From b510d92ff45d5dcaedabea1f7751f7ec3a9c7106 Mon Sep 17 00:00:00 2001
From: "codeflash-ai[bot]"
 <148906541+codeflash-ai[bot]@users.noreply.github.com>
Date: Wed, 30 Jul 2025 01:13:46 +0000
Subject: [PATCH] =?UTF-8?q?=E2=9A=A1=EF=B8=8F=20Speed=20up=20function=20`n?=
 =?UTF-8?q?aive=5Fmatrix=5Fdeterminant`=20by=2020%?=
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The optimization achieves a 19% speedup by making two key improvements to the submatrix creation process:

**1. Replaced nested loops with list comprehension for submatrix creation:**
- **Original**: Used three nested loops to manually build each submatrix element by element, creating empty lists and appending elements one at a time
- **Optimized**: Uses a single list comprehension `[row[:j] + row[j+1:] for row in matrix[1:]]` that leverages Python's efficient slicing operations

Looking at the profiler results, the original code spent 25.1% of time in the innermost `for k in range(n)` loop and additional time in row creation/appending operations. The optimized version eliminates these nested loops entirely, reducing the submatrix creation from ~50% of total time to ~17%.

**2. Replaced exponentiation with bitwise operation for sign calculation:**
- **Original**: `sign = (-1) ** j` uses expensive exponentiation
- **Optimized**: `sign = -1 if (j & 1) else 1` uses fast bitwise AND to check if j is odd/even

The profiler shows the sign calculation went from 2.1% to 8.0% of total time, but this is misleading - the absolute time decreased significantly as the overall runtime improved.

**Why these optimizations work:**
- **List slicing** (`row[:j] + row[j+1:]`) is implemented in C and operates on contiguous memory, making it much faster than Python loops with individual element access and list appends
- **Bitwise operations** are among the fastest CPU instructions, while exponentiation involves multiplication loops
- **Reduced function call overhead** by eliminating the nested loop structure and multiple `append()` calls

**Performance characteristics from test results:**
- The optimization shows consistent 15-20% improvements on larger matrices (5x5 and above), where the recursive nature amplifies the submatrix creation savings
- Smaller matrices (1x1, 2x2) show minimal or no improvement since they hit base cases quickly
- The speedup scales well with matrix size - 10x10 matrices show ~20% improvement, demonstrating that the optimization benefits compound with the recursive depth
---
 src/numpy_pandas/np_opts.py | 16 ++++++----------
 1 file changed, 6 insertions(+), 10 deletions(-)

diff --git a/src/numpy_pandas/np_opts.py b/src/numpy_pandas/np_opts.py
index 7cf690b..8a0677c 100644
--- a/src/numpy_pandas/np_opts.py
+++ b/src/numpy_pandas/np_opts.py
@@ -16,6 +16,7 @@ def binomial_coefficient_recursive(n: int, k: int) -> int:
 
 def naive_matrix_determinant(matrix: List[List[float]]) -> float:
     """Calculate determinant using cofactor expansion."""
+
     n = len(matrix)
 
     if n == 1:
@@ -25,17 +26,12 @@ def naive_matrix_determinant(matrix: List[List[float]]) -> float:
         return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0]
 
     determinant = 0
+    # Only create submatrices with fast list comprehension
     for j in range(n):
-        # Create submatrix by removing first row and column j
-        submatrix = []
-        for i in range(1, n):
-            row = []
-            for k in range(n):
-                if k != j:
-                    row.append(matrix[i][k])
-            submatrix.append(row)
-
-        sign = (-1) ** j
+        # Submatrix: remove row 0, and column j
+        submatrix = [row[:j] + row[j + 1 :] for row in matrix[1:]]
+        # Alternate sign using bit operation for speed
+        sign = -1 if (j & 1) else 1
         determinant += sign * matrix[0][j] * naive_matrix_determinant(submatrix)
 
     return determinant