Replace @test_approx_eq_eps a b ε with @test a ≈ b atol=ε

Note that tolerance of the first Anscombe's quartet test in
test/linalg/generic.jl had to be loosened from 10e-5 to 15e-5
because ≈ uses the norm of the difference instead of the maximum
absolute difference, which is a stricter test.

Closes #19899.

test/complex.jl

@@ -101,7 +101,7 @@ end
             @test exp(x) ≈ exp(big(x))
             @test exp10(x) ≈ exp10(big(x))
             @test exp2(x) ≈ exp2(big(x))
-            @test_approx_eq_eps expm1(x) expm1(big(x)) eps(T)
+            @test expm1(x) ≈ expm1(big(x)) atol=eps(T)
             @test log(x) ≈ log(big(x))
             @test log10(x) ≈ log10(big(x))
             @test log1p(x) ≈ log1p(big(x))

test/linalg/arnoldi.jl

@@ -34,7 +34,7 @@ using Base.Test
             @test a*v[:,2] ≈ d[2]*v[:,2]
             @test norm(v) > testtol # eigenvectors cannot be null vectors
             # (d,v) = eigs(a, b, nev=3, tol=1e-8) # not handled yet
-            # @test_approx_eq_eps a*v[:,2] d[2]*b*v[:,2] testtol
+            # @test a*v[:,2] ≈ d[2]*b*v[:,2] atol=testtol
             # @test norm(v) > testtol # eigenvectors cannot be null vectors
 
             (d,v) = eigs(asym, nev=3)
@@ -47,7 +47,7 @@ using Base.Test
             @test eigs(apd; nev=1, sigma=d[3])[1][1] ≈ d[3]
 
             (d,v) = eigs(apd, bpd, nev=3, tol=1e-8)
-            @test_approx_eq_eps apd*v[:,2] d[2]*bpd*v[:,2] testtol
+            @test apd*v[:,2] ≈ d[2]*bpd*v[:,2] atol=testtol
             @test norm(v) > testtol # eigenvectors cannot be null vectors
 
             @testset "(shift-and-)invert mode" begin
@@ -56,7 +56,7 @@ using Base.Test
                 @test norm(v) > testtol # eigenvectors cannot be null vectors
 
                 (d,v) = eigs(apd, bpd, nev=3, sigma=0, tol=1e-8)
-                @test_approx_eq_eps apd*v[:,1] d[1]*bpd*v[:,1] testtol
+                @test apd*v[:,1] ≈ d[1]*bpd*v[:,1] atol=testtol
                 @test norm(v) > testtol # eigenvectors cannot be null vectors
             end
 
@@ -99,7 +99,7 @@ let A6965 = [
     #          0.8  0.1   0.1
     #          0.7  0.1   0.2 ]
     #d,v,nconv = eigs(T6965,nev=1,which=:LM)
-    #@test_approx_eq_eps T6965*v d[1]*v 1e-6
+    # @test T6965*v ≈ d[1]*v atol=1e-6
 end
 
 # Example from Quantum Information Theory

test/linalg/bidiag.jl

@@ -198,7 +198,7 @@ srand(1)
                     Test.test_approx_eq_modphase(u1, u2)
                     Test.test_approx_eq_modphase(v1, v2)
                 end
-                @test_approx_eq_eps 0 vecnorm(u2*diagm(d2)*v2'-Tfull) n*max(n^2*eps(relty),vecnorm(u1*diagm(d1)*v1'-Tfull))
+                @test 0 ≈ vecnorm(u2*diagm(d2)*v2'-Tfull) atol=n*max(n^2*eps(relty),vecnorm(u1*diagm(d1)*v1'-Tfull))
                 @inferred svdvals(T)
                 @inferred svd(T)
             end

