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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 for the reason that 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 growth efforts roughly twice a yr. We launched Model 13.0 on December 13, 2021. And now, roughly six months later, we’re releasing Model 13.1. As standard, although it’s a “.1” launch, it’s obtained numerous 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 at all times thrilling to see what we mange to ship in every new model. And in Model 13.1 we now have 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 consumer interfaces and the compiler.

The Wolfram Language because it exists as we speak encompasses an unlimited vary of performance. However its nice energy comes not simply from what it accommodates, but additionally from how coherently every thing in it suits collectively. And for almost 36 years I’ve taken it as a private accountability 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 on daily basis in my use of the Wolfram Language, I’m pleased 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 evaluation conferences. It’s an unprecedented stage of openness—and engagement with the group. In designing Model 13.1 we’ve finished 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 additionally of why it was constructed, and the way choices about it had been made.

However, OK, so what lastly is in Model 13.1? Let’s speak about 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 might be added:

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

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

Think about:

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

This makes use of peculiar listability, successfully computing:

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

However what if you’d like as a substitute to “go down a stage” and thread {x,y} into the bottom elements of the primary record? Nicely, now you should utilize Threaded to do this:

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

By itself, Threaded is only a symbolic wrapper:

Threaded

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

Right here’s one other instance. Create an inventory:

Table

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

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

However now we’ve obtained Threaded, and so as a substitute we are able to simply say:

Table

You may give Threaded as an argument to any listable perform, not simply Plus and Instances:

Mod

You should utilize Threaded and peculiar 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 best way, will get its identify from the perform Thread, which explicitly does “threading”, as in:

Thread

By default, Threaded will at all times thread into the bottom stage of an inventory:

{{{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:

ImageData

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

Image3D

More often than not you both wish to use peculiar listability that operates on the high stage of an inventory, otherwise you wish to use the default type of Threaded, that operates on the lowest stage of an inventory. However Threaded has a extra normal type, in which you’ll be able to explicitly say what stage you need it to function at.

Right here’s the default case:

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

Right here’s stage 1, which is rather like peculiar listability:

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

And right here’s threading into stage 2:

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

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

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

Right here now could be “peculiar listability” conduct:

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

For the arrays we’re taking a look at right here, the default conduct is equal to:

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

Generally it’s clearer to jot down this out in a type like

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

which says that the primary stage of the array contained in the Threaded is to be aligned with the second stage of the surface array. Typically, the default case is equal to –1 → –1, specifying that the underside stage of the array contained in the Threaded ought to be aligned with the underside stage of the array exterior.

But Extra Language Comfort Features

In each model of the Wolfram Language we attempt to add new features that can make normal packages simpler to jot down and simpler to learn. In Model 13.1 an important such perform is Threaded. However there are fairly just a few others as effectively.

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

DeleteElements

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

DeleteElements

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

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 record or the opposite, however not each. For instance, what international locations are within the G20 or the EU, however not each?

SymmetricDifference

Let’s say you might have a number of lists, and also you wish to know what components are distinctive to only 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 varied alphabets:

UniqueElements

We’ve had Map and Apply, with brief types /@ and @@, ever since Model 1.0. In Model 4.0 we added @@@ to symbolize Apply[f,expr,1]. However we by no means added a separate perform to correspond to @@@. And through the years, there’ve been fairly just a few events the place I’ve mainly needed, 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 perform. 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—primarily an operator type of SameQ. Why is such a assemble wanted? Nicely, there are at all times 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 that for consistency SameQ[expr] should at all times return True—so it’s not accessible 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:

SameAs

Procedural programming—usually with “variables hanging out”—isn’t the popular fashion for many Wolfram Language code. However generally it’s probably the most handy approach to do issues. And in Model 13.1 we’ve add a small piece of streamlining by introducing the perform 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 received’t ever consider physique. Till[test,body] does issues the opposite approach round: it evaluates physique till check is now not 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 record of latest core language features in Model 13.1 is ReplaceAt. Substitute makes an attempt to use a alternative rule to a whole expression—or an entire stage 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 finished. And that’s what ReplaceAt gives:

ReplaceAt

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

ReplaceAt

Why is that this vital? The reply is that it offers a symbolic approach to specify not simply what alternative is made, but additionally 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 had been represented as 8-bit objects: often ASCII, however you can decide one other “character encoding” (therefore the ChararacterEncoding possibility) when you needed. Then within the early Nineties got here Unicode—which we had been one of many very first firms to assist. Now “characters” may very well be 16-bit constructs, with almost 65536 doable “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 techniques didn’t but have built-in assist for. However we had been betting on Unicode, and so we constructed our personal infrastructure for dealing with it.

