From 02a1747fe1518d831b54b99a294b76a2d46e0a66 Mon Sep 17 00:00:00 2001
From: patternizer <patternizer@gmail.com>
Date: Sat, 18 Jul 2020 22:51:02 +0100
Subject: [PATCH] Add block to loop over 50x50 matrices and perform sensitivity
 analysis

---
 E2.py                    | 359 ++++++++++++++++++++++-----------------
 sensitivity_analysis.png | Bin 0 -> 21764 bytes
 2 files changed, 200 insertions(+), 159 deletions(-)
 create mode 100755 sensitivity_analysis.png

diff --git a/E2.py b/E2.py
index 9e13d40..22ce2d5 100644
--- a/E2.py
+++ b/E2.py
@@ -1,159 +1,200 @@
-import numpy as np
-from functools import reduce
-
-
-# Calculate the norm square of the jth element of the ith eigenvector using the alternate algorithm
-# H is a square np.matrix
-def vij_sq(H, i, j):
-    # Get the size of the matrix
-    n = H.shape[0]
-
-    # Get the eigenvalues of the matrix using numpy
-    eigenvalues = np.linalg.eigvalsh(H)
-
-    # Determine the jth minor
-    Mj = np.delete(np.delete(H, j, 0), j, 1)
-
-    # Get the eigenvalues of the minor
-    minor_eigenvalues = np.linalg.eigvalsh(Mj)
-
-    # Calculate the numerator
-    numerator = 1.
-    for k in range(n - 1):
-        numerator *= eigenvalues[i] - minor_eigenvalues[k]
-
-    # Calculate the denominator
-    denominator = 1.
-    for k in range(n):
-        if k == i:
-            continue
-        denominator *= eigenvalues[i] - eigenvalues[k]
-
-    return numerator / denominator
-
-
-def eig_sq(H):
-    # Get the size of the matrix
-    n = H.shape[0]
-    res = np.zeros((n, n))
-
-    for i in range(n):
-        for j in range(n):
-            res[i, j] = vij_sq(H, j, i)
-
-    return res
-
-
-# Calculate the norm square with performance improvements
-def vij_sq_perf(H, i, j):
-    # Get the size of the matrix
-    n = H.shape[0]
-
-    # Get the eigenvalues of the matrix using numpy
-    eigenvalues = np.linalg.eigvalsh(H)
-
-    # Determine the jth minor
-    Mj = np.delete(np.delete(H, j, 0), j, 1)
-
-    # Get the eigenvalues of the minor
-    minor_eigenvalues = np.linalg.eigvalsh(Mj)
-
-    # Calculate the numerator
-    eigenvaluei = eigenvalues[i]
-    numerator = reduce(lambda x, y: x * y, [eigenvaluei - minor_eigenvalues[k] for k in range(n - 1)])
-
-    # Calculate the denominator
-    denominator = reduce(lambda x, y: x * y, [eigenvaluei - eigenvalues[k] for k in range(n) if k != i])
-
-    return numerator / denominator
-
-
-def eig_sq_perf(H):
-    # Get the size of the matrix
-    n = H.shape[0]
-    res = np.zeros((n, n))
-
-    for i in range(n):
-        for j in range(n):
-            res[i, j] = vij_sq_perf(H, j, i)
-
-    return res
-
-
-# Calculate the norm square of all the elements of all the eigenvectors using the alternate algorithm
-# H is a square np.matrix
-def eig_sq_vect(H):
-    # Get size of the matrix
-    n = H.shape[0]
-
-    # Get eigenvalues of the matrix
-    eigvals = np.linalg.eigvalsh(H)
-
-    # Create a matrix of all the minors minors and a matrix of eigenvalues as required in the algorithm
-    minors = np.empty((n, n - 1, n - 1), dtype=H.dtype)
-    eig_matrix = np.empty((n, n - 1))
-    for i in range(n):
-        minors[i] = np.delete(np.delete(H, i, 0), i, 1)
-        eig_matrix[i] = np.delete(eigvals, i)
-
-    # Get eigenvalues of all the minors
-    minor_eigvals = np.linalg.eigvalsh(minors)
-
-    # Calculate the numerators and the denominators
-    numerator = np.empty((n, n))
-    denominator = np.empty((n,))
-    for i in range(n):
-        numerator[i] = np.prod(eigvals[i] - minor_eigvals, axis=1)
-        denominator[i] = np.prod(eigvals[i] - eig_matrix[i])
-
-    return np.divide(numerator.T, denominator.reshape((1, n)))
-
-
-# Calculate the norm square of all the elements of all the eigenvectors using numpy's built in eigenvector calculator
-# H is a square np.matrix
-def eig_sq_np(H):
-    # Get the eigenvalues and eigenvalues of the matrix
-    eigenvalues, eigenvectors = np.linalg.eigh(H)