test/linalg/bunchkaufman.jl

@@ -57,7 +57,7 @@ bimg  = randn(n,2)/2
                         @test logabsdet(bc1)[2] ≈ sign(det(bc1))
                     end
                     @test inv(bc1)*asym ≈ eye(n)
-                    @test_approx_eq_eps asym*(bc1\b) b 1000ε
+                    @test asym*(bc1\b) ≈ b atol=1000ε
                     @testset for rook in (false, true)
                         @test inv(bkfact(a.'+a, :U, true, rook))*(a.'+a) ≈ eye(n)
                         @test size(bc1) == size(bc1.LD)
@@ -76,7 +76,7 @@ bimg  = randn(n,2)/2
                         @test logabsdet(bc2)[1] ≈ log(abs(det(bc2)))
                         @test logabsdet(bc2)[2] == sign(det(bc2))
                         @test inv(bc2)*apd ≈ eye(n)
-                        @test_approx_eq_eps apd*(bc2\b) b 150000ε
+                        @test apd*(bc2\b) ≈ b atol=150000ε
                         @test ishermitian(bc2) == !issymmetric(bc2)
                     end
                 end

test/linalg/dense.jl

@@ -26,10 +26,10 @@ for elty in (Float32, Float64, Complex64, Complex128)
             a = view(ainit, 1:n, 1:n)
         end
         # cond
-        @test_approx_eq_eps cond(a,1) 4.837320054554436e+02 0.01
-        @test_approx_eq_eps cond(a,2) 1.960057871514615e+02 0.01
-        @test_approx_eq_eps cond(a,Inf) 3.757017682707787e+02 0.01
-        @test_approx_eq_eps cond(a[:,1:5]) 10.233059337453463 0.01
+        @test cond(a,1) ≈ 4.837320054554436e+02 atol=0.01
+        @test cond(a,2) ≈ 1.960057871514615e+02 atol=0.01
+        @test cond(a,Inf) ≈ 3.757017682707787e+02 atol=0.01
+        @test cond(a[:,1:5]) ≈ 10.233059337453463 atol=0.01
         @test_throws ArgumentError cond(a,3)
     end
 end
@@ -75,8 +75,8 @@ debug && println("Solve square general system of equations")
 debug && println("Test nullspace")
             a15null = nullspace(a[:,1:n1]')
             @test rank([a[:,1:n1] a15null]) == 10
-            @test_approx_eq_eps norm(a[:,1:n1]'a15null,Inf) zero(eltya) 300ε
-            @test_approx_eq_eps norm(a15null'a[:,1:n1],Inf) zero(eltya) 400ε
+            @test norm(a[:,1:n1]'a15null,Inf) ≈ zero(eltya) atol=300ε
+            @test norm(a15null'a[:,1:n1],Inf) ≈ zero(eltya) atol=400ε
             @test size(nullspace(b), 2) == 0
             @test nullspace(zeros(eltya,n)) == eye(eltya,1)
         end

test/linalg/diagonal.jl

@@ -47,17 +47,17 @@ srand(1)
         end
 
         for func in (det, trace)
-            @test_approx_eq_eps func(D) func(DM) n^2*eps(relty)*(1+(elty<:Complex))
+            @test func(D) ≈ func(DM) atol=n^2*eps(relty)*(1+(elty<:Complex))
         end
         if relty <: BlasFloat
             for func in (expm,)
-                @test_approx_eq_eps func(D) func(DM) n^3*eps(relty)
+                @test func(D) ≈ func(DM) atol=n^3*eps(relty)
             end
-            @test_approx_eq_eps logm(Diagonal(abs.(D.diag))) logm(abs.(DM)) n^3*eps(relty)
+            @test logm(Diagonal(abs.(D.diag))) ≈ logm(abs.(DM)) atol=n^3*eps(relty)
         end
         if elty <: BlasComplex
             for func in (logdet, sqrtm)
-                @test_approx_eq_eps func(D) func(DM) n^2*eps(relty)*2
+                @test func(D) ≈ func(DM) atol=n^2*eps(relty)*2
             end
         end
     end
@@ -73,18 +73,18 @@ srand(1)
                     U = view(UU, 1:n, 1:2)
                 end
 
-                @test_approx_eq_eps D*v DM*v n*eps(relty)*(1+(elty<:Complex))
-                @test_approx_eq_eps D*U DM*U n^2*eps(relty)*(1+(elty<:Complex))
+                @test D*v ≈ DM*v atol=n*eps(relty)*(1+(elty<:Complex))
+                @test D*U ≈ DM*U atol=n^2*eps(relty)*(1+(elty<:Complex))
 
