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inst/doc/viscomplexr-vignette.R
In viscomplexr: Phase Portraits of Functions in the Complex Number Plane
## ---- include = FALSE---------------------------------------------------------
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
# This option anti-aliases the plots made below under Windows
if(Sys.info()[["sysname"]] == "Windows") {
knitr::opts_chunk$set(dev = "CairoPNG")
}
options(rmarkdown.html_vignette.check_title = FALSE)
## ----setup, echo = FALSE------------------------------------------------------
library(viscomplexr)
## ---- figure_1, fig.width = 5, fig.height = 5, results = 'hide', fig.align='center', cache = FALSE, fig.show = 'hold', fig.cap = 'Phase portrait of the function $f(z)=z$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$.'----
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
xlab = "real", ylab = "imaginary", main = "f(z) = z",
nCores = 2) # Probably not required on your machine (see below)
# Note the argument 'nCores' which determines the number of parallel processes to
# be used. Setting nCores = 2 has been done here and in all subsequent
# examples as CRAN checks do not allow more parallel processes.
# For normal work, we recommend not to define nCores at all which will make
# phasePortrait use all available cores on your machine.
# The progress messages phasePortrait is writing to the console can be
# suppressed by including 'verbose = FALSE' in the call (see documentation).
## ----figure_2, fig.width=5, fig.height=5, results="hide", fig.align='center', fig.show='hold', cache=TRUE, fig.cap= "Different options for including reference lines with the argument `pType`."----
# divide graphics device into four regions and adjust plot margins
op <- par(mfrow = c(2, 2),
mar = c(0.25, 0.55, 1.10, 0.25))
# plot four phase portraits with different choices of pType
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "p",
main = "pType = 'p'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pa",
main = "pType = 'pa'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
main = "pType = 'pm'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pma",
main = "pType = 'pma'", axes = FALSE, nCores = 2)
par(op) # reset the graphics parameters to their previous values
## ----eval=FALSE, figure_3, fig.width=5, fig.height=5, results='hide', fig.align='center', cache=TRUE, fig.show='hold', fig.cap='Phase portrait of the function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$.'----
# op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# # and general text size
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2",
# xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# xlab = "real", ylab = "imaginary",
# nCores = 2) # Increase or leave out for higher performance
# par(op) # reset the graphics parameters to their previous values
## ----eval = FALSE, figure_4, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with three different settings of `pi2Div` and `pType = "pa"`.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, pType = "pa", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div =", n), line = -1.2)
# }
# par(op) # reset graphics parameters to previous values
## ----figure_5, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with three different settings of `pi2Div` and `pType = "pma"`.'----
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div =", n), line = -1.2)
}
par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_6, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with decoupled settings of `pi2Div` and `logBase`.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, logBase = sqrt(3), pType = "pma", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div = ", n, ", logBase = 3^(1/3)", sep = ""), line = -1.2)
# }
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_7, fig.width = 5, fig.height = 5, results = 'hide', fig.align='center', fig.show='hold', cache = TRUE, fig.cap = 'Phase portrait of the function $f(z)=\\mathrm{e}^z$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$ with iso-modulus lines.'----
# op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# # and general text size
# phasePortrait(exp, xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
# xlab = "real", ylab = "imaginary", nCores = 2)
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_8, fig.width=7, fig.height=2.8, results='hide', fig.align='center', fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\mathrm{e}^z$ portrayed with the default coupling of `pi2Div` and `logBase` as implemented in *phasePortrait*.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("exp(z)", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div = ", n, ", logBase = exp(2*pi/pi2Div)", sep = ""),
# line = -1.2, cex.main = 0.9)
# }
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_9, fig.width=5, fig.height=5, fig.show='hold', results='hide', cache=TRUE, fig.cap='Tuning reference zone contrast with the parameters `darkestShade` (column-wise, 0, 0.2, 0.4), and `lambda` (row-wise, 0.1, 1, 10).'----
# op <- par(mfrow = c(3, 3), mar = c(0.2, 0.2, 0.2, 0.2))
# for(lb in c(0.1, 1, 10)) {
# for(dS in c(0, 0.2, 0.4)) {
# phasePortrait("tan(z^2)", xlim = c(-1.7, 1.7), ylim = c(-1.7, 1.7),
# pType = "pm", darkestShade = dS, lambda = lb,
# axes = FALSE, xaxs = "i", yaxs = "i", nCores = 2)
# }
# }
# par(op)
## ----eval=FALSE, figure_10, fig.width=7, fig.height=2.7, results='hide', fig.align='center', fig.show='hold', cache=TRUE, fig.cap='Three phase portraits with branch cuts (dashed line), illustrating the three values of $f(z)=z^{1/3}$, $z \\in \\mathbb{C} \\setminus \\lbrace 0 \\rbrace$. The transitions between the phase portraits are indicated by same-coloured arrows pointing at the branch cuts.'----
# op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
# for(k in 0:2) {
# FUNstring <- paste0("z^(1/3) * exp(1i * 2*pi/3 * ", k, ")")
# phasePortrait(FUN = FUNstring,
# xlim = c(-1.5, 1.5), ylim = c(-1.5, 1.5), pi2Div = 12,
# axes = FALSE, nCores = 2)
# title(sub = paste0("k = ", k), line = -1)
# # emphasize branch cut with a dashed line segment
# segments(-1.5, 0, 0, 0, lwd = 2, lty = "dashed")
# # draw colored arrows
# upperCol <- switch(as.character(k),
# "0" = "black", "1" = "red", "2" = "green")
# lowerCol <- switch(as.character(k),
# "0" = "green", "1" = "black", "2" = "red")
# arrows(x0 = c(-1.2), y0 = c(1, -1), y1 = c(0.2, -0.2),
# lwd = 2, length = 0.1, col = c(upperCol, lowerCol))
# }
# par(op)
## ----eval=FALSE, figure_11, fig.width=7, fig.height=2.7, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Three branches of $\\log z=\\log r+\\mathrm{i}\\cdot(\\varphi + k\\cdot2\\pi), r>0, \\varphi\\in\\left[0,2\\pi\\right[$, with $k=-1,0,1$. The branch cuts are marked with dashed white lines.'