1. EE 3561_Unit_1(c)Al-Dhaifallah 14351 EE 3561 : - Computational Methods in Electrical Engineering Unit 1: Introduction to Computational Methods and Taylor Series Lectures 1-3: Mujahed Al-Dhaifallah (Term 351)

2. EE 3561_Unit_1(c)Al-Dhaifallah 14352 Mujahed Al-Dhaifallah مجاهد آل ضيف الله  Office Hours Office Hours: SMT, 1:30 – 2:30 PM, by appointment  Office: Dean Office.  Tel: 7842983  Email: muja2007hed@gmail.com http://faculty.sau.edu.sa/m.aldhaifallah

3. Prerequisites  Successful completion of Math 1070, Math2040, CS1090 EE 3561_Unit_1(c)Al-Dhaifallah 14353

4. Course Description  EE 3561 is a course on Introduction to computational methods using computer packages, e.g. Matlab, Mathcad or IMSL. Solution of non-linear equations. Solution of large systems of linear equations. Interpolation. Function approximation. Numerical differentiation and integration. Solution of the initial value problem of ordinary differential equations. Applications on Electrical Engineering. EE 3561_Unit_1(c)Al-Dhaifallah 14354

5. Textbooks  “Numerical Methods for Engineers”. By Steven C. Chapra and Raymond P. Canale EE 3561_Unit_1(c)Al-Dhaifallah 14355

6. Attendance  Regular lecture attendance is required. There will be part of the grade on attendance EE 3561_Unit_1(c)Al-Dhaifallah 14356

7. Student Evaluation Exam 1 10 Exam 2 10 Computer Work 5 Quizzes 5 HW 5 Attendance 5 Final Exams 60 Total 100 EE 3561_Unit_1(c)Al-Dhaifallah 14357

8. Quizzes  Announced  After each HW. From HW material EE 3561_Unit_1(c)Al-Dhaifallah 14358

9. Assignment Requirements  Late assignments will not be accepted.  assignments are due at the beginning of lecture.  Sloppy or disorganized work will adversely affect your grade. EE 3561_Unit_1(c)Al-Dhaifallah 14359

10. Exams  Attendance is mandatory.  Make-up exam are not given unless permission is obtained prior to exam day from the instructor a valid, documented emergency has arisen EE 3561_Unit_1(c)Al-Dhaifallah 143510

11. Academic Dishonesty  cheating  fabrication  falsification  multiple submissions  plagiarism  complicity EE 3561_Unit_1(c)Al-Dhaifallah 143511

12. EE 3561_Unit_1(c)Al-Dhaifallah 143512 Rules and Regulations  No make up quizzes  DN grade --- (25%) 8 unexcused absences  Homework Assignments are due to the beginning of the lectures.  Absence is not an excuse for not submitting the Homework.  Late submission may not be accepted

13. EE 3561_Unit_1(c)Al-Dhaifallah 143513 Lecture 1 Introduction to Computational Methods  What are Computational METHODS ?  Why do we need them?  Topics covered in EE3561. Reading Assignment: pages 3-10 of text book

14. EE 3561_Unit_1(c)Al-Dhaifallah 143514 Analytical Solutions  Analytical methods give exact solutions  Example: Analytical method to evaluate the integral  Numerical methods are mathematical procedures to calculate approximate solution (Trapezoid Method)

15. EE 3561_Unit_1(c)Al-Dhaifallah 143515 Computational Methods Computational Methods: Algorithms that are used to obtain approximate solutions of a mathematical problem. Why do we need them? 1. No analytical solution exists, 2. An analytical solution is difficult to obtain or not practical.

16. EE 3561_Unit_1(c)Al-Dhaifallah 143516 What do we need Basic Needs in the Computational Methods: Practical: can be computed in a reasonable amount of time. Accurate:  Good approximate to the true value  Information about the approximation error (Bounds, error order,… )

17. EE 3561_Unit_1(c)Al-Dhaifallah 143517 Outlines of the Course  Taylor Theorem  Number Representation  Solution of nonlinear Equations  Interpolation  Numerical Differentiation  Numerical Integration  Solution of linear Equations  Least Squares curve fitting  Solution of ordinary differential equations

18. EE 3561_Unit_1(c)Al-Dhaifallah 143518 Solution of Nonlinear Equations  Some simple equations can be solved analytically  Many other equations have no analytical solution

19. EE 3561_Unit_1(c)Al-Dhaifallah 143519 Methods for solving Nonlinear Equations o Bisection Method o Newton-Raphson Method o Secant Method

20. EE 3561_Unit_1(c)Al-Dhaifallah 143520 Solution of Systems of Linear Equations

21. EE 3561_Unit_1(c)Al-Dhaifallah 143521 Cramer’s Rule is not practical

22. EE 3561_Unit_1(c)Al-Dhaifallah 143522 Methods for solving Systems of Linear Equations o Naive Gaussian Elimination o Gaussian Elimination with Scaled Partial pivoting

23. EE 3561_Unit_1(c)Al-Dhaifallah 143523 Curve Fitting  Given a set of data  Select a curve that best fit the data.  One choice is find the curve so that the sum of the square of the error is minimized.

