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HomeMathLaunching Model 13.1 of Wolfram Language & Mathematica 🙀🤠🥳—Stephen Wolfram Writings

Launching Model 13.1 of Wolfram Language & Mathematica 🙀🤠🥳—Stephen Wolfram Writings

Exploring Wolfram Language 13.1 with Stephen Wolfram

The Epic Continues…

Final week it was 34 years because the unique launch of Mathematica and what’s now the Wolfram Language. And thru all these years we’ve energetically continued constructing additional and additional, including ever extra capabilities, and steadily extending the area of the computational paradigm.

In recent times we’ve established one thing of a rhythm, delivering the fruits of our improvement efforts roughly twice a 12 months. We launched Model 13.0 on December 13, 2021. And now, roughly six months later, we’re releasing Model 13.1. As normal, despite the fact that it’s a “.1” launch, it’s acquired a whole lot of new (and up to date) performance, a few of which we’ve labored on for a few years however lastly now delivered to fruition.

For me it’s all the time thrilling to see what we handle to ship in every new model. And in Model 13.1 we’ve got 90 fully new features—in addition to 203 current features with substantial updates. And past what seems in particular features, there’s additionally main new performance in Model 13.1 in areas like person interfaces and the compiler.

The Wolfram Language because it exists right now encompasses an enormous vary of performance. However its nice energy comes not simply from what it accommodates, but in addition from how coherently all the pieces in it suits collectively. And for almost 36 years I’ve taken it as a private duty to make sure that that coherence is maintained. It’s taken each nice focus and plenty of deep mental work. However as I expertise them day by day in my use of the Wolfram Language, I’m happy with the outcomes.

And for the previous 4 years I’ve been sharing the “behind the scenes” of the way it’s achieved—by livestreaming our Wolfram Language design evaluate conferences. It’s an unprecedented degree of openness—and engagement with the neighborhood. In designing Model 13.1 we’ve achieved 90 livestreams—lasting greater than 96 hours. And in opening up our course of we’re offering visibility not solely into what was constructed for Model 13.1, but in addition of why it was constructed, and the way selections about it have been made.

However, OK, so what lastly is in Model 13.1? Let’s discuss some highlights….

Past Listability: Introducing Threaded

From the very starting of Mathematica and the Wolfram Language we’ve had the idea of listability: when you add two lists, for instance, their corresponding components shall be added:

{a, b, c} + {x, y, z}

It’s a really handy mechanism, that sometimes does precisely what you’d need. And for 35 years we haven’t actually thought of extending it. But when we take a look at code that will get written, it typically occurs that there are components that mainly implement one thing very very like listability, however barely extra basic. And in Model 13.1 we’ve got a brand new symbolic assemble, Threaded, that successfully lets you simply generalize listability.

Take into account:

{{1, 2}, {3, 4}} + {x, y}

This makes use of bizarre listability, successfully computing:

{{1, 2} + x, {3, 4} + y}

However what if you’d like as an alternative to “go down a degree” and thread {x,y} into the bottom components of the primary listing? Nicely, now you need to use Threaded to try this:

{{1, 2}, {3, 4}} + Threaded

By itself, Threaded is only a symbolic wrapper:


However as quickly because it seems in a operate—like Plus—that has attribute Listable, it specifies that the listability needs to be utilized after what’s specified inside Threaded is “threaded” on the lowest degree.

Right here’s one other instance. Create a listing:


How ought to we then multiply every ingredient by {1,–1}? We might do that with:

(#1*{1, -1} & ) /@ Table

However now we’ve acquired Threaded, and so as an alternative we are able to simply say:


You can provide Threaded as an argument to any listable operate, not simply Plus and Instances:


You should utilize Threaded and bizarre listability collectively:

{{1, 2}, {3, 4}} + Threaded

You may have a number of Threadeds collectively as effectively:

{{1, 2}, {3, 4}} + Threaded

Threaded, by the way in which, will get its identify from the operate Thread, which explicitly does “threading”, as in:


By default, Threaded will all the time thread into the bottom degree of a listing:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded
{{{{4, 5}, {5, 6}}, {{5, 6}, {6, 7}}}, {{{5, 6}, {6, 7}}, {{6, 7}, {7, 8}}}} + Threaded

Right here’s a “real-life” instance of utilizing Threaded like this. The info in a 3D shade picture consists of a rank-3 array of triples of RGB values:


This multiplies each RGB triple by {0,1,2}:


More often than not you both wish to use bizarre listability that operates on the high degree of a listing, otherwise you wish to use the default type of Threaded, that operates on the lowest degree of a listing. However Threaded has a extra basic kind, in which you’ll explicitly say what degree you need it to function at.

Right here’s the default case:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

Right here’s degree 1, which is rather like bizarre listability:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

And right here’s threading into degree 2:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

Threaded gives a really handy option to do all types of array-combining operations. There’s further complexity when the article being “threaded in” itself has a number of ranges. The default on this case is to align the bottom degree within the factor being threaded in with the bottom degree of the factor into which it’s being threaded:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

Right here now could be “bizarre listability” habits:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

For the arrays we’re right here, the default habits is equal to:

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

Typically it’s clearer to write down this out in a kind like

{{{3, 4}, {4, 5}}, {{4, 5}, {5, 6}}} + Threaded

which says that the primary degree of the array contained in the Threaded is to be aligned with the second degree of the skin array. Usually, the default case is equal to –1 → –1, specifying that the underside degree of the array contained in the Threaded needs to be aligned with the underside degree of the array exterior.

But Extra Language Comfort Features

In each model of the Wolfram Language we attempt to add new features that may make basic applications simpler to write down and simpler to learn. In Model 13.1 a very powerful such operate is Threaded. However there are fairly a couple of others as effectively.

First in our assortment for Model 13.1 is DeleteElements, which deletes specified components from a listing. It’s like Complement, besides that it doesn’t reorder the listing (analogous to the way in which DeleteDuplicates removes duplicate components, with out reordering in the way in which that Union does):


DeleteElements additionally permits extra detailed management of what number of copies of a component will be deleted. Right here it’s as much as 2 b’s and three c’s:


Speaking of DeleteDuplicates, one other new operate in Model 13.1 is DeleteAdjacentDuplicates:


We’ve had Union, Intersection and Complement since Model 1.0. In Model 13.1 we’re including SymmetricDifference: discover components that (within the 2-argument case) are in a single listing or the opposite, however not each. For instance, what international locations are within the G20 or the EU, however not each?


Let’s say you’ve a number of lists, and also you wish to know what components are distinctive to simply one among these lists, and don’t happen in a number of lists. The brand new UniqueElements tells one.

For example, this tells us which letters uniquely happen in numerous alphabets:


We’ve had Map and Apply, with quick types /@ and @@, ever since Model 1.0. In Model 4.0 we added @@@ to signify Apply[f,expr,1]. However we by no means added a separate operate to correspond to @@@. And through the years, there’ve been fairly a couple of events the place I’ve mainly wished, for instance, to do one thing like “Fold[@@@, ...]”. Clearly Fold[Apply[#1,#2,1]&,...] would work. Nevertheless it feels as if there’s a “lacking” named operate. Nicely, in Model 13.1, we added it: MapApply is equal to @@@:

f @@@ {{a, b}, {c, d}}

One other small comfort added in Model 13.1 is SameAs—basically an operator type of SameQ. Why is such a assemble wanted? Nicely, there are all the time tradeoffs in language design. And again in Model 1.0 we determined to make SameQ work with any variety of arguments (so you possibly can check whether or not an entire sequence of issues are the identical). However which means for consistency SameQ[expr] should all the time return True—so it’s not obtainable as an operator of SameQ. And that’s why now in Model 13.1 we’re including SameAs, that joins the household of operator-form features like EqualTo and GreaterThan:


Procedural programming—typically with “variables hanging out”—isn’t the popular model for many Wolfram Language code. However typically it’s probably the most handy option to do issues. And in Model 13.1 we’ve add a small piece of streamlining by introducing the operate Till. Ever since Model 1.0 we’ve had Whereas[test,body] which repeatedly evaluates physique whereas check is True. But when check isn’t True even at first, Whereas gained’t ever consider physique. Till[test,body] does issues the opposite manner round: it evaluates physique till check turns into True. So if check isn’t True at first, Till will nonetheless consider physique as soon as, in impact solely trying on the check after it’s evaluated the physique.

