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Date: Wed, 30 Jul 2025 01:57:52 +0000
Subject: [PATCH] =?UTF-8?q?=E2=9A=A1=EF=B8=8F=20Speed=20up=20function=20`m?=
 =?UTF-8?q?atrix=5Fdecomposition=5FLU`=20by=201,015%?=
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The optimized code achieves a **15.9x speedup** by replacing explicit nested loops with vectorized NumPy operations, specifically using `np.dot()` for computing dot products.

**Key Optimizations Applied:**

1. **Vectorized dot products for U matrix computation**: Instead of the nested loop `for j in range(i): sum_val += L[i, j] * U[j, k]`, the optimized version uses `np.dot(Li, U[:i, k])` where `Li = L[i, :i]`.

2. **Pre-computed slices for L matrix computation**: The optimized version extracts `Ui = U[:i, i]` once per iteration and reuses it with `np.dot(L[k, :i], Ui)` instead of recalculating the sum in a loop.

**Why This Creates Significant Speedup:**

The original implementation has O(n³) scalar operations performed in Python loops. From the line profiler, we can see that the innermost loop operations (`sum_val += L[i, j] * U[j, k]` and `sum_val += L[k, j] * U[j, i]`) account for **60.9%** of total runtime (30.7% + 30.2%).

The optimized version leverages NumPy's highly optimized BLAS (Basic Linear Algebra Subprograms) routines for dot products, which:
- Execute in compiled C code rather than interpreted Python
- Use vectorized CPU instructions (SIMD)
- Have better memory access patterns and cache locality

**Performance Characteristics by Test Case:**

- **Small matrices (≤10x10)**: The optimization shows **38-47% slower performance** due to NumPy function call overhead dominating the small computation cost
- **Medium matrices (50x50)**: Shows **3-6x speedup** where vectorization benefits start outweighing overhead
- **Large matrices (≥100x100)**: Demonstrates **7-15x speedup** where vectorized operations provide maximum benefit

The crossover point appears around 20-30x30 matrices, making this optimization particularly effective for larger matrix decompositions commonly encountered in scientific computing and machine learning applications.
---
 src/numpy_pandas/matrix_operations.py | 12 +++++-------
 1 file changed, 5 insertions(+), 7 deletions(-)

diff --git a/src/numpy_pandas/matrix_operations.py b/src/numpy_pandas/matrix_operations.py
index f7d45df..72e5cbe 100644
--- a/src/numpy_pandas/matrix_operations.py
+++ b/src/numpy_pandas/matrix_operations.py
@@ -60,16 +60,14 @@ def matrix_decomposition_LU(A: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:
     L = np.zeros((n, n))
     U = np.zeros((n, n))
     for i in range(n):
+        # Compute the U[i, k] entries using vectorized dot product
+        Li = L[i, :i]
         for k in range(i, n):
-            sum_val = 0
-            for j in range(i):
-                sum_val += L[i, j] * U[j, k]
-            U[i, k] = A[i, k] - sum_val
+            U[i, k] = A[i, k] - np.dot(Li, U[:i, k])
         L[i, i] = 1
+        Ui = U[:i, i]
         for k in range(i + 1, n):
-            sum_val = 0
-            for j in range(i):
-                sum_val += L[k, j] * U[j, i]
+            sum_val = np.dot(L[k, :i], Ui)
             if U[i, i] == 0:
                 raise ValueError("Cannot perform LU decomposition")
             L[k, i] = (A[k, i] - sum_val) / U[i, i]