feat(matrices): add parter

Author: AnzhiZhang

Diff for: src/matrices/index.jl

@@ -30,6 +30,7 @@ export
     Moler,
     Neumann,
     Oscillate,
+    Parter,
     Triw
 
 # include all matrices
@@ -60,6 +61,7 @@ include("minij.jl")
 include("moler.jl")
 include("neumann.jl")
 include("oscillate.jl")
+include("parter.jl")
 include("triw.jl")
 
 # matrix groups
@@ -95,6 +97,7 @@ MATRIX_GROUPS[GROUP_BUILTIN] = Set([
     Moler,
     Neumann,
     Oscillate,
+    Parter,
     Triw,
 ])
 MATRIX_GROUPS[GROUP_USER] = Set([])

Diff for: src/matrices/parter.jl

@@ -0,0 +1,41 @@
+"""
+Parter Matrix
+=============
+The Parter matrix is a Toeplitz and Cauchy matrix
+            with singular values near `π`.
+
+*Input options:*
+
++ dim: the dimension of the matrix.
+
+*References:*
+
+The MathWorks Newsletter, Volume 1, Issue 1,
+            March 1986, page 2. S. V. Parter, On the distribution of the
+            singular values of Toeplitz matrices, Linear Algebra and
+            Appl., 80 (1986), pp. 115-130.
+"""
+struct Parter{T<:Number} <: AbstractMatrix{T}
+    n::Integer
+
+    function Parter{T}(n::Integer) where {T<:Number}
+        n >= 0 || throw(ArgumentError("$n < 0"))
+        return new{T}(n)
+    end
+end
+
+# constructors
+Parter(n::Integer) = Parter{Int}(n)
+Parter{T}(n::Integer) where {T<:Integer} = Parter{Rational{T}}(n)
+
+# metadata
+@properties Parter [:eigen]
+
+# properties
+size(A::Parter) = (A.n, A.n)
+
+# functions
+@inline Base.@propagate_inbounds function getindex(A::Parter{T}, i::Integer, j::Integer) where {T}
+    @boundscheck checkbounds(A, i, j)
+    return one(T) / T(i - j + 0.5)
+end