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ComplexArrayPlot—Wolfram Documentation

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ComplexArrayPlot[array]

generates a plot in which complex values z_ij in an array array are shown in a discrete array of squares with Arg[z_ij] indicated by color and Abs[z_ij] by shading.

• ComplexArrayPlot is used to visualize tables of complex numbers.
• ComplexArrayPlot[array] by default arranges successive rows of the mn dimensional array down the page and successive columns across, just as a table or grid would normally be formatted.
• ComplexArrayPlot uses a cyclic color function over the values of Arg[z_ij] and shading to indicate values of Abs[z_ij]. Complex numbers with large moduli are lighter than numbers with smaller moduli.
• In ComplexArrayPlot[array], array can have the following forms:
• {{z₁₁,z₁₂,…,z_1n},…,{z_m1,z_m2,…,z_mn}} values z_ij SparseArray values as a normal array NumericArray values as a normal array Dataset values as a normal array
• The following special entries can be used for the z_ij:
• None background color color directive specified color
• If array is ragged, shorter rows are treated as padded on the right with background.
• ComplexArrayPlot has the same options as Graphics, with the following additions and changes: [List of all options]
• The rules given by ColorRules are applied to the value of each cell. The rules can involve patterns.
• If none of the rules in ColorRules apply, then ColorFunction is used to determine the color.
• With the default setting ColorRules->Automatic, an explicit setting ColorFunctioncfunc is used instead of ColorRules.
• ColorFunction->{cfunc,sfunc} uses cfunc to generate the base color and sfunc to adjust the color to highlight features.
• Possible named settings for sfunc are:
• Functions in ColorFunction are by default supplied with scaled versions of Re[z], Im[z], Abs[z], Arg[z].
• If the color determined for a particular cell is None, the cell is rendered in the background color.
• If no color is determined for a particular cell, the cell is rendered in a default dark red color.
• With DataReversed->True, the order of rows is reversed, so that rows run from bottom to top, with the last row at the top.
• With the setting FrameTicks->Automatic, ticks are placed at round integers, typically multiples of 5 or 10.
• With the setting FrameTicks->All, ticks are also placed at the minimum and maximum and .
• In explicit FrameTicks specifications, the tick coordinates are taken to refer to and .
• PlotRange can take the following forms:
• z_max show Abs[z_ij] values between 0 and z_max {z_min,z_max} show Abs[z_ij] values between z_min and z_max {range_i,range_j} show z_ij values with i in range_i and j in range_j {range_i,range_j,range_z} show z_ij values with i in range_i, j in range_j, and z_ij in range_z
• The array index ranges range_i and range_j can take the following forms:
• {min,max} include indices between min and max All include all indices
• Typical settings for PlotLegends include:
• None no legend Automatic automatically determine legend Placed[lspec,…] specify placement for legend
• Mesh->True draws mesh lines between each cell in the array.
• Mesh->{m_i,m_j} gives mesh specifications for the and directions, respectively.
• With the default setting Frame->Automatic, a frame is drawn only when Mesh->False.
• For purposes of combining with other graphics, the array element is taken to cover a unit square centered at coordinate position , .
• A setting DataRange->{{x_min,x_max},{y_min,y_max}} specifies that the centers of successive cells should be at equally spaced positions between x_min and x_max in the horizontal direction, and y_min and y_max in the vertical direction. With the default setting DataReversed->False, is centered at {x_min,y_max}.
• With the default setting DataRange->All and DataReversed->False, the array element will be taken to cover a unit square centered at coordinate position , .

Plot an array of numbers:

Give explicit color directives to specify colors for individual cells:

Specify overall color rules:

Include a mesh:

Plot a table of data:

Use a standard blend as a color function:

By default, colors change with argument from (cyan) to (red) to (cyan):

By default, colors are lighter for complex numbers with larger moduli:

Unknown, symbolic or missing values are shown dark red:

Specify explicit colors for cells:

Plot a ragged array, padding on the right:

Cells with value None are rendered like the background:

Plot a sparse array:

Plot a numeric array:

Add a mesh:

Add a legend:

Turn off the default shading:

Change the color function:

Use shading schemes like ComplexPlot:

Choose a color function and a shading scheme:

By default, the ratio of the height to width for the plot is determined automatically:

Use numerical value to specify the height to width ratio:

Make the height the same as the width with AspectRatio1:

AspectRatioFull adjusts the height and width to tightly fit inside other constructs:

By default, ComplexArrayPlot uses a frame instead of axes:

Use axes instead of a frame:

Use AxesOrigin to specify where the axes intersect:

Turn each axis on individually:

No axes labels are drawn by default:

Place a label on the axis:

Specify axes labels:

The position of the axes is determined automatically:

Specify an explicit origin for the axes:

Change the style of the axes:

Specify a style for each axis:

Use different styles for the ticks and the axes:

Use different styles for the labels and the axes:

Background is normally visible only around the edges:

The background "shows through" whenever an explicit entry is None:

Background also by default shows through for values outside the plot range:

ClippingStyle overrides background color:

The default is to show values of with values outside the plot range in the background color:

