives according to the specified tolerance toll (ref).


System`Utilities`ExprLookupAdd[expr] stores expr to memory and returns an integer (not machine sized) key for retrieval.

System`Utilities`ExprLookup[integer] returns the corresponding expr if int is a valid integer key, and returns $Failed otherwise.

System`Utilities`SymbolList[expr] returns a list of atomic Symbols (including heads) in expr.
System`Utilities`SymbolList[expr, f] applies f to each Symbol before evaluating them. System`Utilities`SymbolList[expr, f, {"Context1`", ...}] omits symbols belonging to contexts Context1`, etc.

System`Utilities`Hash* (12 functions) (more info)

Some more stuff

Reduce`FreeVariables[expr] returns a List of Symbols in expr (more info). Unclear. See this for discussion.

GroupTheory`Tools`MultiSubsets[list, {n, m}], if n + m = Length(list), gives the set of subsets of exactly n elements appended to the set of subsets of exactly m elements in reverse order. (equivalent to MultiSubsets[list_, {n, m}] /; Length[list] == n + m := Join @@@ Transpose[{Subsets[list, {m}, Binomial[n + m, n]], Reverse[Subsets[list, {n}, -Binomial[n + m, n]]]}] and not much faster) To figure out: What if n + m ≠ Length(list)?

GroupTheory`Tools`PartitionRagged[list, {n1, n2, ...}] seems to be equivalent to Internal`PartitionRagged[list, {n1, n2, ...}], but works even if n1 + n2 ...Length[list].

GroupTheory`Tools`IntegerPartitionCounts[n] returns a list of lists corresponding to number (counts) of integers appearing in each partition. (the correspondence with IntegerPartitions[n] appears to be reversed).

GroupTheory`Tools`ConsecutiveReplace[expr,{patt1->list1, patt2->list2,...}] replaces elements of expr (Head usually List) that match patt1, patt2, ... with elements of list1, list2 ... in the order they appear in expr. If any of the list1, list2, ... are exhausted, it wraps around.

Integrate`InverseIntegrate[expr, {x, x0, x1}] performs the definite integration by attempting various substitutions of the form u == g[x] where g[x] is an expression in the integrand. (ref) (application) (application)


Posted 2011-12-11T03:45:20.323

Reputation: 18 597

Note that RationalFunctionQ works on trig expressions such as Internal`RationalFunctionQ[Tan[x] + 1/(Csc[x] - 1), {Sin[x], Cos[x]}]. It does not work on all expressions equivalent to a rational function though; for instance, Internal`RationalFunctionQ[Sin[2 ArcTan[x]], x]. – Michael E2 – 2019-10-15T20:47:59.857





  • Bag creates an expression bag, optionally with preset elements.
  • BagPart obtains parts of an expression bag, similar to Part for ordinary expressions. It can also be used on the lhs, e.g. to reset a value. StuffBag appends elements to the end of a bag.
  • We also have a BagLength, which is useful for iterating over a bag.




Internal`Bag, Internal`StuffBag, Internal`BagPart





Posted 2011-12-11T03:45:20.323

Reputation: 285

1For Internal`Bag, you should credit Daniel Lichtblau for that explanation, not me. As I noted in my post, that was just a quote from DL's answer which you linked to in the line above. Unrelated note: Using @username to ping people in an answer won't actually ping them. @-mentions don't work in answers. – rm -rf – 2015-05-11T03:16:23.687

@TheToad, OK, I got it. – xyz – 2015-05-11T05:23:04.123



This one has a usage statement!

Internal`PartitionRagged[Range[14], {3, 5, 2, 4}]
{{1, 2, 3}, {4, 5, 6, 7, 8}, {9, 10}, {11, 12, 13, 14}}

Note that Length[list] must equal n1 + ... + nk.

(* changed the last 4 to 3 *)
Internal`PartitionRagged[Range[14], {3, 5, 2, 3}]
Internal`PartitionRagged[Range[14], {3, 5, 2, 3}]

Internal`S1, Internal`S2, Internal`P2

Is it possible to have a documentation of these frequently-used functions with the help of the users in this community?

These guy's aren't frequently used (and probably aren't used at all), but they're really mysterious looking.

