Implement gaussian elimination in python (#370)

* initial implementation

* Debug ref and rref

* Add to gitbook

* Make requested changes

* Change ref to row echelon form

Author: jasmaa

contents/gaussian_elimination/code/python/gaussian_elimination.py

@@ -0,0 +1,89 @@
+import numpy as np
+
+def gaussian_elimination(A):
+
+    pivot_row = 0
+    
+    # Step 1: Go by column
+    for pivot_col in range(min(A.shape[0], A.shape[1])):
+
+        # Step 2: Swap row with highest element in col
+        max_i = np.argmax(abs(A[pivot_row:, pivot_col])) + pivot_row
+        
+        temp = A[pivot_row, :].copy()
+        A[pivot_row, :] = A[max_i, :]
+        A[max_i, :] = temp
+
+        # Skip on singular matrix,  not actually a pivot
+        if A[pivot_row, pivot_col] == 0:
+            continue
+
+        # Steps 3 & 4: Zero out elements below pivot
+        for r in range(pivot_row + 1,  A.shape[0]):
+            # Step 3: Get fraction
+            frac = -A[r, pivot_col] / A[pivot_row, pivot_col]
+            # Step 4: Add rows
+            A[r, :] += frac * A[pivot_row, :]
+
+        pivot_row += 1
+
+
+# Assumes A is already row echelon form
+def gauss_jordan_elimination(A):
+    
+    col = 0
+
+    # Scan for pivots
+    for row in range(A.shape[0]):
+        while col < A.shape[1] and A[row, col] == 0:
+            col += 1
+            
+        if col >= A.shape[1]:
+            continue
+
+        # Set each pivot to one via row scaling
+        A[row, :] /= A[row, col]
+
+        # Zero out elements above pivot
+        for r in range(row):
+            A[r, :] -= A[r, col] * A[row, :]
+
+
+# Assumes A has a unique solution and A in row echelon form
+def back_substitution(A):
+    
+    sol = np.zeros(A.shape[0]).T
+
+    # Go by pivots along diagonal
+    for pivot_i in range(A.shape[0] - 1,  -1,  -1):
+        s = 0
+        for col in range(pivot_i + 1,  A.shape[1] - 1):
+            s += A[pivot_i, col] * sol[col]
+        sol[pivot_i] = (A[pivot_i, A.shape[1] - 1] - s) / A[pivot_i, pivot_i]
+        
+    return sol
+
+
+def main():
+    A = np.array([[2, 3, 4, 6],
+                  [1, 2, 3, 4,],
+                  [3, -4, 0, 10]], dtype=float)
+
+    print("Original")
+    print(A, "\n")
+
+    gaussian_elimination(A)
+    print("Gaussian elimination")
+    print(A, "\n")
+
+    print("Back subsitution")
+    print(back_substitution(A), "\n")
+
+    gauss_jordan_elimination(A)
+    print("Gauss-Jordan")
+    print(A, "\n")
+
+
+if __name__ == "__main__":
+    main()
+

contents/gaussian_elimination/gaussian_elimination.md

@@ -360,6 +360,8 @@ In code, this looks like:
 [import:41-78, lang:"rust"](code/rust/gaussian_elimination.rs)
 {% sample lang="hs" %}
 [import:10-36, lang:"haskell"](code/haskell/gaussianElimination.hs)
+{% sample lang="py" %}
+[import:3-28, lang:"python"](code/python/gaussian_elimination.py)
 {% endmethod %}
 
 Now, to be clear: this algorithm creates an upper-triangular matrix.
@@ -393,6 +395,8 @@ This code does not exist yet in rust, so here's Julia code (sorry for the inconv
 [import:67-93, lang:"julia"](code/julia/gaussian_elimination.jl)
 {% sample lang="hs" %}
 [import:38-46, lang:"haskell"](code/haskell/gaussianElimination.hs)
+{% sample lang="py" %}
+[import:31-49, lang:"python"](code/python/gaussian_elimination.py)
 {% endmethod %}
 
 ## Back-substitution
@@ -423,6 +427,8 @@ In code, this involves keeping a rolling sum of all the values we substitute in
 [import:79-94, lang:"rust"](code/rust/gaussian_elimination.rs)
 {% sample lang="hs" %}
 [import:48-53, lang:"haskell"](code/haskell/gaussianElimination.hs)
+{% sample lang="py" %}
+[import:52-64, lang:"python"](code/python/gaussian_elimination.py)
 {% endmethod %}
 
 ## Conclusions
@@ -445,6 +451,8 @@ As for what's next... Well, we are in for a treat! The above algorithm clearly h
 [import, lang:"rust"](code/rust/gaussian_elimination.rs)
 {% sample lang="hs" %}
 [import, lang:"haskell"](code/haskell/gaussianElimination.hs)
+{% sample lang="py" %}
+[import, lang:"python"](code/python/gaussian_elimination.py)
 {% endmethod %}