----
# op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
# for(k in -1:1) {
# FUNstring <- paste0("log(Mod(z)) + 1i * (Arg(z) + 2 * pi * ", k, ")")
# phasePortrait(FUN = FUNstring, pi2Div = 36,
# xlim = c(-2, 2), ylim = c(-2, 2), axes = FALSE, nCores = 2)
# segments(-2, 0, 0, 0, col = "white", lwd = 1, lty = "dashed")
# title(sub = paste0("k = ", k), line = -1)
# }
# par(op)
## ----figure_12, fig.width=7, fig.height=3.5, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Mapping the complex number plane on the Riemann sphere. Left: lower (southern) hemisphere; right upper (northern hemisphere). Folding both figures face to face along a vertical line in the middle between them can be imagined as closing the Riemann sphere.'----
op <- par(mfrow = c(1, 2), mar = rep(0.1, 4))
# Southern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, nCores = 2)
riemannMask(annotSouth = TRUE)
# Northern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
riemannMask(annotNorth = TRUE)
par(op)
## ----figure_13, fig.width=7, fig.height=3.7, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Riemann sphere plot of the function $f(z)=\\frac{(z^{2}+\\frac{1}{\\sqrt{2}}+\\frac{\\mathrm{i}}{\\sqrt{2}})\\cdot(z+\\frac{1}{2}+\\frac{\\mathrm{i}}{2})}{z-1}$. Annotated are the zeroes $z_1$, $z_2$, $z_3$, and the poles $z_4$, $z_5$.'----
op <- par(mfrow = c(1, 2), mar = c(0.1, 0.1, 1.4, 0.1))
# Define function
FUNstring <- "(z^2 + 1/sqrt(2) * (1 + 1i)) * (z + 1/2*(1 + 1i)) / (z - 1)"
# Southern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, nCores = 2)
riemannMask()
title("Southern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)/sqrt(2), 1),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)/sqrt(2), 0),
c(expression(z[1]), expression(z[2]), expression(z[3]), expression(z[4])),
pos = c(1, 2, 4, 2), offset = 1, col = "white")
# Northern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
riemannMask()
title("Northern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)*sqrt(2), 1, 0),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)*sqrt(2), 0, 0),
c(expression(z[1]), expression(z[2]), expression(z[3]),
expression(z[4]), expression(z[5])),
pos = c(1, 4, 3, 4, 4), offset = 1,
col = c("white", "white", "black", "white", "white"))
par(op)
## ---- eval=FALSE--------------------------------------------------------------
# x11(width = 8, height = 2/3 * 8) # Open graphics window on screen
# op <- par(mar = c(0, 0, 0, 0)) # Do not leave plot margins
# phasePortrait(mandelbrot, moreArgs = list(itDepth = 30),
# ncores = 1, # Increase or leave out for higher performance
# xlim = c(-2, 1), ylim = c(-1, 1),
# hsvNaN = c(0, 0, 0), # black color for points outside the set
# axes = FALSE, # No coordinate axes
# xaxs = "i", yaxs = "i") # No space between plot region and plot
# par(op) # Set graphics parameters to original
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600 # set resolution to 600 dpi
# # open png graphics device with in DIN A4 format
# # DIN A format has an edge length ratio of sqrt(2)
# png("Mandelbrot Example.png",
# width = 29.7, height = 29.7/sqrt(2), # DIN A4 landscape
# units = "cm",
# res = res) # resolution is required
# op <- par(mar = c(0, 0, 0, 0)) # set graphics parameters - no plot margins
# xlim <- c(-1.254, -1.248) # horizontal (real) plot limits
# # the function below adjusts the imaginary plot limits to the
# # desired ratio (sqrt(2)) centered around the desired imaginary value
# ylim <- ylimFromXlim(xlim, centerY = 0.02, x_to_y = sqrt(2))
# phasePortrait(mandelbrot,
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0), # Black color for NaN results
# xaxs = "i", yaxs = "i", # suppress R's default axis margins
# axes = FALSE, # do not plot axes
# res = res) # resolution is required
# par(op) # reset graphics parameters
# dev.off() # close graphics device and complete the png file
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600
# png("Julia Example 1.png", width = 29.7, height = 29.7/sqrt(2),
# units = "cm", res = res)
# op <- par(mar = c(0, 0, 0, 0))
# xlim <- c(-1.8, 1.8)
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = sqrt(2))
# phasePortrait(juliaNormal,
# # see documentation of juliaNormal about the arguments
# # c and R_esc
# moreArgs = list(c = -0.09 - 0.649i, R_esc = 2),
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0),
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# res = res)
# par(op)
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600
# png("Julia Example 2.png", width = 29.7, height = 29.7/sqrt(2),
# units = "cm", res = res)
# op <- par(mar = c(0, 0, 0, 0))
# xlim <- c(-0.32, 0.02)
# ylim <- ylimFromXlim(xlim, center = -0.78, x_to_y = sqrt(2))
# phasePortrait(juliaNormal,
# # see documentation of juliaNormal about the arguments
# # c and R_esc
# moreArgs = list(c = -0.119 - 0.882i, R_esc = 2),
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0),
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# res = res)
# par(op)
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# # Map the complex plane on itself, show all bwType options
#
# x11(width = 8, height = 8)
# 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",
# nCores = 2) # Increase or leave out for higher performance
# }
# # Add normal phase portrait for comparison
# phasePortrait("z", xlim = c(-2, 2), ylim = c(-2, 2),
# xlab = "real", ylab = "imaginary",
# pi2Div = 18, # Use same angular division as default
# # in phasePortraitBw
# nCores = 2) # Increase or leave out for higher performance
# par(op)
#
## ----eval=FALSE---------------------------------------------------------------
# # A rational function, show all bwType options
#
# x11(width = 8, height = 8)
# 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",
# nCores = 2) # Increase or leave out for higher performance
# }
# # Add normal phase portrait for comparison
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# xlab = "real", ylab = "imaginary",
