Question Choose f(x) appropriately and determine the corresponding difference eq

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Question

Choose f(x) appropriately and determine the corresponding difference equation for finding the rth root of sin. Do five iterations to calulate the solution of xsinx=4 using xo=1 as the initial guess.

Write a simple Matlab program to demonstrate this.

It is desired to determine a recursive expression (difference equation) for finding the rth root of a real number N. We start by writing the Taylor series expansion of function fix) about a point x0: where If the series is truncated after two terms, we have Let x represent the next iterate xn+1, and let x also represent a solution of the equation fix) = 0. Then Eq. (P2-4-2) becomes Equation (P2-4-4) is a difference equation useful for solving f(x) = 0.

Explanation / Answer

procedure Roots_Of_Unity is Root : Complex; begin for N in 2..10 loop Put_Line ("N =" & Integer'Image (N)); for K in 0..N - 1 loop Root := Compose_From_Polar ( Modulus => 1.0, Argument => Float (K), Cycle => Float (N) ); -- Output Put (" k =" & Integer'Image (K) & ", "); if Re (Root) < 0.0 then Put ("-"); else Put ("+"); end if; Put (abs Re (Root), Fore => 1, Exp => 0); if Im (Root) < 0.0 then Put ("-"); else Put ("+"); end if; Put (abs Im (Root), Fore => 1, Exp => 0); Put_Line ("i"); end loop; end loop; end Roots_Of_Unity;

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