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

It’s been a gradual growth. Authentic 16-bit Unicode is “aircraft 0”. Now there are as much as 16 extra planes. Not fairly 32-bit characters, however given the best way computer systems work, the method now could be to permit characters to be represented by 32-bit objects. It’s removed from trivial to do this uniformly and effectively. And for us it’s been an extended course of to improve every thing 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 while you begin to fear about diphthongs like æ. But when there are pretty few of those, it really works to only 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 these items might be fairly difficult. However after a few years of growth, main working techniques 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 like? Utilizing CharacterRange (or FromCharacterCode) we are able to dive in and simply see what’s on the market in “character house”. Right here’s a part of peculiar 16-bit Unicode house:

CharacterRange

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

CharacterRange

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

CharacterRange

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

Expand

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

ToCharacterCode

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

Emojis in notebook

The world of emoji is quickly evolving—and that may generally result in issues. Right here’s an emoji vary that features some very acquainted emoji, however on my pc system additionally consists of emoji that show solely as :

CharacterRange

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

Style

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:

FromCharacterCode

The primary character right here can stand by itself:

FromCharacterCode

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

FromCharacterCode

However as soon as the composite हि has been shaped it acts “textually” similar 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, nonetheless, it could possibly nonetheless be damaged into its constituent Unicode components:

Characters

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

FromCharacterCode

Despite the fact that this seems to be 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”:

Characters

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

CharacterNormalize

It seems to be the identical, however now it actually is only one character:

ToCharacterCode

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

FromCharacterCode

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

ToCharacterCode

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

ToCharacterCode

along with the emoji for a microscope

ToCharacterCode

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

ToCharacterCode

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

FromCharacterCode

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

Grid

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

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

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

Grid

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

Toolbar

It was once that the one “particular key sequence” one completely ought to know as a way to use Wolfram Notebooks was shiftenter. However step by step there have began to be an increasing number of high-profile operations which might be conveniently finished by “urgent a button”. And quite 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 might be displayed by default in each customary 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 straightforward to design the default toolbar (and we hope you’ll like what we got here up with!) The primary downside is that Wolfram Notebooks are very normal, and there are an incredible many issues you are able to do with them—which it’s difficult to arrange right into a manageable toolbar. (Some particular kinds of notebooks have had their very own specialised toolbars for some time, which had 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 just equal to urgent shiftret (as its tooltip says). Menu optionsmeans “Abort”, and can cease a computation. To the appropriate of Menu options is the menu proven above. The primary a part of the menu lets you select what might be evaluated. (Don’t overlook the extraordinarily helpful “Consider In Place” that allows you to consider no matter code you might have 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.

Shifting alongside the toolbar, we subsequent have:

Menu option

In case your cursor isn’t already in a cell, the pulldown lets you choose what kind of cell you wish to insert (it’s much like the Menu options “tongue” that seems throughout 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 fashion that’s getting used, and allow you to reset it.)

Menu options offers you somewhat panel to regulate to look of cells, altering their background colours, frames, dingbats, and many others.

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 usually press repeatedly) is for extending a variety—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 approach 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 straightforward to learn (say by iconizing particulars of choices). (You may as well iconize a variety from the right-click menu, or with ctrlcmd'.)

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

Should you’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 customary. Menu options permits 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 might have chosen within the pocket book, or opening the entrance web page (“root information web page”) of the principle Wolfram Language documentation. Lastly, Menu options permits 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. Count on extra options in later variations. One factor that’s notable in regards to the toolbar usually is that it’s 100% carried out in Wolfram Language. And along with including quite a lot of flexibility, this additionally implies that the toolbar instantly works on all platforms. (By the best way, when you don’t need the toolbar in a selected pocket book—or for all of your notebooks—simply right-click the background of the toolbar to select that possibility.)

