Finding the matrix exponential function using Taylor series.
Here’s an example of a Java program that calculates the matrix exponential function using Taylor series approximation:
import java.util.Arrays;
public class MatrixExponential {
public static void main(String[] args) {
double[][] matrix = {
{ 1, 2 },
{ 3, 4 }
};
double[][] exponential = matrixExponential(matrix, 10);
System.out.println("Matrix Exponential:");
for (double[] row : exponential) {
System.out.println(Arrays.toString(row));
}
}
public static double[][] matrixExponential(double[][] matrix, int terms) {
int n = matrix.length;
double[][] exponential = new double[n][n];
double[][] temp = new double[n][n];
// Initialize exponential matrix as identity matrix
for (int i = 0; i < n; i++) {
exponential[i][i] = 1;
}
// Calculate exponential using Taylor series approximation
for (int k = 1; k <= terms; k++) {
multiplyMatrix(matrix, exponential, temp);
scalarMultiply(1.0 / k, temp);
addMatrix(exponential, temp, exponential);
}
return exponential;
}
public static void multiplyMatrix(double[][] matrixA, double[][] matrixB, double[][] result) {
int n = matrixA.length;
int m = matrixB[0].length;
int p = matrixB.length;
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
double sum = 0;
for (int k = 0; k < p; k++) {
sum += matrixA[i][k] * matrixB[k][j];
}
result[i][j] = sum;
}
}
}
public static void scalarMultiply(double scalar, double[][] matrix) {
int n = matrix.length;
int m = matrix[0].length;
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
matrix[i][j] *= scalar;
}
}
}
public static void addMatrix(double[][] matrixA, double[][] matrixB, double[][] result) {
int n = matrixA.length;
int m = matrixA[0].length;
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
result[i][j] = matrixA[i][j] + matrixB[i][j];
}
}
}
}
In this example, the `matrixExponential` method takes a square matrix and the number of terms in the Taylor series approximation as input. It initializes an identity matrix as the initial approximation for the matrix exponential. Then, it iteratively calculates the matrix exponential using the Taylor series approximation by multiplying the matrix with the previous approximation, scaling it by the reciprocal of the term number, and adding it to the cumulative exponential. Finally, it returns the calculated matrix exponential.
In the `main` method, we create a sample matrix and call the `matrixExponential` method with the matrix and the number of terms as arguments. The resulting exponential matrix is then printed to the console.