Optimize spline coefficient computation with direct sparse methods

[krystophny]

Summary

Improve performance of spline coefficient computation by using direct sparse matrix methods instead of dense operations.

Background

Current spline computation uses dense matrix operations which are inefficient for the sparse structure of spline matrices.

Tasks

• Analyze current spline implementation and identify bottlenecks
• Implement sparse matrix assembly for spline problems
• Use existing sparse solver (UMFPACK) more efficiently
• Add comprehensive spline validation tests
• Benchmark performance improvements
• Test accuracy against original dense implementation
• Update documentation

Implementation Strategy

1. Create new branch from main
2. Implement sparse matrix version alongside dense version
3. Add switch to choose implementation
4. Comprehensive testing against original
5. Performance benchmarking

Expected Benefits

• Significant memory reduction for large spline problems
• Better performance for typical use cases
• Maintains exact same results as dense method

Testing Requirements

• Unit tests for spline computation
• Accuracy comparison with original dense method
• Performance benchmarks
• Golden record validation

Related Work

• Independent of all other issues
• Can use existing sparse solver functionality in the code
• Work can be extracted from bicgstab/faster_spline branches