diff --git a/.gitignore b/.gitignore
index 4f2cc4b7..73664508 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,5 +1,6 @@
 *.jl.*.cov
 *.jl.cov
 *.jl.mem
+.*.swp
 Manifest.toml
 docs/build/
diff --git a/Project.toml b/Project.toml
index d8ea7ab4..254377c9 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "MatrixAlgebraKit"
 uuid = "6c742aac-3347-4629-af66-fc926824e5e4"
 authors = ["Jutho <jutho.haegeman@ugent.be> and contributors"]
-version = "0.1.1"
+version = "0.1.2"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
@@ -18,6 +18,7 @@ ChainRulesCore = "1"
 ChainRulesTestUtils = "1"
 JET = "0.9"
 LinearAlgebra = "1"
+SafeTestsets = "0.1"
 StableRNGs = "1"
 Test = "1"
 TestExtras = "0.2,0.3"
@@ -28,10 +29,11 @@ julia = "1.10"
 Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
 ChainRulesTestUtils = "cdddcdb0-9152-4a09-a978-84456f9df70a"
 JET = "c3a54625-cd67-489e-a8e7-0a5a0ff4e31b"
+SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
 StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 TestExtras = "5ed8adda-3752-4e41-b88a-e8b09835ee3a"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 
 [targets]
-test = ["Aqua", "JET", "Test", "TestExtras","ChainRulesCore", "ChainRulesTestUtils", "StableRNGs", "Zygote"]
+test = ["Aqua", "JET", "SafeTestsets", "Test", "TestExtras","ChainRulesCore", "ChainRulesTestUtils", "StableRNGs", "Zygote"]
diff --git a/ext/MatrixAlgebraKitChainRulesCoreExt.jl b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
index 5ef1fd9b..e21842c6 100644
--- a/ext/MatrixAlgebraKitChainRulesCoreExt.jl
+++ b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
@@ -176,4 +176,4 @@ function ChainRulesCore.rrule(::typeof(right_polar!), A::AbstractMatrix, PWᴴ,
     return PWᴴ, right_polar_pullback
 end
 
-end
\ No newline at end of file
+end
diff --git a/src/algorithms.jl b/src/algorithms.jl
index ea260aa2..fc247317 100644
--- a/src/algorithms.jl
+++ b/src/algorithms.jl
@@ -178,4 +178,4 @@ macro check_size(x, sz, size=:size)
                           string($sz)
                    szx == $sz || throw(DimensionMismatch($err))
                end)
-end
\ No newline at end of file
+end
diff --git a/src/implementations/orthnull.jl b/src/implementations/orthnull.jl
index e134121b..c34df721 100644
--- a/src/implementations/orthnull.jl
+++ b/src/implementations/orthnull.jl
@@ -215,4 +215,4 @@ function right_null!(A::AbstractMatrix, Nᴴ; kwargs...)
     else
         throw(ArgumentError("`right_null!` received unknown value `kind = $kind`"))
     end
-end
\ No newline at end of file
+end
diff --git a/src/implementations/truncation.jl b/src/implementations/truncation.jl
index f135df10..73b59c97 100644
--- a/src/implementations/truncation.jl
+++ b/src/implementations/truncation.jl
@@ -11,10 +11,17 @@ function TruncationStrategy(; atol=nothing, rtol=nothing, maxrank=nothing)
     if isnothing(maxrank) && isnothing(atol) && isnothing(rtol)
         return NoTruncation()
     elseif isnothing(maxrank)
-        @assert isnothing(rtol) "TODO: rtol"
-        return trunctol(atol)
+        atol = @something atol 0
+        rtol = @something rtol 0
+        return TruncationKeepAbove(atol, rtol)
     else
-        return truncrank(maxrank)
+        if isnothing(atol) && isnothing(rtol)
+            return truncrank(maxrank)
+        else
+            atol = @something atol 0
+            rtol = @something rtol 0
+            return truncrank(maxrank) & TruncationKeepAbove(atol, rtol)
+        end
     end
 end
 
@@ -82,6 +89,28 @@ Truncation strategy to discard the values that are larger than `atol` in absolut
 """
 truncabove(atol) = TruncationKeepFiltered(≤(atol) ∘ abs)
 
+"""
+    TruncationComposition(trunc1::TruncationStrategy, trunc2::TruncationStrategy)
+
+Compose two truncation strategies, keeping values common between the two strategies.
+"""
+struct TruncationComposition{T<:Tuple{Vararg{TruncationStrategy}}} <:
+       TruncationStrategy
+    components::T
+end
+function Base.:&(trunc1::TruncationStrategy, trunc2::TruncationStrategy)
+    return TruncationComposition((trunc1, trunc2))
+end
+function Base.:&(trunc1::TruncationComposition, trunc2::TruncationComposition)
+    return TruncationComposition((trunc1.components..., trunc2.components...))
+end
+function Base.:&(trunc1::TruncationComposition, trunc2::TruncationStrategy)
+    return TruncationComposition((trunc1.components..., trunc2))
+end
+function Base.:&(trunc1::TruncationStrategy, trunc2::TruncationComposition)
+    return TruncationComposition((trunc1, trunc2.components...))
+end
+
 # truncate!
 # ---------
 # Generic implementation: `findtruncated` followed by indexing
@@ -147,6 +176,11 @@ function findtruncated(values::AbstractVector, strategy::TruncationKeepAbove)
     return 1:i
 end
 
