Methods for Advanced Mathematics (C3) Coursework Numerical Methods

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2 Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on Change of Sign... 9 Activit 3... Setting up a spreadsheet to do Decimal Search.... Activit Fied point iteration using = g()... 4 Activit... 4 Staircase and Cobweb Diagrams... Activit Eploring cobweb and staircase diagrams... 6 Activit Setting up a spreadsheet to do = g() fied point iteration... 8 Activit When the = g() method fails to converge... 0 Wh the method fails... 0 Coursework Requirements on = g() fied point iteration... Newton-Raphson method... Activit Setting up a spreadsheet for Newton-Raphson method... 3 Activit Investigating wh the Newton-Raphson Method fails... 4 Coursework Requirements on the Newton-Raphson Method... 4 Coursework Requirements on Comparison of the methods... Coursework Requirements on Oral Communication... Change of sign method... 7 = g() method Newton-Raphson method... 3 Comparison of methods Written Communication Oral Communication Methods for Advanced Mathematics (C3) Coursework: Assessment Sheet... 3 Woodhouse College 0 Page

3 Introduction In this coursework ou will investigate numerical methods of solving equations. B the end of the coursework ou should be able to: use the terms equation, function, root and solution appropriatel understand that some equations cannot be solved analticall b, for eample, factorising appl different methods for the numerical solution of such equations to an degree of accurac using computers and calculators compare the methods in terms of their efficienc and ease of use be able to eplain how the methods work with the help of graphs The methods ou will learn are Sstematic search for a change of sign (decimal search, bisection or linear interpolation) Fied point iteration after rearranging the equation f() = 0 into the form = g() Fied point iteration using the Newton-Raphson method This coursework represents 0% of the assessment for this module. Terminolog You must use the following terms correctl: 4 root = root = 0 3 Graph = f() Function f() =.9-4.4²+.6 Equation.9-4.4²+.6 = root = root = Solution ( solution, 4 roots) = -.384, = 0, = , = You can obtain up to one mark for the correct use of notation and terminolog: mark Correct terminolog throughout ½ mark Some errors in terminolog 0 mark Repeated failure to use the correct terminolog Woodhouse College 0 Page 3

4 Activit Find all the mistakes in the following: In order find the three roots of the epression = we could draw the functions = and = then find the points at which the intersect. Solution: = Solution: = Alternativel the function = can be rearranged to give = 0. If we now let f ( ) = we can check if one of the solutions, = 4.488, found before will work: f (0.3) = (4.488) = This gives a value that is close to zero. It is not eactl zero because the -values we obtained were rounded to four decimal places and so were not eact. = f then this curve will cross the -ais at the roots of If we draw the graph of ( ) the function: Solution: = Solution: =4.488 Woodhouse College 0 Page 4

5 Wh use numerical methods? 3 If we want to find the solution to the equation = 0 we can factorise it, so that + = so the three roots are = 0, = and =. ( )( ) 0 If we want to find the solution to the equation = 0 we can use the b ± b 4ac quadratic equation = which gives the two roots = (4 d.p.) a and -9.3 (4 d.p.). 3 Some equations, however, like + 3 = 0 cannot be solved b algebraic or analtical methods (factorising or b a simple equation). To solve these equations we use numerical methods. You will be asked to investigate using three numerical methods and will have to choose our own equations to use. You will lose marks if ou choose equations which can be solved algebraicall or analticall as we should onl use numerical methods when we cannot solve them otherwise. Change of sign The first method that we are going to eplore uses the fact that around a root, the value of a function changes sign: + root As an eample consider the function f ( ) = + 3. f ( ) = f ( ) = ( ) ( ) + 3 = = 9 ( ) ( ) + 3 = = 7 Since the function changes sign between - and - (it goes from -9 to +7) then there must be a root in the interval [-,-]. Woodhouse College 0 Page

