INTRODUCTION
Predictorcorrector method is very essential for finding a suitable stepsize^{1}. This study is concerned with approximating the solution of Initial Value Problems (IVPs) for firstorder ODEs of the form^{1}:
The aim was to formulate a variable stepsize block predictorcorrector method. This technique of continuing in variable stepsize predictorcorrector method started with Milne and it is referred to as Milne’s device^{14}. Other researchers proposed block predictorcorrector method for computing the solution of ODEs in the simple form of Adams type as sited^{512}. Gear’s method known for stiff problems is the Backward Differentiation Formula (BDF) as stated previously^{1315}. In addition, this study possesses a lot of computational advantages as discussed previously^{3,4}. Predictorcorrector techniques constantly provide two estimates at each step, thirdly, they are an efficient device for errorcontrol adaptation which has been reported previously^{7}. To present this process, a variable stepsize block predictorcorrector method apply the explicit AdamsBashforth KStep method as a predictor and the implicit AdamsMoulton K1Step method as a corrector of the same order^{1,10,11}.
Furthermore, it is speculated by Adesanya et al.^{5,6}, Anake et al.^{7}, Bakoji et al.^{8}, James and Adesanya^{9} and Voss and Abbas^{12 }that block predictorcorrector method(s) is a faster method than other nonblock predictorcorrector method(s) with better results and as such, ensure convergence. Again, Adesanya et al.^{5,6}, Anake et al.^{7}, Bakoji et al.^{8}, James and Adesanya^{9} and Voss and Abbas^{12 }suggested that solving block predictorcorrector method(s) simultaneously using fixed step size is sufficient enough to guarantee maximum errors, while others proposed Backward Differentiation Formula (BDF) to provide the solution for stiff ODEs. This study is motivated by the fact that block predictorcorrector method can be extended using the variable stepsize technique to solve nonstiff and mildly stiff ODEs.
Definition 1: bblock, rpoint method. If r denotes the block size and h is the step size, then block size in time is rh. Let m = 0, 1, 2,... represent the block number and let n = mr, then the bblock, rpoint method can be written in the following general form in Eq. 2:
Where:
A_{S} and B_{S} are r×r coefficients matrices^{13}.
MATERIALS AND METHODS
Formulation of the block predictorcorrector method: Newton’s backward difference formula was used to formulate the block predictorcorrector method.
Suppose f(x) has a continuous kth derivative, t_{m} = t_{0}+mh, f_{m} = f (t_{m}) and backward differences are presented by Eq. 3:
where, ∇^{q} f_{m} = f_{m}, then:
where, f^{(k)} (ε) is the kth derivative of f appraised at some point in an interval having t, t_{mk+1} and t_{m}. Assume to fix and m = n1, (1.3) yields:
Where:
Subbing the above in:
to get:
Whenever the last term in Eq. 4 is disregarded, the left behind will be called kstep AdamsBashforth formula which is shown in Eq. 5:
Expressing the backward differences in terms of the values at continuing points by:
Thus, Eq. 5 can be rewritten as:
Continuing with Eq. 6 to generate the block kstep AdamsBashforth Formula^{8}.
Similarly, the implicit multistep methodsthe Adamsmoulton method can be derived by setting m = n in Eq. 3 and putting into:
we get:
Ignoring the error term, gives the method as Eq. 7:
Replacing for ∇^{j}f_{n} in terms of f_{n}, f_{n1}, f_{n2},..., yields the form:
By continuing with Eq. 8, the block k1step Adamsmoulton method can be generated^{16}.
Practical application of block predictorcorrector method: Assuming P defines the application program of the block predictor, C defines block corrector application program, with E as the evaluation application program of f with respect to given values of its parameter. If is computed from the block predictor, is calculated one time and employ the corrector at one time as well to obtain , this describes the computation as PEC. Further appraisal of succeeded by another application program of the corrector gives and thus, denoted by PEC^{(2)}. Implementing the application program of the block corrector m many times can be referred to as PEC^{(m)}. Since m is constant, is accepted as the computational solution at X_{n+k}. At this point, the last computational value for f_{n+k} is preferred as and this will be further decided whether or not to execute . Assuming this concluding execution is done, the mode is denoted by P(EC)^{m} or P(EC)^{m }E. Eventually the decision clearly impacts the next step of the execution, when both predicted and corrected numerical values for y_{n+k+1} will rely on whether f_{n+k} is accepted as or . Finally, for a given m, P(EC)^{m} or P(EC)^{m }E mode utilize the corrector the same number of times; only P(EC)^{m }E requires one more evaluation per step than P(EC)^{m} as expressed^{3,4}.
Theorem 1: If the multistep method 2 is convergent for pth order equations, then the order of 2 is at least p^{16}.
Theorem 2: The order of a predictorcorrector method for first order equations must be >1 if it is convergent^{16}.
Theorem 1 and 2 draw the conclusion that the order and convergence of the method hold.
Implementation of block predictorcorrector method: Concurring to Jain et al.^{17} and Lambert^{3,4}, the implementation in the P(EC)^{m} or P(EC)^{m} E mode becomes substantial for the explicit (predictor) and implicit (corrector) methods if both are separate of like order and this requirement makes it indispensable for the step number of the explicit (predictor) method to be one step higher than that of the implicit (corrector) method. Consequently, the mode P(EC)^{m} or P(EC)^{m} E can be formally examined in Eq. 9 for m = 1, 2,...;
P(EC)^{m}:
P(EC)^{m}E:
Remarking that as m→∞, the result of evaluating with either of the above mode will slope to those given by the mode of correcting to convergence.
