NewtonRaphson Weeds: This is why your trial root sometimes goes off into the rough.
You have an image and wish to fill it with something. If you are new to G`MIC and coming from a paint program like GIMP, then your first thought after encountering this command might possibly be: “Ah HA! the fill command! At last I can fill circles and squares and things with red!”
Well yes. And no. If paint buckets come to mind, then go over to fill_color, next aisle down, the one labeled “Color Manipulation.” That’s OK. Lots of people get lost in here first time through. G’MIC is a big place.
That isn’t to say you can’t use fill to make an image red. It’s just that you will have do it with a little finesse:
gmic input 2,2,1,3 fill. 220,220,220,220,20,20,20,20,60,60,60,60
The command’s numeric parameters inject a carefully arranged data stream into a freshly minted 2 X 2 pixel, single slice, three channel image. The data are so arranged as to fill this tiny image with the X11 color Crimson: 0xdc143c, or decimal: 220, 20, 60. The individual numeric elements take their seats in the target image by first filling channel zero, row zero, starting at column zero, then going to column one, then two and so on. You have to arrange your data stream in a like manner; fill doesn’t know about any other ordering schemes.
G`MIC geeks may find this excessively verbose, preferring:
gmic '(220,220;220,220^20,20;20,20^60,60;60,60)'
which is terser, but dispenses with fill entirely. Instead, the second example implicitly invokes the input command. The input specification to which that command adheres uses commas, semicolons, carets, slashes and parentheses to describe image topology. This makes the terser example somewhat more informative at a glance.
However, since fill is the topic of this article and we maintain a sentimental attachment to our topics, we’ll just send those interested in terse efficiency over to the aisle marked “InputOutput”. They’re having a blueplate special on input streams today. fill itself does not use input's file specification. Its worldview is limited to ordered, one dimensional streams.
You may gather from this that fill is a fairly lowlevel command. Indeed. It fills images with pels. That’s it.
‘Pel’, you ask? It is shorthand for Picture Element, an indivisible quantum of an image. Could be a byte long. Could be longer. Could even be shorter. Might be a float, or an unsigned integer – or even a signed integer. The particulars of a pel’s organization stems from the image format. But a pel is not a pixel. Pixels are vectorlike; they have components. Pels are scalars, Pixels are made of pels. Pels are components of pixels. Careless scriveners may put ‘pixel’ when they mean ‘pel’ but we make the distinction here because the essential item that fill manipulates are pels. An RGBA pixel has four pels, and, we should note, fill could touch upon that particular kind of pixel as many as four distinct times while filling an image. At its most basic, then, fill pours pels into images from some notional stream.
The notional stream can be specified in three ways:
Here are these choices expressed syntactically:
fill
The easiest streaming method to grasp is probably not the most practical – though there are exceptions to this. You can just stream commaseparated pels into an image, an openended argument list.
Notice that, in contrast to the image specification streams used by input, there are no structure markers – No opening or closing parentheses, commas do not denote columns – they just separate pels, the semicolons that separate successive columns into rows, the carets which separate a plane of rows into channels, the slashes which separate a series of channels into slices – none of those appear in fill arguments. Nada. Ziltch. They aren’t needed. Images already have a builtin structure of slices, channels, rows and columns. fill just – well – fills them, following the topology of the target image.
In light of this, order your data stream according to the dimensions of the target image. The largest (implicit) divisions in the pel stream are image slices. The fill command fills the first slice of a selected image entirely before starting on the second and subsequent slices, should they exist. Within a slice, fill populates channel zero completely before beginning on channel one and subsequent channels, if they exist. Within a channel, fill populates the top image row completely before the second and subsequent rows. Finally, fill populates rows from column zero through to column w – 1, with w, the width of the image. So you block out your data first by slices, within slices by channels, within channels, by rows. As a consequence of this, the pels that make up a multichannel pixel are separated by a stride. For an image in the RGB color space, the red component is in channel zero. w×h pels downstream comes the green component. w×h pels further downstream comes the blue component. Revisit our initial example, above, for a tiny stride example.
Data streams need not be properly sized, which gives rise to a class of pattern generation. If fill runs out of a source stream while there is still space left in the target image, it starts at the beginning of the source stream again. fill loops over streams as many times as is necessary to fill a particular image. Once the target image is filled, the command discards the remaining excess.
