Add Code Files

Code/.gitignore

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+__pycache__/
+*.py[cod]
+*$py.class
+
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+build/
+dist/
+downloads/
+eggs/
+.eggs/
+*.egg-info/
+.installed.cfg
+*.egg
+
+# Virtual environments
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# Pyenv
+.python-version
+
+# Hatch build artifacts
+.hatch/
+
+# IDEs and editors
+.vscode/
+.idea/
+*.swp
+*~
+
+# Logs
+*.log

Code/README.md

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+# Serie 1
+## Gleitkommaarithmetik
+
+# Serie 2
+## FixpunktIteration und Heron-Verfahren
+
+# Serie 3
+## NewtonSchema
+
+# Serie 4
+## Neville-Aitken-Schema
+
+# Serie 5
+## Regression
+
+# Serie 6
+## Trapez-Regel, Simpson-Regel
+
+# Serie 7
+## Gauss-Jordan-Verfahren

Code/main.py

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+"""Main script for numerik package."""
+import sys
+import os
+
+# Ensure src directory is on the Python path for local imports
+sys.path.insert(0, os.path.join(os.path.dirname(__file__), "src"))
+
+def main():
+    """Demonstrate polynomial interpolation using divided differences."""
+    from serie_3 import poly_mat
+
+    x = [1, 2, 3, 4, 5]
+    y = [2, 3, 5, 7, 11]
+
+    result = poly_mat.poly_mat(list(zip(x, y)))
+    print(result)
+
+if __name__ == "__main__":
+    main()

Code/pyproject.toml

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+[project]
+name = "numerik"
+version = "0.1.0"
+description = "Add your description here"
+readme = "README.md"
+authors = [
+    { name = "DotNaos", email = "schuetzoliver00@gmail.com" }
+]
+requires-python = ">=3.8"
+dependencies = [
+    "matplotlib>=3.10.1",
+    "numpy>=2.2.4",
+    "pandas>=2.2.3",
+]
+
+[build-system]
+requires = ["hatchling"]
+build-backend = "hatchling.build"
+
+[tool.hatch.build.targets.wheel]
+packages=[
+    "src/serie_1",
+    "src/serie_2",
+    "src/serie_3",
+    "src/serie_4",
+    "src/serie_5",
+    "src/serie_6",
+    "src/serie_7",
+]

Code/src/serie_1/__init__.py

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+"""Series 1: Floating point arithmetic (Gleitkommaarithmetik)."""

Code/src/serie_2/Serie2.py

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+"""Fixed-point iteration visualizer (Series 2).
+
+Provides a decorator to visualize convergence of iterative methods."""
+import logging
+from typing import Callable
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+logger = logging.getLogger(__name__)
+
+def visualize_convergence(x0: float, tol: float) -> Callable[[Callable[[float], float]], Callable[..., float]]:
+    """
+    Decorator to visualize fixed-point iteration convergence.
+
+    Args:
+        x0: Initial guess for the iteration.
+        tol: Convergence tolerance (stop when |x_next - x| <= tol).
+    Returns:
+        A decorator that wraps a function f(x) -> x_next, plotting each step.
+    """
+    def decorator(func: Callable[[float], float]) -> Callable[..., float]:
+        def wrapper(*args, **kwargs) -> float:
+            x = x0
+            x_next = func(x)
+            # Iterate until convergence
+            while abs(x_next - x) > tol:
+                plt.plot([x], [x_next], 'o')
+                logger.debug('Iterate: x=%f -> x_next=%f', x, x_next)
+                x = x_next
+                x_next = func(x)
+            # Plot final point
+            plt.plot([x], [x_next], 'o')
+            return x_next
+        return wrapper
+    return decorator
+
+
+@visualize_convergence(x0=10.0, tol=1e-7)
+def a4(x_n: float) -> float:
+    """Fixed-point function for method A."""
+    return 1 - (1 / 5) * x_n
+
+@visualize_convergence(x0=1.0, tol=1e-7)
+def b4(x_n: float) -> float:
+    """Fixed-point function for method B (cosine)."""
+    return np.cos(x_n)
+
+
+@visualize_convergence(x0=1.0, tol=1e-7)
+def c4(x_n: float) -> float:
+    """Fixed-point function for method C (exponential decay)."""
+    return np.exp(-x_n)
+
+
+if __name__ == '__main__':
+    logging.basicConfig(level=logging.INFO)
+    # Demonstration for c4 method
+    result = c4()
+    print(f'Converged to {result}')
+    plt.show()

Code/src/serie_2/__init__.py

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+"""Series 2: Fixed-point iteration and convergence visualization."""
+from .Serie2 import visualize_convergence, a4, b4, c4