6 Activit 3 Is our number over 0? No. One plaer chooses a number between and 00 and writes it down. The other plaer can ask es/no questions to tr and find out the number. How man questions did ou take? What is the smallest number of questions ou can guarantee finding the number in? What if the number chosen was between and 000? What if the number chosen was between and n? Woodhouse College 0 Page 6

7 Interval Bisection Consider the equation + 3 = 0. We use the function f ( ) = + 3 and we have alread seen that there is a sign change and therefore a root in the interval [-,-]. With interval bisection, we now bisect the interval, in other words we net tr the midpoint of the interval (- + -) / = -. f (.) = (.) (.) + 3 =.9(3 s.f.) f(- ) = Net tr midpoint f(-.) =.9 (3 s.f.) - 0 f(- ) = We now know that there is a sign change from f(-) = -9 to f(-.) =.9 and so the root must lie in the interval [-,-.]. We now repeat the process. The midpoint of the current interval is (- + -.) / = -.7 and f(-.7) = We know that there is a sign change from f(-.7) = to f(-.) =.9 and so the root must lie in the interval [-.7, -.] Our current estimate of the root is ( ) / = -.6 The maimum possible error is -.6 (-.7) = 0. so root = -.6 ± 0. f(-.) =.9 (3 s.f.) - - f(- ) = - 9 f(- ) = 7 f(-.7) = (3 s.f) We can repeat the process until an required degree of accurac is obtained Woodhouse College 0 Page 7

8 Decimal Search Consider the equation + 3 = 0. We use the function f ( ) = + 3 and we have alread seen that there is a sign change and therefore a root in the interval [-,-]. f(- ) = f(- ) = In this method, ou first take increments in of size 0. within the interval and work out the value of the function for each one: f() From the table we can see that there is a sign change from -.7 to -.6 and so this is our new interval in which the root must lie [-.7, -.6] Woodhouse College 0 Page 8

9 We now repeat the process b considering steps of size 0.0 from -.7 to -.6 : f() We can see that there is a sign change from -.6 to -.6 and so this is the new interval in which the root must lie [-.6, -.6] f() We can see that there is a sign change from -.69 to -.68 and so our estimate of the root at this stage is ( ) / = -.68 with a maimum error of ± Coursework Requirements on Change of Sign You will need to demonstrate one change of sign method for our coursework. You can use Interval Bisection or Decimal Search You should find a root of an equation that cannot easil be found using algebraic or analtical methods. You must choose an equation for ourself that does not appear in these notes and must not be the same equation as anone else. ( mark ) You must eplain the method ou have used, and our eplanation must include graphs to illustrate the method. ( ½ mark ) You must correctl state the error bounds for our answer (e.g. ±0.000) ( ½ mark) You must give an eample of an equation where one of the roots cannot be found using our chosen change of sign method. You must include an eplanation with graphs to illustrate wh the method does not work in this case. ( mark ) Eamples of wh change of sign might not work: Woodhouse College 0 Page 9

10 A repeated root causes change of sign to fail. Note that the root must be to dp or it will be easil found Roots that are ver close together will cause the change of sign method to fail Woodhouse College 0 Page 0

11 Activit 3 Setting up a spreadsheet to do Decimal Search. We are going to use an Ecel spreadsheet to do a decimal search for a root of the equation = +. We need to rearrange the equation to give = 0. We will draw the graph = (using Omnigraph or Autograph) in order to locate a root of the function f ( ) = : It appears from the graph that there is a root at = and if we check: f () = = 0, this confirms that one root is. As this root was easil located from the graph, it was not appropriate to use numerical methods to find it. Also looking at the graph we can see there is a root between - and - which we can confirm b looking for a sign change: f ( ) = ( ) = 0. f ( ) = ( ) = 0. The sign change shows that there is a root between - and -. We will set up an Ecel spreadsheet to do a decimal search for this root. The diagram below shows the formulas we need to enter. You will find it easier if ou cop the formulas in cells C and B across the page using the autofill tool. This should give a spreadsheet that looks like this. Set the format of row to four decimal places so that the information can be read easil. Woodhouse College 0 Page