So to make any one of a bazillion or so herringbone patterns, have some fun with discrete math and clock arithmetic:
gmic \ input 20,20,1,3 \ fill. 205,127,63,91,31,16,8 \ resize. 150,150,[1],[1],1 \ output. herr_00.png 

gmic \ depth='{2^161}' \ input 17,17,1,3 \ fill. 189,129,6,207,84,104,239,153,6,78,85 \ resize. 300,300,[1],[1],5 \ normalize 0,'{$depth}' \ output. herr_01.png 
Tweak the numbers following the fill command. Add a few. Take a few away. The game is to choose pel streams with lengths that do not evenly divide the capacity of selected images, or which do not evenly divide various image dimensions. Pel streams with lengths equal to prime numbers furnish a neverending source of delight, should it be your wont for that kind of herringbone pattern.
In these examples, we use resize with different interpolation schemes to upscale and filter the herringbone pattern for additional effect. In the last example, we renormalize the image immediately before output to ensure that the pels range within the limits imposed by the 16 bit/channel Portable Network Graphics format, since the cubic interpolation scheme we specified for resize can carry values outside this range. See Images As Datasets elsewhere on this site for a discussion on why this step is sometimes necessary.
The fill command can populate one image with the contents of another. Instead of a stream of pels, we pass an image selection parameter in square brackets. The images in the selection becomes the fill source; Targets live in the selector decorator adorning the fill command.
Beware: fill is not a cheap version of array. The command effectively flattens the source into a one dimensional stream of pels. The very first pel in the source stream corresponds to column zero, row zero, channel 0 of slice 0 of the target image. The fill command then increments, first by columns, then by rows, then channels, then slices, following the same hierarchy it imposes on literal pel streams. Indeed, as far as fill is concerned, an image is just a convenient way to specify a pel stream. Unless the source and target images happen to have identical dimensions, fill will not preserve image coherency. For that, investigate array.
This notwithstanding, we can still play herringbone games by building a herringbone pattern in stages:
small_herringbone.png big_herringbone.png 
gmic \ input 13,13,1,3 \ fill. 65,71,36,154,27,84,39,233,146,82 \ input 256,256,1,3 \ fill. [2] \ output. big_herringbone.png \ output.. small_herringbone.png 
This example illustrates using fill with a pel stream, making an initial small herringbone image using a handwritten pel stream, then making a large herringbone using the initial pattern as an image source.
Here is an example you don’t want to do – unless you are being devious.
face.png 
face.png is a 72 X 72 pixel image of a happy face with a nice transparent background – it has red, green, blue and alpha channels. 
facetimesfour.png 
When the selected target image has dimensions 144 X 144, the fill command will not make a 2 X 2 array of four 72 X 72 happy faces. That is, instead of the image on the left, the commands: gmic \ input face.png \ input 144,144,1,4 \ fill. [2] \ output funny.png 
funny_faces.png 
produces this strange image, which is probably not what we want. So how did we go off the rails? 
We’ve heard it said that fill is a lowlevel command. It just streams pels into an image. That's really all it does. Essentially fill turns source images into pel streams. Thus, the four channel, 72 X 72 happy face becomes a linear stream of 82,944 pels. Having flattened its source image, fill then populates its selected 144 X 144 X 1 X 4 target image using the precise same hierarchy we stated before: it completely fills the first slice of the target before commencing the second, and within a slice, it completely fills channel zero before commencing on channel one, and so on.
In particular, the fill command took the first 72 pel row of channel 0 of the source image and filled just the left half of the first row of the target image, which is, at 144 pels, twice as long. The second 72 pel row of the source image now fills the right half of the first target row. And so on. That explains the pair of flattened happy faces: the left half of the target image consists of the even rows; the right half contains the odd rows of the source image.
One quarter of the way toward filling the target’s channel zero, the source channel zero taps out. No matter. The fill command just begins streaming in the source’s second channel, copying those second channel pels into the target’s still not entirely filled first channel. Since the dimensions of the target image are exactly twice that of the source, with the area of each target channel exactly four times that of the source, the entire happy face source image exactly fills just channel zero of the target. The source has been entirely used up. Still no matter. The fill command goes back to the beginning of its source and commences to fill channel one of the target image through a second pass of the source.
At the end of the day, the source image is copied four times, each time filling just one channel of the target. That makes the four channels of the target image exact duplicates – hence, the target image is a study in gray of flattened happy faces. Could be they are not that happy.
