A Proof of the Non-Singularity of the D Matrix Used in Deriving the Two—Step Butcher’s Hybrid Scheme for the Solution of Initial Value Problems

In this paper, we state and prove the conditions for the non-singularity of the D matrix used in deriving the continuous form of the Two-step Butcher’s hybrid scheme and from it the discrete forms are deduced. We also show that the discrete scheme gives outstanding results for the solution of stiff and non-stiff initial value problems than the 5 th order Butcher’s algorithm in pre-dictor-corrector form.


Introduction
This paper focuses on finding numerical approximations to stiff and non-stiff initial value problems of the type Numerical methods for the solution of IVPs are vast such as the Runge-Kutta and Backward Differentiation Formulae (BDF). Similar work to this present one was done by [1] and [2] but while both authors used the same D matrix, neither did they investigate its non-singularity nor plotted the region of absolute stability. The non-singularity of the D matrix is necessary not only because only non-singular matrices have inverses but also guides us on permissible values the step size should take. Besides, we used the fast vector-based approach proposed by [3] in calculating the order of the derived discrete schemes unlike in the 0,1 in [2] albeit the availability of computer and software environments like wxMaxima/Maple [4] [5] and octave/Matlab [6] [7]. To the best of our knowledge, this is the first attempt at proving the non-singularity of the D matrix. Existing discrete schemes derived from their continuous counterparts for linear multistep methods in literature [1] [2] [3] [8] [9] [10] [11] [12] only assumed its non-singularity. The assumption on the non-singularity of the D matrix is not only limited to the earlier mentioned articles but also [13]- [18] and a host of others too numerous to mention.

Methodology
In this section, we re-derived 1 the continuous formulation of the Two-Step Butcher's scheme and use it to deduce the discrete ones. We shall find the order, error constant, investigate the zero stability and consistency of the derived discrete schemes and the 5 th order Butcher's algorithm in Lambert [19].  1 We used brute force in doing so in [2], but here we show the Maxima codes for doing so. Journal of Applied Mathematics and Physics

Derivation of Multistep Collocation Methods
The matrix in (5) where t is the number of interpolation points while m is the number of collocation points used. In deriving the continuous and discrete forms of the Two-Step Butcher's Scheme, we took Thus, the matrix D in (5) becomes   2  3  4  5   2  3  4  5  1  1  1  1  1  2  3  4   2  3  4  1  1  1  1  2  3  4  2  2  2  2  2  3  4  3  3  3  3  2  2  2 Since only non-singular matrices have inverses, we need to show that the matrix D is indeed non-singular if the step size h is not too small. Otherwise, we cannot invert the matrix. This is stated in the form of the following theorem.
Replace row two with row two minus row one to give Next, we perform the following row operations with respect to row two as Furthermore, we perform the following row operations with respect to row

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Finally, we replace row six with four times row six minus row five i.e., where det means determinant. The determinant of D is non-zero iff 11 93 4 h is strictly greater than zero. This result implies that if h is too small, then the determinant will be zero or less than macheps and D will be singular and non-invertible.  We have shown that the D matrix is non-singular if the step size is not too small from the above theorem. Now, we can find the inverse matrix C from

DC I =
where I is the six by six identity matrix. We use the following wxMaxima(maple) codes to invert the D matrix as shown in the Appendix.
We are only interested in the first row of the C matrix which are, We evaluated (10)
Proof: In finding the order and error constant of the block scheme, we substituted the above vectors in the following formula.
The above formula showed that 0 This implies that the order of each of the discrete schemes in the Two-step Butcher's scheme in block form is 5.
The block method (11) can be represented as In addition, we need the following matrices in analysing the zero stability of the block method,  Journal of Applied Mathematics and Physics The characteristic polynomial corresponding to (11) is given as Lemma 2.2: The Two-step Butcher's scheme in block form (11) is zero stable, consistent and hence convergent.
Proof: By definition, a Linear Multistep Method is said to be zero-stable if none of the roots of its characteristic polynomial has modulus greater than one and each of the roots with modulus one must be distinct. This is immediate from above. As shown in Lemma 2.1 the order of the Two-step scheme is 5 p = which is greater than one. Therefore, consistency is established [21]. Since it is both zero-stable and consistent, by definition, it is convergent.
Proof: In finding the order and error constant of the block scheme, we substituted the above vectors in the following formula.  α  β  β  β   3  3  1  2  1  2  2  2   3  2  3  2  3   1  3  1  3  2  2  3!  2 The above formula showed that 0 This implies that the order of each of the discrete schemes in the Two-step Butcher's scheme in block form is 3.

