diff --git a/src/algorithms.jl b/src/algorithms.jl
index 6f9a4d49..af183103 100644
--- a/src/algorithms.jl
+++ b/src/algorithms.jl
@@ -77,9 +77,6 @@ passed as the third positional argument in the form of a `NamedTuple`.
 """ select_algorithm
 
 function select_algorithm(f::F, A, alg::Alg=nothing; kwargs...) where {F,Alg}
-    return select_algorithm(f, typeof(A), alg; kwargs...)
-end
-function select_algorithm(f::F, ::Type{A}, alg::Alg=nothing; kwargs...) where {F,A,Alg}
     if isnothing(alg)
         return default_algorithm(f, A; kwargs...)
     elseif alg isa Symbol
@@ -193,10 +190,24 @@ macro functiondef(f)
         end
 
         # define fallbacks for algorithm selection
-        @inline function select_algorithm(::typeof($f), ::Type{A}, alg::Alg;
-                                          kwargs...) where {Alg,A}
+        @inline function select_algorithm(::typeof($f), A, alg::Alg; kwargs...) where {Alg}
             return select_algorithm($f!, A, alg; kwargs...)
         end
+        # define default algorithm fallbacks for out-of-place functions
+        # in terms of the corresponding in-place function
+        @inline function default_algorithm(::typeof($f), A; kwargs...)
+            return default_algorithm($f!, A; kwargs...)
+        end
+        # define default algorithm fallbacks for out-of-place functions
+        # in terms of the corresponding in-place function for types,
+        # in principle this is covered by the definition above but
+        # it is necessary to avoid ambiguity errors with the generic definitions:
+        # ```julia
+        # default_algorithm(f::F, A; kwargs...) where {F} = default_algorithm(f, typeof(A); kwargs...)
+        # function default_algorithm(f::F, ::Type{T}; kwargs...) where {F,T}
+        #     throw(MethodError(default_algorithm, (f, T)))
+        # end
+        # ```
         @inline function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}
             return default_algorithm($f!, A; kwargs...)
         end
diff --git a/src/interface/eig.jl b/src/interface/eig.jl
index 77aa0672..fd75193f 100644
--- a/src/interface/eig.jl
+++ b/src/interface/eig.jl
@@ -99,8 +99,13 @@ for f in (:eig_full!, :eig_vals!)
     end
 end
 
-function select_algorithm(::typeof(eig_trunc!), ::Type{A}, alg; trunc=nothing,
-                          kwargs...) where {A<:YALAPACK.BlasMat}
-    alg_eig = select_algorithm(eig_full!, A, alg; kwargs...)
-    return TruncatedAlgorithm(alg_eig, select_truncation(trunc))
+function select_algorithm(::typeof(eig_trunc!), A, alg; trunc=nothing, kwargs...)
+    if alg isa TruncatedAlgorithm
+        isnothing(trunc) ||
+            throw(ArgumentError("`trunc` can't be specified when `alg` is a `TruncatedAlgorithm`"))
+        return alg
+    else
+        alg_eig = select_algorithm(eig_full!, A, alg; kwargs...)
+        return TruncatedAlgorithm(alg_eig, select_truncation(trunc))
+    end
 end
diff --git a/src/interface/eigh.jl b/src/interface/eigh.jl
index 3ed38789..a650ca44 100644
--- a/src/interface/eigh.jl
+++ b/src/interface/eigh.jl
@@ -100,8 +100,13 @@ for f in (:eigh_full!, :eigh_vals!)
     end
 end
 
-function select_algorithm(::typeof(eigh_trunc!), ::Type{A}, alg; trunc=nothing,
-                          kwargs...) where {A<:YALAPACK.BlasMat}
-    alg_eigh = select_algorithm(eigh_full!, A, alg; kwargs...)
-    return TruncatedAlgorithm(alg_eigh, select_truncation(trunc))
+function select_algorithm(::typeof(eigh_trunc!), A, alg; trunc=nothing, kwargs...)
+    if alg isa TruncatedAlgorithm
+        isnothing(trunc) ||
+            throw(ArgumentError("`trunc` can't be specified when `alg` is a `TruncatedAlgorithm`"))
+        return alg
+    else
+        alg_eig = select_algorithm(eigh_full!, A, alg; kwargs...)
+        return TruncatedAlgorithm(alg_eig, select_truncation(trunc))
+    end
 end
diff --git a/src/interface/lq.jl b/src/interface/lq.jl
index 9de85bae..6f1ed12f 100644
--- a/src/interface/lq.jl
+++ b/src/interface/lq.jl
@@ -77,9 +77,7 @@ function default_lq_algorithm(::Type{T}; kwargs...) where {T<:YALAPACK.BlasMat}
 end
 