The fill command is not a blitter – a block copier. It does not think in rectangles, just streams. To do what you really want to do, try array instead:
gmic input face.png array. 2,2,2 output. facetimesfour.png
The third way to stream pels is really quite different. The source – well, there is hardly a source in the literal sense. The source is what an algorithm encoded in a math expression produces when given a pel. The math expression is a pel processor written in G’MIC’s math scripting language.
fill, along with its cousin input, both furnish hooks for pel processors, enabling finegrained computations at the pel level. When the G’MIC command processor discovers that fill has been given a math expression as an argument, it suspends its own processing and invokes the math expression parser. That math expression now sources a virtual stream of pels. It is invoked once for every pel in the target image and the expression can do whatever it wants to produce that pel  accessing in any manner the current or neighboring pels in the target image, pels in other images on the list, or none of these sources at all, as in the first example following, which simply pulls random numbers out of the ether without regard to the content of the target or any other image on the list. The final computation of the math expression, whatever that result may be, becomes the new value of the pel, an implicit assignment.
Here are a few examples:
colornoise.png (exerpt) 
The pel processor is one character, ‘u’, which is a math expression hooked into a random number generator. It’s invoked 7,415,193,600 times, once for each pel in the image. On each invocation, it produces a random value between the limits zero and one inclusive. Since this is the last computation this processor performs (as well as its first), G’MIC implicitly assigns the resulting random value to the pel. A one character script, and you’ve written a generator for an Ultra High Definition video stream that, for ten seconds, emulates a 1967 color television set tuned between stations. I bet if you thought about it for any length of time, you’d wind up writing a much more involved script for this very same FX, assuming you wanted such a thing. 
bendexter.png 
Draws a one pixel bend dexter. Pels on the diagonal have equal column and row coordinates, so, for those elements, the math expression returns 1 (True), otherwise it returns 0 (False). That is, the pel is set with the result of the comparison. Without further commands, the dynamic range of the image is from zero to one, which may not be what you want. x and y are two of many predefined variables available to math expressions, these providing the current pel's column (x) and row (y) coordinates. Similarly, there are current slice (z) and channel (c) coordinates that are predefined, while i furnishes the value of the pel addressed by these four coordinates. There are many other such variables – too many to list here. Mathematical Expressions of the GMIC Handbook furnishes a complete listing. 
bendsinister.png 
Draws a one pixel bend sinister by means almost identical to the previous example, with just a variation in logic. h and its companion w are more predefined variables which tell the pel processor about the height and width of the current image. 
bendboth.png 
The expectation may be to draw bend dexter and bend sinister by combining the two previous means. To this end, the math expression has two statements, separated by a semicolon. Combining the logic statements, one may suppose, combines the image. Alas, the results do not bear this reasoning out. G'MIC implicitly assigns the results of the last executed statement in the script. The last executed statement is the comparison of the current column coordinate to the height of the image, less the current row coordinate. That means sinister wins over dexter every time. Better luck next season. 
xelevencrimson.png 
Yet Another Way To Fill An Image With X11 Crimson. With newer releases of G`MIC (2.0.x) math expressions acquired Vectors and Matrices. Math expressions that return vectors, such as the minimalist one here, set all of the pels of the pixel in one fell swoop. The number of iterations drop as well. Math expressions which return vectors equal to the channel count iterate over the entire image once – not once per channel. They are true 'pixel processors.' If the length of the return vector does not equal the channel count of the image, G`MIC will fill missing channels with '0' or ignore excess channels, as the case may be. Write vectors as comma separated items enclosed in square brackets. With vectors come a small host of vector and matix functions. G`MIC treats complex numbers as two element vectors, thus 3.5 + 1.27j looks like [3.5, 1.27] and cabs([3.5, 1.27]) finds the hypotenuse (i.e., 'complex absolute value'): 3.7233, or there abouts, so if you have been writing sqrt(a^2+b^2) a lot, you may find cabs([a,b]) a terser equivalent. Skip forward to the closing example of filling images with NewtonRaphson basins of attraction for more working examples of vectors, and vectors as complex numbers. 