Moreover, predictor and corrector pair based on method 2 can be implemented. The mode P(EC)^{m} or P(EC)^{m}E specified by Eq. 9, where h is the step size. Since the predictor and corrector both have the same order p.
Theorem 3 demonstrates adequate condition for the convergence of P(EC)^{m} or P(EC)^{m}E.
Theorem 3: Let be a sequence of approximations of y_{n+1} obtained by a PECE method. If:
(for all y near y_{n+1} including ) where, L satisfies the condition , then the sequence converges to y_{n+1}.
Proof: The numeric solution satisfies the equation:
The corrector satisfies the equation:
Subtracting these two equations, we obtain:
Applying the Lagrange mean value theorem to arrive at:
where, . Thus:
Now:
This means that the conclusion of Theorem 3 holds as seen^{17}.
In cases, where C_{p+1}, are the computed error constant of the predictorcorrector method, respectively. The following consequence holds.
Proposition: Suppose the predictor method have order p* and the corrector method have order p. Then: If p*>p (or p*<p with m> pp*), then the predictorcorrector methods possesses the same order and the same PLTE as the corrector.
If p*<p and m = pp*, then the predictorcorrector method possesses the same order as the corrector, but different PLTE.
If p*<p and m<pp*1, then the predictorcorrector method possesses the same order equal to p*+m (thus less than p).
Specifically, it is observed that, suppose the predictor has order p1 and the corrector has order p, the PEC answers to get a method of order p. Moreover, the P(EC)^{m} or P(EC)^{m}E scheme has always the same order and the same PLTE as discussed^{3,4}.
Combining^{14}, Milne’s device stated that it is viable to estimate the principal local truncation error of the explicit and implicit (predictorcorrector) method without estimating higher derivatives of y(x). Assuming that p = *, where p* and p defines the order of the explicit (predictor) and implicit (corrector) methods with the same order. Directly, for a method of order p, the principal local truncation errors can be written as Eq. 10 and 11:
Also:
where, W_{n+j} and C_{n+j} are called the predicted and corrected approximations are given by the method of order p while and C_{p+1} are independent of h.
Neglecting terms of degree p+2 and above, it is easy to make estimates of the principal local truncation error of the method as Eq. 12:
Noting the fact that and .
However, the estimate of the principal local truncation error (12) is used to determine whether to accept the results of the current step or to reconstruct the step with a smaller step size. The step is accepted based on a test as prescribed by Eq. 12^{18}. Equation 12 is the convergence criteria otherwise called Milne’s estimate for correcting to convergence. Furthermore, Eq. 12 ensures the convergence criterion of the method during the test evaluation^{18}.
Problem tested: Three test problems are employed. These problems are implemented using variablestepsize block predictorcorrector method.
• 
Test problem 1:y' (x)=xyy(0) = 10<x<1 
Solution: 
• 
Test problem 2:y’(x) = xyy(0) = 10<x<1 
Solution: y(x) = x+e^{–x}1 
• 
Test problem 3:y’(x) = 10xyy(0) = 10<x<10 
Solution: 
RESULTS AND DISCUSSION
The numeric results to demonstrate the performance of the variable stepsize block predictorcorrector method in solving firstorder ODEs. The complete result supplied were incurred with the aid of Mathematica 9 Kernel for Microsoft windows (64 bit). The nomenclature utilized are listed in Table 1:
VSSBPCM: 
Variable stepsize block predictorcorrector Method 
TOL: 
Tolerance level 
h: 
Step size 
MTH: 
Method used 
MAXE: 
The magnitude of the maximum errors of VSSBPCM 
ESTM: 
Error in stormercowell method^{2} for test problem 1 and 2 
1BDF: 
r = 1point BDF method^{9} for test problem 3 
2BDF: 
r = 2point BDF method^{9} for test problem 3 
3BDF: 
r = 3point BDF method^{9} for test problem 3 
The process of estimating the maximum errors and determining the tolerance level are defined as follows:
Table 1:  Shows the numeric results of problems 1, 2 and 3 using VSSBPCM with comparison to existing methods 

Observing the fact that ≠ C_{p+1} and P_{n+j}≠C_{n+j·} and C_{p+1} are independent of h.
Where, and C_{p+1} are the estimates of the principal local truncation error of the predictor and corrector method. P_{n+j} and C_{n+j }are called the predicted and corrected approximations are given by the method of order p.
CONCLUSION
Numeric results have demonstrated the VSSBPCM is achieved with the aid of the tolerance level. This tolerance level criteria decide whether the result is accepted or repeated. The results likewise establish the performance of the VSSBPCM is remarked to be quicker than the block StormerCowell method and block backward differentiation formula implemented with fixed step size. Hence, it can be resolved that the method formulated is worthy for solving nonstiff and stiff ODEs.
SIGNIFICANCE STATEMENTS
The significance statements of this study are to:
• 
Extend the block predictorcorrector method 
• 
Introduce the tolerance level otherwise referred to as convergence criteria 
• 
Design a suitable step size 
• 
Control the error with the aid of a suitable step size 
ACKNOWLEDGMENT
The authors would like to thank Covenant University for providing financial support through grants during the study period.