Sharpening 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 really is picture when it’s in a pocket book? Previously, it’s been a reasonably advanced symbolic construction primarily appropriate for analysis. However in Model 13.1 we’ve made it a lot easier. 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 utility, like pure-text e mail. Previously this produced one thing that may 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:

Range
Range

Speaking of huge outputs (in addition to different issues that preserve 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 downside had already been solved, however on the Mac modifications within the OS had led to a state of affairs the place the pocket book entrance finish might mysteriously pop to the entrance in your desktop—a state of affairs that has now been resolved.

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

The fundamental pocket book consumer interface works as a lot as doable with customary interface components on all platforms, in order that when these components are up to date, we at all times mechanically get the “most fashionable” look. However there are elements of the pocket book interface which might be fairly particular to Wolfram Notebooks and are at all times 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:

Preferences

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

Specifically, in Model 13.1, there’s an instance associated to the Wolfram Perform Repository. Say you employ a perform for which an replace is offered. 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

Should you 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 thought is to have an interface the place there are doubtlessly a number of passes, and the place a computation at all times runs to completion, however you might have a simple approach to change the way it’s arrange, after which run it once more.

Giant-Scale Code Modifying

One of many nice issues in regards to the Wolfram Language is that it really works effectively for packages of any scale—from lower than a line lengthy to thousands and thousands of strains lengthy. And for the previous a number of years we’ve been engaged on increasing our assist for very massive Wolfram Language packages. Utilizing LSP (Language Server Protocol) we’ve supplied the potential for most traditional exterior IDEs to mechanically 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 enhancing in notebooks extra handy. The primary—and broadly requested—is block indent and outdent of code. Choose the strains 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 power to work with .wl package deal recordsdata (in addition to .wls script recordsdata) utilizing our pocket book enhancing system. A brand new default characteristic in Model 13.1 is numbering of all code strains that seem within the underlying file (and, sure, we accurately 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 perform, you’ll instantly have the ability to use the road quantity it studies to seek 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 should utilize the drawing instruments within the canvas to create an annotation overlay:

Drawing

Should you 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 elemental Tree assemble in Model 13.1 is the power to call branches at every node, by giving them in an affiliation:

Tree

All tree features now embody assist for associations:

NestTree

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

RandomTree

and right here’s the unlabeled analog:

UnlabeledTree

In Model 12.3 we launched ExpressionTree for deriving bushes from normal symbolic expressions. Our plan is to have a variety of “particular bushes” acceptable 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:

ExpressionTree

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

TreeExpression

(Keep 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:

TreeGraph

The brand new perform TreeLeafCount permits you to depend the whole variety of leaf nodes on a tree (mainly the analog of LeafCount for a normal symbolic expression):

TreeLeafCount

One other new perform in Model 13.1 that’s usually 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 foundation”:

RootTree

It could actually additionally get a subtree that’s “removed from the leaves”, on this case happening to components which might be at stage –2 within the tree:

RootTree

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 selected tree ingredient have an effect on solely that tree ingredient:

Tree

However you may give guidelines that specify how components within the subtree beneath that ingredient are affected:

Tree

In Model 13.1 there’s now detailed management accessible for styling each nodes and edges within the tree. Right here’s an instance that provides styling for mum or dad edges of nodes:

Tree

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

Tree

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

Tree

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

Tree

But Extra Date-Dealing with Particulars

We first launched dates into Wolfram Language in Model 2.0, and we launched fashionable date objects in Model 10.0. However to actually make dates totally computable, there are various detailed instances to think about. And in Model 13.1 we’re coping with yet one more of them. Let’s say you’ve obtained 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:

UnitConvert

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

DatePlus

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

DatePlus

If one’s doing one thing like astronomy, this type 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 a substitute, one needs to land on some calendar date or one other.

Right here’s the default within the Wolfram Language:

DatePlus

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":

DatePlus

No matter technique 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 now have the weirdness of repeating February 28:

DatePlus

With "RollForward" we now have the weirdness of repeating March 1:

DatePlus

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

DatePlus

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 conduct to be.

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

It’s value 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, generally by elements of a whole lot.