24. EE 3561_Unit_1(c)Al-Dhaifallah 143524 Interpolation  Given a set of data  find a polynomial P(x) whose graph passes through all tabulated points.

25. EE 3561_Unit_1(c)Al-Dhaifallah 143525 Methods for Curve Fitting o Least Squares o Linear Regression o Nonlinear least Squares Problems o Interpolation o Newton polynomial interpolation o Lagrange interpolation

26. EE 3561_Unit_1(c)Al-Dhaifallah 143526 Integration  Some functions can be integrated analytically

27. EE 3561_Unit_1(c)Al-Dhaifallah 143527 Methods for Numerical Integration o Upper and Lower Sums o Trapezoid Method o Romberg Method o Gauss Quadrature

28. EE 3561_Unit_1(c)Al-Dhaifallah 143528 Solution of Ordinary Differential Equations

29. EE 3561_Unit_1(c)Al-Dhaifallah 143529 Summary  Computational Methods: Algorithms that are used to obtain numerical solution of a mathematical problem.  We need them when No analytical solution exist or it is difficult to obtain.  Solution of nonlinear Equations  Solution of linear Equations  Curve fitting Least Squares Interpolation  Numerical Integration  Numerical Differentiation  Solution of ordinary differential equations Topics Covered in the Course

30. EE 3561_Unit_1(c)Al-Dhaifallah 143530  Number Representation  Normalized Floating Point Representation  Significant Digits  Accuracy and Precision  Rounding and Chopping  Reading assignment: Chapter 2 Lecture 2 Number Representation and accuracy

31. EE 3561_Unit_1(c)Al-Dhaifallah 143531 Representing Real Numbers  You are familiar with the decimal system  Decimal System Base =10, Digits(0,1,…9)  Standard Representations

32. EE 3561_Unit_1(c)Al-Dhaifallah 143532 Normalized Floating Point Representation  Normalized Floating Point Representation  No integral part,  Advantage Efficient in representing very small or very large numbers

33. EE 3561_Unit_1(c)Al-Dhaifallah 143533 Binary System  Binary System Base=2, Digits{0,1}

34. EE 3561_Unit_1(c)Al-Dhaifallah 143534 7-Bit Representation (sign: 1 bit, Mantissa 3bits,exponent 3 bits)

35. EE 3561_Unit_1(c)Al-Dhaifallah 143535 Fact  Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system  You can never represent 0.1 exactly in any computer

36. EE 3561_Unit_1(c)Al-Dhaifallah 143536 Representation  Hypothetical Machine (real computers use ≥ 23 bit mantissa)  Example: If a machine has 5 bits representation distributed as follows Mantissa 2 bits exponent 2 bit sign 1 bit Possible machine numbers (0.25=00001) (0.375= 01111) (1.5=00111)

37. EE 3561_Unit_1(c)Al-Dhaifallah 143537 Representation Gap near zero

38. EE 3561_Unit_1(c)Al-Dhaifallah 143538 Remarks  Numbers that can be exactly represented are called machine numbers  Difference between machine numbers is not uniform. So, sum of machine numbers is not necessarily a machine number 0.25 +.375 =0.625 (not a machine number)

39. EE 3561_Unit_1(c)Al-Dhaifallah 143539 Significant Digits  Significant digits are those digits that can be used with confidence. 0 1 2 3 4 Length of green rectangle = 3.45 significant

40. EE 3561_Unit_1(c)Al-Dhaifallah 143540 Loss of Significance  Mathematical operations may lead to reducing the number of significant digits 0.123466 E+02 6 significant digits ─ 0.123445 E+02 6 significant digits ────────────── 0.000021E+02 2 significant digits 0. 210000E-02 Subtracting nearly equal numbers causes loss of significance

41. EE 3561_Unit_1(c)Al-Dhaifallah 143541 Accuracy and Precision  Accuracy is related to closeness to the true value  Precision is related to the closeness to other estimated values

42. EE 3561_Unit_1(c)Al-Dhaifallah 143542 Better Precision Better accuracy Accuracy is related to closeness to the true value Precision is related to the closeness to other estimated values Accuracy and Precision

43. EE 3561_Unit_1(c)Al-Dhaifallah 143543 Rounding and Chopping  Rounding: Replace the number by the nearest machine number  Chopping: Throw all extra digits 0 1 2 True 1.1681 Rounding (1.2) Chopping (1.1)

44. EE 3561_Unit_1(c)Al-Dhaifallah 143544 Error Definitions True Error can be computed if the true value is known

45. EE 3561_Unit_1(c)Al-Dhaifallah 143545 Error Definitions Estimated error Used when the true value is not known

46. EE 3561_Unit_1(c)Al-Dhaifallah 143546 Notation We say the estimate is correct to n decimal digits if We say the estimate is correct to n decimal digits rounded if

47. EE 3561_Unit_1(c)Al-Dhaifallah 143547 Summary  Number Representation Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system.  Normalized Floating Point Representation Efficient in representing very small or very large numbers Difference between machine numbers is not uniform Representation error depends on the number of bits used in the mantissa.

48. EE 3561_Unit_1(c)Al-Dhaifallah 143548 Summary  Rounding Chopping  Error Definitions: Absolute true error True Percent relative error Estimated absolute error Estimated percent relative error