Calculate Blaschke products

Description

This function calculates Blaschke products (https://en.wikipedia.org/wiki/Blaschke_product) for a complex number z given a sequence a of complex numbers inside the unit disk, which are the zeroes of the Blaschke product.

Usage

blaschkeProd(z, a)

Arguments

Details

A sequence of points a[n] located inside the unit disk satisfies the Blaschke condition, if sum[1:n] (1 - abs(a[n])) < Inf. For each element a != 0 of such a sequence, B(a, z) = abs(a)/a * (a - z)/(1 - conj(a) * z) can be calculated. For a = 0, B(a, z) = z. The Blaschke product B(z) results as B(z) = prod[1:n] (B(a[n], z)).

Value

The value of the Blaschke product at z.

See Also

Other maths: jacobiTheta(), juliaNormal(), mandelbrot()

Examples

# Generate random vector of 17 zeroes inside the unit disk
n <- 17
a <- complex(modulus = runif(n, 0, 1), argument = runif(n, 0, 2*pi))
# Portrait the Blaschke product
phasePortrait(blaschkeProd, moreArgs = list(a = a),
 xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
 nCores = 1) # Max. two cores on CRAN, not a limit for your use
 

Jacobi theta function

Description

Approximation of "the" Jacobi theta function using the first nn factors in its triple product version

Usage

jacobiTheta(z, tau, nn = 30L)

Arguments

Details

This function approximates the Jacobi theta function theta(z; tau) which is the sum of exp(pi*i*n^2*tau + 2*pi*i*n*z) for n in -Inf, Inf. It uses, however, the function's triple product representation. See https://en.wikipedia.org/wiki/Theta_function for details. This function has been implemented in C++, but it is only slightly faster than well-crafted R versions, because the calculation can be nicely vectorized in R.

Value

The value of the function for z and tau.

See Also

Other maths: blaschkeProd(), juliaNormal(), mandelbrot()

Examples

phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2-1/4),
pType = "p", xlim = c(-2, 2), ylim = c(-2, 2),
nCores = 1) # Max. two cores on CRAN, not a limit for your use
 
phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2-1/2),
pType = "p", xlim = c(-2, 2), ylim = c(-2, 2),
nCores = 1)
 
phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/3+1/3),
pType = "p", xlim = c(-2, 2), ylim = c(-2, 2),
nCores = 1)
 
phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/4+1/2),
pType = "p", xlim = c(-2, 2), ylim = c(-2, 2),
nCores = 1)
 

Julia iteration with a given number of steps

Description

This function is designed as the basis for visualizing normal Julia sets with phasePortrait . In contrast to usual visualizations of Julia sets, this requires coloring the actual member points of the set and not the points outside. Therefore, for numbers that can be identified as not being parts of the Julia set, this function returns NaN+NaNi. All other numbers are mapped to the complex value obtained after a user-defined number of iterations. This function has been implemented in C++; therefore it is fairly fast.

Usage

juliaNormal(z, c, R_esc, itDepth = 500L)

Arguments

Details

Normal Julia sets are closely related to the Mandelbrot set. A normal Julia set comprises all complex numbers z for which the following sequence is bounded for all n > 0: a[n+1] = a[n]^2 + c, starting with a[0] = z. The parameter c is a complex number, and the sequence is certainly unbounded if abs(a[]) >= R with R being an escape Radius which matches the inequality R^2 - R >= abs(c). As the visualization with this package gives interesting pictures (i.e. other than a blank screen) only for c which are elements of the Mandelbrot set, R = 2 is a good choice. For the author's taste, the Julia visualizations become most interesting for c located in the border zone of the Mandelbrot set.

Value

Either NaN+NaNi or the complex number obtained after itDepth iterations

See Also

Other fractals: mandelbrot()

Other maths: blaschkeProd(), jacobiTheta(), mandelbrot()

Examples

# This code visualizes a Julia set with some appeal (for the author's
# taste). Zoom in as you like by adjusting xlim and ylim.
phasePortrait(juliaNormal,
 moreArgs = list(c = -0.09 - 0.649i, R_esc = 2),
 xlim = c(-2, 2),
 ylim = c(-1.3, 1.3),
 hsvNaN = c(0, 0, 0),
 nCores = 1) # Max. two cores on CRAN, not a limit for your use
 

Mandelbrot iteration with a given number of steps

Description

This function is provided as a basis for visualizing the Mandelbrot set with phasePortrait . While usual visualizations color the points outside the Mandelbrot set dependent on the velocity of divergence, this function produces the information required for coloring the Mandelbrot set itself. For numbers that can be identified as not being elements of the Mandelbrot set, we obtain a NaN+NaNi value; for all other numbers, the function gives back the value after a user-defined number of iterations. The function has been implemented in C++; it runs fairly fast.