Obtain ramps by the simple expedient of citing a current coordinate, which G'MIC implicitly assigns to the current pel. The low column coordinates begin at zero and end with w – 1 – dark to light. The dynamic range of the image depends on image geometry, which rarely reconciles with the dynamic range of image formats. Pel processors return values to images exactly as computed, no more, no less. It is on your watch to ensure that the image works within the native dynamic range of your chosen output format. In many cases, the results of simple expedients can be set to rights by the judicious application of normalize See also Images as Datasets for the usual cautionaries. 

Obtain a top to bottom ramp by the same expedience as the prior example, just substituting the current row coordinate, y, for the current column coordinate, x. 

Ramps at arbitrary angles may be obtained by utilizing other math expression features. A pel processor may have multiple statements, these separated by semicolons. In the last example, the first statement converts its argument, the desired orientation of the ramp, from degrees to radians. The second statement references the converted argument, along with current column and row coordinates, to produce a pel value. The math parser then assigns the evaluation of this second statement to the pel addressed by the current values of x, y, c, and z. So far, we've seen just predefined variables, such as pi, the ratio of the circumference of a circle to its diameter. In this expression, we have defined our own, harnessing two species of variables, in fact. The native variant is rad, created in the math expression and then referenced in the math expression. Being native to the math expression, It does not take a $ prefix; adding one may or may not raise a syntax error. The difficulty (see below) is with how the command processor preprocesses math expressions, substituting the contents of command level variables appearing in math expressions for the variables themselves. Since an inadvertent $ prefix on a math expression variable most likely references a neverbefore seen command level variable, G'MIC completes the substitution by supplying the default values for newlyminted command level variables: empty strings, which in numerical contexts are zero. Sosubstituted, the resulting math expression is syntactically correct, more often than not, but likely not behaving as you want. The scope of these math expression variables is limited just to the math expression itself – they furnish a local, nonpersistent scratchpad that disappears when the math expression ceases execution, and their values do not persist from one invocation of the pel processor to the next. The second variant is an import from the command environment. Command level variables like ang, above, can be referenced in the math expression when adorned with their $ prefix. In fact, behind the scenes, the command processor performs a substitution pass on math expressions, looking for variables with $ prefixes and substituting for these their current values. Consequently, command level variables turn into numeric constants before the math expression is invoked. Command level variables, then, do persist between invocations of pel processors, but pel processors have no real knowledge of their existence, because they manifest themselves as numeric constants within math expressions, and cannot harness them as persistent storage. 

turb.png turb_xor.png 
This math expression harnesses two of the many builtin functions available to such. The builtin xor(a,b) performs a bitwise exclusive OR operation on its arguments, rounding them down to the nearest integer, if necessary. i(x,y,z,c,interpolation_type, boundary_conditions) samples the image at a specified set of coordinates, which need not be integral and need not be on the image. The coordinates may have fractional parts, implying subpixel sampling. In this case, the interpolation_type flag chooses between nearest neighbor and linear interpolation. The boundary_condition flag furnishes policy when the sampling coordinates are offimage. Images Have Edges, Now What? reviews the various types, but briefly: Dirichlet (0) means all offimage samples are black, by definition. Neumann (1) extends the values of the images out to infinity; offimage samples reflect the values of those pixels on the nearest edge. Cyclic (2) treat images as torii; offimage samples wrap around to the far edge. Mirror (3) furnishes offimage samples by reflecting the offimage position back into the interior of the image. For terseness, arguments may be omitted, counting from the right, so i(x,y) references a pel at column x and row y. Omitted, slice (z) and channel (c) default to zero, nearest neighbor interpolation and Dirichlet conventions at the boundary. Math expressions are not limited to their operand image. Any predefined variable may be adorned with a #<image selector> suffix to reference any other image on the list. The numbering conventions follow those of the command processor's selector. Positive indices count from the beginning of the list; negative indices count from the end of the list. On a list with four images, iM#1 refers to the maximum intensity of the second image on the list, counting from the beginning (the image list follows the zeroindexing convention). iM#3 references the same image, counting from the end of the list. This convention extends to the image sampling function i() where the initial argument can be a selector. That is the key facet of this example, which samples each pel of a two channel image other than the operand, containing synthesized turbulence, and XOR's them together, producing the corresponding pel of the operand image. 
This has been kind of a Whitman Sampler of G'MIC math expressions. Since this article is about the fill command, it is not a propitious place to lay out a fullblown math expression tutorial – for such would overwhelm the fill command itself. The definitive and complete, if terse, reference to math expressions remains the Mathematical Expressions section in the GMIC Handbook. Everything's there, sans examples and artfully illustrative errors. The key takeaway here is that the fill command lets you hook up a pel or pixel level processor to your image, so you can operate on the raster itself.
