Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Five Curve Fitting

Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 2 Motivation In all practical engineering cases, the sampling data are acquired at discrete points. x x y x = c sampling points interpolation points Curve fitting tries to fit a continuous curve through the sampling data that can then define the function value at any point by interpolation. That means the function values at points other than these sampling points are undefined; but they are wanted in many applications. x y In many cases, it is not required to find a curve that fit exactly every sampling point; instead a curve (e.g. the blue line) that shows the trend of the function is wanted. This is called regression. Example: how to prove g = 9.8 m/s 2 ?

Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 3 Noncomputer Methods for Curve Fitting Visually sketch a line that conforms to the data (inaccurate) Connect the data points consecutively by lines segments (significant errors if the data are not evenly spaced or the underlying relationship is highly curvilinear) Connect the data points consecutively by simple curves (too tedious and difficult to do manually)

Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 4 Simple Statistics

Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 5 In most engineering applications, the sampling data set conforms to the normal distribution if the size of the data set is sufficiently large. For the normal distribution, the range defined by and will encompass approximately 68 percent of the total measurement. Similarly, the range between and will encompass approximately 95%. The Normal Distribution histogram normal distribution

Mech4740 Numerical Methods, Hong Kong University of Science and Technology. 6 Overall Structure