Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose.
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。
wiki
https://en.wikipedia.org/wiki/Cholesky_decomposition
MATLAB
https://www.mathworks.com/help/matlab/ref/chol.html?s_tid=gn_loc_drop#responsive_offcanvas
Cplusplus
https://eigen.tuxfamily.org/dox/classEigen_1_1LLT.html
#include <iostream>
#include <vector>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
MatrixXd A(3,3);
A << 4,-1,2, -1,6,0, 2,0,5;
cout << "The matrix A is" << endl << A << endl;
LLT<MatrixXd> lltOfA(A); // compute the Cholesky decomposition of A
//cout << "lltofA" << endl << lltOfA(A) << endl;
MatrixXd L = lltOfA.matrixL(); // retrieve factor L in the decomposition
// The previous two lines can also be written as "L = A.llt().matrixL()"
cout << "The Cholesky factor L is" << endl << L << endl;
cout << "To check this, let us compute L * L.transpose()" << endl;
cout << L * L.transpose() << endl;
cout << "This should equal the matrix A" << endl;
}
MATLAB Cholesky分解
>> A = [4 -1 2 ; -1 6 0 ; 2 0 5 ]
A =
4 -1 2
-1 6 0
2 0 5
>> L = chol(A,'lower')
L =
2.00000 0.00000 0.00000
-0.50000 2.39792 0.00000
1.00000 0.20851 1.98910
>> B = L * L'
B =
4.0000e+00 -1.0000e+00 2.0000e+00
-1.0000e+00 6.0000e+00 -2.5602e-17
2.0000e+00 -2.5602e-17 5.0000e+00
>>