pinv.jl
# This file is a part of Julia. License is MIT: http://julialang.org/license
#
# Test the pseudo-inverse
#
debug = false
using Base.Test
function hilb(T::Type, n::Integer)
a=Array{T}(n,n)
for i=1:n
for j=1:n
a[j,i]=one(T)/(i+j-one(T))
end
end
return a
end
hilb(n::Integer) = hilb(Float64,n)
function hilb(T::Type, m::Integer, n::Integer)
a=Array{T}(m,n)
for i=1:n
for j=1:m
a[j,i]=one(T)/(i+j-one(T))
end
end
return a
end
hilb(m::Integer, n::Integer) = hilb(Float64,m,n)
function onediag(T::Type, m::Integer, n::Integer)
a=zeros(T,m,n)
for i=1:min(n,m)
a[i,i]=one(T)/(float(i)^5)
end
a[1,1] = 0
a[min(m,n),min(m,n)] = 0
return a
end
onediag(m::Integer, n::Integer) = onediag(Float64, m::Integer, n::Integer)
function onediag_sparse(T::Type, n::Integer)
a=zeros(T,n)
for i=1:n
a[i]=one(T)/(float(i)^5)
end
a[1] = 0
a[n] = 0
return Diagonal(a)
end
onediag_sparse(n::Integer) = onediag_sparse(Float64, n::Integer)
function tridiag(T::Type, m::Integer, n::Integer)
a=zeros(T,m,n)
for i=1:min(n,m)
a[i,i]=one(T)/(float(i)^5)
end
for i=1:min(n,m)-1
a[i+1,i]=2*one(T)/(float(i)^5)
a[1,i+1]=2*one(T)/(float(i)^5)
end
return a
end
tridiag(m::Integer, n::Integer) = tridiag(Float64, m::Integer, n::Integer)
function randn_float64(m::Integer, n::Integer)
a=randn(m,n)
b=Array{Float64}(m,n)
for i=1:n
for j=1:m
b[j,i]=convert(Float64,a[j,i])
end
end
return b
end
function randn_float32(m::Integer, n::Integer)
a=randn(m,n)
b=Array{Float32}(m,n)
for i=1:n
for j=1:m
b[j,i]=convert(Float32,a[j,i])
end
end
return b
end
function test_pinv(a,m,n,tol1,tol2,tol3)
debug && println("=== julia/matlab pinv, default tol=eps(1.0)*max(size(a)) ===")
apinv = @inferred pinv(a)
@test vecnorm(a*apinv*a-a)/vecnorm(a) ≈ 0 atol=tol1
x0 = randn(n); b = a*x0; x = apinv*b
@test vecnorm(a*x-b)/vecnorm(b) ≈ 0 atol=tol1
debug && println(vecnorm(a*apinv*a - a)/vecnorm(a))
debug && println(vecnorm(a*x-b)/vecnorm(b))
debug && println("=== julia pinv, tol=sqrt(eps(1.0)) ===")
apinv = pinv(a,sqrt(eps(real(one(eltype(a))))))
@test vecnorm(a*apinv*a-a)/vecnorm(a) ≈ 0 atol=tol2
x0 = randn(n); b = a*x0; x = apinv*b
@test vecnorm(a*x-b)/vecnorm(b) ≈ 0 atol=tol2
debug && println(vecnorm(a*apinv*a - a)/vecnorm(a))
debug && println(vecnorm(a*x-b)/vecnorm(b))
end
srand(12345)
let
for eltya in (Float64, Complex128)
debug && println("\n\n<<<<<", eltya, ">>>>>")
m = 1000
n = 100
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float64(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e-2,1e-5,1e-5)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-5,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
m = 100
n = 100
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float64(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e-2,1e-5,1e-5)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-5,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
m = 100
n = 1000
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float64(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e-2,1e-5,1e-5)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-5,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
end
end
for eltya in (Float32, Complex64)
debug && println("\n\n<<<<<", eltya, ">>>>>")
m = 1000
n = 100
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float32(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e0,1e-2,1e-2)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-2,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
m = 100
n = 100
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float32(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e0,1e-2,1e-2)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-2,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
m = 100
n = 1000
debug && println("\n\n n = ", n, ", m = ",m)
default_tol = (real(one(eltya))) * max(m,n) * 10
debug && println("\n--- dense/ill-conditioned matrix ---\n")
### a = randn_float32(m,n) * hilb(eltya,n)
a = hilb(eltya,m,n)
test_pinv(a,m,n,1e0,1e-2,1e-2)
debug && println("\n--- dense/diagonal matrix ---\n")
a = onediag(eltya,m,n)
test_pinv(a,m,n,default_tol,default_tol,default_tol)
debug && println("\n--- dense/tri-diagonal matrix ---\n")
a = tridiag(eltya,m,n)
test_pinv(a,m,n,default_tol,1e-2,default_tol)
debug && println("\n--- Diagonal matrix ---\n")
a = onediag_sparse(eltya,m)
test_pinv(a,m,m,default_tol,default_tol,default_tol)
end
# test zero matrices
for eltya in (Float32, Float64, Complex64, Complex128)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- zero constant ---")
a = pinv(zero(eltya))
@test a ≈ 0.0
end
for eltya in (Float32, Float64, Complex64, Complex128)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- zero vector ---")
a = pinv([zero(eltya); zero(eltya)])
@test a[1] ≈ 0.0
@test a[2] ≈ 0.0
end
for eltya in (Float32, Float64, Complex64, Complex128)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- zero Diagonal matrix ---")
a = pinv(Diagonal([zero(eltya); zero(eltya)]))
@test a.diag[1] ≈ 0.0
@test a.diag[2] ≈ 0.0
end
# test sub-normal matrices
for eltya in (Float32, Float64)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- sub-normal constant ---")
a = pinv(realmin(eltya)/100)
@test a ≈ 0.0
debug && println("\n\n<<<<<Complex{", eltya, "}>>>>>")
debug && println("\n--- sub-normal constant ---")
a = pinv(realmin(eltya)/100*(1+1im))
@test a ≈ 0.0
end
for eltya in (Float32, Float64)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- sub-normal vector ---")
a = pinv([realmin(eltya); realmin(eltya)]/100)
@test a[1] ≈ 0.0
@test a[2] ≈ 0.0
debug && println("\n\n<<<<<Complex{", eltya, "}>>>>>")
debug && println("\n--- sub-normal vector ---")
a = pinv([realmin(eltya); realmin(eltya)]/100*(1+1im))
@test a[1] ≈ 0.0
@test a[2] ≈ 0.0
end
for eltya in (Float32, Float64)
debug && println("\n\n<<<<<", eltya, ">>>>>")
debug && println("\n--- sub-normal Diagonal matrix ---")
a = pinv(Diagonal([realmin(eltya); realmin(eltya)]/100))
@test a.diag[1] ≈ 0.0
@test a.diag[2] ≈ 0.0
debug && println("\n\n<<<<<Complex{", eltya, "}>>>>>")
debug && println("\n--- sub-normal Diagonal matrix ---")
a = pinv(Diagonal([realmin(eltya); realmin(eltya)]/100*(1+1im)))
@test a.diag[1] ≈ 0.0
@test a.diag[2] ≈ 0.0
end