ShenCastanMatrix

ShenCastanMatrix[r]
gives a matrix that corresponds to an exponential kernel of radius r.

ShenCastanMatrix[{r, }]
gives a matrix corresponding to an exponential kernel with radius r and region of support specified by .

ShenCastanMatrix[r, {n1, n2}]
gives a matrix formed from the ^(th) derivative of the exponential with respect to rows and the ^(th) derivative with respect to columns.

ShenCastanMatrix[r, {{n11, n12}, {n21, n22}, ...}]
gives a matrix formed from the sums of the and derivatives.

ShenCastanMatrix[{{r1, r2, ...}, }, ...]
gives an array corresponding to an exponential kernel with radius in the i^(th) index direction.

Details and OptionsDetails and Options

  • ShenCastanMatrix[{r, }] gives values proportional to  exp(-TemplateBox[{x}, Abs]/b) at x index positions from the center, where b is proportional to , so that a value of gives approximately 95% of the total area under the exponential.
  • ShenCastanMatrix[r] uses .
  • By default, the elements of ShenCastanMatrix[r] sum to 1.
  • ShenCastanMatrix[..., {n1, n2}] constructs derivatives as finite differences.
  • ShenCastanMatrix[{Automatic, , f}, ...] constructs a matrix just large enough to include at least a fraction f of the discrete integral of an exponential in each direction.
  • ShenCastanMatrix allows any of r, , and f to be lists, specifying different values for different directions.
  • For integer r, ShenCastanMatrix[r, ...] yields a × matrix.
  • For noninteger r, the value of r is effectively rounded to an integer.
  • Options for ShenCastanMatrix include:
  • WorkingPrecisionAutomaticthe precision with which to compute matrix elements
    "Standardization"Truewhether to rescale and shift the matrix to account for truncation
  • With "Standardization"->True, the elements of ShenCastanMatrix[r] will sum to 1. However, the elements of ShenCastanMatrix[r, {n1, n2, ...}] with at least one nonzero will sum to 0, and the sum of the elements, weighted in each direction by times the distance from the origin to the power of , will be 1.

ExamplesExamplesopen allclose all

Basic Examples (3)Basic Examples (3)

Exponential matrix of radius 20:

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MatrixPlot of an exponential matrix:

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1D exponential vector:

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