4.7. Newton s Method. Procedure for Newton s Method HISTORICAL BIOGRAPHY

Transcription

1 4. Newton s Method Newton s Method HISTORICAL BIOGRAPHY Niels Henrik Abel (18 189) One of the basic problems of mathematics is solving equations. Using the quadratic root formula, we know how to find a point (solution) where =. There are more complicated formulas to solve cubic or quartic equations (polnomials of degree 3 or 4), but the Norwegian mathematician Niels Abel showed that no simple formulas eist to solve polnomials of degree equal to five. There is also no simple formula for solving equations like sin =, which involve transcendental functions as well as polnomials or other algebraic functions. In this section we stud a numerical method, called Newton s method or the Newton Raphson method, which is a technique to approimate the solution to an equation ƒsd =. Essentiall it uses tangent lines in place of the graph of = ƒsd near the points where ƒ is zero. (A value of where ƒ is zero is a root of the function ƒ and a solution of the equation ƒsd =. ) Procedure for Newton s Method The goal of Newton s method for estimating a solution of an equation ƒsd = is to produce a sequence of approimations that approach the solution. We pick the first number of the sequence. Then, under favorable circumstances, the method does the rest b moving step b step toward a point where the graph of ƒ crosses the -ais (Figure 4.43). At each

2 3 Chapter 4: Applications of Derivatives (, f( )) Root sought 3 Fourth ( 1, f( 1 )) 1 Third Second APPROXIMATIONS f() (, f( )) First FIGURE 4.43 Newton s method starts with an initial guess and (under favorable circumstances) improves the guess one step at a time. step the method approimates a zero of ƒ with a zero of one of its linearizations. Here is how it works. The initial estimate,, ma be found b graphing or just plain guessing. The method then uses the tangent to the curve = ƒsd at s, ƒs dd to approimate the curve, calling the point 1 where the tangent meets the -ais (Figure 4.43). The number 1 is usuall a better approimation to the solution than is. The point where the tangent to the curve at s 1, ƒs 1 dd crosses the -ais is the net approimation in the sequence. We continue on, using each approimation to generate the net, until we are close enough to the root to stop. We can derive a formula for generating the successive approimations in the following wa. Given the approimation n, the point-slope equation for the tangent to the curve at s n, ƒs n dd is We can find where it crosses the -ais b setting = (Figure 4.44). This value of is the net approimation = ƒs n d + ƒ s n ds - n d. = ƒs n d + ƒ s n ds - n d - ƒs nd ƒ s n d = - n = n - ƒs nd ƒ s n d If ƒ s n d Z n + 1. Here is a summar of Newton s method. f() Point: ( n, f( n )) Slope: f'( n ) Tangent line equation: f( n ) f'( n )( n ) ( n, f( n )) Tangent line (graph of linearization of f at n ) Procedure for Newton s Method 1. Guess a first approimation to a solution of the equation ƒsd =. A graph of = ƒsd ma help.. Use the first approimation to get a second, the second to get a third, and so on, using the formula n + 1 = n - ƒs nd ƒ s n d, if ƒ s nd Z (1) n n 1 n f( n ) f'( n ) FIGURE 4.44 The geometr of the successive steps of Newton s method. From n we go up to the curve and follow the tangent line down to find n + 1. Appling Newton s Method Applications of Newton s method generall involve man numerical computations, making them well suited for computers or calculators. Nevertheless, even when the calculations are done b hand (which ma be ver tedious), the give a powerful wa to find solutions of equations. In our first eample, we find decimal approimations to b estimating the positive root of the equation ƒsd = - =. EXAMPLE 1 Finding the Square Root of Find the positive root of the equation ƒsd = - =.

