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0)*k25... 0)*k14... 0 )*k24... 0)*k34... 0)*k15... 0 )*k25... 0)*k35... 0)*k14... 0 )*k24... 0)*k34... 0)*k44... 0)*k15... 0 )*k25... 0)*k35... 0)*k45... 0)*k14... 0)*k34... 0)*k44... 0)*k15... 0)*k35... 0)*k45... 0)*k55... 49i. We can note the following points: • Clearly there is a substantial degree of repetitive coding that could be streamlined through the use of 1D arrays (particularly in the calculation of k1 to k6 ). • The O(h 4 ) and O(h 5 ) solutions are computed independently, and we will next observe that they can be combined.

27a. 44c). 4 Tumor model of eqs. 48) (or eqs. 3f\n',h); fprintf(... 10f\n',... 2 are in the way that the RK constants are computed and used. In particular, while keeping in mind that y1 is the O(h) (Euler method) and y2 is the O(h 2 ) (modified Euler method), the base point is selected as the running value of y2: % % Store solution at base point yb=y2; tb=t; where the initial value of y2 was set previously as an initial condition. 0; esty1=y2-y1; end Note in this code that: — The estimated error in y1, esty1, is computed by p refinement (subtraction of the O(h) solution from the O(h 2 ) solution).

001 steps, respectively. 001 = 10000 steps produced excessive accuracy. However, 10/1 = 10 steps were inadequate as might be expected. , the thirdorder RK, in reducing the error in the numerical solution of ODEs is clearly evident from this example. To conclude this section, we consider a widely used RK (4, 5) pair, the Runge Kutta Fehlberg (RKF) method (Iserles,2 p. 49h is only O(h 5 ) (the number of derivative evaluations will, in general, be equal to or greater than the order of the final stepping formula).