12 We can see from the table that there is a sign change between -.7 and -.6 so we now need to zoom in on this interval. Cop and paste the first table then change it to look at this interval. You will have to change B4 to be -.7 and remember to change the increment to 0.0 instead of 0. in cell C4 then cop the new formula across. The spreadsheet should then look like this: Now we can see that the sign change is between -.7 and -.69 so we zoom in on this interval. Again cop and paste the table down then change it. Set the increment in the formula in C7 to 0.00 then cop the formula across. Our estimate of the root is ( )/ = and the error bounds are ± (since the root must be between -.69 and -.69) Woodhouse College 0 Page

13 Activit 4 Eplain wh a change of sign method will not work in the following cases:. The equation = 0 is to be solved using a change of sign method. + Show that the function f ( ) = gives a sign change from = to = 3 and eplain with the aid of a graph wh the sign change method fails to find a root in this case The equation = 0 has a root.68 (3 d.p.) but testing for a sign change from = to = 3 fails. 4 3 Show that the function f ( ) = does not give a sign change from = to = 3 and eplain with the aid of a graph wh the sign change method fails to find the root in this case The equation = 0 is to be solved using a change of sign method. 3 Show that the function f ( ) = does not give a sign change from = - to = 0 and eplain with the aid of a graph wh the sign change method fails to find a root in this interval even though one eists. Woodhouse College 0 Page 3

14 Fied point iteration using = g() Activit. Set our calculator in radian mode. Enter the following kes: 0 = cos ANS = Press = repeatedl and describe what happens.. Enter the following kes: 0. = ANS = Press = repeatedl and describe what happens. Repeat the process with other positive values in place of 0.. What happens? 3. Enter the following kes: 0. = (ANS + ) = Press = repeatedl and describe what happens. Repeat the process with other positive values in place of 0.. What happens? 4. A diagram that goes with the first question above is given below: 0 cos The equation that this solves is = cos (wh?) or cos = 0 Check that our final value from question is a root of this equation. Draw similar diagrams for questions and 3. Write down the equations that the solve.. Two graphs connected with question are: =- cos = =cos - - Indicate which point on each of these graphs corresponds to the root ou found. Sketch similar graphs for questions and 3 and indicate which points the roots correspond to. Woodhouse College 0 Page 4

15 Staircase and Cobweb Diagrams To solve the equation cos = 0 we firstl rearrange it so that it is in the form = g() which in this case could be = cos. If we write this as an iterative formula: r+ = cos r This sas that the net is the cosine of the previous If we start with 0 then 0 = = cos 0 = cos0 = = cos = cos = = cos = cos = and so on Drawing the graphs of = and = cos, we can show these values on the graph: = 0. = cos The diagram shows how the iterative process converges in on the root. This tpe of diagram is called a cobweb diagram. Looking at the second eample, to solve = 0 we can rearrange to = Note that this equation would not normall be solved using numerical methods because it can be solved b factorising and would not be suitable for our coursework. If we write this as an iterative formula: r+ = r This sas that the net is the square root of the previous If we start with 0 = 0. then = 0 = 0. = = = = Woodhouse College 0 Page

16 = = and so on 3 = Drawing graphs of = and = we can show these values on a graph: = = cos Activit 6 The diagram shows how the iterative process converges in on the root. This tpe of diagram is called a staircase diagram. Eploring cobweb and staircase diagrams. Below are the graphs of = and = 3 + used to illustrate how the equation = 0 can be solved using fied point iteration. a) Show that the equation = 0 can be rearranged into this form = g() b) Starting with = draw lines to show whether this is a staircase or a cobweb c) What happens if ou started with 0 = 0. 9? Woodhouse College 0 Page 6