Final however not least within the listing of latest core language features in Model 13.1 is ReplaceAt. Change makes an attempt to use a substitute rule to an entire expression—or an entire degree in an expression. ReplaceAll (/.) does the identical factor for all subparts of an expression. However very often one needs extra management over the place replacements are achieved. And that’s what ReplaceAt gives:


An vital characteristic is that it additionally has an operator kind:


Why is that this vital? The reply is that it offers a symbolic option to specify not simply what substitute is made, but in addition the place it’s made. And for instance that is what’s wanted in specifying steps in proofs, say as generated by FindEquationalProof.

Emojis! And Extra Multilingual Assist

What’s a personality? Again when Model 1.0 was launched, characters have been represented as 8-bit objects: normally ASCII, however you can decide one other “character encoding” (therefore the ChararacterEncoding choice) when you wished. Then within the early Nineties got here Unicode—which we have been one of many very first corporations to assist. Now “characters” might be 16-bit constructs, with almost 65,536 potential “glyphs” allotted throughout completely different languages and makes use of (together with some mathematical symbols that we launched). Again within the early Nineties Unicode was a newfangled factor, that working methods didn’t but have built-in assist for. However we have been betting on Unicode, and so we constructed our personal infrastructure for dealing with it.

Thirty years later Unicode is certainly the common normal for representing character-like issues. However someplace alongside the way in which, it turned out the world wanted greater than 16 bits’ price of character-like issues. At first it was about supporting variants and historic writing methods (suppose: cuneiform or Linear B). However then got here emoji. And it grew to become clear that—sure, arguably in a return to the Egyptian hieroglyph model of communication—there was an nearly infinite variety of potential pictorial emoji that might be made, every of them being encoded as their very own Unicode code level.

It’s been a gradual enlargement. Unique 16-bit Unicode is “aircraft 0”. Now there are as much as 16 further planes. Not fairly 32-bit characters, however given the way in which computer systems work, the strategy now could be to permit characters to be represented by 32-bit objects. It’s removed from trivial to try this uniformly and effectively. And for us it’s been a protracted course of to improve all the pieces in our system—from string manipulation to pocket book rendering—to deal with full 32-bit characters. And that’s lastly been achieved in Model 13.1.

However that’s removed from all. In English we’re just about used to with the ability to deal with textual content as a sequence of letters and different characters, with every character being separate. Issues get a bit extra difficult if you begin to fear about diphthongs like æ. But when there are pretty few of those, it really works to simply introduce them as particular person “Unicode characters” with their very own code level. However there are many languages—like Hindi or Khmer—the place what seems in textual content like a person character is mostly a composite of letter-like constructs, diacritical marks and different issues. Such composite characters are usually represented as “grapheme clusters”: runs of Unicode code factors. The principles for dealing with this stuff will be fairly difficult. However after a few years of improvement, main working methods now efficiently do it most often. And in Model 13.1 we’re in a position to make use of this to assist such constructs in notebooks.

OK, so what does 32-bit Unicode appear to be? Utilizing CharacterRange (or FromCharacterCode) we are able to dive in and simply see what’s on the market in “character area”. Right here’s a part of bizarre 16-bit Unicode area:


Right here’s a few of what occurs in “plane-1” above character code 65535, on this case catering to “legacy computations”:


Airplane-0 (under 65535) is just about all full. Above that, issues are sparser. However round 128000, for instance, there are many emoji:


You should utilize these within the Wolfram Language, and in notebooks, identical to another characters. So, for instance, you possibly can have wolf and ram variables:


The 🐏 types earlier than the 🐺 as a result of it occurs to have a numerically smaller character code:


In a pocket book, you possibly can enter emoji (and different Unicode characters) utilizing normal working system instruments—like ctrlcmdarea on macOS:

Emojis in notebook

The world of emoji is quickly evolving—and that may typically result in issues. Right here’s an emoji vary that features some very acquainted emoji, however on a minimum of one among my laptop methods additionally consists of emoji that show solely as :


The rationale that occurs is that my default fonts don’t comprise glyphs for these emoji. However all shouldn’t be misplaced. In Model 13.1 we’re together with a font from Twitter that goals to comprise glyphs for just about all emoji:


Past coping with particular person Unicode characters, there’s additionally the matter of composites, and grapheme clusters. In Hindi, for instance, two characters can mix into one thing that’s rendered (and handled) as one:


The primary character right here can stand by itself:


However the second is mainly a modifier that extends the primary character (on this specific case including a vowel sound):


However as soon as the composite हि has been shaped it acts “textually” identical to a single character, within the sense that, for instance, the cursor strikes via it in a single step. When it seems “computationally” in a string, nevertheless, it could possibly nonetheless be damaged into its constituent Unicode components:


This type of setup can be utilized not just for a language like Hindi but in addition for European languages which have diacritical marks like umlauts:


Although this appears like one character—and in Model 13.1 it’s handled like that for “textual” functions, for instance in notebooks—it’s in the end made up of two distinct “Unicode characters”:


On this specific case, although, this may be “normalized” to a single character:


It appears the identical, however now it actually is only one character:


Right here’s a “mixed character” which you could kind


however for which there’s no single character to which it normalizes:


The idea of composite characters applies not solely to bizarre textual content, but in addition to emojis. For instance, take the emoji for a girl


along with the emoji for a microscope


and mix them with the “zero-width-joiner” character (which, for sure, doesn’t show as something)


and also you get (sure, considerably bizarrely) a lady scientist!


Evidently, you are able to do this computationally—although the “calculus” of what’s been outlined up to now in Unicode is pretty weird:


I’m form of hoping that the way forward for semantics doesn’t find yourself being outlined by the way in which emojis mix 😎.

As one final—arguably hacky—instance of mixing characters, Unicode defines numerous “two-letter” mixtures to be flags. Sort then , and also you get 🇺🇸!

As soon as once more, this may be made computational:


(And, sure, it’s an attention-grabbing query what renders right here, and what doesn’t. In some working methods, no flags are rendered, and we’ve got to tug in a particular font to do it.)


It was once that the one “particular key sequence” one completely ought to know with a view to use Wolfram Notebooks was shiftenter. However progressively there have began to be an increasing number of high-profile operations which can be conveniently achieved by “urgent a button”. And moderately than anticipating individuals to recollect all these particular key sequences (or suppose to look in menus for them) we’ve determined to introduce a toolbar that shall be displayed by default in each normal pocket book. Model 13.1 has the primary iteration of this toolbar. Subsequent variations will assist an rising vary of capabilities.

It’s not been simple to design the default toolbar (and we hope you’ll like what we got here up with!) The primary drawback is that Wolfram Notebooks are very basic, and there are an amazing many issues you are able to do with them—which it’s difficult to prepare right into a manageable toolbar. (Some particular kinds of notebooks have had their very own specialised toolbars for some time, which have been simpler to design by advantage of their specialization.)

So what’s within the toolbar? On the left are a few analysis controls:

Evaluation menu options

Menu options means “Consider”, and is solely equal to urgent shiftret (as its tooltip says). Menu optionsmeans “Abort”, and can cease a computation. To the best of Menu options is the menu proven above. The primary a part of the menu lets you select what shall be evaluated. (Don’t overlook the extraordinarily helpful “Consider In Place” that permits you to consider no matter code you’ve chosen—say to show RGBColor[1,0,0] in your enter into .) The underside a part of the menu offers a few extra detailed (however extremely helpful) analysis controls.

Transferring alongside the toolbar, we subsequent have:

Menu option

In case your cursor isn’t already in a cell, the pulldown lets you choose what sort of cell you wish to insert (it’s just like the Menu options “tongue” that seems inside the pocket book). (In case your cursor is already inside a cell, then like in a typical phrase processor, the pulldown will let you know the model that’s getting used, and allow you to reset it.)