3 4. Newton s Method 31 Solution With ƒsd = - and ƒ sd =, Equation (1) becomes The equation n + 1 = n - n - n = n - n + 1 n = n + 1 n. n + 1 = n + 1 n enables us to go from each approimation to the net with just a few kestrokes. With the starting value = 1, we get the results in the first column of the following table. (To five decimal places, = ) Error Number of correct digits = 1 1 = 1.5 = = Newton s method is the method used b most calculators to calculate roots because it converges so fast (more about this later). If the arithmetic in the table in Eample 1 had been carried to 13 decimal places instead of 5, then going one step further would have given correctl to more than 1 decimal places. FIGURE 4.45 The graph of ƒsd = crosses the -ais once; this is the root we want to find (Eample ). EXAMPLE Using Newton s Method Find the -coordinate of the point where the curve = 3 - crosses the horizontal line = (1.5,.85) (1, 1) FIGURE 4.46 The first three -values in Table 4.1 (four decimal places). Solution The curve crosses the line when 3 - = 1 or =. When does ƒsd = equal zero? Since ƒs1d = -1 and ƒsd = 5, we know b the Intermediate Value Theorem there is a root in the interval (1, ) (Figure 4.45). We appl Newton s method to ƒ with the starting value = 1. The results are displaed in Table 4.1 and Figure At n = 5, we come to the result 6 = 5 = When n + 1 = n, Equation (1) shows that ƒs n d =. We have found a solution of ƒsd = to nine decimals. In Figure 4.4 we have indicated that the process in Eample might have started at the point B s3, 3d on the curve, with = 3. Point B is quite far from the -ais, but the tangent at B crosses the -ais at about (.1, ), so 1 is still an improvement over. If we use Equation (1) repeatedl as before, with ƒsd = and ƒ sd = 3-1, we confirm the nine-place solution = 6 = in seven steps.

4 3 Chapter 4: Applications of Derivatives TABLE 4.1 The result of appling Newton s method to with = 1 n n ƒ( n ) ƒ ( n ) ƒsd = n 1 n ƒs nd ƒ s n d E B 1 (.1, 6.35) B (3, 3) FIGURE 4.4 An starting value to the right of = 1> 3 will lead to the root. f() The curve in Figure 4.4 has a local maimum at = -1> 3 and a local minimum at = 1> 3. We would not epect good results from Newton s method if we were to start with between these points, but we can start an place to the right of = 1> 3 and get the answer. It would not be ver clever to do so, but we could even begin far to the right of B, for eample with = 1. It takes a bit longer, but the process still converges to the same answer as before. Convergence of Newton s Method In practice, Newton s method usuall converges with impressive speed, but this is not guaranteed. One wa to test convergence is to begin b graphing the function to estimate a good starting value for. You can test that ou are getting closer to a zero of the function b evaluating ƒ ƒs n d ƒ and check that the method is converging b evaluating ƒ n - n + 1 ƒ. Theor does provide some help. A theorem from advanced calculus sas that if ƒsdƒ sd ` sƒ sdd ` 6 1 for all in an interval about a root r, then the method will converge to r for an starting value in that interval. Note that this condition is satisfied if the graph of ƒ is not too horizontal near where it crosses the -ais. Newton s method alwas converges if, between r and, the graph of ƒ is concave up when ƒs d and concave down when ƒs d 6. (See Figure 4.48.) In most cases, the speed of the convergence to the root r is epressed b the advanced calculus formula () r ' ƒ n r ƒ ma ƒ ƒ ƒ ƒ min ƒ ƒ n - r ƒ = constant # ƒ n - r ƒ, (++)++* ƒ (+)+* error e n 1 error e n (3) FIGURE 4.48 Newton s method will converge to r from either starting point. where ma and min refer to the maimum and minimum values in an interval surrounding r. The formula sas that the error in step n + 1 is no greater than a constant times the square of the error in step n. This ma not seem like much, but think of what it sas. If the constant is less than or equal to 1 and ƒ then ƒ n r ƒ n - r ƒ 6 1-3,. In a single step, the method moves from three decimal places of accurac to si, and the number of decimals of accurac continues to double with each successive step.

