1. Coursework Requirements Numerical Methods

2. 1.Front Cover indicating that the coursework is about the numerical Solution of Equations. Include your name, candidate number and centre number (12290) on every page as a footer 2.Aim 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion

3. 1.Front Cover 2.Aim – State clearly that you are investigating (3) numerical methods of solving equations 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion

4. 1.Front Cover 2.Aim 3.Introduction – here you summarise the contents and give an overview of the main ideas. Although this is the first section that is read, it is the last section that you should write. 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion

5. 1.Front Cover 2.Aim 3.Introduction 4.Method 1 & Failure Case – Explain the change of sign and decimal search method, showing full calculations and demonstrate an example of this method failing 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion Use short, precise sentences, correct notation and terminology

6. Recommendation: Horizontal tables will save you space

7. You need to manipulate the graph so that the roots are not visible to 1dp

8. In effect, what you are doing is looking at ever greater detail … Looking at the root between 1 and 2 Looking at the root between 1.2 and 1.3

9. In effect, what you are doing is looking at ever greater detail … Looking at the root between 1.27 and 1.28 Looking at the root between 1.2 and 1.3 (error 0.05) Looking at the root between 1.274 and 1.275

10. Marking  Calculations for finding one root to 5dp  Graphical illustration + statement of error bounds (±0.00005)  Case study of an illustrated example where the method fails (and reasons for this) Total : 3 marks

11. 1.Front Cover 2.Aim 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case – Explain the fixed point iteration g(x) method, showing full calculations and demonstrate an example of this method failing 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion

13. If the results are divergent you need to find another formula … Start with estimating a root

14. Illustrate this convergence with a staircase diagram

15. Staircase diagram showing convergence (1,1) (1,0.6667) (0.6667,0.6667) (0.6667,0.432)

16. Case 1Case 2 This method of finding the root will only be successful if … The gradient g ’( x ) of the point near your root is between -1 and +1 Now you should read your notes and extend this because you need to discuss the magnitude of g’( x ) at the end of each section

17. Make sure you have 5 graphs: a. f ( x ) showing intersection of y = f ( x ) and y = 0 … roots b. f ( x ) and intersection of y = g ( x ) and y = x … movement of roots c.Convergent case (staircase diagram) d.Divergent case (spider web diagram) e.Tangent at the intersections

18. Marking  Rearrangement of equation and calculation of root  Graphical illustration of convergence, magnitude of g ’( x ) discussed  Case study showing failure to converge using the same equation  Graphical illustration of divergence, magnitude of g ’( x ) discussed Total : 4 marks

19. 1.Front Cover 2.Aim 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case – Explain the Newton-Raphson method, showing full calculations and demonstrate an example of this method failing 7.Comparison of the 3 methods 8.Conclusion

20. Method 3 : Newton-Raphson Iteration Pages 8-10, 14 &15 Start the search at the closest integer to the root: x 0 = 2 Draw a tangent to the curve at this point x 1 = intersection of the tangent and the x axis x 1 = 1.68571 We can calculate this point using the formula:

21. Starting point where x 0 = 2 This can be confirmed on Autograph for the curve y = x 3 − 5 x +4.2

22. x 0 is the starting value x 1 is the intersection of the tangent and the x axis x 2 is the intersection of the second tangent with the x axis

23. Failure: When f ‘( x ) = 0 If the tangent to the curve does not intersect the x axis, a new estimate of the root cannot be found.

24. Marking  Calculation of one root  Calculations of ALL roots  Graphical illustration for 1 root  Error bounds for 1 root  Case study showing where this method fails and graphical illustration of failure Total : 5 marks

25. 1.Front Cover 2.Aim 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7a Compare the 3 methods in terms of how efficiently the calculations work on the same root with the same accuracy 7b Compare the 3 methods in terms of speed of convergence and ease of use of the hardware and software. 8. Conclusion

26. 1.Front Cover 2.Aim 3.Introduction 4.Method 1 & Failure Case 5.Method 2 & Failure Case 6.Method 3 & Failure Case 7.Comparison of the 3 methods 8.Conclusion – summarise the information in an accurate and comprehensive manner