DHS Numerical Approximation of Roots (9649) (Revision Solutions)

Numerical Methods – Approximation of Roots H2 Double Math (TOPICAL REVISION – SUGGESTED SOLUTIONS) NUMERICAL METHODS (PART 1) APPROXIMATION OF ROOTS OF EQUATIONS 1 Show that the equation 3 5 1 0,xx− + = has exactly one root in (0,1). Use two iterations of linear interpolation between 0x= and 1x= to yield a fractional approximation of this root. [4] Three possible rearrangements of the given equation in the form ()x F x= are 3 5 1,xx=− 31 ( 1),5xx=+ 3 4 1.x x x= − + Only one of these rearrangements will provide an iterative method, of the form 1 ()nnx F x+ = , 0 2,x = which converges to the root between 2 and 3 . Use this rearrangement to find this root correct to 3 significant figures. [4] It is known the correct rearrangement above provides a convergent iterative method by analysing the derivative of F . For this purpose, show that 0 '( ) 1,Fx whenever 2x . [2] [EJC/FM/2018/P1/Q9] [Solution] With 3g( ) 5 1,x x x= − + .x Note that g(0) 1 0,g(1) 3 0,=  =−  thus g(0) g(1) 0 Since g is continuous on [0,1] , the equation g( ) 0x = has at least 1 root in [0,1]. (1) 2g ( ) 3 5 0, [0,1].x x x= −   Thus g( ) 0x = has at most 1 root in [0,1]. Therefore, g( ) 0x = has exactly 1 real root in [0,1]. Using linear interpolation on [0,1] , we have 1x = (0)g(1) (1)g(0) g(1) g(0) − − = 1 31 − −− = 1 4 Since 1 1 1 1 5 16 64 15g 5 1 ,4 64 4 64 64 64 64    = − + = − + =−       < 0 and g(0) = 1 > 0  lies in (0, 1/4).

Numerical Methods – Approximation of Roots 2x = 11(0)g g(0)44 1g g(0)4    −        − = 11 1644 79 715 64 91 64    −      ==−  − 3 5 1,xx=− (This fixed point iterations based on this rearrangement converges to 2.13) 0 2x = 1 2.08x  2 2.11x  3 2.12x  4 2.13x  5 2.13x  Using GC, we can see for the other 2 schemes based on: 31 ( 1),5xx=+ the fixed point iteration does not converge to a value between 2 and 3, but to 0.205. 3 4 1,x x x= − + the fixed point iteration diverges. ( ) ( ) 3 1 2 (1) 3 2 (1) 3 3/2 3/2 ( ) 5 1 51( ) 5 1 0,35 5( ) 5 1 1, 3 5whenever 5 1 , 3 15. ., 1 0.630,53 which is satisfied as ( ) increases, when ever 2>0.630. F x x F x x x F x x x i e x F x x − − =− = −   = −  −    +   

Numerical Methods – Approximation of Roots 2 (a) The curve with equation 2 2e 3xyx= + − has exactly one stationary point in the interval  1, 0− . Use the Newton-Raphson method to find the x-coordinate of the stationary point, correct to 4 decimal places. [5] (b) (i) Show that the equation 3 7 2 0xx − + = has a root,  , in the interval [0, 1]. [1] (ii) Student A uses the recurrence relation ( ) 3 1 1 14 27 nnnx x x+ =− − for finding  . Explain why Student A will fail to find  . [2]