Usage

mandelbrot(z, itDepth = 500L)

Arguments

Details

The Mandelbrot set comprises all complex numbers z for which the sequence a[n+1] = a[n]^2 + z starting with a[0] = 0 remains bounded for all n > 0. This condition is certainly not true, if, at any time, abs(a[]) >= 2. The function mandelbrot performs the iteration for n = 0, ..., itDepth - 1 and permanently checks for abs(a[n+1]) >= 2. If this is the case, it stops the iteration and returns NaN+NaNi. In all other cases, it returns a[itDepth].

Value

Either NaN+NaNi or the complex number obtained after itDepth iterations

See Also

Other fractals: juliaNormal()

Other maths: blaschkeProd(), jacobiTheta(), juliaNormal()

Examples

# This code shows the famous Mandelbrot figure in total, just in the
# opposite way as usual: the Mandelbrot set itself is colored, while the
# points outside are uniformly black.
# Adjust xlim and ylim to zoom in wherever you like.
phasePortrait(mandelbrot,
 xlim = c(-2.3, 0.7),
 ylim = c(-1.2, 1.2),
 hsvNaN = c(0, 0, 0),
 nCores = 1) # Max. two cores on CRAN, not a limit for your use
 

Create phase portraits of complex functions

Description

phasePortrait makes phase portraits of functions in the complex number plane. It uses a technique often (but not quite correctly) called domain coloring (https://en.wikipedia.org/wiki/Domain_coloring). While many varieties of this technique exist, this book relates closely to the standards proposed by E. Wegert in his book Visual Complex Functions (Wegert 2012). In a nutshell, the argument (Arg ) of any complex function value is displayed as a color from the chromatic circle. The fundamental colors red, green, and blue relate to the arguments (angles) of 0, 2/3pi, and 4/3pi (with smooth color transitions in between), respectively. Options for displaying the modulus (Mod ) of the complex values and additional reference lines for the argument are available. This function is designed for being used inside the framework of R base graphics. It makes use of parallel computing, and depending on the desired resolution it may create extensive sets of large temporary files (see Details and Examples).

Usage

phasePortrait(
 FUN,
 moreArgs = NULL,
 xlim,
 ylim,
 invertFlip = FALSE,
 res = 150,
 blockSizePx = 2250000,
 tempDir = NULL,
 nCores = max(1, parallel::detectCores() - 1),
 pType = "pma",
 pi2Div = 9,
 logBase = exp(2 * pi/pi2Div),
 argOffset = 0,
 darkestShade = 0.1,
 lambda = 7,
 gamma = 0.9,
 stdSaturation = 0.8,
 hsvNaN = c(0, 0, 0.5),
 asp = 1,
 deleteTempFiles = TRUE,
 noScreenDevice = FALSE,
 autoDereg = FALSE,
 verbose = TRUE,
 ...
)

Arguments

Details

This function is intended to be used inside the framework of R base graphics. It plots into the active open graphics device where it will display the phase plot of a user defined function as a raster image. If no graphics device is open when called, the function will plot into the default graphics device. This principle allows to utilize the full functionality of R base graphics. All graphics parameters (par ) can be freely set and the function phasePortrait accepts all parameters that can be passed to the plot.default function. This allows all kinds of plots - from scientific representations with annotated axes and auxiliary lines, notation, etc. to poster-like artistic pictures.