Menu options offers you a bit of panel to manage to look of cells, altering their background colours, frames, dingbats, and so forth.

Subsequent come cell-related buttons: Menu options. The primary is for cell construction and grouping:

Cell options

Menu options copies enter from above (cmdL). It’s an operation that I, for one, find yourself doing on a regular basis. I’ll have an enter that I consider. Then I’ll wish to make a modified model of the enter to judge once more, whereas protecting the unique. So I’ll copy the enter from above, edit the copy, and consider it once more.

Menu options copies output from above. I don’t discover this fairly as helpful as copy enter from above, however it may be useful if you wish to edit output for subsequent enter, whereas leaving the “precise output” unchanged.

The Menu options block is all about content material in cells. Menu options (which you’ll typically press repeatedly) is for extending a range—in impact going ever upwards in an expression tree. (You may get the identical impact by urgent ctrl. or by multiclicking, nevertheless it’s much more handy to repeatedly press a single button than to have to exactly time your multiclicks.)

Menu options is the single-button option to get ctrl= for getting into pure language enter:

Natural language input

Menu options iconizes your choice:

Iconize selection

Iconization is one thing we launched in Model 11.3, and it’s one thing that’s proved extremely helpful, significantly for making code simple to learn (say by iconizing particulars of choices). (It’s also possible to iconize a range from the right-click menu, or with ctrlcmd'.)

Menu options is most related for code, and toggles commenting (with Menu options) a range. Menu options brings up a palette for math typesetting. Menu options helps you to enter that shall be transformed to Wolfram Language math typesetting. Menu options brings up a drawing canvas. Menu options inserts a hyperlink (cmdshiftH).

If you happen to’re in a textual content cell, the toolbar will look completely different, now sporting a textual content formatting management: Text menu options

Most of that is pretty normal. Menu options helps you to insert “code voice” materials. Menu optionsand Menu options are nonetheless within the toolbar for inserting math right into a textual content cell.

On the right-hand finish of the toolbar are three extra buttons: Menu options. Menu options offers you a dialog to publish your pocket book to the cloud. Menu options opens documentation, both particularly trying up no matter you’ve chosen within the pocket book, or opening the entrance web page (“root information web page”) of the primary Wolfram Language documentation. Lastly, Menu options helps you to search in your present pocket book.

As I discussed above, what’s in Model 13.1 is simply the primary iteration of our default toolbar. Anticipate extra options in later variations. One factor that’s notable in regards to the toolbar normally is that it’s 100% applied in Wolfram Language. And along with including an excessive amount of flexibility, this additionally signifies that the toolbar instantly works on all platforms. (By the way in which, when you don’t need the toolbar in a specific pocket book—or for all of your notebooks—simply right-click the background of the toolbar to choose that choice.)

Sprucing the Person Interface

We first launched Wolfram Notebooks with Model 1.0 of Mathematica, in 1988. And ever since then, we’ve been progressively sprucing the pocket book interface, doing extra with each new model.

The ctrl= mechanism for getting into pure language (“Wolfram|Alpha-style”) enter debuted in Model 10.0—and in Model 13.1 it’s now accessible from the picture button within the new default pocket book toolbar. However what truly is picture when it’s in a pocket book? Up to now, it’s been a reasonably advanced symbolic construction primarily appropriate for analysis. However in Model 13.1 we’ve made it a lot less complicated. And whereas that doesn’t have any direct impact when you’re simply utilizing picture purely in a pocket book, it does have an impact when you copy picture into one other software, like pure-text e-mail. Up to now this produced one thing that might work if pasted again right into a pocket book, however positively wasn’t significantly readable. In Model 13.1, it’s now merely the Wolfram Language interpretation of your pure language enter:

Natural language input

What occurs if the computation you do in a pocket book generates an enormous output? Ever since Model 6.0 we’ve had some type of “output limiter”, however in Model 13.1 it’s change into a lot sleeker and extra helpful. Right here’s a typical instance:


Speaking of massive outputs (in addition to different issues that maintain the pocket book interface busy), one other change in Model 13.1 is the brand new asynchronous progress overlay on macOS. This doesn’t have an effect on different platforms the place this drawback had already been solved, however on the Mac modifications within the OS had led to a scenario the place the pocket book entrance finish might mysteriously pop to the entrance in your desktop—a scenario that has now been resolved.

One of many barely uncommon person interface options that’s existed ever since Model 1.0 is the Why the Beep? menu merchandise—that permits you to get a proof of any “error beep” that happens whilst you’re operating the system. The operate Beep helps you to generate your personal beep. And now in Model 13.1 you need to use Beep["string"] to arrange a proof of “your beep”, that customers can retrieve via the Why the Beep? menu merchandise.

The fundamental pocket book person interface works as a lot as potential with normal interface components on all platforms, in order that when these components are up to date, we all the time routinely get the “most trendy” look. However there are components of the pocket book interface which can be fairly particular to Wolfram Notebooks and are all the time customized. One which hadn’t been up to date for some time is the Preferences dialog—which now in Model 13.1 will get a full makeover:


Whenever you inform the Wolfram Language to do one thing, it usually simply goes off and does it, with out asking you something (effectively, except it explicitly wants enter, wants a password, and so forth.) However what if there’s one thing that it may be a good suggestion to do, although it’s not strictly essential? What ought to the person interface for this be? It’s difficult, however I feel we now have an excellent resolution that we’ve began deploying in Model 13.1.

Specifically, in Model 13.1, there’s an instance associated to the Wolfram Operate Repository. Say you utilize a operate for which an replace is accessible. What now occurs is {that a} blue field is generated that tells you in regards to the replace—although it nonetheless retains going with the computation, ignoring the replace:

Update option

If you happen to click on the Replace Now button within the blue field you are able to do the replace. After which the purpose is which you could run the computation once more (for instance, simply by urgent shiftenter), and now it’ll use the replace. In a way the core concept is to have an interface the place there are doubtlessly a number of passes, and the place a computation all the time runs to completion, however you’ve a simple option to change the way it’s arrange, after which run it once more.

Giant-Scale Code Enhancing

One of many nice issues in regards to the Wolfram Language is that it really works effectively for applications of any scale—from lower than a line lengthy to hundreds of thousands of traces lengthy. And for the previous a number of years we’ve been engaged on increasing our assist for very massive Wolfram Language applications. Utilizing LSP (Language Server Protocol) we’ve supplied the aptitude for most traditional exterior IDEs to routinely do syntax coloring and different customizations for the Wolfram Language.

In Model 13.1 we’re additionally including a few options that make large-scale code modifying in notebooks extra handy. The primary—and extensively requested—is block indent and outdent of code. Choose the traces you wish to indent or outdent and easily press tab or shifttab to indent or outdent them:

Tab indent

Ever since Model 6.0 we’ve had the flexibility to work with .wl bundle information (in addition to .wls script information) utilizing our pocket book modifying system. A brand new default characteristic in Model 13.1 is numbering of all code traces that seem within the underlying file (and, sure, we appropriately align line numbers accounting for the presence of non-code cells):

Code line numbering

So now, for instance, when you get a syntax error from Get or a associated operate, you’ll instantly have the ability to use the road quantity it experiences to search out the place it happens within the underlying file.

Scribbling on Notebooks

In Model 12.2 we launched Canvas as a handy interface for interactive drawing in notebooks. In Model 13.1 we’re introducing the notion of toggling a canvas on high of any cell.

Given a cell, simply choose it and press picture, and also you’ll get a canvas:

Canvas function

Now you need to use the drawing instruments within the canvas to create an annotation overlay:


If you happen to consider the cell, the overlay will keep. (You may do away with the “canvas wrapper” by making use of Regular.)

Timber Proceed to Develop 🌱🌳

In Model 12.3 we launched Tree as a brand new elementary assemble within the Wolfram Language. In Model 13.0 we added quite a lot of styling choices for bushes, and in Model 13.1 we’re including extra styling in addition to quite a lot of new elementary options.

