Bottom SciencePython Programming
Basics
Numerical Methods
- Bisection Method
- Secant Method
- Regular Falsi (False Position) Method
- Newton Raphson Method
- Gauss Elimination Method
- Gauss Jordan Method
- Gauss-Seidel Method
- Lagrange Interpolation Method
- Newton Divided Difference Interpolation
- Newton Forward Difference Interpolation
- Newton Backward Difference Interpolation
- Trapezoidal Rule
- Simpson 1/3rd Rule
- Simpson 3/8 Rule
- Euler’s Method
- Euler’s Modified Method
- Runge-Kutta 2nd Order Method
- Runge-Kutta 4th Order Method
- Cubic Spline Method
- Bilinear Interpolation Method
- Milne’s Method
- More topics coming soon…
Advance Numerical Methods
The Gauss elimination method is a method for solving systems of linear equations of the form Ax = b, where A is an n x n matrix, x is the unknown variable, and b is a column vector.
Gauss elimination method in Python for equations:
2x + y – z = 8
-3x – y + 2z = -11
-2x + y + 2z = -3
def gauss_elimination(A, b):
"""
Gauss elimination method [By Bottom Science].
A - the coefficient matrix (an n x n matrix)
b - the right-hand side column vector (an n x 1 matrix)
"""
n = len(A)
# Perform Gauss elimination
for i in range(n):
# Find the pivot element
pivot = abs(A[i][i])
pivot_row = i
for j in range(i+1, n):
if abs(A[j][i]) > pivot:
pivot = abs(A[j][i])
pivot_row = j
# Swap the pivot row with the current row (if necessary)
if pivot_row != i:
A[i], A[pivot_row] = A[pivot_row], A[i]
b[i], b[pivot_row] = b[pivot_row], b[i]
# Eliminate the current variable from the other equations
for j in range(i+1, n):
factor = A[j][i] / A[i][i]
for k in range(i, n):
A[j][k] -= factor * A[i][k]
b[j] -= factor * b[i]
# Back-substitute to find the solution
x = [0 for _ in range(n)]
for i in range(n-1, -1, -1):
x[i] = b[i]
for j in range(i+1, n):
x[i] -= A[i][j] * x[j]
x[i] /= A[i][i]
return x
# A - Coefficient Matrix
A = [[2, 1, -1], [-3, -1, 2], [-2, 1, 2]]
b = [8, -11, -3]
x = gauss_elimination(A, b)
print(x) # Output: [1.0, 3.0, -2.0]