                 @test U.'*D ≈ U.'*Array(D)
                 @test U'*D ≈ U'*Array(D)
 
                 if relty != BigFloat
-                    @test_approx_eq_eps D\v DM\v 2n^2*eps(relty)*(1+(elty<:Complex))
-                    @test_approx_eq_eps D\U DM\U 2n^3*eps(relty)*(1+(elty<:Complex))
-                    @test_approx_eq_eps A_ldiv_B!(D,copy(v)) DM\v 2n^2*eps(relty)*(1+(elty<:Complex))
-                    @test_approx_eq_eps A_ldiv_B!(D,copy(U)) DM\U 2n^3*eps(relty)*(1+(elty<:Complex))
-                    @test_approx_eq_eps A_ldiv_B!(D,eye(D)) D\eye(D) 2n^3*eps(relty)*(1+(elty<:Complex))
+                    @test D\v ≈ DM\v atol=2n^2*eps(relty)*(1+(elty<:Complex))
+                    @test D\U ≈ DM\U atol=2n^3*eps(relty)*(1+(elty<:Complex))
+                    @test A_ldiv_B!(D,copy(v)) ≈ DM\v atol=2n^2*eps(relty)*(1+(elty<:Complex))
+                    @test A_ldiv_B!(D,copy(U)) ≈ DM\U atol=2n^3*eps(relty)*(1+(elty<:Complex))
+                    @test A_ldiv_B!(D,eye(D)) ≈ D\eye(D) atol=2n^3*eps(relty)*(1+(elty<:Complex))
                     @test_throws DimensionMismatch A_ldiv_B!(D, ones(elty, n + 1))
                     @test_throws SingularException A_ldiv_B!(Diagonal(zeros(relty,n)),copy(v))
                     b = rand(elty,n,n)

test/linalg/eigen.jl

@@ -65,7 +65,7 @@ aimg  = randn(n,n)/2
             f = eigfact(asym_sg, a_sg'a_sg)
             @test asym_sg*f[:vectors] ≈ (a_sg'a_sg*f[:vectors]) * Diagonal(f[:values])
             @test f[:values] ≈ eigvals(asym_sg, a_sg'a_sg)
-            @test_approx_eq_eps prod(f[:values]) prod(eigvals(asym_sg/(a_sg'a_sg))) 200ε
+            @test prod(f[:values]) ≈ prod(eigvals(asym_sg/(a_sg'a_sg))) atol=200ε
             @test eigvecs(asym_sg, a_sg'a_sg) == f[:vectors]
             @test eigvals(f) === f[:values]
             @test eigvecs(f) === f[:vectors]
@@ -86,7 +86,7 @@ aimg  = randn(n,n)/2
             f = eigfact(a1_nsg, a2_nsg)
             @test a1_nsg*f[:vectors] ≈ (a2_nsg*f[:vectors]) * Diagonal(f[:values])
             @test f[:values] ≈ eigvals(a1_nsg, a2_nsg)
-            @test_approx_eq_eps prod(f[:values]) prod(eigvals(a1_nsg/a2_nsg)) 50000ε
+            @test prod(f[:values]) ≈ prod(eigvals(a1_nsg/a2_nsg)) atol=50000ε
             @test eigvecs(a1_nsg, a2_nsg) == f[:vectors]
             @test_throws KeyError f[:Z]
 

test/linalg/generic.jl

@@ -107,17 +107,17 @@ y = linspace(50, 200, 100)
 # Anscombe's quartet (https://en.wikipedia.org/wiki/Anscombe%27s_quartet)
 x123 = [10.0; 8.0; 13.0; 9.0; 11.0; 14.0; 6.0; 4.0; 12.0; 7.0; 5.0]
 y1 = [8.04; 6.95; 7.58; 8.81; 8.33; 9.96; 7.24; 4.26; 10.84; 4.82; 5.68]
-@test_approx_eq_eps [linreg(x123,y1)...] [3.0,0.5] 10e-5
+@test [linreg(x123,y1)...] ≈ [3.0,0.5] atol=15e-5
 
