Numerical Methods KuangHui Chi Office EL 303 Ext

Numerical Methods Kuang-Hui Chi Office EL 303; Ext. 4245 chikh@yuntech. edu. tw http: //winlab. ee. yuntech. edu. tw/NM. htm http: //140. 125. 21. 43/NM. htm Please bring your laptop PC with MATLAB installed in later classes

Synopsis } Numerical methods (numerical analysis) } Methodology of using computer programs with math concepts to solve problems in data science } Approximate solutions (exact solutions may hardly be obtained on many occasions) } Various fields adopt numerical methods, e. g. } } Regression analysis of data trending and predictions Assessment of physical characteristics of circuits Torque required for electric motor control. . . } Implement with } MATLAB is a useful programing language with many builtin libraries to make coding easier } Other programming languages can surely do 2

Textbook, Grading Policy } Textbook } Steven C. Chapra, Applied Numerical Methods with MATLAB (3 rd Ed. ), Mc. Graw Hill, 2012 } Grading policy } Programming exercises (3 to 5 on-site practice or take-home assignments) (35%) } Self-oriented hands-on project * • To earn extra marks if you do not score well } Mid-term exam (25%) } Final project (30%) } Attendance and involvement (10%) 3

Five Categories of Numerical Methods (1/2) 4

Five Categories of Numerical Methods (2/2) 5

Mathematical Models } Dependent variable } A characteristic usually reflecting the behavior/state of the system } Independent variable } Dimensions like time and space, along which the system’s behavior is determined } Parameters } Constants reflective of the system’s properties or composition } Forcing function } External influences acting upon the system 6

Analytical vs. Numerical Modeling } Once a mathematical model is constructed, one can obtain solutions with: } Analytical methods } Numerical methods } Analytical methods } Produce exact solutions } Not always feasible } May require mathematical sophistication } Numerical methods } Produce an approximate solution } Time to solve a numerical problem is a function of desired accuracy of the approximation 7

Example: Analytical Model } Consider a bungee jumper in midair } Model for its velocity is given by the differential equation } Change in velocity is affected by the gravitational force which pulls it down and are opposed by the drag force } Dependent variable: velocity v } Independent variable: time t } Parameters: mass m, drag coefficient cd } Forcing function: gravitational acceleration g 8

Example: Analytical Solution } The model can be used to generate a graph } Example: 68. 1 kg jumper, drag coefficient 0. 25 kg/m 9

Example: Numerical Solution } Observe that the time rate of change of velocity can be approximated as 10

Example: Numerical Result } The efficiency and accuracy of numerical methods depend upon how the method is applied. } Applying the previous method in 2 s intervals yields 11

Application: COVID-19 Trending } Q: Given statistics of COVID-19 confirmed cases of past several weeks, predict when and how infection drops below 100 patients http: //bit. ly/3 am. Io. He } Also check out http: //bit. ly/3 cok. MUq 12

Application: Making Decisions during Handover } Handover: A process of a mobile handset switching its connection from one base station to another } Which one of the neighboring BSs serves best?

Application: Data Forecasting 14

Example: Hands-on Self-Oriented Project 15

MATLAB } A resourceful website } https: //www. tutorialspoint. com/matlab/index. htm } MATLAB Examples – Math. Works } http: //www. mathworks. com/examples/ } http: //www. mathworks. com/help/matlab/examples/basicmatrix-operations. html 16