+function findtruncated(values::AbstractVector, strategy::TruncationComposition)
+    inds = map(Base.Fix1(findtruncated, values), strategy.components)
+    return intersect(inds...)
+end
+
 """
     TruncatedAlgorithm(alg::AbstractAlgorithm, trunc::TruncationAlgorithm)
 
diff --git a/src/interface/orthnull.jl b/src/interface/orthnull.jl
index de7c3153..236184bd 100644
--- a/src/interface/orthnull.jl
+++ b/src/interface/orthnull.jl
@@ -62,7 +62,7 @@ of `kind`.
     the input matrix `A`. Always use the return value of the function as it may not always be
     possible to use the provided `CV` as output.
 
-See also [`right_orth(!)`](@ref right_orth), [`left_orth(!)`](@ref left_orth), [`right_orth(!)`](@ref right_orth)
+See also [`right_orth(!)`](@ref right_orth), [`left_null(!)`](@ref left_null), [`right_null(!)`](@ref right_null)
 """
 function left_orth end
 function left_orth! end
@@ -117,7 +117,7 @@ of `kind`.
     the input matrix `A`. Always use the return value of the function as it may not always be
     possible to use the provided `CVᴴ` as output.
 
-See also [`left_orth(!)`](@ref left_orth), [`left_orth(!)`](@ref left_orth), [`right_orth(!)`](@ref right_orth)
+See also [`left_orth(!)`](@ref left_orth), [`left_null(!)`](@ref left_null), [`right_null(!)`](@ref right_null)
 """
 function right_orth end
 function right_orth! end
@@ -224,4 +224,4 @@ function right_null!(A::AbstractMatrix; kwargs...)
 end
 function right_null(A::AbstractMatrix; kwargs...)
     return right_null!(copy_input(right_null, A); kwargs...)
-end
\ No newline at end of file
+end
diff --git a/src/pullbacks/polar.jl b/src/pullbacks/polar.jl
index ed6fc17d..2eea389e 100644
--- a/src/pullbacks/polar.jl
+++ b/src/pullbacks/polar.jl
@@ -58,4 +58,4 @@ function right_polar_pullback!(ΔA::AbstractMatrix, PWᴴ, ΔPWᴴ)
         ΔA .+= PΔWᴴ
     end
     return ΔA
-end
\ No newline at end of file
+end
diff --git a/test/chainrules.jl b/test/chainrules.jl
index 47de1c04..ab895a4a 100644
--- a/test/chainrules.jl
+++ b/test/chainrules.jl
@@ -1,6 +1,10 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
 using ChainRulesCore, ChainRulesTestUtils, Zygote
 using MatrixAlgebraKit: diagview, TruncatedAlgorithm, PolarViaSVD
-using LinearAlgebra: UpperTriangular, Diagonal, Hermitian
+using LinearAlgebra: UpperTriangular, Diagonal, Hermitian, mul!
 
 function remove_svdgauge_depence!(ΔU, ΔVᴴ, U, S, Vᴴ;
                                   degeneracy_atol=MatrixAlgebraKit.default_pullback_gaugetol(S))
@@ -356,4 +360,4 @@ end
         test_rrule(config, right_null, A; fkwargs=(; kind=:lqpos), output_tangent=ΔNᴴ,
                    atol=atol, rtol=rtol, rrule_f=rrule_via_ad, check_inferred=false)
     end
-end
\ No newline at end of file
+end
diff --git a/test/eig.jl b/test/eig.jl
index da9c3961..d0dc13e3 100644
--- a/test/eig.jl
+++ b/test/eig.jl
@@ -1,3 +1,10 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: Diagonal
+using MatrixAlgebraKit: diagview
+
 @testset "eig_full! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
     m = 54
diff --git a/test/eigh.jl b/test/eigh.jl
index 8967f665..5a3c5a8a 100644
--- a/test/eigh.jl
+++ b/test/eigh.jl
@@ -1,3 +1,10 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: LinearAlgebra, Diagonal, I
+using MatrixAlgebraKit: diagview
+
 @testset "eigh_full! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
     m = 54
diff --git a/test/lq.jl b/test/lq.jl
index b1643080..10375e50 100644
--- a/test/lq.jl
+++ b/test/lq.jl
@@ -1,3 +1,9 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: diag, I
+
 @testset "lq_compact! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
     m = 54
diff --git a/test/orthnull.jl b/test/orthnull.jl
index 7b926b59..0a93c593 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -1,3 +1,9 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: LinearAlgebra, I
+
 @testset "left_orth and left_null for T = $T" for T in (Float32, Float64, ComplexF32,
                                                         ComplexF64)
     rng = StableRNG(123)
@@ -209,4 +215,4 @@ end
             end
         end
     end
-end
\ No newline at end of file
+end
diff --git a/test/polar.jl b/test/polar.jl
index 064e0e59..513654ea 100644
--- a/test/polar.jl
+++ b/test/polar.jl
@@ -1,3 +1,8 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: LinearAlgebra, I, isposdef
 using MatrixAlgebraKit: PolarViaSVD
 
