3 8 Newtons Method Objective Approximate a zero

3. 8: Newton’s Method

Objective • Approximate a zero of a function using Newton's Method.

Recall the Bisection Method for approximating roots. Need interval where f(endpoints) have different signs. Interval Midpoint(c) f(c) [1, 2] [1, 1. 5] [1. 25, 1. 5] 1. 5 1. 25 1. 375 + - [1. 375, 1. 5] 1. 4375 [ -, +] Converges, but it may take a while.

Another method for finding zeros (besides the Bisection Method) is called Newton’s Method (also known as Newton-Raphson Method). Requirements for Newton’s Method: • Needs an initial “good” estimate • Uses the derivative of the function in the calculation, so the function needs to be differentiable in the open interval (a, b) around the zero

Newton's Method is based on the assumption that the graph of f and the tangent line at (x, f(x)) both cross the x-axis at about the same point. You use that x-intercept to make another estimate (usually better) for the zero.

Newton's Method: Tangent line: set y=0 and solve for x intercept new guess Next estimates:

Using Newton's Method: Let f(c)=0 where f is differentiable on an open interval containing c. 1. Make an initial estimate x 1 (that's "close" to c). 2. Determine a new approximation: 3. If is within the desired accuracy, stop. Otherwise repeat step 2.

Calculate 3 iterations of Newton's Method to approximate a zero of Use x 1=1 as the initial guess. Find x 4.

TI 84 Procedure: 1. Set y 1 = function and y 2 = derivative 2. Type in the initial approximation and hit ENTER. 3. Type in 1. ANS – y 1(ANS) / y 2(ANS) 2. hit ENTER 4. Hit ENTER for next approximation. 5. Repeat step 4 until the desired accuracy is reached.

Use Newton's Method to approximate the zeros of Continue until successive approximations differ by less than 0. 0001. Test some #s: Or look at graph on calculator (x≈-1. 2) x 1=-1. 2 x 2=-1. 23511 x 3=-1. 233754 x 4=-1. 23375 x 1=-1. x 2=-1. 333333 x 3=-1. 243386 x 4=-1. 233855 x 5=-1. 23375 x 6=-1. 23375 x 1=0 x 2=1 x 3=0. 5715286 x 18=-1. 23375

Use Newton's Method to find zeros of Let x 1=0. 1.

There are some limitations to Newton’s method: Looking for this root. Bad guess. Wrong root found Failure to converge

Condition for Convergence:

Example: On the interval (1, 3), it will be <1. So convergence of Newton's method is guaranteed.

Example: For any value of x, it equals 2, so Newton's Method will fail to converge.

Homework 3. 8 (p. 226) #1 -17 odd 27, 29, 33 (#5 -13 only find one root)