17 . Below are the graphs of = and = used to illustrate how the equation = 0 can be solved using fied point iteration. 3 a) Show that the equation = 0 can be rearranged to = g() b) Starting with 0 = draw lines to show whether this is a staircase or a cobweb Below are the graphs of = and =. a) Which equation of the form f() = 0 do these graphs illustrate the solution of? b) Starting with 0 = 0. draw lines to show fied point iteration c) Eplain wh the iteration starts out as a staircase and then becomes a cobweb Woodhouse College 0 Page 7

18 Activit 7 Setting up a spreadsheet to do = g() fied point iteration We are going to set up a spreadsheet to solve (sin + cos ) = 0. Show that this can be rearranged into the form = sin + cos +. Use a graph drawing program to check the graphs below. What do these graphs illustrate? 3. Set up a new spreadsheet as shown below (the top left cell is A). Enter the first formula (for r = ) and then use the autofill tool to cop the formula down the spreadsheet. Woodhouse College 0 Page 8

19 4. You should obtain results like those below: If ou continue the spreadsheet downwards, ou should find that the iterative process converges to the root.7079 ( s.f). Use the sign change method with f ( ) = (sin + cos ) to show that this root is correct to s.f : Calculate f(.7078) Calculate f(.7079) Show that there is a sign change between these and therefore the root must be between.7078 and Since all the values in this range are.7079 ( s.f) then the root must be correct to s.f. 6. Draw lines on the graph below to show whether it is a staircase or cobweb diagram: Woodhouse College 0 Page 9

20 Activit 8 When the = g() method fails to converge Consider the equation + 3 = Show that this can be rearranged to give =. Set up a spreadsheet to do fied point iteration using this rearrangement. Start with = to find a root of the equation between = 0 and = correct to significant figures. On the graph below illustrate this process: Change our starting value to =. 3 to tr and find the root between. and.3. What happens? Show on the diagram above what is happening. Wh the method fails = g(). Gradient of g() > (steeper than = ) and so the sequence does not converge = 0. Gradient of g() < (less steep than = ) and so the sequence converges Woodhouse College 0 Page 0

21 4. Show that another rearrangement of + 3 = 0 is = 3. Use a graph drawing program to show = and = 3 on the same aes. 6. Set up a spreadsheet to iterate = 3 and tr to find each of the three roots of the equation + 3 = 0 correct to s.f. (don t forget to check our roots with a sign change) 7. Using our graph from and looking at the gradient of = 3 around each of the roots, eplain wh some of the roots can be found but not all of them. Coursework Requirements on = g() fied point iteration Using = g() fied point iteration ou should find one root of an equation that cannot easil be found using algebraic or analtical methods. You must choose an equation for ourself that does not appear in these notes, must not be the same equation as anone else and should be different from the equation ou used for the change of sign method. ( mark ) You must show using a graph of our = and our = g() how the convergence works (cobweb or staircase diagram) ( ½ mark ) Eplain using our graphs how the gradient of g() near to the root means that the iteration converges. (½ mark ) You must give an eample, using the same original equation, where = g() fied point iteration fails to converge. This could be using the same =g() rearrangement as above (if there is another root for which it does not converge) or using a different rearrangement of the same original equation. ( mark ) You must show using a graph of our = and our = g() how the convergence fails to work ( ½ mark ) Eplain using our graphs how the gradient of g() near to the root means that the iteration does not converge. (½ mark ) Woodhouse College 0 Page

22 Newton-Raphson method This is another fied point iteration method and, as with = g() iteration, ou need an estimate of the root as a starting place. Consider finding a root of the graph f() = 0 with an initial estimate of = If we draw a tangent to the curve at =, then where that tangent crosses the -ais is a better estimate of the root. 30 Draw a tangent at the estimate = 0 0 Root we are finding 3 Where the tangent crosses the - ais is the net estimate We can then iterate (repeat the procedure) to find better and better estimates: 30 0 Tangent drawn each time 0 Root we are finding - 3 Initial Third Second estimate estimate estimate Woodhouse College 0 Page