After reading this paper, I realized they're submethods used in computing PrimePi.

With[{x = 10^9},
    Internal`S1[x] + Internal`S2[x] + Internal`P2[x] + PrimePi[x^(1/3)] - 1
{50847534, 50847534}


(* Attributes[Internal`Square] = {Listable, NumericFunction, Protected} *)

Test it with a list:

list = RandomReal[{0, 100}, 10^8];

r1 = list*list; // RepeatedTiming
(* 0.118 seconds *)
r2 = list^2; // RepeatedTiming
(* 0.191 seconds *)
r3 = Internal`Square[list]; // RepeatedTiming
(* 0.121 seconds *)

The advantage of this function seems to come when computing higher powers on a list:

lis = RandomReal[{0, 1}, 10^7];

lis*lis*lis*lis; // RepeatedTiming
(* 0.55 seconds *)
lis^4; // RepeatedTiming
(* 0.21 seconds *)
Internal`Square @ Internal`Square @ lis; // RepeatedTiming
(* 0.15 seconds *)

Chip Hurst

Posted 2011-12-11T03:45:20.323

Reputation: 29 735

1+1, It is very convenient to utilize Internal`PartitionRagged[Range[# (# + 1)/2], Range@#] &[5] to achieve the following lists (*{{1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}, {11, 12, 13, 14, 15}}*) – xyz – 2015-05-11T07:49:57.240


@ShutaoTang regarding PartitionRagged I like my own implementation better as it does things PartitionRagged does not. I hope you will give it a look.

– Mr.Wizard – 2015-05-14T15:43:39.013

@Mr.Wizard, I have known your neat implementation last year. – xyz – 2015-05-15T03:25:32.737

@ShutaoTang Okay. :-) – Mr.Wizard – 2015-05-15T03:28:06.803

Thank you very much. It's really hard for me to decide whose answer is the best one. Since @blochwave is the first answer, I give this bounty to his answer. I'm sorry, I can only vote up for yours:) – xyz – 2015-05-16T00:53:09.700

@ShutaoTANG Will you happy this?<code> In[76]:= SymbolicTensorsSymbolicTensorsDumpMultiRange[Range[5]] Out[76]= {{1},{2,3},{4,5,6},{7,8,9,10},{11,12,13,14,15}}</code> – yode – 2015-09-24T14:59:39.330

In[76]:= SymbolicTensorsSymbolicTensorsDumpMultiRange[Range[5]] Out[76]= {{1},{2,3},{4,5,6},{7,8,9,10},{11,12,13,14,15}} – yode – 2015-09-24T15:05:25.300


Properties for SparseArray and InterpolatingFunction objects

SparseArray objects can accept a range of Properties (or Methods) that allow the efficient extraction of certain information, most commonly "AdjacencyLists" or "NonzeroPositions" as a frequently faster alternative to Position. I started this answer to detail them but as it grew I came to believe that it needs a Q&A of its own, so I posted one:

Likewise InterpolatingFunction also supports a number of Methods which I have detailed here:

Undocumented parameters

For Normal: Is there a way to control which special forms Normal converts?

For Return and Break: Is there a Break[] equivalent for short-circuiting in Table?

For Fold and FoldList the two-parameter syntax existed but was undocumented in versions 9.0 through 10.0.1: Shorter syntax for Fold and FoldList?


Posted 2011-12-11T03:45:20.323

Reputation: 259 163



This is the one that initially struck me as interesting since I use compiled functions quite a lot. From the documentation of Do:

Unless an explicit Return is used, the value returned by Do is Null.

