Implemented Lanczos Algorithm

linear_algebra/Lanczos-algorithm.py

@@ -0,0 +1,34 @@
+import numpy as np
+def lanczos(A: np.ndarray) -> ([float], [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))
+    V[:, 0] = np.random.randn(n)
+    V[:, 0] /= np.linalg.norm(V[:, 0])
+    alpha = []
+    beta = []
+    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