# Polynomial Rings and Composition Series

Let R = Q[X,Y] be the polynomial ring in two variables. Show that the R-module R/(X^2 -Y) has no composition series.

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Proof:

Suppose the -module has a composition series as follows.

Then we have

The composition factors are , .

Since each composition factor is simple, then is the proper ...

#### Solution Summary

Polynomial rings and composition series are investigated.

Write a program or programs to interpolate the data given below at the specified points using Neville iteration, Newton's interpolatory divided-difference formula, and a natural cubic spline.

Math 609D Programming Assignment #3 Due Date: March 17, 2008

Write a program or programs to interpolate the data given below at the specified points using

Neville iteration, Newton's interpolatory divided-difference formula, and a natural cubic spline.

Use your programs to do the following.

1. For x = 0.25, x = 0.5, and x = 0.95, construct the full Neville interpolation table and use this

table to calculate the interpolated values using 2, 4, 6, and 11 nodes. These nodes should be

the nodes closest to the given x. You will be calculating the values of the first, third, fifth and

tenth degree interpolating polynomials at each given value of x.

2. Construct a divided difference table for this data set and use the table to compute Newton's

interpolatory divided-difference formula. Use this form of the interpolating polynomial to

calculate the interpolated values for each x value.

3. Construct a natural cubic spline for this data set and use it to calculate interpolated values

for each x value. Print the coefficients of each cubic polynomial in the cubic spline.

4. Compare your interpolated values with the values of the function f(x) =

ex

2

1 + 25x2 .

5. Graph the original function, the full interpolating polynomial, and the natural cubic spline on

a single plot.

Output:

a. Turn in a well-documented program or programs.

b. Print the full divided-difference table.

c. For each x value, print the full Neville interpolation table.

d. Print the interpolated values, clearly labeled, and the exact values.

e. Analyze your output and include a summary of your findings.

Data:

-1.0 0.104549

-0.8 0.111558

-0.6 0.143333

-0.4 0.234702

-0.2 0.520405

0.0 1.000000

0.2 0.520405

0.4 0.234702

0.6 0.143333

0.8 0.111558

1.0 0.104549

Program Procedures

Each assignment must be submitted in PDF format and will consist of three sections.

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The first section will include the computer code you have written. This code should be well documented so that it can be easily understood by someone other than the author.

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The second section will include the output generated by your programs.

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The third section will contain a summary of your calculations.

-- Code may not contain any high-level functions. For example you can't use Maple's fsolve command when trying to solve an equation. Your code must be based on control loops and if-else statements. If you use Maple and a procedure uses a derivative of a function, the derivative must be passed to the procedure as an argument. You may not compute derivatives inside of a procedure.