Gershgorin

GaussJordan.py

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+import numpy as np
+# calculate the inverse of the matrix A using Gauss-Jordan algorithm
+def GaussJordan (A):
+    (n, n) = A.shape
+    B = np.eye(n)
+
+    # Assume A has an inverse matrix
+    success = 1
+
+    if np.linalg.det(A) == 0:
+        # A cannot be reversed
+        B = np.empty(n)
+        success = 0
+        return
+
+    # merge the two matrices
+    Ae = np.concatenate((A, B.T), axis=1)
+    for i in range (0, n):
+        if Ae[i, i] == 0:
+            B = np.empty(n)
+            success = 0
+            return
+         # make the pivot position to one (as in the eye matrix)
+        Ae[i, :] = Ae[i, :] / Ae[i, i]
+
+        #form zeros above and under the main diagonal in the
+        # first half and calculate the inverse in the second half
+        for j in range(0, n):
+            if i != j:
+                Ae[j, :] = Ae[j, :] - Ae[i, :] * Ae[j, i]
+
+    #extract the inverse of matrix A from the augmented matrix
+    B = Ae[:, n : 2 * n ]
+    return (B, success)
+
+
+# example for a 4x4 matrix
+A = np.random.rand(4,4)
+(inv, success) = GaussJordan(A)
+print "The inverse using Gauss-Jordan is:"
+print inv
+print "The inverse using inv from linalg is:"
+print np.linalg.inv(A)

python/Gershgorin.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+def Gershgorin (A):
+    # A must be a square matrix
+    (m,n)  = A.shape
+    if m != n:
+        print "You must introduce a square matrix"
+        return
+
+    plt.axes()
+    # For each row:
+    for i in range (0, n):
+        # the circle has the center in (h, k) where
+        # h is the real part of A(i,i) and k is the real part of A(i,i)
+        h = np.real(A[i,i])
+        k = np.imag(A[i,i])
+        plt.plot(h, k, marker='x', markersize=5, color="blue")
+
+        # the radius of the circle is the sum of 
+        # norm of the elements in the row where i != j
+
+        r = 0
+        for j in range (0,n):
+            if i != j:
+                r = r + np.linalg.norm(A[i,j])
+        # plot the circle
+        circle = plt.Circle((h, k), r, fill = False)
+        plt.gca().add_patch(circle)        
+
+    eigenval = np.linalg.eigvals(A)
+   # plot the eigenvalues of the matrix
+    for x in eigenval:
+        plt.plot(np.real(x), np.imag(x), marker='o', markersize=5, color="red")
+
+    plt.axis('scaled')
+    plt.show()
+# example for a 3x3 matrix
+A = np.random.rand(3,3)
+print A
+Gershgorin(A)