23 The Newton-Raphson iterative formula is: r+ = r f f ( r ) ( ) r Activit 9 Setting up a spreadsheet for Newton-Raphson method Consider the equation + 3 = 0. We want to solve f ( ) = 0 where f ( ) = + 3. Differentiate f ( ) to find f ( ) The Newton-Raphson formula will become: + 3 r + = r 4 and we will start with an initial estimate =. Set up the spreadsheet. The formulas are given below. Once ou have entered the formula into cell B3, cop it down using the autofill tool. 3. You should find that the iterations converge to.77 ( s.f.). Check with a sign change that this is correct: Calculate f(.76) Calculate f(.77) Show that there is a sign change between these values and hence the root lies between them. All values in this interval round to.77 ( s.f.). Woodhouse College 0 Page 3

24 Activit 0 Investigating wh the Newton-Raphson Method fails. Investigate what happens when ou tr to use the Newton-Raphson method to solve the equation + 3 = 0 with an initial value =. Draw a graph to eplain the problem.. Investigate what happens when ou tr to use the Newton-Raphson method to find the root of the equation between 0 and Tr =, = 0 Draw a graph to help eplain what is happening. = 3. Investigate what happens when ou tr to use the Newton-Raphson method to find 3 the root of the equation = 0 with an initial value = Draw a graph to help eplain what is happening. Coursework Requirements on the Newton-Raphson Method Using Newton-Raphson method ou should find all the roots (at least ) of an equation that cannot easil be found using algebraic or analtical methods. You must choose an equation for ourself that does not appear in these notes, must not be the same equation as anone else and should be different from the equation ou used for either of the other methods (change of sign and =g() iteration). One root found ( mark ) All the roots are found (+ mark = marks total) You must show using a graph of our chosen equation how the convergence works for one of the roots ( mark ) You must use the change of sign method to establish error bounds for one of our roots ( mark) You must give an eample of an equation where the Newton-Raphson method fails to find a particular root, despite a starting value close to it (the starting point should be the nearest integer on either side of the root). Using a graph of our chosen function ou must eplain wh the method fails ( mark) Woodhouse College 0 Page 4

25 Coursework Requirements on Comparison of the methods You must select one of the three equations ou have used so far for our eamples of the 3 different methods and use the other two methods on that same equation so that ou can compare the methods. You should use the same starting point for each method and obtain the answer to the same accurac using each method. ( mark) You must compare the three methods in terms of speed of convergence (how man calculations/iterations were involved) ( mark) You must compare the three methods in terms of how eas the were to appl with the software and hardware ou used. ( mark) Coursework Requirements on Oral Communication You will have a short interview with our teacher about our coursework. The will ask ou to eplain what ou have done, and will ask ou to go through the working for one of the methods to check our understanding. ( marks) Woodhouse College 0 Page

26 COURSEWORK OVERVIEW Change of sign method A. Choose an equation, f() = 0, that cannot be solved b analtical methods. Find one root using decimal search or interval bisection 3. Make sure ou include the correct error bounds 4. Eplain our working using graphs that show our function B. Choose an equation for which the change of sign method fails to find a root. Eplain wh the change of sign method fails in this case using a graph of our function = g() method A. Choose a different equation, f() = 0, that cannot be solved b analtical methods. Rearrange the equation into the form = g() 3. Find one root using = g() iteration 4. Establish the accurac of the root using a change of sign. Eplain our working using a graph that shows our function B. Choose a rearrangement of the equation used in A for which the = g() method fails to find a root. Eplain wh the = g() method fails in this case using a graph of our function Newton-Raphson method A. Choose a different equation, f() = 0, with at least roots that cannot be solved b analtical methods. Differentiate our function and set up the Newton-Raphson formula 3. Find all the roots using the Newton-Raphson method 4. Establish the accurac of the roots using the change of sign method. Eplain our working using graphs that show our function B. Choose an equation for which the Newton-Raphson method fails to find a root. Eplain wh the Newton-Raphson method fails in this case using a graph of our function Comparison of methods. Choose one of the equations ou have alread found a root using change of sign, = g() or Newton-Raphson method. Find that same root using the other two methods (using the same starting value and finding the answer to the same accurac) 3. Compare the three methods in terms of speed of convergence (how man iterations to find the root) and ease of use of the hardware / software Woodhouse College 0 Page 6

