# Implementation Notes

## Matrix Operations

We've made several enhancements to the matrix operations for improved numerical stability and gradient computation:

1. **Matrix Inverse**
   - Added regularization for near-singular matrices
   - Improved error handling for singular matrices
   - Enhanced gradient computation stability 
   - Added better debug output

2. **Determinant**
   - Improved robustness for handling near-zero determinants
   - Fixed gradient computation for edge cases
   - Added checks to prevent NaN or infinite values

3. **Linear Solvers**
   - Added regularization for ill-conditioned systems
   - Improved the solver gradient computation
   - Added better error reporting for singular matrices

4. **QR Decomposition**
   - Fixed shape extraction from output tensors
   - Improved gradient computation stability
   - Added better debug information for shape mismatches

5. **Matrix Functions**
   - Improved matrix exponential using proper Frechet derivative computation
   - Enhanced matrix square root stability
   - Added detailed debugging for numerical operations

## Testing Challenges

Due to the current state of the tensor evaluation system in this beta version of the library (0.1.0), we were unable to directly test these enhancements using the high-level tensor API. The current implementation of `Graph::eval_tensors` appears to be a placeholder that always returns zero tensors.

To properly test these enhancements, we need one of the following:
1. A fully implemented evaluation system in the library
2. Direct access to the operations' compute and grad methods (which are currently private)
3. Integration with actual use cases to verify stability and performance

Despite these challenges, the implemented improvements follow best practices for numerical stability and should significantly enhance the robustness of the autograd system once the evaluation capabilities are fully implemented.

## Future Work

1. Once the tensor evaluation system is fully implemented, we should add comprehensive tests for all matrix operations, especially testing the gradient computation stability for near-singular matrices.

2. Consider exposing a testing API to allow direct testing of operations' compute and grad methods without going through the tensor evaluation system.

3. Add more numerical stability enhancements to other operations in the library.