An vital replace to the basic Tree assemble in Model 13.1 is the flexibility to call branches at every node, by giving them in an affiliation:


All tree features now embody assist for associations:


In lots of makes use of of bushes the labels of nodes are essential. However significantly in additional summary purposes one typically needs to take care of unlabeled bushes. In Model 13.1 the operate UnlabeledTree (roughly analogously to UndirectedGraph) takes a labeled tree, and mainly removes all seen labels. Here’s a normal labeled tree


and right here’s the unlabeled analog:


In Model 12.3 we launched ExpressionTree for deriving bushes from basic symbolic expressions. Our plan is to have a variety of “particular bushes” applicable for representing completely different particular sorts of symbolic expressions. We’re starting this course of in Model 13.1 by, for instance, having the idea of “Dataset bushes”. Right here’s ExpressionTree changing a dataset to a tree:


And now right here’s TreeExpression “inverting” that, and producing a dataset:


(Bear in mind the conference that *Tree features return a tree; whereas Tree* features take a tree and return one thing else.)

Right here’s a “graph rendering” of a extra difficult dataset tree:


The brand new operate TreeLeafCount helps you to rely the entire variety of leaf nodes on a tree (mainly the analog of LeafCount for a basic symbolic expression):


One other new operate in Model 13.1 that’s typically helpful in getting a way of the construction of a tree with out inspecting each node is RootTree. Right here’s a random tree:

tree = RandomTree

RootTree can get a subtree that’s “near the basis”:


It could additionally get a subtree that’s “removed from the leaves”, on this case taking place to components which can be at degree –2 within the tree:


In some methods the styling of bushes is just like the styling of graphs—although there are some vital variations on account of the hierarchical nature of bushes. By default, choices inserted into a specific tree ingredient have an effect on solely that tree ingredient:


However you can provide guidelines that specify how components within the subtree under that ingredient are affected:


In Model 13.1 there’s now detailed management obtainable for styling each nodes and edges within the tree. Right here’s an instance that offers styling for father or mother edges of nodes:


Choices like TreeElementStyle decide styling from the positions of components. TreeElementStyleFunction, however, determines styling by making use of a operate to the info at every node:


This makes use of each information and place info for every node:


In analogy with VertexShapeFunction for graphs, TreeElementShapeFunction gives a basic mechanism to specify how nodes of a tree needs to be rendered. This named setting for TreeElementShapeFunction makes each node be displayed as a circle:


But Extra Date-Dealing with Particulars

We first launched dates into Wolfram Language in Model 2.0, and we launched trendy date objects in Model 10.0. However to essentially make dates totally computable, there are a lot of detailed instances to think about. And in Model 13.1 we’re coping with one more of them. Let’s say you’ve acquired the date January 31, 2022. What date is one month later—provided that there’s no February 31, 2022?

If we outline a month “bodily”, it corresponds to a sure fractional variety of days:


And, sure, we are able to use this to determine what’s a month after January 31, 2022:


Barely complicated right here is that we’re coping with date objects of “day” granularity. We are able to see extra if we go all the way down to the extent of minutes:


If one’s doing one thing like astronomy, this sort of “bodily” date computation might be what one needs. But when one’s doing on a regular basis “human” actions, it’s nearly actually not what one needs; as an alternative, one needs to land on some calendar date or one other.

Right here’s the default within the Wolfram Language:


However now in Model 13.1 we are able to parametrize extra exactly what we wish. This default is what we name "RollBackward": wherever we “land” by doing the uncooked date computation, we “roll backward” to the primary legitimate date. An alternate is "RollForward":


No matter methodology one makes use of, there are going to be bizarre instances. Let’s say we begin with a number of consecutive dates:

dates = DateRange

With "RollBackward" we’ve got the weirdness of repeating February 28:


With "RollForward" we’ve got the weirdness of repeating March 1:


Is there any different? Sure, we are able to use "RollOver":


This retains advancing via days, however then has the weirdness that it goes backwards. And, sure, there’s no “proper reply” right here. However in Model 13.1 now you can specify precisely what you need the habits to be.

The identical concern arises not only for months, but in addition, for instance, for years. And it impacts not simply DatePlus, but in addition DateDifference.

It’s price mentioning that in Model 13.1, along with coping with the element we’ve simply mentioned, the entire framework for doing “date arithmetic” in Wolfram Language has been made vastly extra environment friendly, typically by elements of tons of.

Capturing Video & Extra

We’ve had ImageCapture since Model 8.0 (in 2010) and AudioCapture since Model 11.1 (in 2017). Now in Model 13.1 we’ve got VideoCapture. By default VideoCapture[] offers you a GUI that permits you to file out of your digicam:


Clicking the down arrow opens up a preview window that exhibits your present video:


Whenever you’ve completed recording, VideoCapture returns the Video object you created:

Now you possibly can course of or analyze this Video object identical to you’ll another:

VideoCapture[] is a blocking operation that waits till you’ve completed recording, then returns a consequence. However VideoCapture can be used “not directly” as a dynamic management. Thus, for instance

helps you to asynchronously begin and cease recording, whilst you do different issues in your Wolfram Language session. However each time you cease recording, the worth of video is up to date.

VideoCapture information video out of your digicam (and you need to use the ImageDevice choice to specify which one you probably have a number of). VideoScreenCapture, however, information out of your laptop display—in impact offering a video analog of CurrentScreenImage.

VideoScreenCapture[], like VideoCapture[], is a blocking operation so far as the Wolfram Language is worried. However if you wish to watch one thing occurring in one other software (say, an internet browser), it’ll do exactly wonderful. And as well as, you can provide a display rectangle to seize a specific area in your display:

VideoScreenCapture[{{0, 50}, {640, 498}}]

Then for instance you possibly can analyze the time sequence of RGB shade ranges within the video that’s produced:

What if you wish to display file from a pocket book? Nicely, then you need to use the asynchronous dynamic recording mechanism that exists in VideoScreenCapture simply because it does in VideoCapture.

By the way in which, each VideoCapture and VideoScreenCapture by default seize audio. You may change off audio recording both from the GUI, or with the choice AudioInputDeviceNone.

If you wish to get fancy, you possibly can display file a pocket book by which you’re capturing video out of your digicam (which in flip exhibits you capturing a video, and so forth.):


Along with capturing video from real-time goings-on, you can even generate video straight from features like AnimationVideo and SlideShowVideo—in addition to by “touring” a picture utilizing TourVideo. In Model 13.1 there are some vital enhancements to TourVideo.

Take an animal scene and extract bounding packing containers for elephants and zebras:

Now you can also make a tour video that visits every animal:

Outline a path operate of a variable t:

Now we are able to use the trail operate to make a “spiralling” tour video:

Faculty Calculus

Reworking school calculus was one of many early achievements of Mathematica. However even now we’re persevering with so as to add performance to make school calculus ever simpler and smoother to do—and extra instantly connectable to purposes. We’ve all the time had the operate D for taking derivatives at a degree. Now in Model 13.1 we’re including ImplicitD for locating implicit derivatives.

So, for instance, it could possibly discover the spinoff of xy with respect to x, with y decided implicit by the constraint x2 + y2 = 1:


Miss the primary argument and also you’ll get the usual school calculus “discover the slope of the tangent line to a curve”:


Thus far all of this can be a pretty easy repackaging of our longstanding calculus performance. And certainly these sorts of implicit derivatives have been obtainable for a very long time in Wolfram|Alpha. However for Mathematica and the Wolfram Language we wish all the pieces to be as basic as potential—and to assist the sorts of issues that present up in differential geometry, and in issues like asymptotics and validation of implicit options to differential equations. So along with bizarre college-level calculus, ImplicitD can do issues like discovering a second implicit spinoff on a curve outlined by the intersection of two surfaces:


In Mathematica and the Wolfram Language Combine is a operate that simply will get you solutions. (In Wolfram|Alpha you possibly can ask for a step-by-step resolution too.) However significantly for instructional functions—and typically additionally when pushing boundaries of what’s potential—it may be helpful to do integrals in steps. And so in Model 13.1 we’ve added the operate IntegrateChangeVariables for altering variables in integrals.

