Python3 QR因式分解

Question

我对python3相当陌生。在秋天上课之前，我想在业余时间学习它。这个练习是我尝试过的最难的一次。我正在尝试获取代码并使用for循环而不是使用NUMPY以简单的形式重写它。最终的结果是编写以矩阵为参数的代码，并使用改良的schmidt算法计算并打印QR因式分解。任何帮助，将不胜感激。先感谢您。

#sample matrix
A = [[80,75,85],
    [75,80,75],
    [80,80,80]]

#function to find QR factorization of a square matrix
#parameter - matrix to be factorized
def findQRFactorization(A):
#check if A is a matrix
    if any(isinstance(i,list) for i in A):
        print("Is a matrix")
        #number of rows and columns in A
        rows = len(A)
        cols = len(A[0])
        print("rows:",rows," cols:",cols)
        #check if A is square matrix
        if rows != cols:
            print("Not a square matrix. Aborting...")
        else:
            print("A square matrix. proceeding...")
            #create an Identiry matrix of size rows x cols
            I = [[0]*cols for i in range(rows)]
            for i in range(rows):
                I[i][i] = 1
                #print(I)
                #calculation of QR factorization based on householder reflection method
                #QR factorization represents A as A = QR where Q is a orthogonal matrix
                #while R is a upper triangular matrix.
                #Initialize Q and R
                Q = [[0.0] * cols for i in range(rows)]
                R = A
                #print("Q:",Q)
                #print("R:",R)
                #loop to perform QR factorization
                for k in range(rows-1):
                    #calculate x, e
                    x = [row[k] for row in R[k:]]
                    e = [row[k] for row in I[k:]]
                    #calculate norm of x
                    norm_x = math.sqrt(sum([i**2 for i in x]))
                    #print("norm x:",norm_x)
                    #calculate alpha
                    sign = lambda x: (1, -1)[x < 0]
                    alpha = -sign(x[0])*norm_x
                    #print('alpha:',alpha)
                    #calculate minor matrix of u and v
                    u = list(map(lambda i,j: i + alpha * j, x, e))
                    norm_u = math.sqrt(sum([i**2 for i in u]))
                    v = list(map(lambda i: i/norm_u, u))

                    #calculate Q
                    Q_minor = [ [float(i==j) - 2.0 * v[i] * v[j] for i in range(rows-k)] for j in range(cols-k) ]

                    def Q_identity(Q_minor,i,j,k):
                        if i < k or j < k:
                            return float(i == j)
                        else:
                            return Q_minor[i-k][j-k]  

                            Q_padded = [[Q_identity(Q_minor,i,j,k) for i in range(rows)] for j in range(cols)]
                            #update Q and R
                            def multiply(P,Q):
                                return [[sum(elemp * elemq for elemp, elemq in zip(rowp, colq)) for colq in zip(*Q)] for rowp in P]              
                            if k == 0:
                                Q = Q_padded                  
                                R = multiply(Q_padded,A)
                            else:
                                Q = multiply(Q_padded,Q)
                                R = multiply(Q_padded,R)

                                #calculate transpose of Q
                                Q_dash = [[Q[i][j] for i in range(cols)] for j in range(rows)]
                                print("QR factorization of ",A)
                                print("Q: ",Q_dash)
                                print("R: ",R)
    else:
        print("Not a matrix. QR factorization not possible.")

#call function
findQRFactorization(A)