diff --git a/src/algorithms/approximate/approximate.jl b/src/algorithms/approximate/approximate.jl
index f28e67e6b..6967dcef0 100644
--- a/src/algorithms/approximate/approximate.jl
+++ b/src/algorithms/approximate/approximate.jl
@@ -1,13 +1,17 @@
 @doc """
-    approximate(ψ₀, (O, ψ), algorithm, [environments]; kwargs...) -> (ψ, environments)
-    approximate!(ψ₀, (O, ψ), algorithm, [environments]; kwargs...) -> (ψ, environments)
-    approximate(ψ₀, ψ, algorithm, [environments]; kwargs...) -> (ψ, environments)
-    approximate!(ψ₀, ψ, algorithm, [environments]; kwargs...) -> (ψ, environments)
+    approximate(ψ₀, (O, ψ), [environments]; kwargs...) -> (ψ, environments, ϵ)
+    approximate(ψ₀, (O, ψ), algorithm, [environments]) -> (ψ, environments, ϵ)
+    approximate!(ψ₀, (O, ψ), algorithm, [environments]) -> (ψ, environments, ϵ)
+    approximate(ψ₀, ψ, algorithm, [environments]) -> (ψ, environments, ϵ)
+    approximate!(ψ₀, ψ, algorithm, [environments]) -> (ψ, environments, ϵ)
 
 Compute an approximation to the application of an operator `O` to the state `ψ` in the form
 of an MPS `ψ₀`. If only a state `ψ` is supplied instead of the `(O, ψ)` pair, `ψ₀` is
 approximated directly to `ψ` (i.e. `O` is taken to be the identity).
 
+**Not every algorithm supports every combination of arguments below** — see the per-algorithm
+notes at the end of this docstring before picking one.
+
 ## Arguments
 - `ψ₀::AbstractMPS`: initial guess of the approximated state
 - `(O::AbstractMPO, ψ::AbstractMPS)`: operator `O` and state `ψ` to be approximated
@@ -16,17 +20,33 @@ approximated directly to `ψ` (i.e. `O` is taken to be the identity).
 - `[environments]`: MPS environment manager
 
 ## Keywords
+The keyword-based call (no explicit `algorithm`) is a convenience method that picks an
+algorithm for you based on the type of `ψ₀` (`DMRG`/`DMRG2` for a finite MPS, `VOMPS`/`IDMRG`/
+`IDMRG2` for an infinite MPS) and only accepts the `(O, ψ)` tuple form of `toapprox`. Once you
+pass an explicit `algorithm`, keywords are no longer accepted here — configure the algorithm
+struct itself instead (e.g. `DMRG(; tol, maxiter, verbosity)`).
 - `tol::Float64`: tolerance for convergence criterium
 - `maxiter::Int`: maximum amount of iterations
 - `verbosity::Int`: display progress information
+- `trscheme`: if supplied, a truncated two-site sweep (`DMRG2`/`IDMRG2`) is prepended to
+  refine the bond dimension before the single-site algorithm polishes the result.
 
 ## Algorithms
-- `DMRG`: Alternating least square method for maximizing the fidelity with a single-site scheme.
-- `DMRG2`: Alternating least square method for maximizing the fidelity with a two-site scheme.
+Each algorithm below only supports a subset of the general interface. Check this table before
+picking one — in particular, note that **only `DMRG`/`DMRG2` accept a bare state `ψ`**; the
+infinite algorithms always require an explicit `(O, ψ)` tuple, and **`VOMPS` has no in-place
+`approximate!`** at all.
+
+| Algorithm | Scheme                        | State `ψ₀`                        | bare `ψ` allowed? | `approximate!` |
+|:--------- |:----------------------------- |:---------------------------------- |:------------------:|:--------------:|
+| `DMRG`    | single-site, fixes bond dim    | `AbstractFiniteMPS`                | ✅                  | ✅              |
+| `DMRG2`   | two-site, truncates via `trscheme` | `AbstractFiniteMPS`            | ✅                  | ✅              |
+| `IDMRG`   | single-site, thermodynamic limit | `InfiniteMPS` / `MultilineMPS`  | ❌ (tuple only)     | ✅              |
+| `IDMRG2`  | two-site, thermodynamic limit, needs unit cell ≥ 2 | `InfiniteMPS` / `MultilineMPS` | ❌ (tuple only) | ✅ |
+| `VOMPS`   | tangent-space truncation       | `InfiniteMPS` / `MultilineMPS`     | ❌ (tuple only)     | ❌ (out-of-place only) |
 
-- `IDMRG`: Variant of `DMRG` for maximizing fidelity density in the thermodynamic limit.
-- `IDMRG2`: Variant of `DMRG2` for maximizing fidelity density in the thermodynamic limit.
-- `VOMPS`: Tangent space method for truncating uniform MPS.
+`InfiniteMPS`/`InfiniteMPO` inputs are converted internally to `MultilineMPS`/`MultilineMPO`
+for `IDMRG`, `IDMRG2`, and `VOMPS`; you can also pass those types directly.
 """
 approximate, approximate!
 
@@ -35,6 +55,30 @@ approximate, approximate!
 _environment_args(Oϕ::Tuple) = Oϕ
 _environment_args(ϕ) = (ϕ,)
 
+function approximate(
+        ψ::AbstractMPS, toapprox::Tuple{<:AbstractMPO, <:AbstractMPS},
+        envs::AbstractMPSEnvironments = environments(ψ, toapprox...);
+        tol = Defaults.tol, maxiter = Defaults.maxiter,
+        verbosity = Defaults.verbosity, trscheme = nothing
+    )
+    if isa(ψ, InfiniteMPS)
+        alg = VOMPS(; tol, verbosity, maxiter)
+        if !isnothing(trscheme)
+            alg = IDMRG2(; tol = min(1.0e-2, 100tol), verbosity, trscheme) & alg
+        end
+    elseif isa(ψ, AbstractFiniteMPS)
+        alg = DMRG(; tol, maxiter, verbosity)
+        if !isnothing(trscheme)
+            alg = DMRG2(; tol = min(1.0e-2, 100tol), verbosity, trscheme) & alg
+        end
+    else
+        throw(ArgumentError("Unknown input state type"))
+    end
+
+    return approximate(ψ, toapprox, alg, envs)
+end
+
+
 # implementation in terms of Multiline
 function approximate(
         ψ::InfiniteMPS, toapprox::Tuple{<:InfiniteMPO, <:InfiniteMPS}, algorithm,
diff --git a/src/algorithms/unionalg.jl b/src/algorithms/unionalg.jl
index 3e37f67f4..9ef7f722a 100644
--- a/src/algorithms/unionalg.jl
+++ b/src/algorithms/unionalg.jl
@@ -32,3 +32,8 @@ function find_groundstate(state, H, alg::UnionAlg, envs = environments(state, H,
     state, envs = find_groundstate(state, H, alg.alg1, envs)
     return find_groundstate(state, H, alg.alg2, envs)
 end
+
+function approximate(state, toapprox, alg::UnionAlg, envs = enviroments(state, toapprox...))
+    state, envs = approximate(state, toapprox, alg.alg1, envs)
+    return approximate(state, toapprox, alg.alg2, envs)
+end