Mode of operation

After being called, phasePortrait gets the size in inch of the plot region of the graphics device it is plotting into. With the parameter res which is the desired plot resolution in dpi, the horizontal and vertical number of pixels is known. As xlim and ylim are provided by the user, each pixel can be attributed a complex number z from the complex plane. In that way a two-dimensional array is built, where each cell represents a point of the complex plane, containing the corresponding complex number z. This array is set up in horizontal strips (i.e. split along the imaginary axis), each strip containing approximately blockSizePx pixels. In a parallel computing loop, the strips are constructed, saved as temporary files and immediately deleted from the RAM in order to avoid memory overflow. After that, the strips are sequentially loaded and subdivided into a number of chunks that corresponds to the number of registered parallel workers (parameter nCores). By parallely processing each chunk, the function f(z) defined by the user in the argument FUN is applied to each cell of the strip. This results in an array of function values that has exactly the same size as the original strip. The new array is saved as a temporary file, the RAM is cleared, and the next strip is loaded. This continues until all strips are processed. In a similar way, all strips containing the function values are loaded sequentially, and in a parallel process the complex values are translated into colors which are stored in a raster object. While the strips are deleted from the RAM after processing, the color values obtained from each new strip are appended to the color raster. After all strips are processed, the raster is plotted into the plot region of the graphics device. If not explicitly defined otherwise by the user, all temporary files are deleted after that.

Temporary file system

By default, the above-mentioned temporary files are deleted after use. This will not happen, if the parameter deleteTempFiles is set to FALSE or if phasePortrait does not terminate properly. In both cases, you will find the files in the directory specified with the parameter tempDir. These files are .RData files, each one contains a two-dimensional array of complex numbers. The file names follow a strict convention, see the following examples:

0001zmat2238046385.RData
0001wmat2238046385.RData

Both names begin with '0001', indicating that the array's top line is the first line of the whole clipping of the complex number plane where the phase portrait relates to. The array which follows below can e.g. begin with a number like '0470', indicating that its first line is line number 470 of the whole clipping. The number of digits for these line numbers is not fixed. It is determined by the greatest number required. Numbers with less digits are zero-padded. The second part of the file name is either zmat or wmat. The former indicates an array whose cells contain untransformed numbers of the complex number plane. The latter contains the values obtained from applying the function of interest to the first array. Thus, cells at the same position in both arrays exactly relate to each other. The third part of the file names is a ten-digit integer. This is a random number which all temporary files stemming from the same call of phasePortrait have in common. This guarantees that no temporary files will be confounded by the function, even if undeleted temporary files from previous runs are still present.

HSV color model

For color-coding the argument of a complex number, phasePortrait uses the hsv (hue, saturation, value) color model. Hereby, the argument is mapped to a position on the chromatic circle, where the fundamental colors red, green, and blue relate to the arguments (angles) of 0, 2/3pi, and 4/3pi, respectively. This affects only the hue component of the color model. The value component is used for shading modulus and/or argument zones. The saturation component for all colors can be defined with the parameter stdSaturation.

Zone definitions and shading

In addition to displaying colors for the arguments of complex numbers, zones for the modulus and/or the argument are shaded for pType other than "p". The modulus zones are defined in a way that each zone covers moduli whose logarithms to the base logBase have the same integer part. Thus, from the lower edge of one modulus zone to its upper edge, the modulus multiplies with the value of logBase. The shading of a modulus zone depends on the fractional parts x of the above-mentioned logarithms, which cover the interval [0, 1[. This translates into the value component v of the hsv color model as follows:

v = darkestShade + (1 - darkestShade) * x^(1/lambda)

where darkestShade and lambda are parameters that can be defined by the user. Modifying the parameters lambda and darkestShade is useful for adjusting contrasts in the phase portraits. The argument zone definition is somewhat simpler: Each zone covers an angle domain of 2*pi / pi2Div, the "zero reference" for all zones being argOffset. The angle domain of one zone is linearly mapped to a value x from the range [0, 1[. The value component of the color to be displayed is calculated as a function of x with the same equation as shown above. In case the user has chosen pType = "pma", x-values xMod and xArg are calculated separately for the modulus and the argument, respectively. They are transformed into preliminary v-values as follows:

vMod = xMod^(1/lambda) and vArg = xArg^(1/lambda)

From these, the final v value results as

v = darkestShade + (1-darkestShade) * (gamma * vMod * vArg + (1-gamma) * (1 - (1-vMod) * (1-vArg)))

The parameter gamma (range [0, 1]) determines they way how vMod and vArg are combined. The closer gamma is to one, the more the smaller of both values will dominate the outcome and vice versa.