 @testset "left_polar! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
diff --git a/test/qr.jl b/test/qr.jl
index 9fb96408..81c64f0e 100644
--- a/test/qr.jl
+++ b/test/qr.jl
@@ -1,3 +1,9 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: diag, I
+
 @testset "qr_compact! and qr_null! for T = $T" for T in (Float32, Float64, ComplexF32,
                                                          ComplexF64)
     rng = StableRNG(123)
diff --git a/test/runtests.jl b/test/runtests.jl
index 07a54125..541c7da8 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1,44 +1,40 @@
-using MatrixAlgebraKit
-using Test
-using TestExtras
-using ChainRulesTestUtils
-using StableRNGs
-using Aqua
-using JET
-using LinearAlgebra: LinearAlgebra, diag, Diagonal, I, isposdef, diagind, mul!
-using MatrixAlgebraKit: diagview
+using SafeTestsets
 
-@testset "QR / LQ Decomposition" begin
+@safetestset "QR / LQ Decomposition" begin
     include("qr.jl")
     include("lq.jl")
 end
-@testset "Singular Value Decomposition" begin
+@safetestset "Singular Value Decomposition" begin
     include("svd.jl")
 end
-@testset "Hermitian Eigenvalue Decomposition" begin
+@safetestset "Hermitian Eigenvalue Decomposition" begin
     include("eigh.jl")
 end
-@testset "General Eigenvalue Decomposition" begin
+@safetestset "General Eigenvalue Decomposition" begin
     include("eig.jl")
 end
-@testset "Schur Decomposition" begin
+@safetestset "Schur Decomposition" begin
     include("schur.jl")
 end
-@testset "Polar Decomposition" begin
+@safetestset "Polar Decomposition" begin
     include("polar.jl")
 end
-@testset "Image and Null Space" begin
+@safetestset "Image and Null Space" begin
     include("orthnull.jl")
 end
-@testset "ChainRules" verbose = true begin
+@safetestset "ChainRules" begin
     include("chainrules.jl")
 end
 
-@testset "MatrixAlgebraKit.jl" begin
-    @testset "Code quality (Aqua.jl)" begin
+@safetestset "MatrixAlgebraKit.jl" begin
+    @safetestset "Code quality (Aqua.jl)" begin
+        using MatrixAlgebraKit
+        using Aqua
         Aqua.test_all(MatrixAlgebraKit)
     end
-    @testset "Code linting (JET.jl)" begin
+    @safetestset "Code linting (JET.jl)" begin
+        using MatrixAlgebraKit
+        using JET
         JET.test_package(MatrixAlgebraKit; target_defined_modules=true)
     end
 end
diff --git a/test/schur.jl b/test/schur.jl
index 05f952b5..71404e89 100644
--- a/test/schur.jl
+++ b/test/schur.jl
@@ -1,3 +1,9 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: I
+
 @testset "schur_full! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
     m = 54
diff --git a/test/svd.jl b/test/svd.jl
index 974957f0..ee4b74e0 100644
--- a/test/svd.jl
+++ b/test/svd.jl
@@ -1,3 +1,10 @@
+using MatrixAlgebraKit
+using Test
+using TestExtras
+using StableRNGs
+using LinearAlgebra: LinearAlgebra, Diagonal, I, isposdef
+using MatrixAlgebraKit: diagview
+
 @testset "svd_compact! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
     m = 54
@@ -108,3 +115,30 @@ end
         end
     end
 end
+
+@testset "svd_trunc! mix maxrank and tol for T = $T" for T in
+                                                         (Float32, Float64, ComplexF32,
+                                                          ComplexF64)
+    rng = StableRNG(123)
+    if LinearAlgebra.LAPACK.version() < v"3.12.0"
+        algs = (LAPACK_DivideAndConquer(), LAPACK_QRIteration(), LAPACK_Bisection())
+    else
+        algs = (LAPACK_DivideAndConquer(), LAPACK_QRIteration(), LAPACK_Bisection(),
+                LAPACK_Jacobi())
+    end
+    m = 4
+    @testset "algorithm $alg" for alg in algs
+        U = qr_compact(randn(rng, T, m, m))[1]
+        S = Diagonal([0.9, 0.3, 0.1, 0.01])
+        Vᴴ = qr_compact(randn(rng, T, m, m))[1]
+        A = U * S * Vᴴ
+
+        U1, S1, V1ᴴ = svd_trunc(A; alg, trunc=(; rtol=0.2, maxrank=1))
+        @test length(S1.diag) == 1
+        @test S1.diag ≈ S.diag[1:1] rtol = sqrt(eps(real(T)))
+
+        U2, S2, V2ᴴ = svd_trunc(A; alg, trunc=(; rtol=0.2, maxrank=3))
+        @test length(S2.diag) == 2
+        @test S2.diag ≈ S.diag[1:2] rtol = sqrt(eps(real(T)))
+    end
+end