In the runup to G‘MIC 2.0, released last November, 2016, the features and capabilities of the math processor grew considerably. The examples presented thus far, one might say, are ‘classic’ G‘MIC math expressions – capable of a series of operations but lacking looping constructs, dynamic variables, being somewhat sensitive to whitespace and limited to returning just scalars. In the firstgeneration math processor, an RGBA pixel could be visited by a pel processor as often as four times, since the processor could alter only one pel at a time.
Most of these limitations have been lifted. Math expression can now utilize control structures, query other images in the pipeline, support vector data types as well as scalars, so that an entire pixel can be a return value, not just one pel. While it does not have a direct bearing on semantic content, math expressions can now also be written with a liberal amount of white space, so that the structure of the math expression is clear.
To illustrate some of the newer features, we've written a G‘MIC command that plots the basins of attraction that arise from NewtonRaphson root polishing. This computation is inherently iterative, as the only practical way to find the roots of higher degree polynomials is to start with a trial value and, exploiting certain properties of polynomials and their first derivatives, iteratively refine intermediary trials. Iteration was not something that could be built into a math expression until recently.
The NewtonRaphson iteration:
says we may find a better trial value for a root λ_{k} of polynomial f(z), z_{i+1}, by subtracting a “correction factor” from a current trial value, z_{i}. We obtain the correction factor by evaluating the polynomial and its first derivative at z_{i} and taking the ratio of the evaluation and derivative. Graphically, we extend the tangent of the function at the evaluation point to the parameter axis, viz:
In this illustration, f(z) is almost linear in the neighborhood of the root and it is not hard to imagine how extending the tangent from a pretty good guess gives us a new guess even closer to the root. Clearly, evaluating again at z_{i+1}, and extending a new tangent at that point will walk us ever closer to the root λ_{k} and it will only be a matter of taste to decide, at some point, that the trial value is close enough.
But what is a pretty good guess? One way to answer that question is to draw a map: newtonraphson, listed below, when given a set of roots, paints such a map on the complex plane, using a grid of values (say, the position of pixels) as initial trial versions. It performs a series of NewtonRaphson iterations on each grid point until an intermediary is really close to one of the given roots. When an iteration finishes, newtonraphson reports the result by plotting a twochannel pixel in a new image. Channel zero is an index uniquely identifying one of the given roots. Channel one holds the iterations it took to arrive at the root from the initial grid point. newtonraphson plots the pixel in the same position as the grid point, essentially reporting how that point behaves under NewtonRaphson. Stand back from this image, squint, and we should see the aggregate of grid points which converge to particular roots – the basins of attraction.
So a basin of attraction is a set of complex numbers which converge to a particular root under a NewtonRaphson iteration. It happens that the division of the complex plane into the various basins of attraction for a polynomial form striking fractal boundaries. In fact, the fractal boundaries separating the basins of attraction underlie why NewtonRaphson root polishing can sometimes be illbehaved. Pick an initial trial in the midsts of the fractal territories, and convergence, at least initially, might not be that convergent in the early part. As we will see later, the initial trial may not converge at all.
If you are not familiar with how these basins of attraction can be imaged, read the Wikipedia article on Newton fractals, and, in particular, see Simon Tatham’ page, Fractals derived from NewtonRaphson Iteration. We stole borrowed quite a few pointers from Mr. Tatham's page.
To begin, include this G‘MIC command in your Mac OSX, Linux: $HOME/.gmic file, or, for Microsoft Windows your %APP_DATA%/<user>.gmic file, where <user> stands in for your login name:
#@gmic #@cli newtonraphson : [roots],_center_x,center_y,_mag,_maxiter #@cli : Draw on the selected images the basins of attraction of the polynomial #@cli : roots given in the argument image using NewtonRaphson root #@cli : polishing. Argument image is N,1,1,2  an N vector for N, complex, #@cli : number of roots. Channel zero contains real, and channel one #@cli : imaginary components. Default values: 'center_x = 0', 'center_y = 0', #@cli : 'magnification = 1', 'maximum_iterations = 100' newtonraphson : check ${is_image_arg\ $1} skip ${2=0},${3=0},${4=1},${5=100}
verbose 
sig=s
if {$!<2}
sig=
endif
verbose +
echo[^1] "Use NewtonRaphson root polishing on image"$sig$?" to paint basins of attraction associated with the roots given by $1."