Defining more complicated functions as strings with vapply

You might want to write and use functions which require more code than a single statement like (z-3)^2+1i*z. As mentioned in the description of the parameter FUN, we recommend to define such functions as separate objects and hand them over as such. There might be, however, cases, where it is more convenient, to define a function as a single long string, and pass this string to FUN. In order to make this work, vapply should be be used for wrapping the actual code of the function. This is probably not the use of vapply intended by its developers, but it works nicely and performs well. The character string has to have the following structure "vapply(z, function(z, other arguments if required) {define function code in here}, define other arguments here, FUN.VALUE = complex(1))". See examples.

References

Wegert E (2012). Visual Complex Functions. An Introduction with Phase Portraits. Springer, Basel Heidelberg New York Dordrecht London. ISBN 978-3-0348-0179-9.

Examples

# Map the complex plane on itself
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait("z", xlim = c(-2, 2), ylim = c(-2, 2),
 xlab = "real", ylab = "imaginary",
 verbose = FALSE, # Suppress progress messages
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
# A rational function
# x11(width = 10, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait("(2-z)^2*(-1i+z)^3*(4-3i-z)/((2+2i+z)^4)",
 xlim = c(-8, 8), ylim = c(-6.3, 4.3),
 xlab = "real", ylab = "imaginary",
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Different pType options by example of the sine function.
# Note the different equivalent definitions of the sine
# function in the calls to phasePortrait
# x11(width = 9, height = 9) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mfrow = c(2, 2), mar = c(2.1, 2.1, 2.1, 2.1))
phasePortrait("sin(z)", xlim = c(-pi, pi), ylim = c(-pi, pi),
 pType = "p", main = "pType = 'p'", axes = FALSE,
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
phasePortrait("sin(z)", xlim = c(-pi, pi), ylim = c(-pi, pi),
 pType = "pm", main = "pType = 'pm'", axes = FALSE,
 nCores = 2)
 
phasePortrait("sin", xlim = c(-pi, pi), ylim = c(-pi, pi),
 pType = "pa", main = "pType = 'pa'", axes = FALSE,
 nCores = 2)
 
phasePortrait(sin, xlim = c(-pi, pi), ylim = c(-pi, pi),
 pType = "pma", main = "pType = 'pma'", axes = FALSE,
 nCores = 2)
 
