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fix project 1 q1 bug

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Zhe Yuan
2025-10-21 16:25:27 -04:00
parent 5acb5d6900
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project1/Project1.md
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@@ -16,15 +16,15 @@ As $\epsilon \rightarrow 0$, the normal equation method numerically unstable. A
Results:
| $\epsilon$ | Normal Equation Method `proj(A, b)` | SVD Method `proj_SVD(A, b)` | Difference (Normal Eq - SVD) |
|:---:|:---|:---|:---|
| **1.0** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 1.78e-15 -2.22e-15 -8.88e-16 8.88e-16 0.00e+00]` |
| **0.1** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 7.28e-14 -4.44e-16 -2.66e-14 -1.62e-14 -5.28e-14]` |
| **0.01** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[-5.45e-12 -7.28e-12 -3.45e-13 -4.24e-12 -4.86e-12]` |
| **1e-4** | `[1.85714297 1.00000012 3.14285716 1.28571442 3.85714302]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.11e-07 1.19e-07 1.92e-08 1.36e-07 1.67e-07]` |
| **1e-8** | **Error:** `LinAlgError: Singular matrix` | `[1.85714286 1. 3.14285714 1.28571428 3.85714286]` | `Could not compute difference due to previous error.` |
| **1e-16**| **Error:** `ValueError: Matrix A must be full rank` | `[1.81820151 1. 3.18179849 2.29149804 2.89030045]` | `Could not compute difference due to previous error.` |
| **1e-32**| **Error:** `ValueError: Matrix A must be full rank` | `[2. 1. 3. 2.5 2.5]` | `Could not compute difference due to previous error.` |
| **1.0** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 2.22e-16 1.55e-15 -4.44e-16 2.22e-16 -8.88e-16]` |
| **0.1** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 8.08e-14 5.87e-14 -8.88e-16 4.82e-14 -1.38e-14]` |
| **0.01** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[-4.77e-12 -1.84e-14 5.54e-13 -4.00e-12 1.50e-12]` |
| **1e-4** | `[1.85714282 1. 3.14285716 1.28571427 3.85714291]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[-3.60e-08 9.11e-13 1.94e-08 -1.20e-08 4.80e-08]` |
| **1e-8** | `[-1.87500007 0.99999993 -3.12499997 -2.62500007 3.49999996]` | `[1.85714286 1. 3.14285714 1.28571427 3.85714286]` | `[-3.73e+00 -7.45e-08 -6.27e+00 -3.91e+00 -3.57e-01]` |
| **1e-16**| **Error:** `ValueError: Matrix A must be full rank` | `[3.4 1. 1.6 1.8 1.8]` | `Could not compute difference due to previous error.` |
| **1e-32**| **Error:** `ValueError: Matrix A must be full rank` | `[3.4 1. 1.6 1.8 1.8]` | `Could not compute difference due to previous error.` |
Numerical experiments show that as $\epsilon$ becomes small, the difference between the two methods increases. When $\epsilon$ becomes very small (e.g., 1e-8), the normal equation method fails while the SVD method continues to provide a stable solution.
Numerical experiments show that as $\epsilon$ becomes small, the difference between the two methods increases. When $\epsilon$ becomes very small (e.g., 1e-8), the normal equation method can't give a valid result and fails when $\epsilon$ continues to decrease (e.g., 1e-16), while the SVD method continues to provide a stable solution.
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