BCA3010 – COMPUTER OREINETED NUMERICAL METHODS

(ii) Explain truncation errors.

(iii) Find the truncation error in the result of the following function for x= (N/2)when we use(a) first three terms(b) first four terms(c) first five terms ex = (1+x²/2+x³/3+x⁴/4+x⁵/5+x⁶/6)

Answer :- State Lagrange’s mean value theorem:- Suppose is a function defined on a closed interval (with ) such that the following two conditions hold:

1. f is a continuous function on the closed interval (i.e., it is right continuous at , left continuous at , and two-sided

(ii) Explain truncation errors: Suppose we have a continuous differential equation

y’ = f(t,y), \q y(t_0) = y_0, \ t \ t_0

and we wish to compute an approximation y_n of the true solution y(t_n) at discrete time steps t_1,t_2,\l ,t_N . For simplicity, assume the time steps are equally spaced:

h = t_n – t_{n-1}, \qquad n=1,2,\ldots,N.

Suppose we compute the sequence y_n with a one-step method of the form

y_n = y_{n-1} + h A(t_{n-1}, y_{n-1}, h, f).

The function A is called the increment function, and can be interpreted as an estimate of the slope of y_n .

Local truncation error

The local truncation error \tau_n is the error that our increment function, A , causes during a single iteration, assuming perfect knowledge of the true solution at the previous iteration.

More formally, the local truncation error, \tau_n , at step n is computed from the difference between the left- and the right-hand side of the equation for the increment y_n \approx y_{n-1} + h A(t_{n-1}, y_{n-1}, h, f) :

\tau_n = y(t_n) – y(t_{n-1}) – h A(t_{n-1}, y(t_{n-1}), h, f).

The numerical method is consistent if the local truncation error is o(h) (this means that for every \varepsilon> 0 there exists an H such that |\tau_n| < \varepsilon h for all h < H ; see big O notation). If the increment function A is differentiable, then the method is consistent if, and only if, A(t,y,0,f) = f(t,y) .

Furthermore, we say that the numerical method has order p if for any sufficiently smooth solution of the initial value problem, the local truncation error is O(h^{p+1}) (meaning that there exist constants C and H such that |\tau_n| <Ch^{p+1} for all h < H ).

Global truncation error

The global truncation error is the accumulation of the local truncation error over all of the iterations, assuming perfect knowledge of the true solution at the initial time step.[citation needed]

More formally, the global truncation error, e_n , at time t_n is defined by:

\begin{align} e_n&= y(t_n) – y_n \\ &= y(t_n) – \Big( y_0 + h A(t_0,y_0,h,f) + h A(t_1,y_1,h,f) + \cdots + h A(t_{n-1},y_{n-1},h,f) \Big). \end{align}

The numerical method is convergent if global truncation error goes to zero as the step size goes to zero; in other words, the numerical solution converges to the exact solution: \lim_{h\to0} \max_n |e_n| = 0 .

(iii) Find the truncation error in the result of the following function for x= (N/2)when we use(a) first three terms(b) first four terms(c) first five terms ex = (1+x²/2+x³/3+x⁴/4+x⁵/5+x⁶/6)

To determine the local truncation error for the three-step Adams-Bash forth m

Q2 Find the Taylors series of the function

f(x)  = 3x⁵– 2x⁴+ 15x³+ 13x²– 12x- 5  atthepoint c=  2

Estimate y (0.8) by Adam Moulton’s method.