par(op)
# I called this one 'nuclear fusion'
# x11(width = 16/9*8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mar = c(0, 0, 0, 0), omi = c(0.2, 0.2, 0.2, 0.2), bg = "black")
phasePortrait("cos((z + 1/z)/(1i/2 * (z-1)^10))",
 xlim = 16/9*c(-2, 2), ylim = c(-2, 2),
 axes = FALSE, xaxs = "i", yaxs = "i",
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
par(op)
# Passing function objects to phasePortrait:
# Two mathematical celebrities - Riemann's zeta function
# and the gamma function, both from the package pracma.
# R's built-in gamma is not useful, as it does not work
# with complex input values.
if(requireNamespace("pracma", quietly = TRUE)) {
# x11(width = 16, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mfrow = c(1, 2))
phasePortrait(pracma::zeta, xlim = c(-35, 15), ylim = c(-25, 25),
 xlab = "real", ylab = "imaginary",
 main = expression(zeta(z)), cex.main = 2,
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
phasePortrait(pracma::gammaz, xlim = c(-10, 10), ylim = c(-10, 10),
 xlab = "real", ylab = "imaginary",
 main = expression(Gamma(z)), cex.main = 2,
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
}
# Using vapply for defining a whole function as a string.
# This is a Blaschke product with a sequence a of twenty numbers.
# See the documentation of the function vector2String for a more
# convenient space-saving definition of a.
# But note that a C++ version of the Blaschke product is available
# in this package (function blaschkeProd()).
# x11(width = 10, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait("vapply(z, function(z, a) {
 fct <- ifelse(abs(a) != 0,
 abs(a)/a * (a-z)/(1-Conj(a)*z), z)
 return(prod(fct))
 },
 a = c(0.12152611+0.06171533i, 0.53730315+0.32797530i,
 0.35269601-0.53259644i, -0.57862039+0.33328986i,
 -0.94623221+0.06869166i, -0.02392968-0.21993132i,
 0.04060671+0.05644165i, 0.15534449-0.14559097i,
 0.32884452-0.19524764i, 0.58631745+0.05218419i,
 0.02562213+0.36822933i, -0.80418478+0.58621875i,
 -0.15296208-0.94175193i, -0.02942663+0.38039250i,
 -0.35184130-0.24438324i, -0.09048155+0.18131963i,
 0.63791697+0.47284679i, 0.25651928-0.46341192i,
 0.04353117-0.73472528i, -0.04606189+0.76068461i),
 FUN.VALUE = complex(1))",
 pType = "p",
 xlim = c(-4, 2), ylim = c(-2, 2),
 xlab = "real", ylab = "imaginary",
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Much more elegant: Define the function outside.
# Here comes a Blaschke product with 200 random points.
# define function for calculating blaschke products, even
# possible as a one-liner
blaschke <- function(z, a) {
 return(prod(ifelse(abs(a) != 0, abs(a)/a * (a-z)/(1-Conj(a)*z), z)))
}
# define 200 random numbers inside the unit circle
n <- 200
a <- complex(modulus = runif(n), argument = runif(n)*2*pi)
# Plot it
# x11(width = 10, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait(blaschke,
 moreArgs = list(a = a),
 pType = "p",
 xlim = c(-2.5, 2.5), ylim = c(-1.7, 1.7),
 xlab = "real", ylab = "imaginary",
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# A hybrid solution: A one-liner expression given as a character string
# can be provided additional arguments with moreArgs
n <- 73
a <- complex(modulus = runif(n), argument = runif(n)*2*pi)
# x11(width = 10, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait("prod(ifelse(abs(a) != 0,
 abs(a)/a * (a-z)/(1-Conj(a)*z), z))",
 moreArgs = list(a = a),
 pType = "p",
 xlim = c(-2.5, 2.5), ylim = c(-1.7, 1.7),
 xlab = "real", ylab = "imaginary",
 nCores = 1) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Note the difference in performance when using the C++ defined
# function blaschkeProd() provided in this package
n <- 73
a <- complex(modulus = runif(n), argument = runif(n)*2*pi)
# Plot it
# x11(width = 10, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortrait(blaschkeProd,
 moreArgs = list(a = a),
 pType = "p",
 xlim = c(-2.5, 2.5), ylim = c(-1.7, 1.7),
 xlab = "real", ylab = "imaginary",
 nCores = 1) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Interesting reunion with Benoit Mandelbrot.
# The function mandelbrot() is part of this package (defined
# in C++ for performance)
# x11(width = 11.7, height = 9/16*11.7) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mar = c(0, 0, 0, 0), bg = "black")
phasePortrait(mandelbrot,
 moreArgs = list(itDepth = 100),
 xlim = c(-0.847, -0.403), ylim = c(0.25, 0.50),
 axes = TRUE, pType = "pma",
 hsvNaN = c(0, 0, 0), xaxs = "i", yaxs = "i",
 nCores = 1) # Max. two cores allowed on CRAN
 # not a limit for your own use
par(op)
 
# Here comes a Julia set.
# The function juliaNormal() is part of this package (defined
# in C++ for performance)
# x11(width = 11.7, height = 9/16*11.7) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mar = c(0, 0, 0, 0), bg = "black")
phasePortrait(juliaNormal,
 moreArgs = list(c = -0.09 - 0.649i, R_esc = 2),
 xlim = c(-2, 2),
 ylim = c(-1.3, 1.3),
 hsvNaN = c(0, 0, 0),
 nCores = 1) # Max. two cores allowed on CRAN
 # not a limit for your own use
par(op)
 

Create two-color phase portraits of complex functions

Description

phasePortraitBw allows for creating two-color phase portraits of complex functions based on a polar chessboard grid (cf. Wegert (2012), p. 35). Compared to the full phase portraits that can be made with phasePortrait , two-color portraits omit information. Especially in combination with full phase portraits they can be, however, very helpful tools for interpretation. Besides, two-color phase portraits have a special aesthetic appeal which is worth exploring for itself. In its parameters and its mode of operation, phasePortraitBw is very similar to phasePortrait .

