共轭梯度法与QR法解矩阵方程

版权声明：本文为博主原创文章，未经博主允许不得转载。 https://blog.csdn.net/m0_37717751/article/details/81050182

import numpy as np
import numpy.linalg as nl

def ConjugateGradient(A, b, n, x0):
    x = x0
    r = b - np.dot(A, x0)
    p = r
    for i in range(0, n):
        a = np.dot(r.transpose(), r) / np.dot(np.dot(p.transpose(), A), p)
        tmp1 = a * p
        x = x + tmp1
        tmp2 = np.dot(a * A, p)
        rr = r - tmp2
        check = np.dot(A, x)
        check = check - b
        if (nl.norm(check) / nl.norm(b) < 1e-6):
            break
        B = np.dot(rr.transpose(), rr) / np.dot(r.transpose(), r)
        p = B * p
        p = p + rr
        r = rr
    return x

def QR(A, b, n):
    Q = np.zeros([n, n])
    cnt = 0
    for a in A.transpose():
        u = np.copy(a)
        for i in range(0, cnt):
            u = u - np.dot(np.dot(Q[:, i].transpose(), a), Q[:, i])
        e = u / nl.norm(u)
        Q[:, cnt] = e
        cnt = cnt + 1
    R = np.dot(Q.transpose(), A)
    x = np.dot(np.dot(nl.inv(R), Q.transpose()), b)
    return x

if __name__ == "__main__":
    A = np.zeros([100, 100])
    b = np.zeros([100,1])
    x0 = np.zeros([100,1])
    for i in range(0, 100):
        A[i][i] = 2
        b[i][0] = 1
        if i == 0:
            A[i][i + 1] = -1
        elif i == 99:
            A[i][i - 1] = -1
        else:
            A[i][i + 1] = A[i][i - 1] = -1
    result1 = ConjugateGradient(A, b, 100, x0)
    print("共轭梯度法：")
    print(result1)
    result2 = QR(A, b, 100)
    print("QR：")
    print(result2)