Corrector Equations in Implicit Runge-Kutta Methods

2 Corrector Equations in Implicit Runge-Kutta Methods 

In order to solve the set of ns nonlinear equations (4.3.3a) Newton s method is applied. Again, like in the multistep case, Newton s method is implemented in a simplified manner, which saves evaluations (and decompositions) of the Jacobian With J being an approximation to the Jacobian 

Using the matrix multiplication rule for Kronecker products 

the constant, method dependent matrix A is separated from the state and problem dependent matrix J. A can be transformed by means of a similarity transformation into a simpler form, i.e. a block diagonal form A 

The coefficient matrix of the three stage Radau Ila method (4.3.5) has one real 

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