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 now have VideoCapture. By default VideoCapture[] offers you a GUI that allows you to file out of your digicam:

VideoCapture

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

VideoCapture

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

Now you possibly can course of or analyze this Video object similar to you’d another:

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

permits 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 data video out of your digicam (and you should utilize the ImageDevice choice to specify which one you probably have a number of). VideoScreenCapture, then again, data out of your pc 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 utility (say, an internet browser), it’ll just do positive. And as well as, you may give a display rectangle to seize a selected area in your display:

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

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

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

By the best way, each VideoCapture and VideoScreenCapture by default seize audio. You may swap off audio recording both from the GUI, or with the choice AudioInputDevice→None.

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

VideoScreenCapture[EvaluationNotebook[]]

Along with capturing video from real-time goings-on, you too can generate video immediately 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 containers for elephants and zebras:

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

Outline a path perform of a variable t:

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

Faculty Calculus

Reworking faculty calculus was one of many early achievements of Mathematica. However even now we’re persevering with so as to add performance to make faculty calculus ever simpler and smoother to do—and extra instantly connectable to purposes. We’ve at all times had the perform 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:

ImplicitD

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

ImplicitD

To date all of it is a pretty simple repackaging of our longstanding calculus performance. And certainly these sorts of implicit derivatives have been accessible for a very long time in Wolfram|Alpha. However for Mathematica and the Wolfram Language we wish every thing to be as normal as doable—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 peculiar college-level calculus, ImplicitD can do issues like discovering a second implicit spinoff on a curve outlined by the intersection of two surfaces:

ImplicitD

In Mathematica and the Wolfram Language Combine is a perform 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 generally additionally when pushing boundaries of what’s doable—it may be helpful to do integrals in steps. And so in Model 13.1 we’ve added the perform IntegrateChangeVariables for altering variables in integrals.

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

Integrate

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

Inactive

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

IntegrateChangeVariables

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

Activate

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

IntegrateChangeVariables

Though the fundamental “structural” transformation of variables in integrals is kind of simple, the entire story of IntegrateChangeVariables is significantly extra difficult. “Faculty-level” modifications of variables are often rigorously organized to return out simply. However within the extra normal case, IntegrateChangeVariables finally ends up having to do nontrivial transformations of geometric areas, tough 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:

DSolveChangeVariables

Generally a change of variables can simply be a comfort. However generally (suppose Common Relativity) it could possibly lead one to an entire completely different view of a system. Right here, for instance, an exponential transformation converts the same old Cauchy–Euler equation to a type with fixed coefficients:

DSolveChangeVariables

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:

FractionalD

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

FractionalD

Within the extra normal case we now have:

FractionalD

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

FractionalD

It may be no less than as tough to compute a fractional spinoff as an integral. However FractionalD can nonetheless usually do it

FractionalD

although the outcome can rapidly change into fairly difficult:

FractionalD

Why is FractionalD a separate perform, quite than simply being a part of a generalization of D? We mentioned this for fairly some time. And the rationale we launched the express FractionalD is that there isn’t a singular definition of fractional derivatives. The truth is, 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:

CaputoD

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

CaputoD

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 additionally do integral transforms and clear up equations that contain it.

Right here’s a -order differential equation

DSolve

and a -order one

DSolve

in addition to a πth order one:

DSolve

Word the looks of MittagLefflerE. This perform (which we launched in Model 9.0) performs the identical type of function for fractional derivatives that Exp performs for peculiar derivatives.

Extra Math—Some Lengthy Awaited

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

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

Think about the differential equation (the Clairaut equation):

DSolve

What DSolve does by default is to offer the generic resolution to this equation, when it comes to the parameter 𝕔1. However the refined level (which in optics is related to caustics) is that the household of options for various values of 𝕔1 has an envelope which isn’t itself a part of the household of options, however can also be an answer:

Plot

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

DSolve

And right here’s a plot of it:

Plot

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

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

ResidueSum

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

ResidueSum

Create Your Personal “Information to Features” Pages

p>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 offer an general “cognitive map” and abstract of some space:

Sample guide page

In Model 13.1 it’s now straightforward to create your personal customized information pages. You may record built-in features or different constructs, in addition to issues from the Wolfram Perform 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 nonetheless you need, then use Deploy to deploy the web page both regionally, or to your cloud account. Both approach, the web page will now present up within the menu from the highest of the foundation information web page (and so they’ll additionally present up in search):

Documentation Center

You may find yourself creating only one customized information web page to your favourite features. Otherwise you may create a number of, say one for every process or matter you generally cope with. 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 perform referred to as?”) reminders and “big-picture” maps—leveraging all the particular perform and different documentation that already exists.