 y2 = [9.14; 8.14; 8.74; 8.77; 9.26; 8.10; 6.12; 3.10; 9.13; 7.26; 4.74]
-@test_approx_eq_eps [linreg(x123,y2)...] [3.0,0.5] 10e-3
+@test [linreg(x123,y2)...] ≈ [3.0,0.5] atol=10e-3
 
 y3 = [7.46; 6.77; 12.74; 7.11; 7.81; 8.84; 6.08; 5.39; 8.15; 6.42; 5.73]
-@test_approx_eq_eps [linreg(x123,y3)...] [3.0,0.5] 10e-3
+@test [linreg(x123,y3)...] ≈ [3.0,0.5] atol=10e-3
 
 x4 = [8.0; 8.0; 8.0; 8.0; 8.0; 8.0; 8.0; 19.0; 8.0; 8.0; 8.0]
 y4 = [6.58; 5.76; 7.71; 8.84; 8.47; 7.04; 5.25; 12.50; 5.56; 7.91; 6.89]
-@test_approx_eq_eps [linreg(x4,y4)...] [3.0,0.5] 10e-3
+@test [linreg(x4,y4)...] ≈ [3.0,0.5] atol=10e-3
 
 # test diag
 let A = eye(4)

test/linalg/lq.jl

@@ -59,18 +59,18 @@ bimg  = randn(n,2)/2
                     @test full(copy(lqa)) ≈ a
                 end
                 @testset "Binary ops" begin
-                    @test_approx_eq_eps a*(lqa\b) b 3000ε
-                    @test_approx_eq_eps lqa*b qra[:Q]*qra[:R]*b 3000ε
-                    @test_approx_eq_eps A_mul_Bc(eye(eltyb,size(q.factors,2)),q)*full(q,thin=false) eye(n) 5000ε
+                    @test a*(lqa\b) ≈ b atol=3000ε
+                    @test lqa*b ≈ qra[:Q]*qra[:R]*b atol=3000ε
+                    @test A_mul_Bc(eye(eltyb,size(q.factors,2)),q)*full(q,thin=false) ≈ eye(n) atol=5000ε
                     if eltya != Int
                         @test eye(eltyb,n)*q ≈ convert(AbstractMatrix{tab},q)
                     end
-                    @test_approx_eq_eps q*b full(q,thin=false)*b 100ε
-                    @test_approx_eq_eps q.'*b full(q,thin=false).'*b 100ε
-                    @test_approx_eq_eps q'*b full(q,thin=false)'*b 100ε
-                    @test_approx_eq_eps a*q a*full(q,thin=false) 100ε
-                    @test_approx_eq_eps a*q.' a*full(q,thin=false).' 100ε
-                    @test_approx_eq_eps a*q' a*full(q,thin=false)' 100ε
+                    @test q*b ≈ full(q,thin=false)*b atol=100ε
+                    @test q.'*b ≈ full(q,thin=false).'*b atol=100ε
+                    @test q'*b ≈ full(q,thin=false)'*b atol=100ε
+                    @test a*q ≈ a*full(q,thin=false) atol=100ε
+                    @test a*q.' ≈ a*full(q,thin=false).' atol=100ε
+                    @test a*q' ≈ a*full(q,thin=false)' atol=100ε
                     @test_throws DimensionMismatch q*b[1:n1 + 1]
                     @test_throws DimensionMismatch Ac_mul_B(q,ones(eltya,n+2,n+2))
                     @test_throws DimensionMismatch ones(eltyb,n+2,n+2)*q