An instantaneous concern is that if you specify an integral with Combine[...], Combine will simply go forward and do the integral:


However for IntegrateChangeVariables you want an “undone” integral. And you will get this utilizing Inactive, as in:


And given this inactive kind, we are able to use IntegrateChangeVariables to do a “trig substitution”:


The result’s once more an inactive kind, now stating the integral otherwise. Activate goes forward and really does the integral:


IntegrateChangeVariables can take care of a number of integrals as effectively—and with named coordinate methods. Right here it’s remodeling a double integral to polar coordinates:


Though the fundamental “structural” transformation of variables in integrals is sort of easy, the entire story of IntegrateChangeVariables is significantly extra difficult. “Faculty-level” modifications of variables are normally rigorously organized to come back out simply. However within the extra basic case, IntegrateChangeVariables finally ends up having to do nontrivial transformations of geometric areas, troublesome simplifications of integrands topic to sure constraints, and so forth.

Along with altering variables in integrals, Model 13.1 additionally introduces DSolveChangeVariables for altering variables in differential equations. Right here it’s remodeling the Laplace equation to polar coordinates:


Typically a change of variables can simply be a comfort. However typically (suppose Basic Relativity) it could possibly lead one to an entire completely different view of a system. Right here, for instance, an exponential transformation converts the standard Cauchy–Euler equation to a kind with fixed coefficients:


Fractional Calculus

The primary spinoff of x2 is 2x; the second spinoff is 2. However what’s the spinoff? It’s a query that was requested (for instance by Leibniz) even within the first years of calculus. And by the 1800s Riemann and Liouville had given a solution—which in Model 13.1 can now be computed by the brand new FractionalD:


And, sure, do one other spinoff and also you get again the 1st spinoff:


Within the extra basic case we’ve got:


And this works even for detrimental derivatives, in order that, for instance, the (–1)st spinoff is an bizarre integral:


It may be a minimum of as troublesome to compute a fractional spinoff as an integral. However FractionalD can nonetheless typically do it


although the consequence can shortly change into fairly difficult:


Why is FractionalD a separate operate, moderately than simply being a part of a generalization of D? We mentioned this for fairly some time. And the rationale we launched the specific FractionalD is that there isn’t a singular definition of fractional derivatives. Actually, in Model 13.1 we additionally assist the Caputo fractional spinoff (or differintegral) CaputoD.

For the spinoff of x2, the reply remains to be the identical:


However as quickly as a operate isn’t zero at x = 0 the reply will be completely different:


CaputoD is a very handy definition of fractional differentiation when one’s coping with Laplace transforms and differential equations. And in Model 13.1 we are able to now solely compute CaputoD but in addition do integral transforms and resolve equations that contain it.

Right here’s a -order differential equation


and a -order one


in addition to a πth-order one:


Observe the looks of MittagLefflerE. This operate (which we launched in Model 9.0) performs the identical type of position for fractional derivatives that Exp performs for bizarre derivatives.

Extra Math—Some Lengthy Awaited

In February 1990 an inner bug report was filed in opposition to the still-in-development Model 2.0 of Mathematica:

Bug report

It’s taken a very long time (and related points have been reported many instances), however in Model 13.1 we are able to lastly shut this bug!

Take into account the differential equation (the Clairaut equation):


What DSolve does by default is to present the generic resolution to this equation, when it comes to the parameter 1. However the refined level (which in optics is related to caustics) is that the household of options for various values of 1 has an envelope which isn’t itself a part of the household of options, however can also be an answer:


In Model 13.1 you possibly can request that resolution with the choice IncludeSingularSolutionsTrue:


And right here’s a plot of it:


DSolve was a brand new operate (again in 1991) in Model 2.0. One other new operate in Model 2.0 was Residue. And in Model 13.1 we’re additionally including an extension to Residue: the operate ResidueSum. And whereas Residue finds the residue of a posh operate at a selected level, ResidueSum finds a sum of residues.

This computes the sum of all residues for a operate, throughout the entire advanced aircraft:


This computes the sum of residues inside a specific area, on this case the unit disk:


Create Your Personal “Information to Features” Pages

An vital a part of the built-in documentation for the Wolfram Language are what we name “information pages”—pages like the next that manage features (and different constructs) to present an total “cognitive map” and abstract of some space:

Sample guide page

In Model 13.1 it’s now simple to create your personal customized information pages. You may listing built-in features or different constructs, in addition to issues from the Wolfram Operate Repository and different repositories.

Go to the “root web page” of the Documentation Heart and press the icon:

Documentation Center

You’ll get a clean customized information web page:

Sample guide page teamplate

Fill within the information web page nevertheless you need, then use Deploy to deploy the web page both regionally, or to your cloud account. Both manner, the web page will now present up within the menu from the highest of the basis information web page (and so they’ll additionally present up in search):

Documentation Center

You would possibly find yourself creating only one customized information web page in your favourite features. Otherwise you would possibly create a number of, say one for every process or subject you generally take care of. Information pages aren’t about placing within the effort to create full-scale documentation; they’re rather more light-weight, and aimed extra at offering fast (“what was that operate referred to as?”) reminders and “big-picture” maps—leveraging all the precise operate and different documentation that already exists.

Visible Results & Beautification

At first it appeared like a minor characteristic. However as soon as we’d applied it, we realized it was rather more helpful than we’d anticipated. Simply as you possibly can model a graphics object with its shade (and, as of Model 13.0, its filling sample), now in Model 13.1 you possibly can model it with its drop shadowing:

Drop shadowing seems to be a pleasant option to “carry graphics to life”

or to emphasise one ingredient over others:

It really works effectively in geo graphics as effectively:

DropShadowing permits detailed management over the shadows: what course they’re in, how blurred they’re and what shade they’re:

Drop shadowing is extra difficult “below the hood” than one may think. And when potential it truly works utilizing {hardware} GPU pixel shaders—the identical expertise that we’ve used since Model 12.3 to implement material-based floor textures for 3D graphics. In Model 13.1 we’ve explicitly uncovered some well-known underlying kinds of 3D shading. Right here’s a geodesic polyhedron (sure, that’s one other new operate in Model 13.1), with its floor normals added (utilizing the once more new operate EstimatedPointNormals):

Right here’s probably the most primary type of shading: flat shading of every aspect (and the specularity on this case doesn’t “catch” any sides):

Right here now could be Gouraud shading, with a somewhat-faceted glint:

After which there’s Phong shading, trying considerably extra pure for a sphere:

Ever since Model 1.0, we’ve had an interactive option to rotate—and zoom into—3D graphics. (Sure, the mechanism was a bit primitive 34 years in the past, nevertheless it quickly acquired to roughly its trendy kind.) However in Model 13.1 we’re including one thing new: the flexibility to “dolly” right into a 3D graphic, imitating what would occur when you truly walked right into a bodily model of the graphic, versus simply zooming your digicam:

And, sure, issues can get a bit surreal (or “treky”)—right here dollying in after which zooming out:

3D Voronoi!

There are some capabilities that—over the course of years—have been requested again and again. Up to now these have included infinite undo, excessive dpi shows, a number of axis plots, and others. And I’m completely happy to say that almost all of those have now been taken care of. However there’s one—seemingly obscure—“straggler” that I’ve heard about for effectively over 25 years, and that I’ve truly additionally wished myself fairly a couple of instances: 3D Voronoi diagrams. Nicely, in Model 13.1, they’re right here.

Arrange 25 random factors in 3D:

pts = RandomReal

Now make a Voronoi mesh for these factors:


To “see inside” we are able to use opacity:


Why was this so laborious? In a Voronoi there’s a cell that surrounds every unique level, and consists of in every single place that’s nearer to that time than to another. We’ve had 2D Voronoi meshes for a very long time:


However there’s one thing simpler in regards to the 2D case. The problem shouldn’t be a lot the algorithm for producing the cells as it’s how the cells will be represented in such a manner that they’re helpful for subsequent computations. Within the 2D case every cell is only a polygon.

