plot::SparseMatrixplot – sparsity pattern of a matrix

plot::SparseMatrixplot(A) creates a 2D plot with the axes representing the rows and columns of the matrix . For each nonzero entry of a point is plotted, thus displaying sparsity patterns in the matrix.

→ Examples

Calls:

plot::SparseMatrixplot(A, Options)

plot::SparseMatrixplot(A, x = $`x_{min}`$ .. $`x_{max}`$ , y = $`y_{min}`$ .. $`y_{max}`$ , <a = amin .. amax>, Options)

plot::SparseMatrixplot([row1, row2, ...], Options)

plot::SparseMatrixplot([row1, row2, ...], x = $`x_{min}`$ .. $`x_{max}`$ , y = $`y_{min}`$ .. $`y_{max}`$ , <a = amin .. amax>, Options)

Parameters:

A:	a matrix of category Cat::Matrix or an array containing real numerical values or expressions of the animation parameter a. A is equivalent to the attribute Data.
row1, row2, ...:	the matrix rows: each row must be a list of real numerical values or expressions of the animation parameter a. All rows must have the same length. row1, row2, ... is equivalent to the attribute Data.
x:	name of the horizontal coordinate: an identifier or an indexed identifier. It is used as the title of the coordinate axis in direction. x is equivalent to the attribute XName.
$`x_{min}`$ .. $`x_{max}`$ :	the range of the horizontal coordinate: $`x_{min}`$ , $`x_{max}`$ must be numerical real value or expressions of the animation parameter . $`x_{min}`$ .. $`x_{max}`$ is equivalent to the attributes XRange, XMin, XMax.
y:	name of the vertical coordinate: an identifier or an indexed identifier. It is used as the title of the coordinate axis in direction. y is equivalent to the attribute YName.
$`y_{min}`$ .. $`y_{max}`$ :	the range of the vertical coordinate: $`y_{min}`$ , $`y_{max}`$ must be numerical real value or expressions of the animation parameter . $`y_{min}`$ .. $`y_{max}`$ is equivalent to the attributes YRange, YMin, YMax.

Details:

plot::SparseMatrixplot interprets the indices of a matrix as and coordinates, respectively. The indices are ordered according to the standard orientation of the axes, i.e., low matrix indices are found in the lower left corner of the plot.

If $x = `x_{min}`..`x_{max}`$ is specified, the -th column of an matrix corresponds to the coordinate .

If $y = `y_{min}`..`y_{max}`$ is specified, the -th row corresponds to the coordinate .

If no coordinate range is specified, $`x_{min}` = 1$ , $`x_{max}` = n$ , and $`y_{min}` = 1$ , $`y_{max}` = m$ is used, i.e., the coordinate corresponds to the -th column, the coordinate corresponds to the -th row.

A point is plotted for each non-zero matrix entry .
By default, the attribute PointColorType = Flat is used. The color of all points is given by PointColor.

With PointColorType = Dichromatic, a color blend from PointColor to PointColor2 is used to indicate the size of the non-zero matrix entries. The color of points corresponding to small entries is PointColor. Large entries are colored with PointColor2.
Animations are triggered by specifying a range $a = `a_{min}`..`a_{max}`$ for a parameter a that is different from the variables x, y. Thus, in animations, both the ranges $x = `x_{min}`..`x_{max}`$ , $y = `y_{min}`..`y_{max}`$ as well as the animation range $a = `a_{min}`..`a_{max}`$ must be specified.

Example 1

We create a random matrix of dimension with 1000 nonzero entries:

A := matrix::random(100, 200, 1000, frandom):
plot(plot::SparseMatrixplot(A))

MuPAD graphics

With PointColorType = Dichromatic, the color of the points indicates the size of the matrix entries:

plot(plot::SparseMatrixplot(A, PointColorType = Dichromatic)):

MuPAD graphics

delete A:

Example 2

Choosing appropriate coordinate ranges, we let two sparse matrix plots overlap each other. The red points correspond to a sparse matrix with 50 random entries. The blue points indicate the corresponding upper triangular form obtained by Gaussian elimination:

A := matrix::random(10, 10, 50, random(1..5)):
B := A::dom::gaussElim(A)[1]:
plot(plot::SparseMatrixplot(A, x = 1..10, y = 1..10,
                            Color = RGB::Red),
     plot::SparseMatrixplot(B, x = 1.3..10.3, y = 1..10,
                            Color = RGB::Blue,
                            PointStyle = FilledCircles),
     PointSize = 2*unit::mm, Scaling = Constrained,
     Axes = Frame)

MuPAD graphics

delete A, B:

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