PHASE 3 vs PHASE 4 PERFORMANCE COMPARISON
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Configuration          Phase 3 (Hessian Cache)    Phase 4 (Line Search)    Improvement
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n= 2000, d=1          0.0910s (1.94x)            0.0519s (2.34x)          43% faster
n= 5000, d=1          0.1361s (1.33x)            0.1083s (1.51x)          20% faster
n=10000, d=1          0.2634s (1.38x)            0.2205s (1.69x)          16% faster
n=20000, d=1          0.4861s                    0.4120s                  15% faster

n= 2000, d=2          0.0288s (4.50x)            0.0259s (5.22x)          10% faster
n= 5000, d=2          0.0642s (2.96x)            0.0606s (3.36x)           6% faster
n=10000, d=2          0.1743s (2.28x)            0.1257s (3.22x)          28% faster

n= 2000, d=4          0.0596s (3.79x)            0.0583s (4.86x)           2% faster
n= 5000, d=4          0.1709s (2.74x)            0.1490s (3.49x)          13% faster

n= 2000, d=8          0.2010s (2.47x)            0.1873s (3.08x)           7% faster
n= 5000, d=8          0.4752s (1.68x)            0.4691s (1.90x)           1% faster

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KEY IMPROVEMENTS:

1. Line Search Optimizations (Phase 4):
   - Armijo condition for early stopping: accepts "sufficient decrease"
   - Adaptive max_half: 10-30 iterations based on gradient magnitude
   - Reduces REML evaluations per line search from ~5-10 to ~2-3

2. Performance Gains:
   - Small-medium problems (n<5000): 2-43% faster
   - Large problems (n=10000): 16-28% faster
   - Most effective for d=1,2 (fewer smooths, faster convergence)

3. vs R's mgcv:
   - Speedup improvements across all cases
   - n=10000, d=2: 2.28x → 3.22x (+41% relative speedup)
   - Maintained stability and correctness

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