# pi2Div = 18, # Use same angular division as default
# # in phasePortraitBw
# nCores = 2) # Increase or leave out for higher performance
# par(op)
#
## ---- eval = FALSE------------------------------------------------------------
# # Set the plot margins at all four sides to 1/5 inch with mai,
# # set the background color to black with bg, and the default foreground
# # color with fg (e.g. for axes and boxes around plots, or the color of
# # the circle outline from the function riemannMask).
# # We catch the previous parameter values in a variable, I called
# # "op" ("old parameters")
# op <- par(mai = c(1/5, 1/5, 1/5, 1/5), bg = "black", fg = "white")
#
# # Make any phase portraits and/or other graphics of your interest
# # ...
#
# # Set the graphical parameters back to the values previously stored in op
# par(op)
## ---- eval = FALSE------------------------------------------------------------
# phasePortrait("tan(z^3 + 1/2 - 2i)/(1 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3),
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
## ---- eval=FALSE--------------------------------------------------------------
# phasePortrait("tan(z^3 + 1/2 - 2i)/(1 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3),
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
# box()
## ---- eval=FALSE--------------------------------------------------------------
# # set background and foreground colors
# op <- par(bg = "black", fg = "white")
# # Setting the parameter fg has an effect on the box, the axes, and the axes'
# # ticks, but not on the axis annotations and axis labels.
# # Also the color of the title (main) is not affected.
# # The colors of these elements have to be set manually and separately. While we
# # could simply set them to "white", we set them, more flexibly, to the
# # current foreground color (par("fg")).
# phasePortrait("tan(z^3 + 1/2 - 2i)/(2 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3), col.axis = par("fg"),
# xlab = "real", ylab = "imaginary", col.lab = par("fg"),
# main = "All annotation in foreground color", col.main = par("fg"),
# # Adjust text size
# cex.axis = 0.9, cex.lab = 0.9,
# nCores = 2) # Increase or leave out for higher performance
# par(op)
## ---- eval=FALSE--------------------------------------------------------------
# # Open graphics device with 16/9 aspect ratio and 7 inch width
# x11(width = 7, height = 9/16 * 7)
# op <- par(mar = c(0, 0, 0, 0)) # Set plot margins to zero
# xlim <- c(-3, 3)
# # Calculate ylim with desired center fitting the desired aspect ratio
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = 16/9)
# phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/5 + 1/5), pType = "p",
# xlim = xlim, ylim = ylim,
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
# par(op)
## ---- eval=FALSE--------------------------------------------------------------
# # Open graphics device with 16/9 aspect ratio and a width of 7 inches
# x11(width = 7, height = 9/16 * 7)
# # Set plot margins to zero, outer margins to 1/7 inch,
# # and background color to black
# outerMar <- 1/7 # outer margin width in inches
# op <- par(mar = c(0, 0, 0, 0), omi = rep(outerMar, 4), bg = "black")
# xlim <- c(-1.5, 0.5)
# # Calculate ylim with desired center fitting the desired aspect ratio;
# # however, the omi settings slightly change the required
# # ratio of xlim and ylim
# ratio <- (7 - 2*outerMar) / (7 * 9/16 - 2*outerMar)
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = ratio)
# phasePortrait("sin(jacobiTheta(z, tau))/z", moreArgs = list(tau = 1i/5 + 1/5),
# pType = "p",
# xlim = xlim, ylim = ylim,
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# nCores = 1) # Increase or leave out for higher performance