 for f in (:lq_full!, :lq_compact!, :lq_null!)
-    @eval begin
-        function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}
-            return default_lq_algorithm(A; kwargs...)
-        end
+    @eval function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}
+        return default_lq_algorithm(A; kwargs...)
     end
 end
diff --git a/src/interface/polar.jl b/src/interface/polar.jl
index 111a6e3a..87346ff2 100644
--- a/src/interface/polar.jl
+++ b/src/interface/polar.jl
@@ -61,10 +61,7 @@ end
 # Algorithm selection
 # -------------------
 default_polar_algorithm(A; kwargs...) = default_polar_algorithm(typeof(A); kwargs...)
-function default_polar_algorithm(T::Type; kwargs...)
-    throw(MethodError(default_polar_algorithm, (T,)))
-end
-function default_polar_algorithm(::Type{T}; kwargs...) where {T<:YALAPACK.BlasMat}
+function default_polar_algorithm(::Type{T}; kwargs...) where {T}
     return PolarViaSVD(default_algorithm(svd_compact!, T; kwargs...))
 end
 
diff --git a/src/interface/qr.jl b/src/interface/qr.jl
index a1c12b7a..62d87080 100644
--- a/src/interface/qr.jl
+++ b/src/interface/qr.jl
@@ -77,10 +77,7 @@ function default_qr_algorithm(::Type{T}; kwargs...) where {T<:YALAPACK.BlasMat}
 end
 
 for f in (:qr_full!, :qr_compact!, :qr_null!)
-    @eval begin
-        function default_algorithm(::typeof($f), ::Type{A};
-                                   kwargs...) where {A<:YALAPACK.BlasMat}
-            return default_qr_algorithm(A; kwargs...)
-        end
+    @eval function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}
+        return default_qr_algorithm(A; kwargs...)
     end
 end
diff --git a/src/interface/svd.jl b/src/interface/svd.jl
index e6f9021a..fd4eb5a5 100644
--- a/src/interface/svd.jl
+++ b/src/interface/svd.jl
@@ -104,8 +104,13 @@ for f in (:svd_full!, :svd_compact!, :svd_vals!)
     end
 end
 
-function select_algorithm(::typeof(svd_trunc!), ::Type{A}, alg; trunc=nothing,
-                          kwargs...) where {A<:YALAPACK.BlasMat}
-    alg_svd = select_algorithm(svd_compact!, A, alg; kwargs...)
-    return TruncatedAlgorithm(alg_svd, select_truncation(trunc))
+function select_algorithm(::typeof(svd_trunc!), A, alg; trunc=nothing, kwargs...)
+    if alg isa TruncatedAlgorithm
+        isnothing(trunc) ||
+            throw(ArgumentError("`trunc` can't be specified when `alg` is a `TruncatedAlgorithm`"))
+        return alg
+    else
+        alg_svd = select_algorithm(svd_compact!, A, alg; kwargs...)
+        return TruncatedAlgorithm(alg_svd, select_truncation(trunc))
+    end
 end
diff --git a/test/algorithms.jl b/test/algorithms.jl
index 49524a89..9f9c4542 100644
--- a/test/algorithms.jl
+++ b/test/algorithms.jl
@@ -2,7 +2,7 @@ using MatrixAlgebraKit
 using Test
 using TestExtras
 using MatrixAlgebraKit: LAPACK_SVDAlgorithm, NoTruncation, PolarViaSVD, TruncatedAlgorithm,
-                        default_algorithm, select_algorithm
+                        TruncationKeepBelow, default_algorithm, select_algorithm
 
 @testset "default_algorithm" begin
     A = randn(3, 3)
@@ -50,6 +50,12 @@ end
                                  NoTruncation())
     end
 