Numerical Experiments
In this section, we applied both the derived discrete scheme in block form and the schemes given in Lambert [19] in Predictor-Corrector form on some initial value problems.
For 2 n = , we have the same matrix as above but different right hand side This process is continued for 4, 6,8, , 30 n = and the results are as shown in     scale and the result is as shown in Figure 3. It can be observed that as n increases, the norm of the residual decreases as expected. In the same vein, Figure 4 shows a plot of the norm of the Error between the Exact and Approximate values of n y using the Two-Step Butcher's scheme in block form on a semilogy scale.
In addition, we used the 5 th Order Butcher's algorithm to solve the same initial problem. Since this algorithm is non self starting, we made two different plots shown by the red and blue lines of Figure 5 and Figure 6. In Figure 5, we used the exact value of 1 y as starting guess in solving the IVP, while in Figure 6, we used the 1 y obtained from the Two-Step Butcher's scheme in solving the IVP and the absolute errors are plotted on semilogy scales. We observed that except For 2 n = , we obtained the following system of equations This process is continued for 4, 6,8, , 30 n = and the results are as shown in Since we are solving a linear system of equations in each iteration, we plotted the values of n against the norm of the residual = − r b Ay on a semilogy scale and the result is as shown in Figure 9. We observed that as n increases, the norm of the residual decreases as expected. In the same vein, Figure 10 shows a plot of We seek numerical approximations to the initial value problem 9 y y ′ = − with initial condition ( ) 0 e y = , We followed the same steps in solving Examples 3.1 and 3.2. We started with      Figure   11 that the Two-Step Butcher's scheme in block form performed at par with the exact solution. Since we are solving a linear system of equations in each iteration, we plotted the values of n against the norm of the residual = − r b Ay on a semilogy scale and the result is as shown in Figure 12. It can be observed that as n increases, the norm of the residual decreases as expected. In the same vein, Figure 13 shows a plot of the norm of the Error between the Exact and Approximate values of n y using the Two-Step Butcher's scheme in block form on a semilogy scale.
In addition, we used the 5 th Order Butcher's algorithm to solve the same initial problem. Since this algorithm is non self starting, we made two different plots shown by the red and blue lines of Figure 14 and Figure 15. In Figure 14, we used the exact value of 1 y as starting guess in solving the IVP, while in Figure   15, we used the 1 y obtained from the Two-Step Butcher's scheme and the absolute errors are plotted on semilogy scales. We observed that except for      Figure 14 and Figure 15, while the minimum absolute error obtained using the Two-Step Butcher's scheme is approximately 10 −25 , that of the 5 th Order Butcher's algorithm is approximately 10 −10 of course in the absence of round off errors. Since the 5 th Order Butcher's algorithm uses the exact value of 1 y , one would have expected that it will give better approximations to the exact solution. However, for both 3 2 n y + and 2 n y + , the Two-Step Butcher's scheme in block form outperformed the former.

Conclusion
We showed that if the step size h is not too small the matrix D will be invertible.
Nowhere in literature has there been any proof on the necessary conditions for the invertibility of the D matrix which was our main aim. In addition, ordinarily speaking one would have expected the 5 th Order Butcher's algorithm which uses the exact solution 1 y as the starting value to give more accurate results than the self-starting Two-step Butcher's scheme in block form, but to our greatest surprise, the reverse was the case as depicted by the Figure of the absolute errors in the preceding section. The accuracy could be due to the fact that all the discrete schemes used in the Two-step Butcher's scheme in block form are of uniform order. From the figures in the last section, we can confidently say that the Two-step Butcher's hybrid scheme performed better than its counterpart. In addition, the former performs well for both stiff and non-stiff IVPs.