Usage

phasePortraitBw(
 FUN,
 moreArgs = NULL,
 xlim,
 ylim,
 invertFlip = FALSE,
 res = 150,
 blockSizePx = 2250000,
 tempDir = NULL,
 nCores = max(1, parallel::detectCores() - 1),
 bwType = "ma",
 pi2Div = 18,
 logBase = exp(2 * pi/pi2Div),
 argOffset = 0,
 bwCols = c("black", "gray95", "gray"),
 asp = 1,
 deleteTempFiles = TRUE,
 noScreenDevice = FALSE,
 autoDereg = FALSE,
 verbose = TRUE,
 ...
)

Arguments

Details

This function is intended to be used inside the framework of R base graphics. It plots into the active open graphics device where it will display the phase plot of a user defined function as a raster image. If no graphics device is open when called, the function will plot into the default graphics device. This principle allows to utilize the full functionality of R base graphics. All graphics parameters (par ) can be freely set and the function phasePortrait accepts all parameters that can be passed to the plot.default function. This allows all kinds of plots - from scientific representations with annotated axes and auxiliary lines, notation, etc. to poster-like artistic pictures. The general mode of operation, including the usage of parallel processing is exactly the same as with phasePortrait , see details section there.

References

Wegert E (2012). Visual Complex Functions. An Introduction with Phase Portraits. Springer, Basel Heidelberg New York Dordrecht London. ISBN 978-3-0348-0179-9.

Examples

# Map the complex plane on itself
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortraitBw("z", xlim = c(-2, 2), ylim = c(-2, 2),
 xlab = "real", ylab = "imaginary",
 verbose = FALSE, # Suppress progress messages
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
# Sinus with default colors and default bwType ("ma")
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortraitBw("sin(z)",
 xlim = c(-pi, pi),
 ylim = c(-pi, pi),
 verbose = FALSE,
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Sinus with custom colors and bwType "a"
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortraitBw("sin(z)",
 xlim = c(-pi, pi),
 ylim = c(-pi, pi),
 bwType = "a",
 bwCols = c("darkgreen", "green", "gray"),
 verbose = FALSE,
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Sinus with custom colors and bwType "m"
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
phasePortraitBw("sin(z)",
 xlim = c(-pi, pi),
 ylim = c(-pi, pi),
 bwType = "m",
 bwCols = c("darkblue", "skyblue", "gray"),
 verbose = FALSE,
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
 
# Map the complex plane on itself, show all bwType options
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mfrow = c(2, 2), mar = c(4.1, 4.1, 1.1, 1.1))
for(bwType in c("ma", "a", "m")) {
 phasePortraitBw("z", xlim = c(-2, 2), ylim = c(-2, 2),
 bwType = bwType,
 xlab = "real", ylab = "imaginary",
 verbose = FALSE, # Suppress progress messages
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
}
# Add normal phase portrait for comparison
phasePortrait("z", xlim = c(-2, 2), ylim = c(-2, 2),
 xlab = "real", ylab = "imaginary",
 verbose = FALSE,
 pi2Div = 18, # Use same angular division as default
 # in phasePortraitBw
 nCores = 2)
par(op)
 
# A rational function, show all bwType options
# x11(width = 8, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
funString <- "(z + 1.4i - 1.4)^2/(z^3 + 2)"
op <- par(mfrow = c(2, 2), mar = c(4.1, 4.1, 1.1, 1.1))
for(bwType in c("ma", "a", "m")) {
 phasePortraitBw(funString, xlim = c(-2, 2), ylim = c(-2, 2),
 bwType = bwType,
 xlab = "real", ylab = "imaginary",
 verbose = FALSE, # Suppress progress messages
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
}
# Add normal phase portrait for comparison
phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
 xlab = "real", ylab = "imaginary",
 verbose = FALSE,
 pi2Div = 18, # Use same angular division as default
 # in phasePortraitBw
 nCores = 2)
par(op)
 

Plot a Riemann sphere mask over a phase portrait

Description

The function riemannMask can be used for laying a circular mask over an existing phasePortrait (as generated with the function phasePortrait ). This mask shades the plot region outside the unit circle. The unshaded area is a projection on the southern or northern Riemann hemisphere. The standard projection used by phasePortrait , i.e. invertFlip = FALSE hereby corresponds to the southern Riemann hemisphere with the origin being the south pole. If phasePortrait was called with invertFlip = TRUE, then the unit circle contains the northern Riemann hemisphere with the point at infinity in the center (see the vignette for more details). Options for adding annotation, landmark points are available (see Wegert (2012), p. 41). Several parameters are on hand for adjusting the mask's transparency, color, and similar features. some details, this function behaves less nicely under Windows than under Linux (see Details).