27 Change of sign method A. Choose an equation, f() = 0, that cannot be solved b analtical methods You should not choose a quadratic or linear function as these are easil solved using methods ou alread know A cubic or higher degree polnomial would be suitable as long as it does not factorise easil or have obvious roots from the graph o For eample = 0 is no good because it can be factorised ( ) = 0 then one solution is = 0 and the quadratic can be solved using the quadratic equation.. Find one root using decimal search or interval bisection If our equation has more than one root, ou onl have to find one. Draw a graph of our function to see where the root is approimatel o See which two integer values our root is between to use as starting points for our method o You could check that these two values give ou a sign change as a starting point Appl the method of decimal search or interval bisection to find the root (see 3 for details of error bounds) o You will probabl use a spreadsheet to do this see the eample in the coursework book 3. Make sure ou include the correct error bounds Your error bounds should be ±0.000 Suppose the last sign change interval ou find is from.46 to.47 then ou know the root is somewhere in this interval. If ou use.46 (the midpoint of the interval) as our estimate then the maimum error (the most ou could be wrong b) is ± Eplain our working using graphs that show our function 3 If our function was f ( ) = then ou would start with 3 a graph of = You will probabl use a graph drawing program to produce this. You ma need to zoom in and out to obtain a good graph. You will then need further graphs for the subsequent stages. Your eplanation should combine graphs and words to eplain what ou have done see the eamples in the coursework book for help. Woodhouse College 0 Page 7

28 B. Choose an equation for which the change of sign method fails to find a root. Eplain wh the change of sign method fails in this case using a graph of our function The equation must not be trivial (the must require a graph to be drawn in order to determine wh the don t work) Eamples of trivial equations are = 0 3, = 0 ( ), ( ) = 0 Three possible was of designing an equation that fails with change of sign to find one of the roots are given below - ou onl need o Designing an equation with repeated roots (done in Polnomials chapter of C) will give a function which touches the -ais and thus the method will fail as no root is detected. You should, however, choose the repeated root to be a number to at least DECIMAL PLACES. Otherwise the first table in the decimal search procedure will solve the equation on its own. = For eample 0( + )( 0.43) 0 will give the graph below: The 0 factor in the eample is simpl to sharpen the verte at 0.43 and emphasise the fact that the graph merel touches at this point. Note that although the initial design of the failure is artificial and uses a squared factor, when writing up the coursework give ALL the equations in an epanded form (without brackets) otherwise the solution is obvious b algebraic techniques. Woodhouse College 0 Page 8

29 o Designing an equation with close roots is similar to the repeated roots eample above. For eample = ( )( )( ) 0 o Designing an equation with a discontinuit can be done with a function that is a fraction where the denominator has solutions. For eample = 0 3 The function h ( ) = in the dominator has a root between 3 and 4 so the whole fraction equation has an asmptote between 3 and 4 causing the change of sign method to think there is a root in the interval when no root eists Woodhouse College 0 Page 9

