feat(matrices): add sampling

Author: AnzhiZhang

Diff for: src/matrices/index.jl

@@ -40,6 +40,7 @@ export
     RandSVD,
     Rohess,
     Rosser,
+    Sampling,
     Toeplitz,
     Triw
 
@@ -81,6 +82,7 @@ include("rando.jl")
 include("randsvd.jl")
 include("rohess.jl")
 include("rosser.jl")
+include("sampling.jl")
 include("toeplitz.jl")
 include("triw.jl")
 
@@ -127,6 +129,7 @@ MATRIX_GROUPS[GROUP_BUILTIN] = Set([
     RandSVD,
     Rohess,
     Rosser,
+    Sampling,
     Toeplitz,
     Triw,
 ])

Diff for: src/matrices/sampling.jl

@@ -0,0 +1,48 @@
+"""
+Matrix with Application in Sampling Theory
+==========================================
+A nonsymmetric matrix with eigenvalues 0, 1, 2, ... n-1.
+
+*Input options:*
+
++ vec: `vec` is a vector with no repeated elements.
+
++ dim: `dim` is the dimension of the matrix.
+            `vec = [1:dim;]/dim`.
+
+*References:*
+
+**L. Bondesson and I. Traat**, A nonsymmetric matrix
+            with integer eigenvalues, linear and multilinear algebra, 55(3)
+            (2007), pp. 239-247
+"""
+struct Sampling{T<:Number} <: AbstractMatrix{T}
+    n::Integer
+    vec::Vector{T}
+
+    function Sampling(vec::Vector{T}) where {T<:Number}
+        n = length(vec)
+        return new{T}(n, vec)
+    end
+end
+
+# constructors
+Sampling(n::Integer) = Sampling{Float64}(n)
+Sampling{T}(n::Integer) where {T<:Number} = Sampling(T[1:n;] / n)
+
+# metadata
+@properties Sampling [:eigen]
+
+# properties
+size(A::Sampling) = (A.n, A.n)
+
+# functions
+@inline Base.@propagate_inbounds function getindex(A::Sampling{T}, i::Integer, j::Integer) where {T}
+    @boundscheck checkbounds(A, i, j)
+    if i == j
+        element = sum([A[i, index] for index = 1:A.n if index != i])
+    else
+        element = A.vec[i] / (A.vec[i] - A.vec[j])
+    end
+    return element
+end