Visible Results & Beautification

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

Drop shadowing seems to be a pleasant approach to “convey 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 route they’re in, how blurred they’re and what shade they’re:

Drop shadowing is extra difficult “beneath the hood” than one may think. And when doable it really works utilizing {hardware} GPU pixel shaders—the identical know-how 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 perform in Model 13.1), with its floor normals added (utilizing the once more new perform EstimatedPointNormals):

Right here’s probably the most primary type of shading: flat shading of every side (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 approach to rotate—and zoom into—3D graphics. (Sure, the mechanism was a bit primitive 34 years in the past, nevertheless it quickly obtained to roughly its fashionable type.) However in Model 13.1 we’re including one thing new: the power to “dolly” right into a 3D graphic, imitating what would occur when you really 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 time and again. Previously these have included infinite undo, excessive dpi shows, a number of axis plots, and others. And I’m comfortable 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 really additionally needed myself fairly just a few instances: 3D Voronoi diagrams. Nicely, in Model 13.1, they’re right here.

Arrange 25 random factors in 3D:

pts = RandomReal
Graphics3D

Now make a Voronoi mesh for these factors:

VoronoiMesh

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

Graphics3D

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

VoronoiMesh

However there’s one thing simpler in regards to the 2D case. The difficulty just isn’t a lot the algorithm for producing the cells as it’s how the cells might be represented in such a approach 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 now have 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 easier case based mostly purely on cells which might be at all times 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 might be any (convex) polyhedral form.

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

Row

Reconstructing Geometry from Level Clouds

Choose 500 random factors inside an annulus:

pts = RandomPoint
Graphics

Model 13.1 now has a normal perform reconstructing geometry from a cloud of factors:

ReconstructionMesh

(After all, given solely a finite variety of factors, the reconstruction can’t be anticipated to be excellent.)

The perform additionally works in 3D:

ReconstructionMesh
ReconstructionMesh

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

ReconstructionMesh

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

ListVectorPlot

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 perform 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:

TernaryListPlot

The plot exhibits the “vitality combination” for various international locations, with those on the underside axis being these with zero nuclear. Inserting colours for every axis, together with grid strains, helps clarify the best way to learn the plot:

TernaryListPlot

More often than not plots are plotting numbers, or no less than 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 potential for 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:

ListPlot

OrdinalScale permits 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—through which completely different symbolic values correspond to completely different “classes”, however in no explicit order.

Representing Quantities of Chemical substances

Molecule lets one symbolically symbolize a molecule. Amount lets one symbolically symbolize 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 symbolize a sure amount of a sure chemical.

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

ChemicalInstance

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

ChemicalInstance

ChemicalConvert lets us do a conversion returning explicit items:

ChemicalConvert

Right here’s as a substitute a conversion to moles:

ChemicalConvert

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

ChemicalInstance

This generates a sequence of straight-chain hydrocarbons:

GraphicsBox

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

Table

ChemicalInstance permits you to specify not simply the quantity of a substance, but additionally 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:

ChemicalConvert

Chemistry as Rule Utility: Symbolic Sample Reactions

On the core of the Wolfram Language is the summary thought of making use of transformations to symbolic expressions. And at some stage 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 a substitute 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 normal symbolic expressions.

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

PatternReaction

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

PatternReaction output

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

ApplyReaction

Listed below are plots of the ensuing product molecules:

MoleculePlot

The molecule patterns within the sample response are matched towards subparts of the concrete molecules, then the transformation is finished, leaving the opposite elements 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 useful programming” with ApplyReaction and PatternReaction. Right here’s an instance the place we’re primarily build up a polymer by successive nesting of a response:

NestList

 

MoleculePlot

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

PatternReaction

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 have been a lot requested, and are vital for fixing explicit 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:

DiffusionPDETerm

However now you possibly can say you’re coping with an axisymmetric system, along with your coordinates being interpreted as radius and peak, and every thing assumed to be symmetrical within the azimuthal route:

DiffusionPDETerm

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

RegionPlot3D

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 identical phenomenon that damps out movement in solids. It’s extra of an engineering story, and it’s often described when it comes to two parameters: mass damping and stiffness damping. And now in Model 13.1 we assist this type of so-called Rayleigh damping in our modeling framework.

