java实现CholeskyDecomposition的逆

Cholesky分解是将对称正定矩阵分解为下三角矩阵和它的转置的乘积的方法。其逆矩阵可以通过以下步骤实现：

1. 对原矩阵进行Cholesky分解，得到下三角矩阵L。
2. 求解Ly = I，其中I为单位矩阵，y为未知矩阵。
3. 求解L^T x = y，其中x为所求逆矩阵。

Java代码实现如下：

import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.CholeskyDecomposition;
import org.apache.commons.math3.linear.RealMatrix;

public class CholeskyInverse {

public static RealMatrix inverse(RealMatrix matrix) {
CholeskyDecomposition cholesky = new CholeskyDecomposition(matrix);
RealMatrix l = cholesky.getL();
RealMatrix lt = l.transpose();
int n = matrix.getColumnDimension();
RealMatrix identity = new Array2DRowRealMatrix(n, n);
for (int i = 0; i < n; i++) {
identity.setEntry(i, i, 1.0);
}
RealMatrix y = new Array2DRowRealMatrix(n, n);
for (int j = 0; j < n; j++) {
RealMatrix b = identity.getColumnMatrix(j);
RealMatrix x = solveLower(l, b);
y.setColumnMatrix(j, x);
}
RealMatrix inverse = new Array2DRowRealMatrix(n, n);
for (int j = 0; j < n; j++) {
RealMatrix b = y.getColumnMatrix(j);
RealMatrix x = solveUpper(lt, b);
inverse.setColumnMatrix(j, x);
}
return inverse;
}

private static RealMatrix solveLower(RealMatrix l, RealMatrix b) {
int n = l.getRowDimension();
RealMatrix x = new Array2DRowRealMatrix(n, 1);
for (int i = 0; i < n; i++) {
double sum = 0.0;
for (int k = 0; k < i; k++) {
sum += l.getEntry(i, k) * x.getEntry(k, 0);
}
double diag = l.getEntry(i, i);
double value = (b.getEntry(i, 0) - sum) / diag;
x.setEntry(i, 0, value);
}
return x;
}

private static RealMatrix solveUpper(RealMatrix lt, RealMatrix b) {
int n = lt.getRowDimension();
RealMatrix x = new Array2DRowRealMatrix(n, 1);
for (int i = n - 1; i >= 0; i--) {
double sum = 0.0;
for (int k = i + 1; k < n; k++) {
sum += lt.getEntry(i, k) * x.getEntry(k, 0);
}
double diag = lt.getEntry(i, i);
double value = (b.getEntry(i, 0) - sum) / diag;
x.setEntry(i, 0, value);
}
return x;
}
}

其中，solveLower和solveUpper分别是求解下三角矩阵和上三角矩阵的线性方程组的方法。这里使用了Apache Commons Math库中的CholeskyDecomposition类。