5 4. Newton s Method 33 ( n, f( n )) n FIGURE 4.49 If ƒ s n d =, there is no intersection point to define n + 1. r 1 f() But Things Can Go Wrong Newton s method stops if ƒ s n d = (Figure 4.49). In that case, tr a new starting point. Of course, ƒ and ƒ ma have the same root. To detect whether this is so, ou could first find the solutions of ƒ sd = and check ƒ at those values, or ou could graph ƒ and ƒ together. Newton s method does not alwas converge. For instance, if - r -, ƒsd = e 6 r - r, Ú r, the graph will be like the one in Figure 4.5. If we begin with = r - h, we get 1 = r + h, and successive approimations go back and forth between these two values. No amount of iteration brings us closer to the root than our first guess. If Newton s method does converge, it converges to a root. Be careful, however. There are situations in which the method appears to converge but there is no root there. Fortunatel, such situations are rare. When Newton s method converges to a root, it ma not be the root ou have in mind. Figure 4.51 shows two was this can happen. f() Root Starting found point FIGURE 4.5 Newton s method fails to 1 1 converge. You go from to 1 and back to Root found, never getting an closer to r. Root sought f() Starting point FIGURE 4.51 If ou start too far awa, Newton s method ma miss the root ou want. Fractal Basins and Newton s Method The process of finding roots b Newton s method can be uncertain in the sense that for some equations, the final outcome can be etremel sensitive to the starting value s location. The equation = is a case in point (Figure 4.5a). Starting values in the blue zone on the -ais lead to root A. Starting values in the black lead to root B, and starting values in the red zone lead to root C. The points ; > give horizontal tangents. The points ; 1> ccle, each leading to the other, and back (Figure 4.5b). The interval between 1> and > contains infinitel man open intervals of points leading to root A, alternating with intervals of points leading to root C (Figure 4.5c). The boundar points separating consecutive intervals (there are infinitel man) do not lead to roots, but ccle back and forth from one to another. Moreover, as we select points that approach 1> from the right, it becomes increasingl difficult to distinguish which lead to root A and which to root C. On the same side of 1>, we find arbitraril close together points whose ultimate destinations are far apart. If we think of the roots as attractors of other points, the coloring in Figure 4.5 shows the intervals of the points the attract (the intervals of attraction ). You might think that points between roots A and B would be attracted to either A or B, but, as we see, that is not the case. Between A and B there are infinitel man intervals of points attracted to C. Similarl between B and C lie infinitel man intervals of points attracted to A. We encounter an even more dramatic eample of such behavior when we appl Newton s method to solve the comple-number equation z 6-1 =. It has si solutions: 1, -1, and the four numbers ;s1>d ; A 3>Bi. As Figure 4.53 suggests, each of the

6 34 Chapter 4: Applications of Derivatives Root A 1 Root B Root C (a) (b) 1 (c) FIGURE 4.5 (a) Starting values in A - q, - >B, A - 1>, 1>B, and A >, q B lead respectivel to roots A, B, and C. (b) The values = ;1> lead onl to each other. (c) Between 1> and >, there are infinitel man open intervals of points attracted to A alternating with open intervals of points attracted to C. This behavior is mirrored in the interval A - >, - 1>B. FIGURE 4.53 This computer-generated initial value portrait uses color to show where different points in the comple plane end up when the are used as starting values in appling Newton s method to solve the equation z 6-1 =. Red points go to 1, green points to s1>d + A 3>Bi, dark blue points to s -1>d + A 3>Bi, and so on. Starting values that generate sequences that do not arrive within.1 unit of a root after 3 steps are colored black.

7 4. Newton s Method 35 si roots has infinitel man basins of attraction in the comple plane (Appendi 5). Starting points in red basins are attracted to the root 1, those in the green basin to the root s1>d + A 3>Bi, and so on. Each basin has a boundar whose complicated pattern repeats without end under successive magnifications. These basins are called fractal basins.