# par(op)
## ---- eval = FALSE------------------------------------------------------------
# # Note that 'FUN =' is not required if the argument to FUN is handed to
# # phasePortrait in the first position
# phasePortrait(FUN = "1/(1 - z^2)", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
# phasePortrait("sin((z - 2)/(z + 2))", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
# phasePortrait("tan(z)", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
## ---- eval = FALSE------------------------------------------------------------
# phasePortrait("-1 * sum(z^c(-k:k))", moreArgs = list(k = 11),
# xlim = c(-2, 2), ylim = c(-1.5, 1.5),
# pType = "p",
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# funString <- "vapply(z, FUN = function(z) {
# n <- 9
# k <- z^(c(1:n))
# rslt <- sum(sin(k))
# return(rslt)
# },
# FUN.VALUE = complex(1))"
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# funString <- "vapply(z, FUN = function(z, fct) {
# n <- 9
# k <- z^(fct * c(1:n))
# rslt <- sum(sin(k))
# return(rslt)
# },
# fct = -1,
# FUN.VALUE = complex(1))"
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Define function
# tryThisOne <- function(z, fct, n) {
# k <- z^(fct * c(1:n))
# rslt <- prod(cos(k))
# return(rslt)
# }
#
# # Call function by its name only, provide additional arguments via "moreArgs"
# phasePortrait("tryThisOne", moreArgs = list(fct = 1, n = 5),
# xlim = c(-2.5, 2.5), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the grey area black
# phasePortrait(function(z) {
# for(j in 1:20) {
# z <- z * sin(z) - 1 + 1/2i
# }
# return(z)
# },
# xlim = c(-3, 3), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the grey area black
# phasePortrait(function(z, n) {
# for(j in 1:n) {
# z <- z * cos(z)
# }
# return(z)
# },
# moreArgs = list(n = 27),
# xlim = c(-3, 3), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # atan from package base
# phasePortrait(atan, xlim = c(-pi, pi), ylim = c(-pi, pi),
# nCores = 2)
#
# # gammaz from package pracma (the package must be installed on your machine
# # if you want this example to be working)
# phasePortrait(pracma::gammaz, xlim = c(-9, 9), ylim = c(-5, 5),
# nCores = 2)
#
# # blaschkeProd from this package (moreArgs example)
# # make random vector of zeroes
# n <- 12
# a <- complex(modulus = runif(n), argument = 2 * pi * runif(n))
# # plot the actual phase portrait
# phasePortrait(blaschkeProd, moreArgs = list(a = a),
# xlim = c(-1.3, 1.3), ylim = c(-1.3, 1.3),
# nCores = 2)
#
# # User function example
# tryThisOneToo <- function(z, n, r) {
# for(j in 1:n) {
# z <- r * (z + z^2)
# }
# return(z)
# }
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the gray areas black
# phasePortrait(tryThisOneToo, moreArgs = list(n = 50, r = 1/2 - 1/2i),
# xlim = c(-3, 2), ylim = c(-2.5, 2.5),
# nCores = 2)
#
## ---- eval = FALSE------------------------------------------------------------
# res <- 300 # Define desired resolution in dpi
# png("Logistic_Function.png", width = 40, height = 40 * 3/4,
# units = "cm", res = res)
# phasePortrait("1/(1+exp(-z))", xlim = c(-25, 25), ylim = c(-15, 15), res = res,
# xlab = "real", ylab = "imaginary",
# nCores = 2) # Increase or leave out for higher performance
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# switch(1 + trunc(runif(1, 0, 6)),
# "... at all?",
# "... in a quick-and-dirty way?",
# "... in Hadley-Wickham-style?",
# "... without a loop?",
# "... without nested loops?",
# "... in a way somebody can understand?")
## ---- include = FALSE---------------------------------------------------------
foreach::registerDoSEQ()
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## ---- include = FALSE---------------------------------------------------------
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
# This option anti-aliases the plots made below under Windows
if(Sys.info()[["sysname"]] == "Windows") {
knitr::opts_chunk$set(dev = "CairoPNG")
}
options(rmarkdown.html_vignette.check_title = FALSE)
## ----setup, echo = FALSE------------------------------------------------------
library(viscomplexr)
## ---- figure_1, fig.width = 5, fig.height = 5, results = 'hide', fig.align='center', cache = FALSE, fig.show = 'hold', fig.cap = 'Phase portrait of the function $f(z)=z$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$.'----
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
xlab = "real", ylab = "imaginary", main = "f(z) = z",
nCores = 2) # Probably not required on your machine (see below)
