Fix condskeel definition (#34512)

* Fix condskeel definition

Skeel's condition number is a relative condition number, however the definition of `condskeel(A,x)` missed to divide by the norm of `x`. The definition is for example in Higham "Accuracy and stability of numerical algorithms" eq. (7.13) (p. 123).

* Test condskeel definition fix

Author: saschatimme

stdlib/LinearAlgebra/src/generic.jl

@@ -1125,7 +1125,7 @@ condskeel(A::AbstractMatrix, p::Real=Inf) = opnorm(abs.(inv(A))*abs.(A), p)
 
 ```math
 \\kappa_S(M, p) = \\left\\Vert \\left\\vert M \\right\\vert \\left\\vert M^{-1} \\right\\vert \\right\\Vert_p \\\\
-\\kappa_S(M, x, p) = \\left\\Vert \\left\\vert M \\right\\vert \\left\\vert M^{-1} \\right\\vert \\left\\vert x \\right\\vert \\right\\Vert_p
+\\kappa_S(M, x, p) = \\frac{\\left\\Vert \\left\\vert M \\right\\vert \\left\\vert M^{-1} \\right\\vert \\left\\vert x \\right\\vert \\right\\Vert_p}{\\left \\Vert x \\right \\Vert_p}
 ```
 
 Skeel condition number ``\\kappa_S`` of the matrix `M`, optionally with respect to the
@@ -1137,7 +1137,9 @@ Valid values for `p` are `1`, `2` and `Inf` (default).
 This quantity is also known in the literature as the Bauer condition number, relative
 condition number, or componentwise relative condition number.
 """
-condskeel(A::AbstractMatrix, x::AbstractVector, p::Real=Inf) = norm(abs.(inv(A))*(abs.(A)*abs.(x)), p)
+function condskeel(A::AbstractMatrix, x::AbstractVector, p::Real=Inf)
+    norm(abs.(inv(A))*(abs.(A)*abs.(x)), p) / norm(x, p)
+end
 
 issymmetric(A::AbstractMatrix{<:Real}) = ishermitian(A)
 

stdlib/LinearAlgebra/test/generic.jl

@@ -462,4 +462,9 @@ end
     end
 end
 
+@testset "condskeel #34512" begin
+    A = rand(3, 3)
+    @test condskeel(A) ≈ condskeel(A, [8,8,8])
+end
+
 end # module TestGeneric