But that doesn't seem to be the case for Compile`InnerDo!

f1 = Compile[{{x}},
      Module[{a}, a = x; Compile`InnerDo[a++, {i, 10^8}]]

f2 = Compile[{{x}},
      Module[{a}, a = x; Do[a++, {i, 10^8}]]

f1[0] // AbsoluteTiming
(* 1.63 seconds, 99999999 *)

f2[0] // AbsoluteTiming
(* 1.63 seconds, Null *)

Essentially it adds an extra line into the result of CompilePrint:

enter image description here


Seems to be just that, and is listable. In fact, if you write a compilable function that contains Mod[x, 1] then it gets compiled down to Compile`Mod1.

f1 = Compile[{{x}}, Compile`Mod1[x]];
f2 = Compile[{{x}}, Mod[x, 1]];

CompilePrint@f1 == CompilePrint@f2
(* True *)

Compile`DLLFunctionLoad / Compile`DLLLoad

These seem to perform the same functions as LibraryFunctionLoad:

fun1 = LibraryFunctionLoad["demo", "demo_I_I", {Integer}, Integer]
fun2 = Compile`DLLFunctionLoad["demo", "demo_I_I", {Integer}, Integer]
fun1[10] == fun2[10]
(* True *)


Posted 2011-12-11T03:45:20.323

Reputation: 8 483


No so much a function as an option...

Problem: You embedd a CDF on a web page but the content is rendered as grey boxes.

Cause: This is a security issue, the same as when you open a notebook with dynamic content from an untrusted path on your computer.

Solution: On your desktop you are asked if you want to enable dynamic content. You press the button and everything in your notebook works. By using the "option" {fullscreen:'true'} an embedded CDF will open in "full screen mode" meaning that the enabled content warning will appear and therefore provide your viewers with the button to enable dynamic content.


<script src="http://www.wolfram.com/cdf-player/plugin/v2.1/cdfplugin.js" type="text/javascript">
<script type="text/javascript">
var cdf = new cdf_plugin();
cdf.embed("http://path.to/myCDF.cdf", 500, 600,{fullscreen:'true'});

Mike Honeychurch

Posted 2011-12-11T03:45:20.323

Reputation: 36 211


Hey Mike, I found this very useful. Could you write an answer to my question found here? I tried what you mentioned here and it seems it could work, but now I face another problem, my CDF now says Initialization timed out after I click on the button to enable the dynamic content.

– jmlopez – 2012-05-27T07:00:56.070


System`Private`*Entry* functions

Since version 10.0 System`Private`SetNoEntry and System`Private`EntryQ functions are available.


Changes internal representation of given expression so that it is considered atomic, it returns "atomized" version of expression. This change affects all references to given expression, but not new expressions even if they are the same.

ClearAll[f, x, y, tmp];
tmp = tmp2 = f[x, y];
System`Private`SetNoEntry@tmp (* f[x, y] *)

Ordinary f[x, y] is not atomic, but all references to f[x, y] that was passed to SetNoEntry are considered atomic:

f[x, y] // AtomQ (* False *)
tmp // AtomQ (* True *)
tmp2 // AtomQ (* True *)

Those expressions are still considered same:

tmp === f[x, y] (* True *)

Head of expression (part 0) is accessible but not other parts:

tmp // Head
(* f *)
(* f *)
(* Part::partd: Part specification f[x,y][[2]] is longer than depth of object. *)
(* f[x, y][[2]] *)

Part accessing/modifying functions treat it as other atomic objects, so they either complain:

Append[tmp, z]
(* Append::normal: Nonatomic expression expected at position 1 in Append[f[x,y],z]. *)
(* Append[f[x, y], z] *)

or ignore our expression:

ReplacePart[tmp, 1 -> z]
(* f[x, y] *)

Pattern matching works as before and can still "go inside" this expression:

Replace[tmp, head_[arg1_, arg2_] :> {head, arg1, arg2}]
(* {f, x, y} *)

I've learned about this function from Leonid's answer to "Make my data-structure atomic" post.


Tests whether given expression was set as "no entry" expression:

tmp = f[x, y];
tmp // System`Private`NoEntryQ (* False *)
tmp // System`Private`NoEntryQ (* True *)

Some built-in atomic data structures use this mechanism, e.g. SparseArrays evaluate to "no entry" atomic form:

SparseArray@{0, 1} // Unevaluated // System`Private`NoEntryQ (* False *)
SparseArray@{0, 1} // System`Private`NoEntryQ (* True *)

Since version 10.4 there are five additional "Entry-related" functions.


Is the opposite of NoEntryQ:

tmp = f[x, y];
tmp // System`Private`EntryQ (* True *)
tmp // System`Private`EntryQ (* False *)




Are variants of SetNoEntry, NoEntryQ and EntryQ with HoldAllComplete attribute.


