NUMERICAL METHODS PROBLEM SET 9

PS9-1 Statistics

Given the data

determine (a) the mean, (b) the standard deviation, (c) the variance, (d) the coefficient of
variation .

PS9-2 Least- Square Regression

Suppose you want to fit a quadratic curve ( parabola ) to the following data points using least
square regression .

(1) What is the function S_r to be minimized to find the least square regression curve?

(2) What are the three conditions to find the coefficients of the quadratic curve of the least
square regression? Write the matrix equation of the three conditions.

PS9-3 Two Dimensional Quadratic Regression

Read and understand section 17.3 of the textbook. The mathematical background of this
programming assignment is very similar to the one in section 17.3.

The goal of two dimensional quadratic regression is to find the “best” values of the coefficients
of a quadratic function y of x₁ and x₂

The coefficients , and can be determined such that they define a function y
that minimizes the sum of the squares of the errors, or residuals,

Differentiation of S _r with respect to each of the unknown coefficients, , and
, gives six linear algebraic equations , or a 6 x 6 matrix equation, [M]{a}={b}, similar to
equation (17.22) on p.461 of the textbook.

(1) Derive the matrix equation [M]{a}={b} for two-dimensional quadrilateral regression. Show
all the derivation process.

(2) Given the following data points write a program for finding coefficients
, and of the quadratic regression function.

To show that your program is properly implemented, your program has to generate: (1) a text
file, output.txt, that contains the 6 x 6 matrix [M], and values of coefficients
, and ; and (2) a VRML file, output.wrl, that illustrates the data
points and the quadratic regression surface.

Your VRML file should look something like Figure 1. To minimize your work to generate
the VRML output the following subroutines and a sample code are provided.

In your hand-in directory on AFS, make a new directory called ps9 (in lower case ) and hand
in the following in the directory. Don’t copy object files (*.obj, *.o).
• Source code files and header file
• Executable file
• Text file name output.txt
• Output VRML file named output.wrl.

Also hand in a printout of the following:
• Source code files and header files
• Text file, ourput.txt
• Two images of the VRML files that look like Figure 1.