Show values outside the plot range in red:

Show low values in black and high values in red:

By default, complex values are colored by Arg[z] and shaded by Abs[z]:

Turn off the shading based on Abs[z]:

Map modulus values from 0 to 1 onto colors according to Hue:

Use a pure function as the color function:

Use a named color gradient from ColorData:

Specify a shading scheme:

Specify a color function and a shading scheme:

With ColorFunctionScalingTrue, the values are first scaled to lie between 0 and 1:

Use a color function that colors complex values by Re[z], Im[z], Abs[z] or Arg[z]:

By default, values are scaled to lie between 0 to 1 before applying the color function:

Choose to not scale the argument prior to applying the color function:

Some color functions are not cyclic so the colors only change in a small band:

Specify color rules for explicit values:

Specify color rules for patterns:

ColorFunction is used if no color rules apply:

The array can contain symbolic values:

Implement a "default color" by adding a rule for _:

Rules are used in the order given:

By default, data is plotted at integer coordinates:

Specify a range for the data:

Specify a range for the data with a pair of complex numbers:

Specify the bottom-left corner of the data range:

Specify the top-right corner of the data range:

Reverse the order of rows:

The frame ticks give the original row numbers:

Reverse the order of rows and columns:

Reverse the order of columns:

Use Epilog to superimpose other graphics:

Epilog uses the standard Graphics coordinate system:

The graphics can be translucent:

Use MaxPlotPoints to limit the number of elements explicitly plotted in each direction:

Insert mesh lines between all cells:

Insert 1 row mesh line and 3 column mesh lines:

Insert mesh lines around the first 4 columns:

Specify styles for the mesh lines:

Style the mesh lines:

Style the row lines differently than the column lines:

No legend is used by default:

Generate a legend automatically:

PlotLegends automatically picks up a named ColorFunction:

PlotLegends does not pick up a user-defined ColorFunction, but the user can still construct an appropriate legend:

Use Placed to change the location of the legend:

Plot all elements:

Plot values of for which 1≤Abs[z]≤2:

Plot values of for which Abs[z]≤2:

The first two entries in PlotRange specify the range of rows and columns to include:

Use a theme with detailed ticks and a legend:

Move the legend below the plot:

Generate a pure discrete two-dimensional sinusoid:

Compute the two-dimensional Fourier transform:

Display the original data and its Fourier transform. In the second plot, the bright square in row 4 and column 6 (measured from the top-left corner) tells us that the corresponding frequencies are 3 and 5, respectively:

Generate a linear combination of discrete sinusoids:

Identify the component frequencies in the Fourier transform:

Display two-dimensional data and its Fourier transform:

Use ColorRules to threshold a Fourier transform:

Display only the top-left quarter of the Fourier transform:

Model the intensity of light diffracted by a small circular aperture:

Display the diffraction pattern and its Fourier transform:

Create data and modify it by shifting the columns left:

The magnitude of Fourier transforms are graphed, but that does not reveal potentially useful phase information. Observe the extra information in the middle plot:

Note the change in the ComplexArrayPlot and the lack of change in the ArrayPlot of the Fourier transform:

Plot a sparse matrix with complex entries:

Plot a random complex-valued matrix:

Visually verify that the matrix is diagonalized:

Visualize the matrices in a matrix factorization:

Visually compare Kronecker products:

Define a complex valued function with roots at 1, ±^{2π /3}:

Compute the corresponding Newton map:

Compute approximations to the roots of for various starting values:

Display the basins of attraction:

Define a complex-valued function with an infinite number of roots, ,:

Compute the corresponding Newton map:

Note that the roots only have three distinct argument values, and :

Compute approximations to the roots of for various starting values:

Display the basins of attraction, using shading to highlight the dependence on the modulus of the roots:

Define a complex-valued function with roots at 1, ±^{2π /3}:

Newton's method has quadratic convergence, but Halley's method has cubic convergence:

Compute approximations to the roots of for various starting values:

Display the basins of attraction for Halley's method:

Display eigenvalues from a two-parameter family of matrices:

Define a matrix:

Compute its eigenvalues:

The ϵ-pseudospectrum of the matrix is the set of values in the complex plane for which the norm of (m-λ I)^-1 is greater than . Graph the ϵ-pseudospectra for ϵ=1,1/2,1/4,1/8,1/16,1/32 and note the eigenvalues at the white dots:

If the largest eigenvalue is used instead of the norm of (m-λ I)^-1, then argument information can be added:

Create a Toeplitz matrix:

Make a plot similar to an ϵ-pseudospectral plot:

Use a named matrix:

Make a graph similar to a spectral portrait:

Visualize complex cellular automata:

Compute iterates of a complex-valued function:

Display the data on a width of 10:

The iterates appear to converge for a different constant:

Iterates are periodic for carefully chosen constants:

Let be the logistic function with a complex coefficient . Consider the sequence , for . Approximate the values of for which is unbounded and visualize the region:

Highlight Gaussian primes in black:

Plot a Fourier matrix:

Highlight the purely real and purely imaginary entries in black and gray, respectively:

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