Creates new "no-entry" expression using first argument as head, and rest of arguments as arguments of created expression:

System`Private`ConstructNoEntry[f, x, y] (* f[x, y] *)
% // System`Private`NoEntryQ (* True *)


Posted 2011-12-11T03:45:20.323

Reputation: 14 388



Mathematica has a nice library TetGenLink to produce irregular 3D meshes. Original TetGen has a lot of features and not all of them available by TetGenLink. One of the features is the setting up the vertex metrics to produce non-uniform grids

enter image description here

Fortunately, the corresponding function is implemented but not documented

TetGenSetPointMetricTensors[tetGenInstance, {{x1, y1, z1}, {x2, y2, z2}, ...}]

The mesh size depends only on the first element of the tensors (x1, x2, x3, ...).


Beveled fractions ${}^a/_b$ available with the undocumented option Beveled in the FractionBox.


"SegmentationMode" option can improve TextRecognize.


Posted 2011-12-11T03:45:20.323

Reputation: 41 907

I only saw this now: The reason it's not documented is that I never could come up with a use case for this TetGen function. If you could point that out to me then it could be documented. – user21 – 2015-10-15T11:32:02.450

@user21 I think now with new V10 functional TetGenLink is no longer relevant. – ybeltukov – 2015-10-15T18:05:58.617



f[z_] := Piecewise[{{0, 0 < z < 30}, {1, 30 < z < 60}, {0, 60 < z < 120}, {-1, 120 < z < 150}}]


-(1 - UnitStep[120 - z]) (1 - UnitStep[-150 + z]) + (1 - UnitStep[30 - z]) (1 - UnitStep[-60 + z])


Posted 2011-12-11T03:45:20.323

Reputation: 22 814

1And it has an inverse too! Let s = Simplify`PWToUnitStep @ f[z] then you can get back to the Piecewise with FullSimplify[s]. – bill s – 2016-12-14T20:21:54.837

1@bill, or use PiecewiseExpand[]. – J. M.'s ennui – 2016-12-15T01:53:51.590


Undocumented option PlotPoints->{n,{p}}

with 2(!) parameters n (number of points or Automatic) and p (list of critical points)


f[x_?NumericQ] := Piecewise[{{1, x == 1}, {0, True}}];
Plot[f[x], {x, 0, 1.1}, PlotPoints -> {Automatic, { 1}}]

enter image description here

Ulrich Neumann

Posted 2011-12-11T03:45:20.323

Reputation: 25 490


Not sure if a new question should be started, anyway, here's some undocumented syntax (sorted alphabetically):


Indexed variable can be used as independent variable of Compiled function:

<< CompiledFunctionTools`
cf = Compile[a[1], Sin@a[1]];
cf // CompilePrint
(* cf is fully compiled *)


D[expr] evaluates to expr.


Derivative[][u] evaluates to u.



ErrorBar isn't necessary if there are only symmetric y error bars:

What's the simplest way to plot an ErrorListPlot with only y error bars?


PlusMinus(±) can be used in ErrorListPlot:



Exit and Quit can work without brackets, Return can work without brackets when it's inside a dialog:

Lists for built-in functions that can work without brackets


it owns a simple syntax for function relationship that returns number or list of number:

Mapping multiple functions


FindRoot[x \[Function] Sin[x] + Exp[x], {0}]
(* {-0.588533} *)
FindRoot[{x, y} \[Function] {Exp[x - 2] - y, y^2 - x}, {{1}, {1}}]
(* {0.019026, 0.137935} *)


They can handle InterpolatingFunction directly since v9:

Easy way to plot ODE solutions from NDSolve?



