Cholesky分解

一、简介

Cholesky 分解是把一个对称正定的矩阵表示成一个下三角矩阵L和其转置的乘积的分解。它要求矩阵的所有特征值必须大于零，故分解的下三角的对角元也是大于零的。Cholesky分解法又称平方根法，是当A为实对称正定矩阵时，LU三角分解法的变形。

如果矩阵A为n阶对称正定矩阵，则存在一个对角元素为正数的下三角实矩阵L，当限定L的对角元素为正时，这种分解是唯一的，称为Cholesky分解。
详解

二、实现

# -*- coding: utf-8 -*-
"""
Created on Wed Dec 14 19:43:58 2016

    Cholesky分解

@author: Administrator
"""

from numpy import *

def cholesky_decomp(A):
    m,n = A.shape
    if m != n:
        return
    temp_A = A.copy()
    L = zeros((n,n),dtype=float)
    for i in range(0,n):
        for j in range(0,n):
            if i >= j + 1:
                if j > 0:
                    L[i,j] = (temp_A[i,j] - sum(L[i,0:j] * L[j,0:j])) / L[j,j]
                else:
                    L[i,j] = temp_A[i,j] / L[j,j]
            else:
                if j > 0:
                    L[j,j] = sqrt(temp_A[j,j] - sum(L[j,0:j] * L[j,0:j]))
                else:
                    L[j,j] = sqrt(A[j,j])
    return L

def cal_cholesky_decomp(A,b):
    m,n = A.shape
    L = cholesky_decomp(A)
    y = array([0 for i in range(n)],dtype=float)
    x = y.copy()
    for i in range(0,n):
        if i == 0:
            y[i] = b[i] / L[i,i]
        else:
            y[i] = (b[i] - sum(L[i,0:i] * y[0:i].transpose())) / L[i,i]
    for i in range(n-1,-1,-1):
        if i == n-1:
            x[i] = y[i] / L[i,i]
        else:
            x[i] = (y[i] - sum(L[i+1:n,i] * x[i+1:n].transpose())) / L[i,i]
    return x

if __name__ == '__main__':
    n = 10
    A = ones((n,n),dtype=float)
    b = arange(0,n)
    for i in range(n):
        for j in range(n):
            A[i,j] = 1. / (i + j + 1) # i +j - 1 + 1 + 1
    print 'n=',n,':',cal_cholesky_decomp(A,b)