-    return abs(eigenvectors) ** 2
-
-
-# Prints true for each element that agrees between the E2 method and the method built into numpy
-# H is a square np.matrix
-# mode determines comparison with normal or improved function
-def compare(H, mode=0):
-    # The tolerance for comparison purposes
-    tol = 1e-12
-
-    # Select function depending upon the value of mode
-    if mode == 0:
-        res = eig_sq(H)
-    elif mode == 1:
-        res = eig_sq_perf(H)
-    elif mode == 2:
-        res = eig_sq_vect(H)
-    else:
-        raise IndexError
-
-    # Return the comparison matrix
-    return abs(res - eig_sq_np(H)) < tol
-
-
-if __name__ == "__main__":
-    # Generate a random Hermitian matrix
-    # Dimension of the matrix
-    n = 8
-    H = np.random.rand(n, n) + 1j * np.random.rand(n, n)
-    H = 0.5 * (H + H.T)
-
-    # Print the second element of the first eigenvector as calculated in both methods
-    print("\n--- using the algorithm ---")
-    basic = eig_sq(H)
-    print("basic implementation: ", basic[0, 1])
-    perf = eig_sq_perf(H)
-    print("performant implementation: ", perf[0, 1])
-    vect = eig_sq_vect(H)
-    print("vectorized implementation implementation: ", vect[0, 1])
-
-    print("\n--- using numpy ---")
-    nump = eig_sq_np(H)
-    print(nump[0, 1])
+import numpy as np
+from functools import reduce
+from numpy import linalg as LA
+import matplotlib.pyplot as plt        
+
+
+# Calculate the norm square of the jth element of the ith eigenvector using the alternate algorithm
+# H is a square np.matrix
+def vij_sq(H, i, j):
+    # Get the size of the matrix
+    n = H.shape[0]
+
+    # Get the eigenvalues of the matrix using numpy
+    eigenvalues = np.linalg.eigvalsh(H)
+
+    # Determine the jth minor
+    Mj = np.delete(np.delete(H, j, 0), j, 1)
+
+    # Get the eigenvalues of the minor
+    minor_eigenvalues = np.linalg.eigvalsh(Mj)
+
+    # Calculate the numerator
+    numerator = 1.
+    for k in range(n - 1):
+        numerator *= eigenvalues[i] - minor_eigenvalues[k]
+
+    # Calculate the denominator
+    denominator = 1.
+    for k in range(n):
+        if k == i:
+            continue
+        denominator *= eigenvalues[i] - eigenvalues[k]
+
+    return numerator / denominator
+
+
+def eig_sq(H):
+    # Get the size of the matrix
+    n = H.shape[0]
+    res = np.zeros((n, n))
+
+    for i in range(n):
+        for j in range(n):
+            res[i, j] = vij_sq(H, j, i)
+
+    return res
+
+
+# Calculate the norm square with performance improvements
+def vij_sq_perf(H, i, j):
+    # Get the size of the matrix
+    n = H.shape[0]
+
+    # Get the eigenvalues of the matrix using numpy
+    eigenvalues = np.linalg.eigvalsh(H)
+
+    # Determine the jth minor
+    Mj = np.delete(np.delete(H, j, 0), j, 1)
+
+    # Get the eigenvalues of the minor
+    minor_eigenvalues = np.linalg.eigvalsh(Mj)
+
+    # Calculate the numerator
+    eigenvaluei = eigenvalues[i]
+    numerator = reduce(lambda x, y: x * y, [eigenvaluei - minor_eigenvalues[k] for k in range(n - 1)])
+
+    # Calculate the denominator
+    denominator = reduce(lambda x, y: x * y, [eigenvaluei - eigenvalues[k] for k in range(n) if k != i])
+
+    return numerator / denominator
+
+
+def eig_sq_perf(H):
+    # Get the size of the matrix
+    n = H.shape[0]
+    res = np.zeros((n, n))
+
+    for i in range(n):
+        for j in range(n):
+            res[i, j] = vij_sq_perf(H, j, i)
+
+    return res
+
+
+# Calculate the norm square of all the elements of all the eigenvectors using the alternate algorithm
+# H is a square np.matrix
+def eig_sq_vect(H):
+    # Get size of the matrix
+    n = H.shape[0]
+
+    # Get eigenvalues of the matrix
+    eigvals = np.linalg.eigvalsh(H)
+
+    # Create a matrix of all the minors minors and a matrix of eigenvalues as required in the algorithm
+    minors = np.empty((n, n - 1, n - 1), dtype=H.dtype)
+    eig_matrix = np.empty((n, n - 1))