verbose 
ppu=255.5
ox=$2
oy=$3
mag=$4
lim=$5
cx={$oxw/(2*$ppu*$mag)}
cy={$oy+h/(2*$ppu*$mag)}
err=1e10
pass$1 1
repeat {$!1}
local[$>,1]
fill[0] "Q = vectorw#1();
ic = 0;
zz = ["$cx"+x/("$ppu"*"$mag"),"$cy"y/("$ppu"*"$mag)"];
dowhile(
acc = [0,0];
for(
k=0, k < w#1, k++,
Q[k] = cabs(zzI(#1,k,0));
acc += [1,0]//(zzI(#1,k,0))
);
dif = [1,0]//acc;ic+=1;
zz = zz  dif,
cabs(dif) > "$err" && ic < "$lim"
);
if(
ic < "$lim",
[argmin(Q),ic],
[w#  1,"$lim"]
)"
endlocal
done
rm.
verbose +
If you look at the math expression which we pass to fill, you might wonder how it is we're doing NewtonRaphson. We're in the business of plotting basins of attraction, not finding roots – in fact, we start from known roots. Since we have the roots, we can rephrase the NewtonRaphson process to suit our convenience. And it is convenient for us to write the polynomial in factored form, each factor being a difference between the trial value, z_{i}, and a member of the set of given roots, {λ_{n}}:
$$f({z}_{i})=\prod _{k=0}^{n1}{z}_{i}{\mathrm{\lambda}}_{k}$$
$${f}^{\text{'}}({z}_{i})=\sum _{j=0}^{n1}\prod _{k=0}^{n\mathrm{1,}j\ne k}({z}_{i}{\mathrm{\lambda}}_{k})$$
$$\frac{{f}^{\text{'}}({z}_{i})}{f({z}_{i})}=\sum _{k=0}^{n1}\frac{1}{({z}_{i}{\mathrm{\lambda}}_{k})}$$
$${z}_{i+1}={z}_{i}\frac{f({z}_{i})}{{f}^{\text{'}}({z}_{i})}$$→ $${z}_{i+1}={z}_{i}\frac{1}{\sum _{k=0}^{n1}\frac{1}{({z}_{i}{\mathrm{\lambda}}_{k})}}$$
In this form, the denominator of the correction factor has an interesting structure. It is a sum of reciprocals. Should, at some iteration i, the trial value z_{i} approaches any one of the roots λ_{k}, the difference between the trial and that root commences to vanish. That drives the corresponding reciprocal to a very large value. The sum of reciprocal differences itself grows very large, being dominated by that one exploding term, so that the correction that is applied to the trial value for the next iteration becomes vanishingly small. On that condition, we exit the dowhile() loop – as we have found a pretty near enough approximation to a root.
Upon that exit, we prudently check if the overall iteration count has exceeded the limit set in $lim. If it does, we return a failure vector. The failure vector uses the root count, n, as an index, which is not assigned to any root. These are indexed from 0 to n – 1. For an iteration count, the failure vector contains the value of $lim. Downstream rendering can mark pixels with these two special channel values however they see fit.
Otherwise, we return a normal success vector. To correctly identify which root the trial value approximates, we keep off to one side a check vector, Q, with components (z_{i} – λ_{k} ). Applying argmin() to this vector returns the index of the smallest component in the vector, identifying the root to which z_{i} converges. This index becomes the first component of the return vector. The number of iterations is recorded in the expression variable ic and becomes the second component of the return vector.
Why would this expression fail to converge? Our game relies on extending tangents to the parameter axis, then walking straight up until we hit the curve, our new evaluation point. Is it possible that this new evaluation point could be the same as an earlier trial value? Oh, yes, indeed, in a significant class of polynomials, there are honeypot traps for those taking the NewtonRaphson route: trial values z_{i} transform to z_{i – k}  a prior trial value tested k steps before the present trial. In that case, we are in a circular iteration of k steps – and we will never get out.