+    alg = TruncatedAlgorithm(LAPACK_Simple(), TruncationKeepBelow(0.1, 0.0))
+    for f in (eig_trunc!, eigh_trunc!, svd_trunc!)
+        @test @constinferred(select_algorithm(eig_trunc!, A, alg)) === alg
+        @test_throws ArgumentError select_algorithm(eig_trunc!, A, alg; trunc=(; maxrank=2))
+    end
+
     @test @constinferred(select_algorithm(svd_compact!, A)) === LAPACK_DivideAndConquer()
     @test @constinferred(select_algorithm(svd_compact!, A, nothing)) ===
           LAPACK_DivideAndConquer()
diff --git a/test/eig.jl b/test/eig.jl
index cd73d94e..cdaec9dc 100644
--- a/test/eig.jl
+++ b/test/eig.jl
@@ -3,7 +3,7 @@ using Test
 using TestExtras
 using StableRNGs
 using LinearAlgebra: Diagonal
-using MatrixAlgebraKit: diagview
+using MatrixAlgebraKit: TruncatedAlgorithm, diagview
 
 @testset "eig_full! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
@@ -57,3 +57,17 @@ end
         @test V2 * ((V2' * V2) \ (V2' * V1)) ≈ V1
     end
 end
+
+@testset "eig_trunc! specify truncation algorithm T = $T" for T in
+                                                              (Float32, Float64, ComplexF32,
+                                                               ComplexF64)
+    rng = StableRNG(123)
+    m = 4
+    V = randn(rng, T, m, m)
+    D = Diagonal([0.9, 0.3, 0.1, 0.01])
+    A = V * D * inv(V)
+    alg = TruncatedAlgorithm(LAPACK_Simple(), truncrank(2))
+    D2, V2 = @constinferred eig_trunc(A; alg)
+    @test diagview(D2) ≈ diagview(D)[1:2] rtol = sqrt(eps(real(T)))
+    @test_throws ArgumentError eig_trunc(A; alg, trunc=(; maxrank=2))
+end
diff --git a/test/eigh.jl b/test/eigh.jl
index 5a3c5a8a..6e785158 100644
--- a/test/eigh.jl
+++ b/test/eigh.jl
@@ -3,7 +3,7 @@ using Test
 using TestExtras
 using StableRNGs
 using LinearAlgebra: LinearAlgebra, Diagonal, I
-using MatrixAlgebraKit: diagview
+using MatrixAlgebraKit: TruncatedAlgorithm, diagview
 
 @testset "eigh_full! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
@@ -62,3 +62,19 @@ end
         @test V2 * (V2' * V1) ≈ V1
     end
 end
+
+@testset "eigh_trunc! specify truncation algorithm T = $T" for T in
+                                                               (Float32, Float64,
+                                                                ComplexF32,
+                                                                ComplexF64)
+    rng = StableRNG(123)
+    m = 4
+    V = qr_compact(randn(rng, T, m, m))[1]
+    D = Diagonal([0.9, 0.3, 0.1, 0.01])
+    A = V * D * V'
+    A = (A + A') / 2
+    alg = TruncatedAlgorithm(LAPACK_QRIteration(), truncrank(2))
+    D2, V2 = @constinferred eigh_trunc(A; alg)
+    @test diagview(D2) ≈ diagview(D)[1:2] rtol = sqrt(eps(real(T)))
+    @test_throws ArgumentError eigh_trunc(A; alg, trunc=(; maxrank=2))
+end
diff --git a/test/svd.jl b/test/svd.jl
index eb6a7805..40de0897 100644
--- a/test/svd.jl
+++ b/test/svd.jl
@@ -3,7 +3,7 @@ using Test
 using TestExtras
 using StableRNGs
 using LinearAlgebra: LinearAlgebra, Diagonal, I, isposdef
-using MatrixAlgebraKit: TruncationKeepAbove, diagview
+using MatrixAlgebraKit: TruncatedAlgorithm, TruncationKeepAbove, diagview
 
 @testset "svd_compact! for T = $T" for T in (Float32, Float64, ComplexF32, ComplexF64)
     rng = StableRNG(123)
@@ -152,3 +152,18 @@ end
         end
     end
 end
+
+@testset "svd_trunc! specify truncation algorithm T = $T" for T in
+                                                              (Float32, Float64, ComplexF32,
+                                                               ComplexF64)
+    rng = StableRNG(123)
+    m = 4
+    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
+    alg = TruncatedAlgorithm(LAPACK_DivideAndConquer(), TruncationKeepAbove(0.2, 0.0))
+    U2, S2, V2ᴴ = @constinferred svd_trunc(A; alg)
+    @test diagview(S2) ≈ diagview(S)[1:2] rtol = sqrt(eps(real(T)))
+    @test_throws ArgumentError svd_trunc(A; alg, trunc=(; maxrank=2))
+end