# Note the argument 'nCores' which determines the number of parallel processes to
# be used. Setting nCores = 2 has been done here and in all subsequent
# examples as CRAN checks do not allow more parallel processes.
# For normal work, we recommend not to define nCores at all which will make
# phasePortrait use all available cores on your machine.
# The progress messages phasePortrait is writing to the console can be
# suppressed by including 'verbose = FALSE' in the call (see documentation).
## ----figure_2, fig.width=5, fig.height=5, results="hide", fig.align='center', fig.show='hold', cache=TRUE, fig.cap= "Different options for including reference lines with the argument `pType`."----
# divide graphics device into four regions and adjust plot margins
op <- par(mfrow = c(2, 2),
mar = c(0.25, 0.55, 1.10, 0.25))
# plot four phase portraits with different choices of pType
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "p",
main = "pType = 'p'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pa",
main = "pType = 'pa'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
main = "pType = 'pm'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pma",
main = "pType = 'pma'", axes = FALSE, nCores = 2)
par(op) # reset the graphics parameters to their previous values
## ----eval=FALSE, figure_3, fig.width=5, fig.height=5, results='hide', fig.align='center', cache=TRUE, fig.show='hold', fig.cap='Phase portrait of the function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$.'----
# op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# # and general text size
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2",
# xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# xlab = "real", ylab = "imaginary",
# nCores = 2) # Increase or leave out for higher performance
# par(op) # reset the graphics parameters to their previous values
## ----eval = FALSE, figure_4, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with three different settings of `pi2Div` and `pType = "pa"`.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, pType = "pa", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div =", n), line = -1.2)
# }
# par(op) # reset graphics parameters to previous values
## ----figure_5, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with three different settings of `pi2Div` and `pType = "pma"`.'----
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div =", n), line = -1.2)
}
par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_6, fig.width=7, fig.height=2.8, results='hide', fig.align='center', , fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\frac{(3+2\\mathrm{i}+z)(-5+5\\mathrm{i}+z)}{(-2-2\\mathrm{i}+z)^2}$ portrayed with decoupled settings of `pi2Div` and `logBase`.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, logBase = sqrt(3), pType = "pma", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div = ", n, ", logBase = 3^(1/3)", sep = ""), line = -1.2)
# }
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_7, fig.width = 5, fig.height = 5, results = 'hide', fig.align='center', fig.show='hold', cache = TRUE, fig.cap = 'Phase portrait of the function $f(z)=\\mathrm{e}^z$ in the window $\\left|\\Re(z)\\right| < 8.5$ and $\\left|\\Im(z)\\right| < 8.5$ with iso-modulus lines.'----
# op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# # and general text size
# phasePortrait(exp, xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
# xlab = "real", ylab = "imaginary", nCores = 2)
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_8, fig.width=7, fig.height=2.8, results='hide', fig.align='center', fig.show='hold', cache=TRUE, fig.cap='The function $f(z)=\\mathrm{e}^z$ portrayed with the default coupling of `pi2Div` and `logBase` as implemented in *phasePortrait*.'----
# # divide graphics device into three regions and adjust plot margins
# op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
# for(n in c(6, 9, 18)) {
# phasePortrait("exp(z)", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
# pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# # separate title call (R base graphics) for nicer line adjustment, just cosmetics
# title(paste("pi2Div = ", n, ", logBase = exp(2*pi/pi2Div)", sep = ""),
# line = -1.2, cex.main = 0.9)
# }
# par(op) # reset graphics parameters to previous values
## ----eval=FALSE, figure_9, fig.width=5, fig.height=5, fig.show='hold', results='hide', cache=TRUE, fig.cap='Tuning reference zone contrast with the parameters `darkestShade` (column-wise, 0, 0.2, 0.4), and `lambda` (row-wise, 0.1, 1, 10).'----
# op <- par(mfrow = c(3, 3), mar = c(0.2, 0.2, 0.2, 0.2))
# for(lb in c(0.1, 1, 10)) {
# for(dS in c(0, 0.2, 0.4)) {
# phasePortrait("tan(z^2)", xlim = c(-1.7, 1.7), ylim = c(-1.7, 1.7),
# pType = "pm", darkestShade = dS, lambda = lb,
# axes = FALSE, xaxs = "i", yaxs = "i", nCores = 2)
# }
# }
# par(op)
## ----eval=FALSE, figure_10, fig.width=7, fig.height=2.7, results='hide', fig.align='center', fig.show='hold', cache=TRUE, fig.cap='Three phase portraits with branch cuts (dashed line), illustrating the three values of $f(z)=z^{1/3}$, $z \\in \\mathbb{C} \\setminus \\lbrace 0 \\rbrace$. The transitions between the phase portraits are indicated by same-coloured arrows pointing at the branch cuts.'----
# op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
# for(k in 0:2) {
# FUNstring <- paste0("z^(1/3) * exp(1i * 2*pi/3 * ", k, ")")
# phasePortrait(FUN = FUNstring,
# xlim = c(-1.5, 1.5), ylim = c(-1.5, 1.5), pi2Div = 12,
# axes = FALSE, nCores = 2)
# title(sub = paste0("k = ", k), line = -1)
# # emphasize branch cut with a dashed line segment
# segments(-1.5, 0, 0, 0, lwd = 2, lty = "dashed")
# # draw colored arrows
# upperCol <- switch(as.character(k),
# "0" = "black", "1" = "red", "2" = "green")
# lowerCol <- switch(as.character(k),
# "0" = "green", "1" = "black", "2" = "red")
# arrows(x0 = c(-1.2), y0 = c(1, -1), y1 = c(0.2, -0.2),
# lwd = 2, length = 0.1, col = c(upperCol, lowerCol))
# }
# par(op)
## ----eval=FALSE, figure_11, fig.width=7, fig.height=2.7, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Three branches of $\\log z=\\log r+\\mathrm{i}\\cdot(\\varphi + k\\cdot2\\pi), r>0, \\varphi\\in\\left[0,2\\pi\\right[$, with $k=-1,0,1$. The branch cuts are marked with dashed white lines.'----
# op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
# for(k in -1:1) {
# FUNstring <- paste0("log(Mod(z)) + 1i * (Arg(z) + 2 * pi * ", k, ")")
# phasePortrait(FUN = FUNstring, pi2Div = 36,
# xlim = c(-2, 2), ylim = c(-2, 2), axes = FALSE, nCores = 2)
# segments(-2, 0, 0, 0, col = "white", lwd = 1, lty = "dashed")
# title(sub = paste0("k = ", k), line = -1)
# }
# par(op)
## ----figure_12, fig.width=7, fig.height=3.5, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Mapping the complex number plane on the Riemann sphere. Left: lower (southern) hemisphere; right upper (northern hemisphere). Folding both figures face to face along a vertical line in the middle between them can be imagined as closing the Riemann sphere.'----
op <- par(mfrow = c(1, 2), mar = rep(0.1, 4))