Usage

riemannMask(
 colMask = "white",
 alphaMask = 0.5,
 circOutline = TRUE,
 circLwd = 1,
 circleSteps = 360,
 circleCol = par("fg"),
 gridCross = FALSE,
 annotSouth = FALSE,
 annotNorth = FALSE,
 xlim = NULL,
 ylim = NULL
)

Arguments

Details

There is, unfortunately, a somewhat different behavior of this function under Linux and Windows systems. Under Windows, the region outside the unit circle is only shaded if the whole unit circle fits into the plot region. If only a part of the unit circle is to be displayed, the shading is completely omitted under Windows (annotation etc. works correctly, however), while it works properly on Linux systems. Obviously, the function polypath , which we are using for creating the unit circle template, is interpreted differently on both systems.

References

Wegert E (2012). Visual Complex Functions. An Introduction with Phase Portraits. Springer, Basel Heidelberg New York Dordrecht London. ISBN 978-3-0348-0179-9.

Examples

# Tangent with fully annotated Riemann masks.
# The axis tick marks on the second diagram (Northern hemisphere)
# have to be interpreted as the real and imaginary parts of 1/z
# (see vignette). The axis labels in this example have been adapted
# accordingly.
# x11(width = 16, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mfrow = c(1, 2), mar = c(4.7, 4.7, 3.5, 3.5))
phasePortrait("tan(z)", pType = "pma",
 main = "Southern Riemann Hemisphere",
 xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
 xlab = "real", ylab = "imaginary",
 xaxs = "i", yaxs = "i",
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
riemannMask(annotSouth = TRUE, gridCross = TRUE)
phasePortrait("tan(z)", pType = "pma",
 main = "Northern Riemann Hemisphere",
 invertFlip = TRUE,
 xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
 xlab = "real (1/z)", ylab = "imaginary (1/z)",
 xaxs = "i", yaxs = "i",
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
riemannMask(annotNorth = TRUE, gridCross = TRUE)
par(op)
# Rational function with Riemann masks without annotation.
# The axis tick marks on the second diagram (Northern hemisphere)
# have to be interpreted as the real and imaginary parts of 1/z
# (see vignette). The axis labels in this example have been adapted
# accordingly.
# x11(width = 16, height = 8) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mfrow = c(1, 2), mar = c(4.7, 4.7, 3.5, 3.5))
phasePortrait("(-z^17 - z^15 - z^9 - z^7 - z^2 - z + 1)/(1i*z - 1)",
 pType = "pma",
 main = "Southern Riemann Hemisphere",
 xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
 xlab = "real", ylab = "imaginary",
 xaxs = "i", yaxs = "i",
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
riemannMask(annotSouth = FALSE, gridCross = FALSE, circOutline = FALSE,
 alphaMask = 0.7)
phasePortrait("(-z^17 - z^15 - z^9 - z^7 - z^2 - z + 1)/(1i*z - 1)",
 pType = "pma",
 main = "Northern Riemann Hemisphere",
 invertFlip = TRUE,
 xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
 xlab = "real (1/z)", ylab = "imaginary (1/z)",
 xaxs = "i", yaxs = "i",
 nCores = 2) # Max. two cores on CRAN, not a limit for your use
 
riemannMask(annotNorth = FALSE, gridCross = FALSE, circOutline = FALSE,
 alphaMask = 0.7)
par(op)

Convert a vector into a comma-separated string

Description

A simple utility function that transforms any vector into a single character string, where the former vector elements are separated by commas. This is can be useful, in some circumstances, for feeding a series of constant numeric values to phasePortrait (see examples). For most applications we recommend, however, to use phasePortrait 's parameter moreArgs instead.

Usage

vector2String(vec)

Arguments

Value

A string, where the former vector elements are separated by commas, enclosed between "c(" and ")".