However within the 3D case the cells are polyhedra, and to make a Voronoi mesh we’ve got to have a polyhedral mesh the place all of the polyhedra match collectively. And it’s taken us a few years to construct the massive tower of computational geometry essential to assist this. There’s a considerably less complicated case based mostly purely on cells which can be all the time both simplices or hexahedra—that we’ve used for finite-element options to PDEs for some time. However in a real 3D Voronoi that’s not sufficient: the cells will be any (convex) polyhedral form.

Listed here are the “puzzle piece” cells for the 3D Voronoi mesh we made above:


Reconstructing Geometry from Level Clouds

Choose 500 random factors inside an annulus:

pts = RandomPoint

Model 13.1 now has a basic operate reconstructing geometry from a cloud of factors:


(In fact, given solely a finite variety of factors, the reconstruction can’t be anticipated to be good.)

The operate additionally works in 3D:


ReconstructionMesh is a basic superfunction that makes use of quite a lot of strategies, together with prolonged variations of the features ConcaveHullMesh and GradientFittedMesh that have been launched in Model 13.0. And along with reconstructing “stable objects”, it could possibly additionally reconstruct lower-dimensional issues like curves and surfaces:


A associated operate new in Model 13.1 is EstimatedPointNormals, which reconstructs not the geometry itself, however regular vectors to every ingredient within the geometry:


New in Visualization

In each new model for the previous 30 years we’ve steadily expanded our visualization capabilities, and Model 13.1 is not any exception. One operate we’ve added is TernaryListPlot—an analog of ListPlot that conveniently plots triples of values the place what one’s making an attempt to emphasise is their ratios. For instance let’s plot information from our knowledgebase on the sources of electrical energy for various international locations:


The plot exhibits the “power combination” for various international locations, with those on the underside axis being these with zero nuclear. Inserting colours for every axis, together with grid traces, helps clarify find out how to learn the plot:


More often than not plots are plotting numbers, or a minimum of portions. In Model 13.0, we prolonged features like ListPlot to additionally settle for dates. In Model 13.1 we’re going a lot additional, and introducing the opportunity of plotting what quantity to purely symbolic values.

Let’s say our information consists of letters A via C:

data = RandomChoice

How can we plot these? In Model 13.1 we simply specify an ordinal scale:


OrdinalScale helps you to specify that sure symbolic values are to be handled as if they’re in a specified order. There’s additionally the idea of a nominal scale—represented by NominalScale—by which completely different symbolic values correspond to completely different “classes”, however in no specific order.

Representing Quantities of Chemical substances

Molecule lets one symbolically signify a molecule. Amount lets one symbolically signify a amount with items. In Model 13.1 we now have the brand new assemble ChemicalInstance that’s in impact a merger of those, permitting one to signify a sure amount of a sure chemical.

This offers a symbolic illustration of 1 liter of acetone (by default at normal temperature and stress):


We are able to ask what the mass of this occasion of this chemical is:


ChemicalConvert lets us do a conversion returning specific items:


Right here’s as an alternative a conversion to moles:


This straight offers the quantity of substance that 1 liter of acetone corresponds to:


This generates a sequence of straight-chain hydrocarbons:


Right here’s the quantity of substance comparable to 1 g of every of those chemical compounds:


ChemicalInstance helps you to specify not simply the quantity of a substance, but in addition its state, specifically temperature and stress. Right here we’re changing 1 kg of water at 4° C to be represented when it comes to quantity:


Chemistry as Rule Software: Symbolic Sample Reactions

On the core of the Wolfram Language is the summary concept of making use of transformations to symbolic expressions. And at some degree one can view chemistry and chemical reactions as a bodily instantiation of this concept, the place one’s not coping with summary symbolic constructs, however as an alternative with precise molecules and atoms.

In Model 13.1 we’re introducing PatternReaction as a symbolic illustration for courses of chemical reactions—in impact offering an analog for chemistry of Rule for basic symbolic expressions.

Right here’s an instance of a “sample response”:


The primary argument specifies a pair of “reactant” molecule patterns to be reworked into “product” molecule patterns. The second argument specifies which atoms by which reactant molecules map to which atoms by which product molecules. If you happen to mouse over the ensuing sample response, you’ll see corresponding atoms “mild up”:

PatternReaction output

Given a sample response, we are able to use ApplyReaction to use the response to concrete molecules:


Listed here are plots of the ensuing product molecules:


The molecule patterns within the sample response are matched in opposition to subparts of the concrete molecules, then the transformation is completed, leaving the opposite components of the molecules unchanged. In a way it’s the direct analog of one thing like

{{a, b}, {c, d}} /. b -> {u, v}

the place the b within the symbolic expression is changed, and the result’s “knitted again” to fill in the place the b was once.

You are able to do what quantities to numerous sorts of “chemical practical programming” with ApplyReaction and PatternReaction. Right here’s an instance the place we’re basically build up a polymer by successive nesting of a response:




It’s typically handy to construct sample reactions symbolically utilizing Wolfram Language “chemical primitives”. However PatternReaction additionally helps you to specify reactions as SMARTS strings:


PDEs for Rods, Rubber and Extra

It’s been a 25-year journey, steadily rising our built-in PDE capabilities. And in Model 13.1 we’ve added a number of (admittedly considerably technical) options which were a lot requested, and are vital for fixing specific sorts of real-world PDE issues. The primary characteristic is with the ability to arrange a PDE as axisymmetric. Usually a 2D diffusion time period can be assumed Cartesian:


However now you possibly can say you’re coping with an axisymmetric system, together with your coordinates being interpreted as radius and peak, and all the pieces assumed to be symmetrical within the azimuthal course:


What’s vital about this isn’t simply that it makes it simple to arrange sure sorts of equations, but in addition that in fixing equations axial symmetry will be assumed, permitting rather more environment friendly strategies for use:


Additionally in Model 13.1 is an extension to the stable mechanics modeling framework launched in Model 13.0. Simply as there’s viscosity that damps out movement in fluids, so there’s an analogous phenomenon that damps out movement in solids. It’s extra of an engineering story, and it’s normally described when it comes to two parameters: mass damping and stiffness damping. And now in Model 13.1 we assist this sort of so-called Rayleigh damping in our modeling framework.

One other phenomenon included in Model 13.1 is hyperelasticity. If you happen to bend one thing like metallic past a sure level (however not up to now that it breaks), it’ll keep bent. However supplies like rubber and foam (and a few organic tissues) can “bounce again” from mainly any deformation.

Let’s think about that we’ve got a sq. of rubber-like materials. We anchor it on the left, after which we pull it on the best with a sure drive. What does it do?

This defines the properties of our materials:


We outline variables for the issue, representing x and y displacements by u and v:


Now we are able to arrange our complete drawback, and resolve the PDEs for it for every worth of the drive:


Then one can plot the outcomes, and see the rubber being nonlinearly stretched:


There’s in the long run appreciable depth in our dealing with of PDE-based modeling, and our rising capability to do “multiphysics” computations that span a number of kinds of physics (mechanical, thermal, electromagnetic, acoustic, …). And by now we’ve acquired almost 1000 pages of documentation purely about PDE-based modeling. And for instance in Model 13.1 we’ve added a monograph particularly about hyperelasticity, in addition to expanded our assortment of documented PDE fashions:

PDE models overview

Interpretable Machine Studying

Let’s say you’ve educated a machine studying mannequin and also you apply it to a specific enter. It offers you some consequence. However why? What have been the vital options within the enter that led it to that consequence? In Model 13.1 we’re introducing a number of features that attempt to reply such questions.

Right here’s some easy “coaching information”:

data = Flatten

We are able to use machine studying to make a predictor for this information:

pf = Predict

Making use of the predictor to a specific enter offers us a prediction:


What was vital in making this prediction? The "SHAPValues" property launched in Model 12.3 tells us what contribution every characteristic made to the consequence; on this case v was extra vital than u in figuring out the worth of the prediction:


However what about normally, for all inputs? The brand new operate FeatureImpactPlot offers a visible illustration of the contribution or “impression” of every characteristic in every enter on the output of the predictor:


What does this plot imply? It’s mainly displaying how typically there are what contributions from values of the 2 enter options. And with this specific predictor we see that there’s a variety of contributions from each options.