One other phenomenon included in Model 13.1 is hyperelasticity. Should you bend one thing like steel 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 now have a sq. of rubber-like materials. We anchor it on the left, after which we pull it on the appropriate with a sure power. What does it do?

This defines the properties of our materials:

pars

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

vars

Now we are able to arrange our complete downside, and clear up the PDEs for it for every worth of the power:

GraphicsBox

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

VectorDisplacementPlot

There’s ultimately appreciable depth in our dealing with of PDE-based modeling, and our rising means to do “multiphysics” computations that span a number of kinds of physics (mechanical, thermal, electromagnetic, acoustic, …). And by now we’ve obtained 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 might have skilled a machine studying mannequin and also you apply it to a selected enter. It offers you some outcome. However why? What had been the vital options within the enter that led it to that outcome? 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 selected enter offers us a prediction:

pf

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

pf

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

FeatureImpactPlot

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

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

FeatureImpactPlot

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

FeatureImpactPlot

FeatureImpactPlot offers a type of fowl’s-eye view of the impression of various options. FeatureValueImpactPlot offers extra element, displaying as a perform 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 below are based mostly on information simulated on the idea of the distribution inferred by the predictor; the precise information is often too huge to wish to carry round, no less than by default):

FeatureValueImpactPlot

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

CumulativeFeatureImpactPlot

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

Mannequin Predictive Management

One space that leverages many algorithmic capabilities of the Wolfram Language is management techniques. We first began creating management techniques 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 techniques.

Over the previous decade we’ve progressively been including extra built-in management techniques capabilities, and in Model 13.1 we’re now introducing mannequin predictive controllers (MPCs). Many easy management techniques (like PID controllers) take an advert hoc method through which they successfully simply “watch what a system does” with out making an attempt to have a selected mannequin for what’s happening 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 explicit 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:

clr

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

ListStepPlot

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:

ListStepPlot

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

clr

What really is that this controller? In the end it’s a group of piecewise features that will depend on the values of states x1[t] and x2[t]:

clr

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

Region

Algorithmic and Randomized Quizzes

In Model 13.0 we launched our query and evaluation framework that lets you creator 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 immediately in pure Wolfram Language code, or you should utilize them via the Query Pocket book authoring GUI.

When you choose Insert Query within the GUI, you now get a selection between Mounted 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 is able to 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” approach to specify multiple-choice questions:

Question GUI

The ExprStruct Knowledge Construction

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

TreeExtract

and has a worth:

1 + 1

Within the regular operation of the Wolfram Language, the evaluator is mechanically utilized to all expressions, and primarily the one approach 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, nonetheless, there’s a brand new approach to deal with “unevaluated” expressions: the "ExprStruct" information construction. ExprStructs symbolize expressions as uncooked information constructions which might be by no means immediately 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 record, however does no analysis:

ds2 = ds

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

ds2

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

Normal

One can do quite a lot of primarily structural operations immediately 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:

dsf

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

ds

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

Import

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

Select

Right here’s a visualization of the primary one:

First

Tremendous-Environment friendly Compiler-Based mostly Exterior Code Interplay

Let’s say you’ve obtained 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 approach to name any perform in a dynamic library immediately from throughout the Wolfram Language.

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

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

C declaration

In Model 13.1 we now have a symbolic approach to symbolize such a declaration immediately within the Wolfram Language:

dec = LibraryFunctionDeclaration

(Typically we’d additionally must specify the library that this perform 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 just a few new issues happening within the declaration. First, as a part of our assortment of compiled sorts, we’re including ones like "CInt" and "CChar" that confer with uncooked C language sorts (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 sorts, just like the char* right here, that’s described in Wolfram Language as "CArray”::[“CChar"].

Having arrange the declaration, we now have to create an precise perform that may take an argument from Wolfram Language, convert it to one thing appropriate for the library perform, then name the library perform, and convert the outcome again to Wolfram Language type. Right here’s a approach to do this on this case:

cf = FunctionCompile

What we get again is a compiled code perform that we are able to immediately use, and that works by very effectively calling the library perform (it returns a uncooked numeric array, so we now have to make use of Regular to make it a traditional expression):

Normal

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

It’s value emphasizing that many of the complexity right here has to do with dealing with information sorts and conversions between them—one thing that the Wolfram Language goes to numerous bother to keep away from often exposing the consumer to. However once we’re connecting to exterior languages that make elementary use of sorts, there’s no selection however to cope with them, and the complexity they contain.

