docs: fix incorrect claim about NonlinearSystem in initialization tutorial

[ChrisRackauckas-Claude]

Summary

• Fixed documentation in initialization.md that incorrectly claimed a "well-conditioned" initialization system would produce a NonlinearSystem
• The actual behavior is that a NonlinearLeastSquaresProblem is returned because the pendulum system is structurally singular (has unassigned variables)
• Changed the solver example from TrustRegion() (which is for NonlinearProblem) to GaussNewton() (which works with NonlinearLeastSquaresProblem)

Details

Line 483 of the original documentation stated:

notice that we instead obtained a NonlinearSystem

However, running the code shows that InitializationProblem returns a NonlinearLeastSquaresProblem for this case because:

1. The system is structurally singular (has unassigned variables like xˍtt(t))
2. Per src/initialization.jl, only systems that are both fully determined AND structurally non-singular get NonlinearProblem/SCCNonlinearProblem

Test plan

• Verified the code examples run correctly with the updated solver

Fixes #4024

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

Updated PR Description

The PR has been updated with an actual code fix (not just documentation changes).

Changes

1. src/initialization.jl: Modified get_initialization_problem_type to return NonlinearProblem for square initialization systems (neqs == nunknown) even if structurally singular. Only SCCNonlinearProblem now requires structural non-singularity.

2. docs/src/tutorials/initialization.md: Fixed typo "NonlinearSystem" → "NonlinearProblem"

Root Cause

The structural analysis performs bipartite graph matching without numerical substitution. With the pendulum constraint differentiated to -2xˍt*x - 2yˍt*y = 0, the structural analysis sees both xˍt and yˍt as variables. However, when y=0 is substituted numerically, the term -2yˍt*y vanishes, leaving xˍt uniquely determined.

Solution

For initialization systems, structural singularity is informational only. A square system may still be solvable even if structurally singular because numerical values can eliminate terms.

Testing

• ODEProblem(pend, [x => 1, y => 0, g => 1], ...) now returns NonlinearProblem ✓
• ODE solves correctly with x=1, y=0, D(x)=0, D(y)=0, λ=0 ✓
• Underdetermined systems still return NonlinearLeastSquaresProblem ✓

Note: Please update the PR title to: fix: return NonlinearProblem for structurally singular but square initialization systems

[ChrisRackauckas-Claude]

PR Updated - Docs-only fix

After discussion, the fix has been simplified to a documentation-only change:

Change

Changed the initialization example from:

[x => 1, y => 0.0, g => 1], guesses = [λ => 1]

To:

[x => 1, D(x) => 0.0, g => 1], guesses = [λ => 1, y => 0]

Rationale

• Specifying D(x) => 0 instead of y => 0 is the correct way to create a well-conditioned system
• With x=1 on the unit circle, y is determined by the constraint x² + y² = 1
• With D(x)=0 specified, the system is fully determined and returns SCCNonlinearProblem (the optimal case)
• Also fixed "NonlinearSystem" → "NonlinearProblem" (the correct type name)

The code changes to src/initialization.jl have been reverted as they are not needed with this documentation fix.