Add truncation error output to truncated decompositions

[Copilot]

Closes #72

This PR extends the truncated decomposition functions (svd_trunc, eig_trunc, eigh_trunc) to additionally return the truncation error, defined as the 2-norm of the discarded singular values or eigenvalues. This provides a quantitative measure of the approximation quality when using truncated factorizations.

Changes

New return signatures:

• svd_trunc(A; trunc=...) -> U, S, Vᴴ, ϵ (previously returned only U, S, Vᴴ)
• eig_trunc(A; trunc=...) -> D, V, ϵ (previously returned only D, V)
• eigh_trunc(A; trunc=...) -> D, V, ϵ (previously returned only D, V)

where ϵ is the truncation error computed as norm(discarded_values).

Example usage:

using MatrixAlgebraKit, LinearAlgebra

A = randn(100, 100)
U, S, Vᴴ, ϵ = svd_trunc(A; trunc=truncrank(50))

# ϵ gives the 2-norm of the 50 smallest singular values
println("Truncation error: ", ϵ)

# Verify: the error matches the norm of discarded singular values
S_all = svd_vals(A)
expected_error = norm(S_all[51:end])
@assert ϵ ≈ expected_error

Implementation details:

• Added _compute_truncerr function to compute the 2-norm of discarded values
• Updated all truncation functions in src/implementations/truncation.jl
• Modified ChainRules integration to properly handle the additional return value in automatic differentiation
• Updated all tests and documentation

Backward compatibility:
This is a breaking change as the return signatures have changed. Users will need to update their code to handle the additional return value:

# Old code:
U, S, Vᴴ = svd_trunc(A; trunc=truncrank(10))

# New code:
U, S, Vᴴ, ϵ = svd_trunc(A; trunc=truncrank(10))
# or if you don't need the error:
U, S, Vᴴ, _ = svd_trunc(A; trunc=truncrank(10))

Verification

All existing tests pass with updated expectations. Added new tests to verify:

• Truncation error is computed correctly as the 2-norm of discarded values
• When using notrunc(), the truncation error is zero
• Error values match expected calculations for various truncation strategies

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Original prompt

Can you try and start tackling issue #72? I want the truncated decompositions to additionally output the truncation error, which I define as the 2-norm of the discarded singular or eigenvalues. The docstrings and documentation sections should be updated accordingly.

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[Jutho]

Closing this in favor of #75