If we use a unique methodology to create the predictor, the outcomes will be fairly completely different. Right here we’re utilizing linear regression, and it seems that with this methodology v by no means has a lot impression on predictions:


If we make a predictor utilizing a choice tree, the characteristic impression plot exhibits the splitting of impression comparable to completely different branches of the tree:


FeatureImpactPlot offers a type of hen’s-eye view of the impression of various options. FeatureValueImpactPlot offers extra element, displaying as a operate of the particular values of enter options the impression factors with these values would have on the ultimate prediction (and, sure, the precise factors plotted listed here are based mostly on information simulated on the premise of the distribution inferred by the predictor; the precise information is normally too massive to wish to carry round, a minimum of by default):


CumulativeFeatureImpactPlot offers a visible illustration of how “successive” options have an effect on the ultimate worth for every (simulated) information level:


For predictors, characteristic impression plots present impression on predicted values. For classifiers, they present impression on (log) chances for specific outcomes.

Mannequin Predictive Management

One space that leverages many algorithmic capabilities of the Wolfram Language is management methods. We first began creating management methods performance greater than 25 years in the past, and by Model 8.0 ten years in the past we began to have built-in features like StateSpaceModel and BodePlot particularly for working with management methods.

Over the previous decade we’ve progressively been including extra built-in management methods capabilities, and in Model 13.1 we’re now introducing mannequin predictive controllers (MPCs). Many easy management methods (like PID controllers) take an advert hoc strategy by which they successfully simply “watch what a system does” with out making an attempt to have a selected mannequin for what’s occurring contained in the system. Mannequin predictive management is about having a selected mannequin for a system, after which deriving an optimum controller based mostly on that mannequin.

For instance, we might have a state-space mannequin for a system:

ssm = StateSpaceModel

Then in Model 13.1 we are able to derive (utilizing our parametric optimization capabilities) an optimum controller that minimizes a sure set of prices whereas satisfying specific constraints:

clr = ModelPredictiveController

The SystemsModelControllerData that we get right here accommodates quite a lot of components that enable us to automate the management design and evaluation workflow. For example, we are able to get a mannequin that represents the controller operating in a closed loop with the system it’s controlling:


Now let’s think about that we drive this complete system with the enter:


Now we are able to compute the output response for the system, and we see that each output variables are pushed to zero via the operation of the controller:


Inside the SystemsModelControllerData object generated by ModelPredictiveController is the precise controller computed on this case—utilizing the brand new assemble DiscreteInputOutputModel:


What truly is that this controller? Finally it’s a group of piecewise features that is dependent upon the values of states x1[t] and x2[t]:


And this exhibits the completely different state-space areas by which the controller has:


Algorithmic and Randomized Quizzes

In Model 13.0 we launched our query and evaluation framework that lets you writer issues like quizzes in notebooks, along with evaluation features, then deploy these to be used. In Model 13.1 we’re including capabilities to allow you to algorithmically or randomly generate questions.

The 2 new features QuestionGenerator and QuestionSelector allow you to specify inquiries to be generated in accordance with a template, or randomly chosen from a pool. You may both use these features straight in pure Wolfram Language code, or you need to use them via the Query Pocket book authoring GUI.

When you choose Insert Query within the GUI, you now get a selection between Fastened Query, Randomized Query and Generated Query:

Question GUI

Choose Randomized Query and also you’ll get

Question selector

which then lets you enter questions, and finally produce a QuestionSelector—which can choose newly randomized questions for each copy of the quiz that’s produced:

Question selector

Model 13.1 additionally introduces some enhancements for authoring questions. An instance is a pure-GUI “no-code” option to specify multiple-choice questions:

Question GUI

The ExprStruct Information Construction

Within the Wolfram Language expressions usually have two facets: they’ve a construction, and so they have a that means. Thus, for instance, Plus[1,1] has each a particular tree construction


and has a worth:

1 + 1

Within the regular operation of the Wolfram Language, the evaluator is routinely utilized to all expressions, and basically the one option to keep away from analysis by the evaluator is to insert “wrappers” like Maintain and Inactive that essentially change the construction of expressions.

In Model 13.1, nevertheless, there’s a brand new option to deal with “unevaluated” expressions: the "ExprStruct" information construction. ExprStructs signify expressions as uncooked information buildings which can be by no means straight seen by the evaluator, however can however be structurally manipulated.

This creates an ExprStruct comparable to the expression {1,2,3,4}:

ds = CreateDataStructure

This structurally wraps Complete across the listing, however does no analysis:

ds2 = ds

One may also see this by “visualizing” the info construction:


Regular takes an ExprStruct object and converts it to a standard expression, to which the evaluator is routinely utilized:


One can do quite a lot of basically structural operations straight on an ExprStruct. This is applicable Plus, then maps Factorial over the ensuing ExprStruct:

dsf = ds

The result’s an ExprStruct representing an unevaluated expression:


With "MapImmediateEvaluate" there’s an analysis achieved every time the mapping operation generates an expression:


One highly effective use of ExprStruct is in doing code transformations. And in a typical case one would possibly wish to import expressions from, say, a .wl file, then manipulate them in ExprStruct kind. In Model 13.1 Import now helps an ExprStructs import ingredient:


This selects expressions that correspond to definitions, within the sense that they’ve SetDelayed as their head:


Right here’s a visualization of the primary one:


Tremendous-Environment friendly Compiler-Primarily based Exterior Code Interplay

Let’s say you’ve acquired exterior code that’s in a compiled C-compatible dynamic library. An vital new functionality in Model 13.1 is a super-efficient and really streamlined option to name any operate in a dynamic library straight from inside the Wolfram Language.

It’s one of many accelerating stream of developments which can be being made potential by the large-scale infrastructure build-out that we’ve been doing in reference to the brand new Wolfram Language compiler—and specifically it typically leverages our refined new type-handling capabilities.

As a primary instance, let’s take into account the RAND_bytes (“cryptographically safe pseudorandom quantity generator”) operate in OpenSSL. The C declaration for this operate is:

C declaration

In Model 13.1 we now have a symbolic option to signify such a declaration straight within the Wolfram Language:

dec = LibraryFunctionDeclaration

(Usually we’d additionally must specify the library that this operate is coming from. OpenSSL occurs to be a library that’s loaded by default with the Wolfram Language so that you don’t want to say it.)

There are fairly a couple of new issues occurring within the declaration. First, as a part of our assortment of compiled varieties, we’re including ones like "CInt" and "CChar" that discuss with uncooked C language varieties (right here int and char). There’s additionally CArray which is for declaring C arrays. Discover the brand new ::[ ... ] syntax for TypeSpecifier that enables compact specs for parametrized varieties, just like the char* right here, that’s described in Wolfram Language as "CArray"::["CChar"].

Having arrange the declaration, we now must create an precise operate that may take an argument from Wolfram Language, convert it to one thing appropriate for the library operate, then name the library operate, and convert the consequence again to Wolfram Language kind. Right here’s a manner to try this on this case:

cf = FunctionCompile

What we get again is a compiled code operate that we are able to straight use, and that works by very effectively calling the library operate:


The FunctionCompile above makes use of a number of constructs which can be new in Model 13.1. What it basically does is to take a Wolfram Language integer (which it assumes to be a machine integer), solid it right into a C integer, then cross this to the library operate, together with a specification of a C char * into which the library operate will put its consequence, and from which the ultimate Wolfram Language consequence shall be retrieved.

It’s price emphasizing that many of the complexity right here has to do with dealing with information varieties and conversions between them—one thing that the Wolfram Language goes to a whole lot of bother to keep away from normally exposing the person to. However after we’re connecting to exterior languages that make elementary use of varieties, there’s no selection however to take care of them, and the complexity they contain.

Within the FunctionCompile above the primary new assemble we encounter is


The fundamental function of that is to create the buffer into which the exterior operate will write its outcomes. The buffer is an array of bytes, declared in C as char *, or right here as "CArray"::["CChar"]. There’s an precise wrinkle although: who’s going to handle the reminiscence related to this array? The "Managed":: sort specifier says that the Wolfram Language wrapper will do reminiscence administration for this object.