30 = g() method A. Choose a different equation, methods f() = 0, that cannot be solved b analtical The equations that ou use for the three methods must all be different to each other and to everbod else doing the coursework (no coping) As with change of sign, a quadratic or linear function is no good. Cubics or higher degree polnomials are ok if the do not factorise easil.. Rearrange the equation into the form = g() You ma have to tr several rearrangements until ou find one that will converge with this method. If ou find rearrangements that don t work then ou can use one of these to demonstrate failure (see B below) 3. Find one root using = g() iteration Draw a graph of our function, = f(), to establish an integer close to a root to use as a starting point You will probabl use a spreadsheet to do the calculations see the eample in the coursework booklet. 4. Establish the accurac of the root using a change of sign Suppose ou obtain a root ( d.p.) then ou should check that the original f() (not the rearrangement g() formula) gives a sign change from to As all numbers in this interval would round to ou can conclude that this is the root to d.p.. Eplain our working using a graph that shows our function You should draw the graph of our = f() to establish a starting point (see 3 above) You should draw = and our chosen = g() on the same aes and use our graph to show whether it is a cobweb diagram or a staircase diagram that converges on the root ou have found. You have to eplain wh the method converges b comparing the gradient of = with the gradient of = g() around the root ou are finding. The gradient of g() should be less than the gradient of = (i.e. between - and ) if it converges. The graph drawing program has Woodhouse College 0 Page 30

31 a gradient function which ou can use to draw the gradient graph = g () which will help with our eplanation. Your eplanation should combine graphs and words to eplain what ou have done see the eamples in the coursework book for help B. Choose a rearrangement of the equation used in A for which the = g() method fails to find a root. Starting with the same f() = 0 equation, tr rearranging it in a different wa to obtain = g(). Usuall an alternative rearrangement will fail to find the root that ou found in A.. Eplain wh the = g() method fails in this case using a graph of our function Draw = and = g() graphs on the same aes for our rearrangement that doesn t work and draw lines on it to show that the method diverges rather than converging on the root You have to eplain wh the method diverges b comparing the gradient of = with the gradient of = g() around the root ou are finding. The gradient of g() should be more than the gradient of = (i.e. less than - or more than ) if it diverges. The graph drawing program has a gradient function which ou can use to draw the gradient graph = g () which will help with our eplanation. Your eplanation should combine graphs and words to eplain what ou have done see the eamples in the coursework book for help Woodhouse College 0 Page 3

32 Newton-Raphson method A. Choose a different equation, f() = 0, with at least roots that cannot be solved b analtical methods With the Newton-Raphson method our equation must have at least roots and ou have to find all of them The equations that ou use for the three methods must all be different to each other and to everbod else doing the coursework (no coping) As with change of sign, a quadratic or linear function is no good. Cubics or higher degree polnomials are ok if the do not factorise easil.. Differentiate our function and set up the Newton-Raphson formula Check our differentiation to make sure it is correct. The Newton-Raphson iterative formula is: r+ = r f f ( r ) ( ) r You do not have to derive this formula You should give our own version of it in our coursework with our f() and f () substituted into it 3. Find all the roots using the Newton-Raphson method Draw a graph of the function ( =f() ) to establish roughl where the roots are. You will start the search for each root at the closest integer to that root. You will probabl use a spreadsheet for this see the coursework booklet for an eample 4. Establish the accurac of the roots using the change of sign method Suppose ou obtain a root ( d.p.) then ou should check that the original f() (not the rearrangement g() formula) gives a sign change from to As all numbers in this interval would round to ou can conclude that this is the root to d.p. Woodhouse College 0 Page 3