test/linalg/pinv.jl

@@ -92,19 +92,19 @@ 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_approx_eq_eps vecnorm(a*apinv*a-a)/vecnorm(a) 0 tol1
+    @test vecnorm(a*apinv*a-a)/vecnorm(a) ≈ 0 atol=tol1
     x0 = randn(n); b = a*x0; x = apinv*b
-    @test_approx_eq_eps vecnorm(a*x-b)/vecnorm(b) 0 tol1
+    @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_approx_eq_eps vecnorm(a*apinv*a-a)/vecnorm(a) 0 tol2
+    @test vecnorm(a*apinv*a-a)/vecnorm(a) ≈ 0 atol=tol2
     x0 = randn(n); b = a*x0; x = apinv*b
-    @test_approx_eq_eps vecnorm(a*x-b)/vecnorm(b) 0 tol2
+    @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

test/linalg/qr.jl

@@ -52,9 +52,9 @@ debug && println("QR decomposition (without pivoting)")
                 @test full(q, thin=false)'q ≈ eye(n)
                 @test eye(n)'q' ≈ full(q, thin=false)'
                 @test q*r ≈ a
-                @test_approx_eq_eps a*(qra\b) b 3000ε
+                @test a*(qra\b) ≈ b atol=3000ε
                 @test full(qra) ≈ a
-                @test_approx_eq_eps A_mul_Bc(eye(eltyb,size(q.factors,2)),q)*full(q,thin=false) eye(n) 5000ε
+                @test A_mul_Bc(eye(eltyb,size(q.factors,2)),q)*full(q,thin=false) ≈ eye(n) atol=5000ε
                 if eltya != Int
                     @test eye(eltyb,n)*q ≈ convert(AbstractMatrix{tab},q)
                     ac = copy(a)
@@ -69,11 +69,11 @@ debug && println("Thin QR decomposition (without pivoting)")
                 @test q'*full(q, thin=false) ≈ eye(n)
                 @test q'*full(q) ≈ eye(n,n1)
                 @test q*r ≈ a[:,1:n1]
-                @test_approx_eq_eps q*b[1:n1] full(q)*b[1:n1] 100ε
-                @test_approx_eq_eps q*b full(q,thin=false)*b 100ε
+                @test q*b[1:n1] ≈ full(q)*b[1:n1] atol=100ε
+                @test q*b ≈ full(q,thin=false)*b atol=100ε
                 @test_throws DimensionMismatch q*b[1:n1 + 1]
                 @test_throws DimensionMismatch b[1:n1 + 1]*q'
-                @test_approx_eq_eps A_mul_Bc(UpperTriangular(eye(eltyb,size(q.factors,2))),q)*full(q,thin=false) eye(n1,n) 5000ε
+                @test A_mul_Bc(UpperTriangular(eye(eltyb,size(q.factors,2))),q)*full(q,thin=false) ≈ eye(n1,n) atol=5000ε
                 if eltya != Int
                     @test eye(eltyb,n)*q ≈ convert(AbstractMatrix{tab},q)
                 end
@@ -92,7 +92,7 @@ debug && println("(Automatic) Fat (pivoted) QR decomposition")
                 @test q*r ≈ (isa(qrpa,QRPivoted) ? a[1:n1,p] : a[1:n1,:])
                 @test q*r[:,invperm(p)] ≈ a[1:n1,:]
                 @test q*r*qrpa[:P].' ≈ a[1:n1,:]
-                @test_approx_eq_eps a[1:n1,:]*(qrpa\b[1:n1]) b[1:n1] 5000ε
+                @test a[1:n1,:]*(qrpa\b[1:n1]) ≈ b[1:n1] atol=5000ε
                 @test full(qrpa) ≈ a[1:5,:]
                 @test_throws DimensionMismatch q*b[1:n1+1]
                 @test_throws DimensionMismatch b[1:n1+1]*q'
@@ -112,7 +112,7 @@ debug && println("(Automatic) Thin (pivoted) QR decomposition")
                 @test full(qrpa) ≈ a[:,1:5]
                 @test_throws DimensionMismatch q*b[1:n1+1]
                 @test_throws DimensionMismatch b[1:n1+1]*q'
-                @test_approx_eq_eps A_mul_Bc(UpperTriangular(eye(eltyb,size(q.factors,2))),q)*full(q,thin=false) eye(n1,n) 5000ε
+                @test A_mul_Bc(UpperTriangular(eye(eltyb,size(q.factors,2))),q)*full(q,thin=false) ≈ eye(n1,n) atol=5000ε
                 if eltya != Int
                     @test eye(eltyb,n)*q ≈ convert(AbstractMatrix{tab},q)
                 end

test/linalg/triangular.jl

@@ -224,8 +224,8 @@ for elty1 in (Float32, Float64, BigFloat, Complex64, Complex128, Complex{BigFloa
         end
 
