feat(matrices): add clement

Author: AnzhiZhang

src/matrices/clement.jl

@@ -0,0 +1,45 @@
+"""
+Clement Matrix
+==============
+The Clement matrix is a tridiagonal matrix with zero
+        diagonal entries. If k = 1, the matrix is symmetric.
+
+*Input options:*
+
++ dim, k: `dim` is the dimension of the matrix.
+        If `k = 0`, the matrix is of type `Tridiagonal`.
+        If `k = 1`, the matrix is of type `SymTridiagonal`.
+
++ dim: `k = 0`.
+
+*References:*
+
+**P. A. Clement**, A class of triple-diagonal
+        matrices for test purposes, SIAM Review, 1 (1959), pp. 50-52.
+"""
+abstract type Clement{T<:Number} <: AbstractMatrix{T} end
+
+# constructors
+Clement(n::Integer) = Clement(n, 0)
+Clement(n::Integer, k::Integer) = Clement{Float64}(n, k)
+Clement{T}(n::Integer) where {T<:Number} = Clement{T}(n, 0)
+function Clement{T}(n::Integer, k::Integer) where {T}
+    # 1x1 case
+    if n == 1
+        return zeros(T, 1, 1)
+    end
+
+    # create the matrix
+    n = n - 1
+    x = T[n:-1:1;]
+    z = T[1:n;]
+    if k == 0
+        return Tridiagonal(x, zeros(T, n + 1), z)
+    else
+        y = sqrt.(x .* z)
+        return SymTridiagonal(zeros(T, n + 1), y)
+    end
+end
+
+# metadata
+@properties Clement [:symmetric, :inverse, :eigen]

src/matrices/index.jl

@@ -9,6 +9,7 @@ export
     ChebSpec,
     Chow,
     Circulant,
+    Clement,
     Companion,
     DingDong,
     Fiedler,
@@ -53,6 +54,7 @@ include("cauchy.jl")
 include("chebspec.jl")
 include("chow.jl")
 include("circulant.jl")
+include("clement.jl")
 include("companion.jl")
 include("dingdong.jl")
 include("fiedler.jl")
@@ -102,6 +104,7 @@ MATRIX_GROUPS[GROUP_BUILTIN] = Set([
     ChebSpec,
     Chow,
     Circulant,
+    Clement,
     Companion,
     DingDong,
     Fiedler,