If the error test is satisfied then the estimated value of the stepsize, , obtained from. In equation 35 , R is order of the predeictor used. Suppose that h is the current stepsize. This strategy is just the same as discussed by Khiyal [6]. However, we were not able to find a corresponding perfect cube iteration scheme for the three-stage sixth order implicit Runge-Kutta method.

Moreover, the perfect square iteration scheme for the fourth order method is not always reliable. For these reasons, in this section, we consider an iteration scheme proposed by Cooper and Butcher [5] which sacrifices the superlinear convergence of the modified Newton method for reduced linear algebra costs.

This iteration scheme requires the solution of s systems of n linear equations at each step of the iteration and the coefficient matrix is the same for each system.

## Linear multistep method

We have shown that for computational convenience the nonautonomous system 2 can be written in the form of an autonomous system and then we write the Runge-Kutta method in autonomous form. Suppose the first order autonomous initial value problem is.

We may simplify our notation by using direct sums and direct products. Let and be column vectors and let Then equation 38 may be represented by. Where, A q I is the direct product of A with the sxs identity matrix and, in general,. The equation 39 may be solved by the modified Newton method but because the Jacobian of F is expensive to evaluate, it is computed only occasionally. Suppose that is an approximation for this Jacobian. Then the modified Newton iteration scheme is. At each iteration, we need to solve a system of linear equations.

Let B and S be real invertible matrices and let L be a strictly lower triangular matrix and r a real constant. Cooper and Butcher [5] propose the following iteration scheme for solving Note that is block diagonal and that LqI is strictly lower triangular. This means that each step of the iteration requires the solution of s sets of n linear equations and when the solution of the jth set of equations is being found, the solutions of the previous j-1 sets of equations may be used. For large s, the linear algebra costs for the Cooper and Butcher iteration scheme are much less per iteration than for the modified Newton method, for which a system of sn linear equations has to be solved at each step of the iteration.

There are three matrices L, B and S and the parameter r which may be chosen in many ways. The submatrices are chosen to have the form.

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Also have the same block diagonal form. We take the matrices L and B to be the same as those given by Cooper and Butcher [5]. That is, we take. Then Following Cooper and Butcher, we take. The implementation of these methods is discussed later. The elements of may be obtained by noting that. Then we have and By numerical evaluation these give.

The matrix S may be obtained by noting that its columns are a linearly independent set of eigenvectors of the matrix. These eigenvectors are obtained by numerical calculation. Again we take when we consider the implementation of this method later in this section. The elements of the matrix may be obtained by. Thus we have By numerical evaluation, we obtain. Then the matrix S is obtained by noting that its first two columns are a linearly independent set of eigenvectors of and the third and the fourth columns are a linearly independent set of eigenvectors of.

Again these eigenvectors are obtained by numerical calculation. As suggested by Cooper and Butcher, we take to obtain a better rate of convergence. Implementation aspects: Now we consider the implementation of the s-stage methods with this iteration scheme for second order systems of the form 1. Note that in Let where and. For a second order system, we have where J is an approximation for the Jacobian of f with respect y and 44 can be written in the form.

Now premultiplying 46 by we obtain. Next we calculate. First consider the residual given by Substituting the values of from Finally we calculate Consider the equation Then we have. We would like to avoid having to evaluate these functions. Two-stage fourth order method: The two-stage fourth order implicit Runge-Kutta method is given by 6. For this method equations 55 become. We predict w 1 0 and w 2 0 by using the interpolating polynomial Q 3,n t of Khiyal [6]. That is, we take This predictor cannot be used on the first step and so on this step, we propose to use.

This gives. In our codes, we use the same convergence strategy and action on divergence of the iteration step as discussed for the fourth order implicit Runge-Kutta method with a perfect square iteration scheme. To form the error estimate we use the formula given by 9 while the predictor is given by Note that the cost of the Cooper and Butcher iteration scheme is the same as the cost of the perfect square iteration scheme assuming both schemes require the same number of iterations because we have to solve two systems of n linear equations at each step of the integration for both schemes.

However, the matrix is the same for both systems for both schemes so we do not need to perform more than one matrix factorisation per step. For this method equation 55 and 56 becomes. To find f w 1 we eliminate f w 2 and f w 3 from the set of three equations 70 , 72 and Similarly, the second derivative of and in 9 gives.

The third derivative of and in 9 gives.

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It is noted that the general fourth order odes involve the first, second and third derivatives. The derivatives can be obtained by imposing that:. By using 14 and evaluating 11 , 12 and 13 at we obtain the first, second and the third derivative scheme as follows:. By combining the schemes 10 , the first, second, third derivatives schemes 15 together and write them in block form, using the definition of implicit block method in [9] to obtain the block formula describe as follows:.

This equation is solved and we obtained values for and as follows:. By expanding and in Taylor series, 21 becomes:.

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The term is called the error constant and implies that the local truncation error is given by: Since but see [10] ; then the main scheme is of order 7 and the error constant is:. The block 17 is said to be Zero stable if the roots of the characteristic polynomial , satisfies and the root has multiplicity not exceeding the order of the differential equation. Moreover as. From main method 10c , the first and second characteristics polynomials of the method are given by:. The order of the method is which is obvious. For the method , , , and , thus. The necessary and sufficient condition for a numerical method to be convergent is for it to be consistent and Zero stable.

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Thus since it has been successfully shown from the above condition, it could be seen that our method is convergent. The stability polynomial of the main method 10c becomes:. By inserting the values of and into 27 and evaluate, we obtain the following results as displayed in the table below:. From here, it could be seen that the region of absolute stability of the method is given by which satisfies the condition for A-stability, similarly the interval of periodicity lies in interval.

To test the accuracy, workability and suitability of the method, I adopted our method to solving some initial value problems of fourth order ordinary differential equations. My method was used to solve the problem and result compared with [6].

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The result is as shown in Table 1. My method was used to solve the problem and result compared with [8]. The result is as shown in Table 2. In this paper, I propose an accurate five off-step points modified implicit block algorithm for the numerical solution of initial value problems of fourth order ordinary differential equations. For better performance of the method, step size is chosen within the stability interval. Table 1.

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Showing results for problem 1. Table 2. Showing results for problem 2. The order of my method is of order 7 higher than that of [6] of order 4, which collaborates the principle, that the higher the order of a method is, the more accurate it is. The absolute errors in [6] are more than those of the new methods; this also means that the new methods are accurate than [6] which is of order 4 and implemented in block mode.

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The results of my new method when also compared with the block method proposed by [8] showed that my method is more accurate. International Journal of Computer Mathematics, 41, Journal of Computational and Applied Mathematics, , This is a preview of subscription content, log in to check access. Axelsson, Global integration of differential equations through Lobatto quadrature , BIT 4 , 69— Google Scholar.