Within the FunctionCompile above the primary new assemble we encounter is

CreateTypeInstance

The fundamental goal of that is to create the buffer into which the exterior perform 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":: kind specifier says that the Wolfram Language wrapper will do reminiscence administration for this object.

The subsequent new assemble we see within the FunctionCompile is

Cast

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

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

The final step within the perform compiled by FunctionCompile is to extract information from the C array and return it as a Wolfram Language record. To do that requires the brand new perform FromRawPointer, which really retrieves information from a specified location in reminiscence. (And, sure, it is a uncooked dereferencing operation that can trigger a crash if it isn’t finished accurately.)

All of this will likely at first appear quite difficult, however for what it’s doing, it’s remarkably easy—and drastically leverages the entire symbolic construction of the Wolfram Language. It’s additionally value realizing that on this explicit instance, we’re simply dipping into compiled code after which returning outcomes. In larger-scale instances we’d be doing many extra operations—usually specified immediately by top-level Wolfram Language code—inside compiled code, and so kind declaration and conversion operations can be a smaller fraction of the code we now have to jot down.

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

typedef

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

divtdec = TypeDeclaration

(The "ReferenceSemantics"False possibility specifies that this sort will really be handed round as a worth, quite than only a pointer to a worth.)

Now the declaration for the ldiv perform can use this new customized kind:

divdec = LibraryFunctionDeclaration

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

divQR = FunctionCompile

And now we are able to use the perform (and, sure, it will likely be as environment friendly as if we’d immediately written every thing in C):

divQR

The examples we’ve given listed below are very small ones. However the entire construction for exterior perform calls that’s now in Model 13.1 is about 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 usually wanted in additional advanced conditions is extra subtle reminiscence administration, and our new “Managed” kind gives a handy and streamlined approach to do that.

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

cf = FunctionCompile

Operating the perform successfully “leaks” reminiscence:

With

However now outline a model of the perform through which the array is “managed”:

cf = FunctionCompile

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

With

Straight Compiling Perform Definitions

When you have an express pure perform (Perform[...]) you should utilize FunctionCompile to provide a compiled model of it. However what you probably have a perform that’s outlined utilizing downvalues, as in:

fac

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

dec = FunctionDeclaration

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

cfac = FunctionCompile

The compiled perform runs considerably quicker than the peculiar symbolic definition:

RepeatedTiming
RepeatedTiming

The power 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 immediately, and in compiled code.

Manipulating Expressions in Compiled Code

An early focus of the Wolfram Language compiler is dealing with low-level “machine” sorts, 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" kind for representing any Wolfram Language expression inside compiled code.

Whenever you use one thing like FunctionCompile, it’ll 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 kind "InertExpression". This units up a compiled perform that constructs an expression (implicitly of kind "InertExpression"):

cf = FunctionCompile

Evaluating the perform constructs after which returns the expression:

cf

Usually, throughout the compiler, an "InertExpression" object might be handled in a purely structural approach, with none analysis (and, sure, it’s intently associated to the "ExprStruct" information construction). However generally 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:

cf

The power to deal with expressions immediately within the compiler may appear to be some type of element. Nevertheless it’s really vastly vital in opening up potentialities for future growth 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 means 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 every thing else within the Wolfram Language.

And Nonetheless Extra…

Even past all of the issues I’ve mentioned up to now, there are all kinds 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:

RandomInstance

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

GraphProduct

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

BuckyballGraph

You may combine pictures immediately into Graphics (and Graphics3D):

Graphics

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

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

FindClusters now permits 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 “fashionable” 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 13.0 we’re starting the introduction of structured matrices which might be saved and computed with in particular, optimized methods. Examples embody PermutationMatrix and LowerTriangularMatrix.

We’ve had intensive 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 means 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 customary textual content types like daring and italic eternally, however now we now have a normal approach to deal with struckthrough textual content as effectively:

Style

Along with all these “inside-the-system” enhancements, we’ve additionally completed making it doable 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, quite than simply on a per-user foundation.

So—as of as we speak, documentation or not—you may get Model 13.1 in your pc. 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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