Code/src/serie_3/Serie3.py

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+from typing import Sequence
+
+"""Newton's divided difference polynomial series (Series 3).
+
+Provides poly() function to compute divided difference coefficients."""
+"""Newton's divided difference polynomial series (Series 3).
+
+Provides poly() to compute divided difference coefficients for interpolation."""
+def poly(items: Sequence[tuple[float, float]]) -> list[float]:
+    """
+    Compute divided difference coefficients for Newton interpolation.
+
+    Parameters:
+        items: Sequence of (x, y) data points.
+    Returns:
+        List of divided difference coefficients [f[x0], f[x0,x1], ..., f[x0,...,xN]].
+    """
+    x_vals, y_vals = zip(*items)
+    N = len(x_vals)
+    # Working copy of y values
+    diff_values = list(y_vals)
+    # First coefficient f[x0]
+    coeffs: list[float] = [diff_values[0]]
+
+    # Compute divided differences in-place
+    for j in range(1, N):
+        for i in range(N - 1, j - 1, -1):
+            diff_values[i] = (diff_values[i] - diff_values[i - 1]) / (x_vals[i] - x_vals[i - j])
+        coeffs.append(diff_values[j])
+
+    return coeffs
+
+
+if __name__ == "__main__":
+    # Demonstration of poly() with sample data
+    sample_points = [(-1.0, -5.0), (0.0, -1.0), (1.0, -1.0), (2.0, 1.0)]
+    coeffs = poly(sample_points)
+    print("Divided difference coefficients:", coeffs)

Code/src/serie_3/__init__.py

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+"""Series 3: Newton's divided difference interpolation."""
+from .poly_mat import poly_mat
+from .Serie3 import poly

Code/src/serie_3/poly_mat.py

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+from typing import Sequence
+import numpy as np
+
+"""Compute divided difference table in matrix form for Newton interpolation (Series 3)."""
+def poly_mat(items: Sequence[tuple[float, float]]) -> np.ndarray:
+    """
+    Compute the divided differences table (Newton interpolation) as a NumPy array.
+
+    Parameters:
+        items: Sequence of (x, y) data points.
+    Returns:
+        A 2D NumPy array of shape (N, N) containing the divided differences table.
+    """
+    x_vals, y_vals = zip(*items)
+    N = len(items)
+    table = np.zeros((N, N), dtype=float)
+    # Zeroth column: y values
+    table[0, :] = y_vals
+
+    # Iterate over every column
+    for col in range(1, N):
+        for row in range(1, col + 1):
+            numerator = table[row - 1, col] - table[row - 1, col - 1]
+            denominator = x_vals[col] - x_vals[col - row]
+            table[row, col] = numerator / denominator
+    return table

Code/src/serie_4/Serie4.py

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+import logging
+from typing import Sequence
+import numpy as np
+
+logger = logging.getLogger(__name__)
+
+def neville_mat(x_: Sequence[float], y_: Sequence[float], x_F: float) -> np.ndarray:
+    """
+    Compute the Neville-Aitken interpolation table for given points and evaluation x_F.
+
+    Parameters:
+        x_: Sequence of x coordinates.
+        y_: Sequence of y coordinates.
+        x_F:  The point at which to evaluate the interpolation.
+
+    Returns:
+        A 2D NumPy array representing the Neville interpolation table.
+    """
+    N = len(x_)
+    table = np.zeros((N, N), dtype=float)
+    table[0, :] = y_
+
+    for col in range(1, N):
+        for row in range(1, col + 1):
+            x_a = x_[col - 1]
+            x_b = x_[col]
+            A_val = table[row - 1, col - 1]
+            B_val = table[row - 1, col]
+            numerator = B_val * (x_F - x_a) - A_val * (x_F - x_b)
+            denominator = x_b - x_a
+            table[row, col] = numerator / denominator
+            logger.debug(
+                "table[%d,%d] = (%f * (x_F - %f) - %f * (x_F - %f)) / %f = %f",
+                row, col, B_val, x_a, A_val, x_b, denominator, table[row, col],
+            )
+
+    return table
+
+
+if __name__ == "__main__":
+    logging.basicConfig(level=logging.INFO)
+    # Example usage for Neville interpolation
+    points1 = {"x_": [-1, 1, 2], "y_": [15, 5, 9], "x_F": 3.0}
+    table1 = neville_mat(**points1)
+    print("Neville table for points1:")
+    print(table1)
+
+    points2 = {"x_": [-1, 0, 1, 2], "y_": [-5, -1, -1, 1], "x_F": 3.0}
+    table2 = neville_mat(**points2)
+    print("\nNeville table for points2:")
+    print(table2)

Code/src/serie_4/__init__.py

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+"""Series 4: Neville-Aitken interpolation."""
+from .Serie4 import neville_mat

Code/src/serie_5/__init__.py

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+"""Series 5: Regression methods."""