See Also

Other helpers: xlimFromYlim(), ylimFromXlim()

Examples

# Make a vector of 77 complex random numbers inside the unit circle
n <- 77
a <- complex(n, modulus = runif(n), argument = 2*pi*runif(n))
a <- vector2String(a)
print(a)
# Use this for portraying a Blaschke product
# x11(width = 9.45, height = 6.30) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
op <- par(mar = c(1, 1, 1, 1), bg = "black")
n <- 77
a <- complex(n, modulus = runif(n), argument = 2*pi*runif(n))
a <- vector2String(a)
FUN <- paste("vapply(z, function(z, a){
 return(prod(abs(a)/a * (a-z)/(1-Conj(a)*z)))
 }, a =", a,
 ", FUN.VALUE = complex(1))", sep = "")
phasePortrait(FUN, pType = "p", axes = FALSE,
 xlim = c(-3, 3), ylim = c(-2.0, 2.0),
 nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
par(op)
 

Adjust xlim to ylim

Description

This simple function is useful for adjusting x and y coordinate ranges xlim and ylim in order to maintain a desired display ratio. The latter must be given, the former will be adjusted.

Usage

xlimFromYlim(ylim, centerX = 0, x_to_y = 16/9)

Arguments

Details

For certain purposes, e.g. producing a graph that exactly matches a screen, the x and y coordinates must be adjusted to match a given display ratio. If the vertical range, ylim, the desired ratio, x_to_y and the desired center of the x-range, centerX, are provided, this function returns an adpated vertical range, that can be used as ylim in any plot including phasePortrait .

Value

A numeric vector of length 2; the lower and upper boundary of the resulting vertical coordinate range

See Also

Other helpers: vector2String(), ylimFromXlim()

Examples

# Make a phase portrait of a pretty function that fully covers a
# plot with a display aspect ratio of 5/4.
# 9 inch wide window with 5/4 display ratio (x/y)
# x11(width = 9, height = 9 * 4/5) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
ylim <- c(-8, 7)
xlim <- xlimFromYlim(ylim, centerX = 0, x_to_y = 5/4)
op <- par(mar = c(0, 0, 0, 0), bg = "black") # Omit all plot margins
phasePortrait("exp(cosh(1/(z - 2i + 2)^2 * (1/2i - 1/4 + z)^3))", pType = "pm",
xlim = xlim, ylim = ylim, # Apply the coordinate ranges
xaxs = "i", yaxs = "i", # Allow for now room between plot and axes
nCores = 2) # Max. two cores allowed on CRAN
 # not a limit for your own use
par(op)
 

Adjust ylim to xlim

Description

This simple function is useful for adjusting x and y coordinate ranges xlim and ylim in order to maintain a desired display ratio. The former must be given, the latter will be adjusted.

Usage

ylimFromXlim(xlim, centerY = 0, x_to_y = 16/9)

Arguments

Details

For certain purposes, e.g. producing a graph that exactly matches a screen, the x and y coordinates must be adjusted to match a given display ratio. If the horizontal range, xlim, the desired ratio, x_to_y and the desired center of the y-range, centerY are provided, this function returns an adapted vertical range, that can be used as ylim in any plot including phasePortrait .

Value

A numeric vector of length 2; the lower and upper boundary of the resulting vertical coordinate range

See Also

Other helpers: vector2String(), xlimFromYlim()

Examples

# Make a phase portrait of a Jacobi theta function that fully covers a
# plot with a display aspect ratio of 4/3.
# 10 inch wide window with 4/3 display ratio (x/y)
# x11(width = 10, height = 10 * 3/4) # Screen device commented out
 # due to CRAN test requirements.
 # Use it when trying this example
xlim <- c(-3, 3)
ylim <- ylimFromXlim(xlim, centerY = -0.3, x_to_y = 4/3)
op <- par(mar = c(0, 0, 0, 0), bg = "black") # Omit all plot margins
phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2 - 1/3),
 xlim = xlim, ylim = ylim, # Apply the coordinate ranges
 xaxs = "i", yaxs = "i", # Allow for now room between plot and axes
 nCores = 1) # Max. two cores allowed on CRAN
 # not a limit for your own use
par(op)