+    for i in range(n):
+        minors[i] = np.delete(np.delete(H, i, 0), i, 1)
+        eig_matrix[i] = np.delete(eigvals, i)
+
+    # Get eigenvalues of all the minors
+    minor_eigvals = np.linalg.eigvalsh(minors)
+
+    # Calculate the numerators and the denominators
+    numerator = np.empty((n, n))
+    denominator = np.empty((n,))
+    for i in range(n):
+        numerator[i] = np.prod(eigvals[i] - minor_eigvals, axis=1)
+        denominator[i] = np.prod(eigvals[i] - eig_matrix[i])
+
+    return np.divide(numerator.T, denominator.reshape((1, n)))
+
+
+# Calculate the norm square of all the elements of all the eigenvectors using numpy's built in eigenvector calculator
+# H is a square np.matrix
+def eig_sq_np(H):
+    # Get the eigenvalues and eigenvalues of the matrix
+    eigenvalues, eigenvectors = np.linalg.eigh(H)
+    return abs(eigenvectors) ** 2
+
+
+# Prints true for each element that agrees between the E2 method and the method built into numpy
+# H is a square np.matrix
+# mode determines comparison with normal or improved function
+def compare(H, mode=0):
+    # The tolerance for comparison purposes
+    tol = 1e-12
+
+    # Select function depending upon the value of mode
+    if mode == 0:
+        res = eig_sq(H)
+    elif mode == 1:
+        res = eig_sq_perf(H)
+    elif mode == 2:
+        res = eig_sq_vect(H)
+    else:
+        raise IndexError
+
+    # Return the comparison matrix
+    return abs(res - eig_sq_np(H)) < tol
+
+
+if __name__ == "__main__":
+    
+    # FPE as a function of matrix size or not
+    sensitivity_analysis = True
+
+    if not sensitivity_analysis:
+        
+        # Generate a random Hermitian matrix
+        # Dimension of the matrix
+            
+        n = 8
+        H = np.random.rand(n, n) + 1j * np.random.rand(n, n)
+        H = 0.5 * (H + H.T)
+
+        # Print the second element of the first eigenvector as calculated in both methods
+        print("\n--- using the algorithm ---")
+        basic = eig_sq(H)
+        print("basic implementation: ", basic[0, 1])
+        perf = eig_sq_perf(H)
+        print("performant implementation: ", perf[0, 1])
+        vect = eig_sq_vect(H)
+        print("vectorized implementation implementation: ", vect[0, 1])
+        
+        print("\n--- using numpy ---")
+        nump = eig_sq_np(H)
+        print(nump[0, 1])
+    
+    else:
+        nmin = 2
+        nmax = 50
+        fpe_basic_array = np.zeros([nmax-nmin+1])
+        fpe_perf_array = np.zeros([nmax-nmin+1])
+        fpe_vect_array = np.zeros([nmax-nmin+1])
+        for i in range(nmin,nmax+1):    
+            n = i
+            H = np.random.rand(n, n) + 1j * np.random.rand(n, n)
+            H = 0.5 * (H + H.T)
+            basic = eig_sq(H)
+            perf = eig_sq_perf(H)
+            vect = eig_sq_vect(H)
+            nump = eig_sq_np(H)
+            fpe_basic = 100.*(LA.norm(basic)-LA.norm(nump))/LA.norm(basic)
+            fpe_perf = 100.*(LA.norm(perf)-LA.norm(nump))/LA.norm(perf)
+            fpe_vect = 100.*(LA.norm(vect)-LA.norm(nump))/LA.norm(vect)
+            fpe_basic_array[i-nmin] = fpe_basic
+            fpe_perf_array[i-nmin] = fpe_perf
+            fpe_vect_array[i-nmin] = fpe_vect
+
+        fig,ax = plt.subplots(figsize=(15,10))
+        plt.scatter(np.arange(nmin, nmax+1), fpe_basic_array, label='f(basic-numpy)', marker='o', color='black')
+        plt.scatter(np.arange(nmin, nmax+1), fpe_perf_array, label='f(perf-numpy)', marker='.', color='red')
+        plt.scatter(np.arange(nmin, nmax+1), fpe_vect_array, label='f(vect-numpy)', marker='o', color='grey')
+        plt.legend()
+        plt.xlabel('Matrix size, n')
+        plt.ylabel('FPE, %')
+        plt.title('Sensitivity analysis')
+        plt.savefig('sensitivity_analysis.png')
+        
diff --git a/sensitivity_analysis.png b/sensitivity_analysis.png
new file mode 100755
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