Code/src/serie_6/__init__.py

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+"""Series 6: Numerical integration (Trapezoidal and Simpson's rule)."""

Code/src/serie_7/__init__.py

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+"""Series 7: Gauss-Jordan elimination."""
+from .gauss import gauss_jordan, extract_solution

Code/src/serie_7/gauss.py

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+"""
+Gaussian elimination (Gauss-Jordan) demonstration for solving linear systems.
+"""
+import logging
+from typing import Optional
+import numpy as np
+
+TOLERANCE = 1e-8
+logger = logging.getLogger(__name__)
+
+def gauss_jordan(A: np.ndarray, B: np.ndarray, tolerance: float = TOLERANCE) -> np.ndarray:
+    """
+    Perform Gauss-Jordan elimination on the linear system Ax = B.
+
+    Parameters:
+        A: Coefficient matrix of shape (n, n).
+        B: Constant vector of shape (n, 1) or (n,).
+        tolerance: Threshold for zero comparisons.
+
+    Returns:
+        Augmented matrix in Reduced Row Echelon Form.
+    """
+    # Ensure B is a column vector
+    B_col = B.reshape(-1, 1) if B.ndim == 1 else B
+    if A.shape[0] != B_col.shape[0]:
+        raise ValueError(
+            f"A and B must have the same number of rows, got {A.shape[0]} and {B_col.shape[0]}"
+        )
+
+    # Create augmented matrix
+    mat = np.hstack([A.astype(float), B_col.astype(float)])
+    rows, cols = mat.shape
+    pivot_row = 0
+
+    for pivot_col in range(cols - 1):
+        if pivot_row >= rows:
+            break
+
+        # Partial pivoting
+        max_row = pivot_row + np.argmax(np.abs(mat[pivot_row:, pivot_col]))
+        if max_row != pivot_row:
+            mat[[pivot_row, max_row]] = mat[[max_row, pivot_row]]
+            logger.debug("Swapped rows %d and %d", pivot_row, max_row)
+
+        pivot = mat[pivot_row, pivot_col]
+        if abs(pivot) < tolerance:
+            logger.debug("Pivot too small at row %d, col %d", pivot_row, pivot_col)
+            continue
+
+        # Normalize pivot row
+        mat[pivot_row] = mat[pivot_row] / pivot
+        mat[pivot_row, pivot_col] = 1.0
+        logger.debug("Normalized row %d with pivot %f", pivot_row, pivot)
+
+        # Eliminate other rows
+        for r in range(rows):
+            if r != pivot_row:
+                factor = mat[r, pivot_col]
+                if abs(factor) > tolerance:
+                    mat[r] -= factor * mat[pivot_row]
+                    mat[r, pivot_col] = 0.0
+                    logger.debug("Eliminated row %d using row %d", r, pivot_row)
+
+        pivot_row += 1
+
+    # Cleanup near-zero entries
+    mat[np.abs(mat) < tolerance] = 0.0
+    return mat
+
+def extract_solution(rref: np.ndarray, tolerance: float = TOLERANCE) -> Optional[np.ndarray]:
+    """
+    Extract solution vector if left part of rref is identity.
+    """
+    n_rows, n_cols = rref.shape
+    n_vars = n_cols - 1
+    if n_rows == n_vars and np.allclose(rref[:, :n_vars], np.eye(n_vars), atol=tolerance):
+        return rref[:, -1].reshape(-1, 1)
+    return None
+
+if __name__ == "__main__":
+    logging.basicConfig(level=logging.INFO)
+    # Sample system: Ax = B
+    A = np.array(
+        [[2.0, 3.0, 1.0], [1.0, -2.0, -1.0], [4.0, 1.0, -3.0]]
+    )
+    B = np.array([8.0, -3.0, 6.0])
+
+    print("INPUT AUGMENTED MATRIX:")
+    print(np.hstack([A, B.reshape(-1, 1)]), end="\n\n")
+
+    rref = gauss_jordan(A, B)
+    print("--- Final Matrix (RREF) ---")
+    print(rref, end="\n\n")
+
+    solution = extract_solution(rref)
+    if solution is not None:
+        print("SOLUTION (x):")
+        print(solution)
+    else:
+        print("No unique solution could be extracted.")

Code/uv.lock

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