COMPUTING AND DATA ANALYSIS WITH EXCEL. Numerical Methods of Solving Equations

Transcription

1 COMPUTING AND DATA ANALYSIS WITH EXCEL Numerical Methods of Solving Equations

2 Estimating Roots of Equations 1 By graphing and inspection Important iterativemethods Bisection method Newton-Raphsonmethod Excel built-intools Goal Seektool Solver tool

3 Estimating Roots ofequations by graphing and inspection Whenfaced with finding the roots to an equation that can t be solved by hand, the first thing to consider is a graph of the equation to obtain valuable insight into the nature of the equation and where the roots lie. We have seen from previous lectures how easy it is to chart values from simple equations of the form y = mx + c where we obtained corresponding y- values for selected x-values. Now, consider the following problem: Estimate the roots of the following polynomial by graphing and inspection. y x 3 x 2 5x 5 for x between +4 and-4

4 Finding roots of equations(cont d) By graphing andinspection Step 1: Create a table of points y x 3 x 2 5x 5 for x values between 4 and 4 In this example, my chosen x values are in column A (increments of 0.5) and the corresponding y-values in column B, see below: To obtain the corresponding value of y for x = -4, we type the formula or equation of the polynomial in cell B7, press enter to display the y value. Next, use the autofill handle to complete the y-values below.

5 Finding roots of equations(cont d) By graphing andinspection Step 2: Graph the values obtained in step 1. We want a visual representation of our equation so that we c a n create a single chart orgraph. To do this, select all the x and y-values including the headings, cells A6:B23

6 Finding roots of equations(cont d) By graphing and inspection Next, from the INSERT Tab, under the CHART Group, click on the down arrow to select the option that says, INSERT SCATTER (x, y) or BUBBLE CHART. From the chart sub-types displayed, point your mouse to the options available to have a preview of what the final chart will look like before making a final choice.

7 Finding roots of equations(cont d) By graphing andinspection From the graph we are able to estimate where the roots lie as follows: 1 st root of our equation lies between x = -2.0 and-3.0, 2 nd root lies at x = rd root lies between x= 2.0 and3.0

8 (for finding roots of equations) BisectionMethod Problem1: Find the roots (or zeros) of the function by estimating the x-value at which the curve crosses the x-axis or between which two x-values y changes polarity. Using the bisection method. Example: y x 3 x 1 We will start with a visual (graphical) representation of our function togain a fair idea of where are rootslie. Do you remember trial and improvement from KS 3? We are going to employ that tactic in excel.

9 (for finding roots of equations) BisectionMethod Example: The graph of thefunction shows that the root of the function lies between 0 and -1. We proceed with the bisection method to determine the actual ornear enough value of the root as follows:

10 BisectionMethod (for finding roots of equations) The two values of x within which we expect the root of the equation are our upper and lower limits and the midpoint value is halfway point between these two x values. Step 1: Evaluate the function at the upper, lower and midpoint values as in row 27 in the diagram tep 2: rom the results of evaluation, determine the new upper, lower and idpoint values, this narrows the range of x within which the root lies.

11 (for finding roots of equations) BisectionMethod We use the IF operator in Excel to determine if the value of f(c) in cell G27 is of the same sign (positive or negative) as f(a). To do this, we multiply f(a) by f(c) and check if the result is positive. Positive results means f(a) and f(c) are of the same sign while negative means they are of different signs. If TRUE that they are of the same sign, the midpoint value c, replaces guess (a). The same logical test is appliedat the upper bound or guess (b) and if True, we replace guess (b) with the midpoint (c). Refer to the excel sheet for a demo.

12 BisectionMethod (for finding roots of equations) Next, we determinethe midpoint value for the new range and evaluate the function at thesepoints. This process continues until either f(c) equals to zero, or the values of guess (a) and (b) are the same or until we have done a number of iteration or until we have attained a set condition e.g. a given error term. efer to the demo excel workbook for ther examples on when you may stop he iteration and report the root of the unction. In this example, guess (a) and guess (b) in the last three rows are equal ( ). This is the root of the function.

13 (for finding roots of equations) BisectionMethod Finally, we decide what value of f(c) is small enough to stop the process. In our example, f(c) in cell G52 is 2.82exp(-8). If we decide to stop at this point, then our root is x=

14 (for finding roots of equations) BisectionMethod Problem2: Find the roots (zeros) where the functionroot lies between two positive x- values. E.g.where a= 1 and f(a)= -3 and b=2 such that f(b)= 1 Using the bisection method. Wewill repeatall of step 1 in the previous example, drawing a graph to help theprocess.

15 Newton-Raphson Method The Newton-Raphson method is an iterative method to numerically estimate the roots of a polynomial. It requires the first differential f (x) of the function as an input to the formula. The only drawback is that it does not always converge to the root. The Newton-Raphson formula: f( xn ) x n 1 x n f '( x ) n Where Xn is the first approximation for a root. We evaluation the function and and its first differential equation at the first approximation and then Apply the Newton-Raphson (N-R) formula to find the next improved approximation Xn+1. The improved approximation then becomes an input for N-R formula by first evaluating f(xn+1) and f (Xn+1). The process continues until we obtain recurring value of Xn+i.

16 Newton-Raphson Method Given that f(x) x 3 x 1 has a root between 0 and -1, use the Newton-Raphson method to estimate the root to 3 decimal places. Steps 1 and 2 We start with a table of values for x and the corresponding f(x) from which we draw the graph of f(x) to show the where the graph crosses the x-axis. In this very example the graph is not required because we are told that it crosses the x-axis between 0 and -1.

17 Newton-Raphson Method Steps 3 and 4 From the curve we can visual inspection estimate an approximate value for the root. This becomes our first approximation, Xn to start the iteration. Next, we begin the iteration until we attain convergence.

18 Newton-Raphson Method Problem: Find the roots (zeros)of the following equationusing the Newton-Raphson method. f x x x 2 ( ) 4 5 6

19 Excel built-in tools Goal Seek and Solver tools Question: Use the Goal Seek tool in excel to find the root of the following function f x x x 3 ( ) 1 Step 1: Select a cell, e.g. B5 and type in the function, f(x). Assume the x value for the function is found in cell B4 and press the enter key. Your typed in function in B5 will look like: =B4^3+B4+1

20 Excel built-in tools Goal Seek and Solver tools Step 2: From the DATA tab, under the Forecast group, click on thedropdown arrow on What if analysis and select Goal Seek. In the dialog box that appears, place the cursor in the text field next to Set cell. Next, click in cell B5 where you typed the function or f(x). You may also just type the cell reference, B5 from the keyboard. Next, type in 0 in the text field next to To value. Finally, type in the B4, the cell ref for the x value to be varied.

21 Excel built-in tools Goal Seek and Solver tools Step 3: Click the OK button to start the search for the root. The value in B5 is the result of evaluating the function at x = the value in B4, the root. In this case, the value in B5 isn t equal to the target zero but is near enough. We therefore click OK to accept as the root for our function.