The Jacobian matrix is required by many algorithms that solve the ordinary differential equations numerically, particular when the ODEs are stiff. The Hessian matrix and the adjoint are required for four-dimensional variational data assimilation (4D-Var).
The Jacobian matrix is crucial for solving numerical ODEs, particularly in situations where the equations are stiff. The Hessian matrix and the adjoint are similarly important tools used in 4D-Var.