Added the implementation of Lanczos algorithm

DIRECTORY.md

@@ -561,6 +561,7 @@
 ## Linear Algebra
   * [Gaussian Elimination](linear_algebra/gaussian_elimination.py)
   * [Jacobi Iteration Method](linear_algebra/jacobi_iteration_method.py)
+  * [Lanczos Algorithm](linear_algebra/lanczos_algorithm.py)
   * [Lu Decomposition](linear_algebra/lu_decomposition.py)
   * Src
     * [Conjugate Gradient](linear_algebra/src/conjugate_gradient.py)

linear_algebra/lanczos_algorithm.py

@@ -0,0 +1,37 @@
+import numpy as np
+
+
+def lanczos(a: np.ndarray) -> tuple[list[float], list[float]]:
+    """
+    Implements the Lanczos algorithm for a symmetric matrix.
+
+    Parameters:
+    -----------
+    matrix : numpy.ndarray
+        Symmetric matrix of size (n, n).
+
+    Returns:
+    --------
+    alpha : [float]
+        List of diagonal elements of the resulting tridiagonal matrix.
+    beta : [float]
+        List of off-diagonal elements of the resulting tridiagonal matrix.
+    """
+    n = a.shape[0]
+    v = np.zeros((n, n))
+    rng = np.random.default_rng()
+    v[:, 0] = rng.standard_normal(n)
+    v[:, 0] /= np.linalg.norm(v[:, 0])
+    alpha: list[float] = []
+    beta: list[float] = []
+    for j in range(n):
+        w = np.dot(a, v[:, j])
+        alpha.append(np.dot(w, v[:, j]))
+        if j == n - 1:
+            break
+        w -= alpha[j] * v[:, j]
+        if j > 0:
+            w -= beta[j - 1] * v[:, j - 1]
+        beta.append(np.linalg.norm(w))
+        v[:, j + 1] = w / beta[j]
+    return alpha, beta