33 . Eplain our working using graphs that show our function On a graph of our curve = f() draw the appropriate tangents to show the method converging on the root. Your eplanation should combine graphs and words to eplain what ou have done see the eamples in the coursework book for help B. Choose an equation for which the Newton-Raphson method fails to find a root. Eplain wh the Newton-Raphson method fails in this case using a graph of our function The failure should not be because ou have started too far awa from the root ou must start at one of the integers either side of the required root One possible wa of designing an equation that fails with the Newton- Raphson method to find one of the roots is given below. Other eamples are given in the Coursework booklet and could be used to find other eamples that fail with this method. Start with an equation f() = 0 with a repeated root, then subtract a constant from the function f(): Eample ( )( ) 3 = 0 has a repeated root at = 3 If we subtract a (small) constant from the left hand side: ( )( ) = 0 then the equation will have a root close to 3 but starting with = 3 using the Newton-Raphson method will not work as the tangent is horizontal at this point: Note that although the initial design of the failure is artificial and uses a squared factor, when writing up the coursework give ALL the equations in an epanded form (without brackets) otherwise the solution is obvious b algebraic techniques. Woodhouse College 0 Page 33

34 Comparison of methods. Choose one of the equations ou have used alread for change of sign, = g() or Newton-Raphson method It does not matter which one ou choose but remember ou will have to be able to differentiate in order to do the Newton-Raphson method and the gradient of the g() in the rearrangement ou use for = g() method must be less than.. Find that same root using the other two methods (using the same starting value and finding the answer to the same accurac) You must start from the same starting value for all three methods (and it must converge to the same root for all three methods) You must obtain the same root to the same degree of accurac (and check that accurac using a sign change where necessar) You must use the same technolog (e.g. a spreadsheet) for each method so that ou can compare how eas each of the methods was using that technolog 3. Compare the three methods in terms of speed of convergence (how man iterations to find the root) and ease of use of the hardware / software How man iterations did each of the methods take? Talk about how eas each of the methods was to set up in our case (e.g. how eas to find a rearrangement of the equation for = g(), how eas the function was to differentiate for Newton-Raphson) and how these considerations would influence our choice of method for other equations. Talk about how eas each of the methods was to implement using the technolog ou chose Written Communication Check through to make sure ou have used the correct terminolog throughout our work. Be particularl careful with equation, function, graph, root and solution. Oral Communication You will be given a short interview on our work in which ou will be asked to eplain what ou did in general and eplain one of the methods in detail. Woodhouse College 0 Page 34

35 Methods for Advanced Mathematics (C3) Coursework: Assessment Sheet Task: Candidates will investigate the solution of equations using the following three methods: Sstematic search for change of sign using one of the three methods: decimal search, bisection or linear interpolation. Fied point iteration using the Newton-Raphson method Fied point iteration after rearranging the equation f() = 0 into the form = g(). Coursework Title Candidate Name Candidate Number Centre Number 9 0 Date Domain Mark Description Comment Mark Change of sign method (3) Newton- Raphson method () Rearranging f() = 0 in the form = g() (4) Comparison of methods (3) The method is applied successfull to find one root of an equation. Error bounds are stated and the method is illustrated graphicall. An eample is given of an equation where one of the roots cannot be found b the chosen method. There is an illustrated eplanation of wh this is the case. The method is applied successfull to find one root of a second equation. All the roots of the equation are found. The method is illustrated graphicall for one root. Error bounds are established for one of the roots. An eample is given of an equation where this method fails to find a particular root despite a starting value close to it. There is an illustrated eplanation of wh this has happened. A rearrangement is applied successfull to find a root of a third equation. Convergence of this rearrangement to a root is demonstrated graphicall and the magnitude of g () is discussed. A rearrangement of the same equation is applied in a situation where the iteration fails to converge to the required root. This failure is demonstrated graphicall and the magnitude of g () is discussed. One of the equations used above is selected and the other two methods are applied successfull to find the same root. There is a sensible comparison of the relative merits of the three methods in terms of speed of convergence. There is a sensible comparison of the relative merits of the three methods in terms of ease of use with available hardware and software. Written communication () Correct notation and terminolog are used. Oral communication () Presentation Interview Discussion Please tick at least one bo and give a brief report. Half marks ma be awarded but the overall total must be an integer. Please report overleaf on an help that the candidate has received beond the guidelines. TOTAL /8 Woodhouse College 0 Page 3

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