The subsequent new assemble we see within the FunctionCompile is


Solid is one among a household of latest features that may seem in compilable code, however haven’t any significance exterior the compiler. Solid is used to specify that information needs to be transformed to a kind in keeping with a specified sort (right here a C int sort).

The core of the FunctionCompile is using LibraryFunction, which is what truly calls the exterior library operate that we declared with the library operate declaration.

The final step within the operate compiled by FunctionCompile is to extract information from the C array and return it as a Wolfram Language listing. To do that requires the brand new operate FromRawPointer, which truly retrieves information from a specified location in reminiscence. (And, sure, this can be a uncooked dereferencing operation that may trigger a crash if it isn’t achieved appropriately.)

All of this may increasingly at first appear moderately difficult, however for what it’s doing, it’s remarkably easy—and enormously leverages the entire symbolic construction of the Wolfram Language. It’s additionally price realizing that on this specific instance, we’re simply dipping into compiled code after which returning outcomes. In larger-scale instances we’d be doing many extra operations—sometimes specified straight by top-level Wolfram Language code—inside compiled code, and so sort declaration and conversion operations can be a smaller fraction of the code we’ve got to write down.

One characteristic of the instance we’ve simply checked out is that it solely makes use of built-in varieties. However in Model 13.1 it’s now potential to outline customized varieties, such because the analog of C structs. For example, take into account the operate ldiv from the C normal library. This operate returns an object of sort ldiv_t, outlined by the next typedef:


Right here’s the Wolfram Language model of this declaration, based mostly on organising a "Product" sort named "CLDivT":

divtdec = TypeDeclaration

(The "ReferenceSemantics"False choice specifies that this sort will truly be handed round as a worth, moderately than only a pointer to a worth.)

Now the declaration for the ldiv operate can use this new customized sort:

divdec = LibraryFunctionDeclaration

The ultimate definition of the decision to the exterior ldiv operate is then:

divQR = FunctionCompile

And now we are able to use the operate (and, sure, it will likely be as environment friendly as if we’d straight written all the pieces in C):


The examples we’ve given listed here are very small ones. However the entire construction for exterior operate calls that’s now in Model 13.1 is ready as much as deal with massive and complicated conditions—and certainly we’ve been utilizing it internally with nice success to arrange vital new built-in items of the Wolfram Language.

One of many components that’s typically wanted in additional advanced conditions is extra refined reminiscence administration, and our new "Managed" sort gives a handy and streamlined manner to do that.

This makes a compiled operate that creates an array of 10,000 machine integers:

cf = FunctionCompile

Working the operate successfully “leaks” reminiscence:


However now outline a model of the operate by which the array is “managed”:

cf = FunctionCompile

Now the reminiscence related to the array is routinely freed when it’s now not referenced:


Immediately Compiling Operate Definitions

If in case you have an express pure operate (Operate[...]) you need to use FunctionCompile to provide a compiled model of it. However what you probably have a operate that’s outlined utilizing downvalues, as in:


In Model 13.1 you possibly can straight compile operate definitions like this. However—as is the character of compilation—you’ve declare what varieties are concerned. Here’s a declaration for the operate fac that claims it takes a single machine integer, and returns a machine integer:

dec = FunctionDeclaration

Now we are able to create a compiled operate that computes fac[n]:

cfac = FunctionCompile

The compiled operate runs considerably quicker than the bizarre symbolic definition:


The flexibility to declare and use downvalue definitions in compilation has the vital characteristic that it lets you write a definition simply as soon as, after which use it each straight, and in compiled code.

Manipulating Expressions in Compiled Code

An early focus of the Wolfram Language compiler is dealing with low-level “machine” varieties, corresponding to integers or reals of sure lengths. However one of many advances within the Model 13.1 compiler is direct assist for an "InertExpression" sort for representing any Wolfram Language expression inside compiled code.

Whenever you use one thing like FunctionCompile, it’s going to explicitly attempt to compile no matter Wolfram Language expressions it’s given. However when you wrap the expressions with InertExpression the compiler will then simply deal with the expressions as inert structural objects of sort "InertExpression". This units up a compiled operate that constructs an expression (implicitly of sort "InertExpression"):

cf = FunctionCompile

Evaluating the operate constructs after which returns the expression:


Usually, inside the compiler, an "InertExpression" object shall be handled in a purely structural manner, with none analysis (and, sure, it’s intently associated to the "ExprStruct" information construction). However typically it’s helpful to carry out analysis on it, and you are able to do this with InertEvaluate:

cf = FunctionCompile

Now the InertEvaluate does the analysis earlier than wrapping Maintain across the inert expression:


The flexibility to deal with expressions straight within the compiler would possibly look like some type of element. Nevertheless it’s truly massively vital in opening up potentialities for future improvement with the Wolfram Language. For the previous 35 years, we’ve internally been in a position to write low-level expression manipulation code as a part of the C language core of the Wolfram Language kernel. However the capability of the Wolfram Language compiler to deal with expressions now opens this up—and lets anybody write maximally environment friendly code for manipulating expressions that interoperate with all the pieces else within the Wolfram Language.

And Nonetheless Extra…

Even past all of the issues I’ve mentioned up to now, there are all types of additional additions and enhancements in Model 13.1, dotted all through the system.

InfiniteLineThrough and CircularArcThrough have been added for geometric computation, and geometric scene specification. Geometric scenes can now be styled for customized presentation:


There are new graph features: GraphProduct, GraphSum and GraphJoin:


And there are new built-in households of graphs: TorusGraph and BuckyballGraph:


You may combine photographs straight into Graphics (and Graphics3D):


AbsoluteOptions now resolves many extra choices in Graphics, telling you what express worth was used if you gave an choice simply as Automated.

The operate LeafCount now has a Heads choice, to rely expression branches inside heads. Splice works with any head, not simply Listing. Features like IntersectingQ now have SameTest choices. You may specify TimeZone choices utilizing geographic entities (like cities).

FindClusters now helps you to specify precisely what number of clusters you wish to partition your information into, in addition to supporting UpTo[n].

In neural nets, ElementwiseLayer now helps “trendy” nonconvex nonmonotonic activation features like Mish and GELU, AttentionLayer helps dropout and native masking, ReplicateLayer now helps integer arrays, and RandomArrayLayer helps additional statistical distributions. NetTrain now handles multioutput and nonscalar losses. Picture encoders and decoders assist resampling and padding, and there’s now assist for nuclear sampling. Our assist for the ONNX switch format continues to develop, with internet operators added in Model 13.1.

CenteredInterval—launched in Model 13.0—now helps 36 additional particular features (and, sure, each wants theorems proved to make this work).

There’ll be extra approaching this in subsequent variations, however in Model 13.1 we’re starting the introduction of structured matrices which can be saved and computed with in particular, optimized methods. Examples embody PermutationMatrix and LowerTriangularMatrix.

We’ve had in depth assist for computational microscopy for some time. However in Model 13.1 the "BioImageFormat" Import format now provides importing of the greater than 160 uncooked picture codecs utilized by completely different sorts of microscopes.

Model 13.0 dramatically expanded our capability to import PDF. We’ve additional enhanced this in Model 13.1, for instance permitting positioned textual content to be imported into graphics as Textual content primitives.

We’ve supported normal textual content kinds like daring and italic without end, however now we’ve got a typical option to deal with struckthrough textual content as effectively:


Along with all these “inside-the-system” enhancements, we’ve additionally completed making it potential to obtain desktop variations of Wolfram Language on all platforms (together with Linux) whereas leaving documentation on the net. Documentation set up may also now be configured globally on a per-machine foundation, moderately than simply on a per-user foundation.

So—as of right now, documentation or not—you will get Model 13.1 in your laptop. Oh, and the Wolfram Cloud has additionally now been up to date to make use of Model 13.1. I hope you take pleasure in the brand new options, and this subsequent step on the epic journey of Mathematica and the Wolfram Language.



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