This maladay is manifested by socalled connected Fatou sets that can also appear in our maps. Entwined among the basins of attraction of every degree n polynomial, f_{n}(z) is a Fatou set of one lower degree. It is typically disconnected, the set forming Cantor dust., but can be connected. That is, the set is bounded by a connected Julia set which cleaves the complex plane into disjoint sets. One set, the trial values that are parts of honeypot loops, form the interiors of the Fatou sets. Trial values within these sets never converge on a root, they just boomerang around. With disconnected Fatou sets, convergence is almost always possible; it is vanishingly unlikely to choose a dust mote of Cantor dust and bounce around forever.
Failure of convergence because we're gooed to a honeypot is not catastrophic for our purposes. We are painters of basins of attraction, and, by extension, basins of nonattraction. If a trial value runs to the weeds, we just paint it another pretty color, an additional bit for the rikka.
The image that newtonraphson produces is not strikingly beautiful. It is intended to be rendered. By design, we separate computation and presentation into distinct tasks. A basic rendering script might employ map to associate colors to the rootassignment channel, then alter this coloration as a function of the iteration values. Here's a use of newtonraphson that goes along those lines:
$ gmic \ wp='{2^161}' \ input 300,300,1,2 \ input '(1,0.5,0.5^0.0,0.866025403784,0.866025403784)' \ newtonraphson[2] [1],0.75,0,3.5 \ rm. \ split. c \ input '(150,250,200^30,260,150^180,7,220)' \ map[3] [1] \ rm. \ mul. 1 \ normalize. 0,1 \ pow. 0.75 \ mul[2,1] \ normalize. 0,'{$wp}' \ output. nr_00.png
Here, we are depicting the basins of attraction for the cubic polynomial z^{3} – 1 which has three roots in the complex plane at $1+0j$ , $\frac{1}{2}+\sqrt{\frac{3}{4}}j$ and $\frac{1}{2}\sqrt{\frac{3}{4}}j$.
And this is the result:
Creating a map of how a grid of values behaves under a NewtonRaphson iteration was not possible with earlier versions of G'MIC; the basic "do until these trial values converges to a root", which lies at the heart of newtonraphson, requires a proximity test at the end of each iteration. Now the language offers dowhile(...) and for(...) for such openended, "until suchandso prevails" testing. dowhile(...) executes an iteration, then tests; for(...) tests, then executes an iteration. whiledo(...) is a convenience variant of for(...) with an argument list that mirrors whiledo(...).
With these, math expressions can peform conditional iteration, opening up new processing capabilities. To accommodate longer scripts, the math expression parser has a more flexible means of handling whitespace, so that expressions may be formatted in a way to illuminate their workflow.
Expanded vector and matrix datatypes turn math expressions into true pixel processors, as math expressions which return vectors can set the entire pixel in one atomic assignment. This reduces iteration counts as it is not necessary to iterate over individual channels.
The math expression control loop constructs are implemented as functions which take expressions as arguments and return the result of the last executing statement, or, in the case of for(...), the Not_A_Number token for the possibility that an initial terminating condition is positive at the outset, so that none of the other expression arguments execute. for(...) tests and executes; dowhile(...) executes and tests. whiledo(...) is a convenience function that, like for(...), tests and executes, but has an argument list that mirrors dowhile, and, in certain circumstances, expresses logic more clearly and is easier to set up, having two arguments instead of four. Like for(...), whiledo(...) can return NAN if its condition expression initally terminates without executing the procedure body at all.
If we take a deep breath and squint sideways, we see that the form of the fill command is no difference in quality than what has gone on before:
fill[<targets>] “<math_expression>”
which can be rendered at a slightly higher resolution:
fill[<targets>] “<math_partial_expression>”$<command_variable>“<mathpartial_expression…>”
Here, a doublequotation marks alert the math expression parser that white space may be used to "prettyformat" the math expression. Historically, G'MIC math expressions had all the charm of Perl Regular Expressions, which, from a casual glance, had the appearance of those final sputter of characters before communication lines drop. Double quoted math expressions may be broken into several lines, indented, and, with care, have white space inserted around operator tokens.
Doublequoted math expressions require special precautions when mixing with command level variables, as the sequence $<variable_name> will be protected from substitution when such appears within double quotes. Mixing in command level variables entails furnishing a closing quote to the partial math expression immediately preceding the command level variable, then an opening quote immediately following the command level variable. One can easily, and mistakenly, invert this context and think of the command line variable being quoted instead. When performing its substitutions, the command parser concatenates the substituted variables and the math subexpressions, making one complete math expression, which is then given to the math expression parser.
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Garry Osgood