# Southern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, nCores = 2)
riemannMask(annotSouth = TRUE)
# Northern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
riemannMask(annotNorth = TRUE)
par(op)
## ----figure_13, fig.width=7, fig.height=3.7, fig.align='center', results='hide', fig.show='hold', cache=TRUE, fig.cap='Riemann sphere plot of the function $f(z)=\\frac{(z^{2}+\\frac{1}{\\sqrt{2}}+\\frac{\\mathrm{i}}{\\sqrt{2}})\\cdot(z+\\frac{1}{2}+\\frac{\\mathrm{i}}{2})}{z-1}$. Annotated are the zeroes $z_1$, $z_2$, $z_3$, and the poles $z_4$, $z_5$.'----
op <- par(mfrow = c(1, 2), mar = c(0.1, 0.1, 1.4, 0.1))
# Define function
FUNstring <- "(z^2 + 1/sqrt(2) * (1 + 1i)) * (z + 1/2*(1 + 1i)) / (z - 1)"
# Southern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, nCores = 2)
riemannMask()
title("Southern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)/sqrt(2), 1),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)/sqrt(2), 0),
c(expression(z[1]), expression(z[2]), expression(z[3]), expression(z[4])),
pos = c(1, 2, 4, 2), offset = 1, col = "white")
# Northern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
riemannMask()
title("Northern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)*sqrt(2), 1, 0),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)*sqrt(2), 0, 0),
c(expression(z[1]), expression(z[2]), expression(z[3]),
expression(z[4]), expression(z[5])),
pos = c(1, 4, 3, 4, 4), offset = 1,
col = c("white", "white", "black", "white", "white"))
par(op)
## ---- eval=FALSE--------------------------------------------------------------
# x11(width = 8, height = 2/3 * 8) # Open graphics window on screen
# op <- par(mar = c(0, 0, 0, 0)) # Do not leave plot margins
# phasePortrait(mandelbrot, moreArgs = list(itDepth = 30),
# ncores = 1, # Increase or leave out for higher performance
# xlim = c(-2, 1), ylim = c(-1, 1),
# hsvNaN = c(0, 0, 0), # black color for points outside the set
# axes = FALSE, # No coordinate axes
# xaxs = "i", yaxs = "i") # No space between plot region and plot
# par(op) # Set graphics parameters to original
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600 # set resolution to 600 dpi
# # open png graphics device with in DIN A4 format
# # DIN A format has an edge length ratio of sqrt(2)
# png("Mandelbrot Example.png",
# width = 29.7, height = 29.7/sqrt(2), # DIN A4 landscape
# units = "cm",
# res = res) # resolution is required
# op <- par(mar = c(0, 0, 0, 0)) # set graphics parameters - no plot margins
# xlim <- c(-1.254, -1.248) # horizontal (real) plot limits
# # the function below adjusts the imaginary plot limits to the
# # desired ratio (sqrt(2)) centered around the desired imaginary value
# ylim <- ylimFromXlim(xlim, centerY = 0.02, x_to_y = sqrt(2))
# phasePortrait(mandelbrot,
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0), # Black color for NaN results
# xaxs = "i", yaxs = "i", # suppress R's default axis margins
# axes = FALSE, # do not plot axes
# res = res) # resolution is required
# par(op) # reset graphics parameters
# dev.off() # close graphics device and complete the png file
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600
# png("Julia Example 1.png", width = 29.7, height = 29.7/sqrt(2),
# units = "cm", res = res)
# op <- par(mar = c(0, 0, 0, 0))
# xlim <- c(-1.8, 1.8)
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = sqrt(2))
# phasePortrait(juliaNormal,
# # see documentation of juliaNormal about the arguments
# # c and R_esc
# moreArgs = list(c = -0.09 - 0.649i, R_esc = 2),
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0),
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# res = res)
# par(op)
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# res <- 600
# png("Julia Example 2.png", width = 29.7, height = 29.7/sqrt(2),
# units = "cm", res = res)
# op <- par(mar = c(0, 0, 0, 0))
# xlim <- c(-0.32, 0.02)
# ylim <- ylimFromXlim(xlim, center = -0.78, x_to_y = sqrt(2))
# phasePortrait(juliaNormal,
# # see documentation of juliaNormal about the arguments
# # c and R_esc
# moreArgs = list(c = -0.119 - 0.882i, R_esc = 2),
# nCores = 1, # Increase or leave out for higher performance
# xlim = xlim, ylim = ylim,
# hsvNaN = c(0, 0, 0),
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# res = res)
# par(op)
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# # Map the complex plane on itself, show all bwType options
#
# x11(width = 8, height = 8)
# 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",
# nCores = 2) # Increase or leave out for higher performance
# }
# # Add normal phase portrait for comparison
# phasePortrait("z", xlim = c(-2, 2), ylim = c(-2, 2),
# xlab = "real", ylab = "imaginary",
# pi2Div = 18, # Use same angular division as default
# # in phasePortraitBw
# nCores = 2) # Increase or leave out for higher performance
# par(op)
#
## ----eval=FALSE---------------------------------------------------------------
# # A rational function, show all bwType options
#
# x11(width = 8, height = 8)
# 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",
# nCores = 2) # Increase or leave out for higher performance
# }
# # Add normal phase portrait for comparison
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# xlab = "real", ylab = "imaginary",