         # Determinant
-        @test_approx_eq_eps det(A1) det(lufact(full(A1))) sqrt(eps(real(float(one(elty1)))))*n*n
-        @test_approx_eq_eps logdet(A1) logdet(lufact(full(A1))) sqrt(eps(real(float(one(elty1)))))*n*n
+        @test det(A1) ≈ det(lufact(full(A1))) atol=sqrt(eps(real(float(one(elty1)))))*n*n
+        @test logdet(A1) ≈ logdet(lufact(full(A1))) atol=sqrt(eps(real(float(one(elty1)))))*n*n
 
         # Matrix square root
         @test sqrtm(A1) |> t -> t*t ≈ A1
@@ -237,14 +237,14 @@ for elty1 in (Float32, Float64, BigFloat, Complex64, Complex128, Complex{BigFloa
         if !(elty1 in (BigFloat, Complex{BigFloat})) # Not handled yet
             vals, vecs = eig(A1)
             if (t1 == UpperTriangular || t1 == LowerTriangular) && elty1 != Int # Cannot really handle degenerate eigen space and Int matrices will probably have repeated eigenvalues.
-                @test_approx_eq_eps vecs*diagm(vals)/vecs full(A1) sqrt(eps(float(real(one(vals[1])))))*(norm(A1,Inf)*n)^2
+                @test vecs*diagm(vals)/vecs ≈ full(A1) atol=sqrt(eps(float(real(one(vals[1])))))*(norm(A1,Inf)*n)^2
             end
         end
 
         # Condition number tests - can be VERY approximate
         if elty1 <:BlasFloat
             for p in (1.0, Inf)
-                @test_approx_eq_eps cond(A1,p) cond(A1,p) (cond(A1,p)+cond(A1,p))
+                @test cond(A1,p) ≈ cond(A1,p) atol=(cond(A1,p)+cond(A1,p))
             end
             @test cond(A1,2) == cond(full(A1),2)
         end

test/linalg/tridiag.jl

@@ -173,7 +173,7 @@ for elty in (Float32, Float64, Complex64, Complex128, Int)
         end
 
     # issue #1490
-    @test_approx_eq_eps det(ones(elty,3,3)) zero(elty) 3*eps(real(one(elty)))
+    @test det(ones(elty,3,3)) ≈ zero(elty) atol=3*eps(real(one(elty)))
 
     @test det(SymTridiagonal(elty[],elty[])) == one(elty)
 
@@ -216,7 +216,7 @@ function test_approx_eq_vecs{S<:Real,T<:Real}(a::StridedVecOrMat{S}, b::StridedV
         ev1, ev2 = a[:,i], b[:,i]
         deviation = min(abs(norm(ev1-ev2)),abs(norm(ev1+ev2)))
         if !isnan(deviation)
-            @test_approx_eq_eps deviation 0.0 error
+            @test deviation ≈ 0.0 atol=error
         end
     end
 end
@@ -267,7 +267,7 @@ let n = 12 #Size of matrix problem to test
 
         debug && println("Simple unary functions")
         for func in (det, inv)
-            @test_approx_eq_eps func(A) func(fA) n^2*sqrt(eps(relty))
+            @test func(A) ≈ func(fA) atol=n^2*sqrt(eps(relty))
         end
 
         debug && println("Rounding to Ints")
@@ -385,7 +385,7 @@ let n = 12 #Size of matrix problem to test
 
         debug && println("Simple unary functions")
         for func in (det, inv)
-            @test_approx_eq_eps func(A) func(fA) n^2*sqrt(eps(relty))
+            @test func(A) ≈ func(fA) atol=n^2*sqrt(eps(relty))
         end
 
         debug && println("Rounding to Ints")