algorithm-in-python/math/numericalAnalysis/linear_equation.py

''' mbinary
#########################################################################
# File : linear_equation.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2018-10-02  21:14
# Description:
#########################################################################
'''

#coding: utf-8
'''************************************************************************
    > File Name: doolittle.py
    > Author: mbinary
    > Mail: zhuheqin1@gmail.com 
    > Blog: https://mbinary.xyz
    > Created Time: 2018-04-20  08:32
 ************************************************************************'''




import  numpy as np
def getLU(A):
    '''doolittle :     A = LU, 
        L is in  down-triangle form, 
        U is in  up-triangle form
    '''
    m, n = A.shape
    if m != n:
        raise Exception("this matrix is not inversable")

    L = np.zeros([m, m])
    U = np.zeros([m, m])
    L = np.matrix(L)
    U = np. matrix(U)
    U[0] = A[0]
    L[:, 0] = A[:, 0] / A[0, 0]
    for i in range(1, m):
        for j in range(i, m):
            U[i, j] = A[i, j] - sum(L[i, k]*U[k, j] for k in range(i))
            L[j, i] = (A[j, i] - sum(L[j, k]*U[k, i]
                                     for k in range(i)))/U[i, i]
    print(L)
    print(U)
    return L, U


def gauss_prior_elimination(A):
    '''using guass elimination,get up_trianglge form of A'''
    m, n = A.shape
    if m != n:
        raise Exception("[Error]: matrix is not inversable")
    # necessary,otherwise when the dtype of A is int, then it will be wrong
    B = np.matrix(A, dtype=float)
    for i in range(m-1):
        # note using abs value,  return a matrix in (m-i)x1 form
        col = abs(B[i:, i])
        mx = col.max()
        if mx == 0:
            raise Exception("[Error]: matrix is not inversable")
        pos = i+col.argmax()
        if pos != i:
            B[[pos, i], :] = B[[i, pos], :]  # note how to swap cols/rows
        B[i, :] = 1/mx*B[i, :]
        for j in range(i+1, m):
            # print(B)
            B[j, :] -= B[j, i] * B[i, :]
    print(B)
    return B


def solveDown(A, b):
    '''A is a matrix in down-triangle form'''
    sol = np.zeros(b.shape)
    for i in range(b.shape[0]):
        sol[i, 0] = (b[i, 0]-sum(A[i, j]*sol[j, 0] for j in range(i)))/A[i, i]
    return sol


def solveUp(A, b):
    '''A is a matrix in up-triangle form'''
    sol = np.zeros(b.shape)
    n = b.shape[0]
    for i in range(n-1, -1, -1):
        sol[i, 0] = (b[i, 0]-sum(A[i, j]*sol[j, 0]
                                 for j in range(n-1, i, -1)))/A[i, i]
    return sol


def doolittle(A, b):
    L, U = getLU(A)
    y = solveDown(L, b)
    x = solveUp(U, y)
    print(y)
    print(x)
    return x


def ldlt(A, b):
    L, U = getLU(A)
    D = np.diag(np.diag(U))
    print(D, "D")
    z = np.linalg.solve(L, b)
    print(z, "z")
    y = np.linalg.solve(D, z)
    print(y, "y")
    x = np.linalg.solve(L.T, y)
    print(x, "x")
    return x


if __name__ == '__main__':
    A = np.matrix([[10, 5, 0, 0],
                   [2, 2, 1, 0],
                   [0, 10, 0, 5],
                   [0, 0, 2, 1]])
    b = np.matrix([[5], [3], [27], [6]])
    gauss_prior_elimination(A)

    '''ldlt
    A = np.matrix([[-6,3,2],
                     [3,5,1],
                     [2,1,6]])
    b = np.matrix([[-4],[11],[-8]])
    ldlt(A,b)
    '''
'''
    A = np.matrix([[2,1,1],
                     [1,3,2],
                     [1,2,2]])
    b = np.matrix([[4],[6],[5]])
    doolittle(A,b)
'''