# pi2Div = 18, # Use same angular division as default
# # in phasePortraitBw
# nCores = 2) # Increase or leave out for higher performance
# par(op)
#
## ---- eval = FALSE------------------------------------------------------------
# # Set the plot margins at all four sides to 1/5 inch with mai,
# # set the background color to black with bg, and the default foreground
# # color with fg (e.g. for axes and boxes around plots, or the color of
# # the circle outline from the function riemannMask).
# # We catch the previous parameter values in a variable, I called
# # "op" ("old parameters")
# op <- par(mai = c(1/5, 1/5, 1/5, 1/5), bg = "black", fg = "white")
#
# # Make any phase portraits and/or other graphics of your interest
# # ...
#
# # Set the graphical parameters back to the values previously stored in op
# par(op)
## ---- eval = FALSE------------------------------------------------------------
# phasePortrait("tan(z^3 + 1/2 - 2i)/(1 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3),
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
## ---- eval=FALSE--------------------------------------------------------------
# phasePortrait("tan(z^3 + 1/2 - 2i)/(1 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3),
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
# box()
## ---- eval=FALSE--------------------------------------------------------------
# # set background and foreground colors
# op <- par(bg = "black", fg = "white")
# # Setting the parameter fg has an effect on the box, the axes, and the axes'
# # ticks, but not on the axis annotations and axis labels.
# # Also the color of the title (main) is not affected.
# # The colors of these elements have to be set manually and separately. While we
# # could simply set them to "white", we set them, more flexibly, to the
# # current foreground color (par("fg")).
# phasePortrait("tan(z^3 + 1/2 - 2i)/(2 - 1i - z)",
# xlim = c(-6, 6), ylim = c(-3, 3), col.axis = par("fg"),
# xlab = "real", ylab = "imaginary", col.lab = par("fg"),
# main = "All annotation in foreground color", col.main = par("fg"),
# # Adjust text size
# cex.axis = 0.9, cex.lab = 0.9,
# nCores = 2) # Increase or leave out for higher performance
# par(op)
## ---- eval=FALSE--------------------------------------------------------------
# # Open graphics device with 16/9 aspect ratio and 7 inch width
# x11(width = 7, height = 9/16 * 7)
# op <- par(mar = c(0, 0, 0, 0)) # Set plot margins to zero
# xlim <- c(-3, 3)
# # Calculate ylim with desired center fitting the desired aspect ratio
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = 16/9)
# phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/5 + 1/5), pType = "p",
# xlim = xlim, ylim = ylim,
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# nCores = 2) # Increase or leave out for higher performance
# par(op)
## ---- eval=FALSE--------------------------------------------------------------
# # Open graphics device with 16/9 aspect ratio and a width of 7 inches
# x11(width = 7, height = 9/16 * 7)
# # Set plot margins to zero, outer margins to 1/7 inch,
# # and background color to black
# outerMar <- 1/7 # outer margin width in inches
# op <- par(mar = c(0, 0, 0, 0), omi = rep(outerMar, 4), bg = "black")
# xlim <- c(-1.5, 0.5)
# # Calculate ylim with desired center fitting the desired aspect ratio;
# # however, the omi settings slightly change the required
# # ratio of xlim and ylim
# ratio <- (7 - 2*outerMar) / (7 * 9/16 - 2*outerMar)
# ylim <- ylimFromXlim(xlim, centerY = 0, x_to_y = ratio)
# phasePortrait("sin(jacobiTheta(z, tau))/z", moreArgs = list(tau = 1i/5 + 1/5),
# pType = "p",
# xlim = xlim, ylim = ylim,
# xaxs = "i", yaxs = "i",
# axes = FALSE,
# nCores = 1) # Increase or leave out for higher performance
# par(op)
## ---- eval = FALSE------------------------------------------------------------
# # Note that 'FUN =' is not required if the argument to FUN is handed to
# # phasePortrait in the first position
# phasePortrait(FUN = "1/(1 - z^2)", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
# phasePortrait("sin((z - 2)/(z + 2))", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
# phasePortrait("tan(z)", xlim = c(-5, 5), ylim = c(-5, 5), nCores = 2)
## ---- eval = FALSE------------------------------------------------------------
# phasePortrait("-1 * sum(z^c(-k:k))", moreArgs = list(k = 11),
# xlim = c(-2, 2), ylim = c(-1.5, 1.5),
# pType = "p",
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# funString <- "vapply(z, FUN = function(z) {
# n <- 9
# k <- z^(c(1:n))
# rslt <- sum(sin(k))
# return(rslt)
# },
# FUN.VALUE = complex(1))"
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# funString <- "vapply(z, FUN = function(z, fct) {
# n <- 9
# k <- z^(fct * c(1:n))
# rslt <- sum(sin(k))
# return(rslt)
# },
# fct = -1,
# FUN.VALUE = complex(1))"
# phasePortrait(funString, xlim = c(-2, 2), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Define function
# tryThisOne <- function(z, fct, n) {
# k <- z^(fct * c(1:n))
# rslt <- prod(cos(k))
# return(rslt)
# }
#
# # Call function by its name only, provide additional arguments via "moreArgs"
# phasePortrait("tryThisOne", moreArgs = list(fct = 1, n = 5),
# xlim = c(-2.5, 2.5), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the grey area black
# phasePortrait(function(z) {
# for(j in 1:20) {
# z <- z * sin(z) - 1 + 1/2i
# }
# return(z)
# },
# xlim = c(-3, 3), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the grey area black
# phasePortrait(function(z, n) {
# for(j in 1:n) {
# z <- z * cos(z)
# }
# return(z)
# },
# moreArgs = list(n = 27),
# xlim = c(-3, 3), ylim = c(-2, 2),
# nCores = 2) # Increase or leave out for higher performance
## ---- eval = FALSE------------------------------------------------------------
# # atan from package base
# phasePortrait(atan, xlim = c(-pi, pi), ylim = c(-pi, pi),
# nCores = 2)
#
# # gammaz from package pracma (the package must be installed on your machine
# # if you want this example to be working)
# phasePortrait(pracma::gammaz, xlim = c(-9, 9), ylim = c(-5, 5),
# nCores = 2)
#
# # blaschkeProd from this package (moreArgs example)
# # make random vector of zeroes
# n <- 12
# a <- complex(modulus = runif(n), argument = 2 * pi * runif(n))
# # plot the actual phase portrait
# phasePortrait(blaschkeProd, moreArgs = list(a = a),
# xlim = c(-1.3, 1.3), ylim = c(-1.3, 1.3),
# nCores = 2)
#
# # User function example
# tryThisOneToo <- function(z, n, r) {
# for(j in 1:n) {
# z <- r * (z + z^2)
# }
# return(z)
# }
# # Use argument "hsvNaN = c(0, 0, 0)" if you want the gray areas black
# phasePortrait(tryThisOneToo, moreArgs = list(n = 50, r = 1/2 - 1/2i),
# xlim = c(-3, 2), ylim = c(-2.5, 2.5),
# nCores = 2)
#
## ---- eval = FALSE------------------------------------------------------------
# res <- 300 # Define desired resolution in dpi
# png("Logistic_Function.png", width = 40, height = 40 * 3/4,
# units = "cm", res = res)
# phasePortrait("1/(1+exp(-z))", xlim = c(-25, 25), ylim = c(-15, 15), res = res,
# xlab = "real", ylab = "imaginary",
# nCores = 2) # Increase or leave out for higher performance
# dev.off()
## ---- eval=FALSE--------------------------------------------------------------
# switch(1 + trunc(runif(1, 0, 6)),
# "... at all?",
# "... in a quick-and-dirty way?",
# "... in Hadley-Wickham-style?",
# "... without a loop?",
# "... without nested loops?",
# "... in a way somebody can understand?")
## ---- include = FALSE---------------------------------------------------------
foreach::registerDoSEQ()
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