ShenCastanMatrix

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

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

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,},σ},]
gives an array corresponding to an exponential kernel with radius in the i^(th) index direction.

Details and OptionsDetails and Options

  • ShenCastanMatrix[{r,σ}] gives values proportional to  exp(-TemplateBox[{x}, Abs]/b) at x index positions from the center, where b is proportional to σ, 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,σ,f},] constructs a matrix just large enough to include at least a fraction f of the discrete integral of an exponential in each direction.
  • Any of the r, σ, and f can 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.
  • The following options can be specified:
  • 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.
Introduced in 2012
(9.0)