The 1st argument can be a nested list, no Flatten or Join is needed:

eq = y''[x] + Sin[y[x]] y[x] == 0;
bc = {y[0] == 1, y'[0] == 0};

NDSolve[{eq, bc}, y, {x, 0, 30}]


The 2nd argument can almost be anything:

NDSolveValue[{y'[x] == y@x Cos[x + y@x], y@0 == 1}, y[x] + y'[x] + Sin[x], {x, 0, 30}]


"Coordinates" sub-option of TensorProductGrid can be set in the range:

grid = {0, 1, 1.1, 3.2, 2.1, 4.6, 5};

seq = Sequence[{D[u[t, x], t] == D[u[t, x], x, x], u[0, x] == 0, u[t, 0] == Sin[t], 
    u[t, 5] == 0}, u, {t, 0, 10}];

NDSolve[seq, Flatten@{x, grid}] == 
 NDSolve[seq, {x, 0, 5}, 
  Method -> {"MethodOfLines", 
    "SpatialDiscretization" -> {"TensorProductGrid", "Coordinates" -> grid}}]
(* True *)


The 1st argument doesn't need to be equation(s):

polys = {x^2 + y^2 - 1, 2 x + 3 y - 4};
NSolve[polys, {x, y}]


a[[]], or equivalently Part[a], evaluates to a.


PlotRange can be a function, which returns the plot range of Graphics/Graphics3D:

Plot[Sin[x], {x, 0, 6 Pi}]
% // Head
(* Graphics *)
%% // PlotRange
(* {{0., 18.8496}, {-0.999999, 0.999999}} *)
Plot3D[x^2 + y^2, {x, -2, 2}, {y, -2, 2}] // PlotRange
% // Head
(* Graphics3D *)
%% // PlotRange
(* {{-2., 2.}, {-2., 2.}, {0., 8.}} *)


The 3rd argument can be a list of variables to be eliminated. This syntax is almost hidden since v8:



Undocumented since v6:

ReplacePart[{a, b, c, d, e}, xxx, 3]
(* {a, b, xxx, d, e} *)
ReplacePart[{a, b, c, d, e}, xx, {{2}, {5}}]
(* {a, xx, c, d, xx} *)

You need the syntax if you want to compile them because Rule isn't compilable:

error when defining a compiled function with ReplacePart


The option value of Method can be a user-defined function:



Table[a] evaluates to a.


Posted 2011-12-11T03:45:20.323

Reputation: 44 878


Pure functions as arguments to FindRoot are shown in this earlier Q&A: https://mathematica.stackexchange.com/q/25745/4999

– Michael E2 – 2018-01-08T17:00:46.997


LabeledSlider is a version of Slider first mentioned by @ybeltukov in Can Manipulate controls have the option Appearance -> "Labeled" by default?.

If you find yourself expanding the sliders in manipulate in order to see the value, just use this form in order to see the value without having to expand the slider:

Manipulate[x, {x, 1, 10, LabeledSlider}]

Mathematica graphics

Gustavo Delfino

Posted 2011-12-11T03:45:20.323

Reputation: 6 847



This function (ref#1, ref#2, ref#3) can be used to ensure that some clean-up code will always be executed, even if an abort or other non-local exit occurs within some protected code. To illustrate:

  Print["opening a file"]
, Print["doing something with the file"]
; Abort[]
; Print["never gets here"]
, Print["closing the file"]

(* During evaluation of In[1]:= opening a file
   During evaluation of In[1]:= doing something with the file
   During evaluation of In[1]:= closing the file
   Out[1]= $Aborted

Earlier versions of this function did not handle Catch/Throw properly, but this has been corrected and the function now seems to the most reliable way to protect against the unwinding of the evaluation stack.


In a similar vein, the CheckAll function (ref) ensures that a function is called whenever a body of code was exited, whether normally or abnormally. Unlike Internal`WithLocalSettings, the handler code completely intercepts the exit. To illustrate:

CheckAll["normal exit", myHandlerFunction]
(* myHandlerFunction["normal exit", Hold[]] *)

CheckAll[Throw[3], myHandlerFunction]
(* myHandlerFunction[$Aborted, Hold[Throw[3]]] *)

The second example shows how the handler function is informed about the exact nature of the exit. It is up to that function to decide whether to re-evaluate the expression that triggered the exit, thus continuing to pass control up the stack.


Posted 2011-12-11T03:45:20.323

Reputation: 62 787



It basically does what it says. See here for one of its applications.

Random`Private`MapThreadMin and Random`Private`MapThreadMax






Henrik Schumacher

Posted 2011-12-11T03:45:20.323

Reputation: 85 430