# Matrix formulation for interpolation of data

In the attached article, the author proposes a way to interpolate quarterly values when only annual values are available. The only page of this article which is useful for this problem is page 66 (or the second page in this attachment).

I am trying to understand the very first step, which is how to get from the statement of the problem in equations (2.1) and (2.2), to its matrix formulation in equation (2.4).

If one of you understands how this matrix formulation is derived, I would be grateful if you could explain it step-by-step, in the case where we want to interpolate from quarterly to monthly data.

I would be very grateful to anyone who is able to help.

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#### Solution Preview

Please see the attached file for the complete answer to your question.

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Update: What I presented in my attachment is the case where interpolation was performed from yearly data to quarterly data. ...

#### Solution Summary

A detailed explanation of how to get from the statement of the problem (of interpolation of quarterly values from annual values) given in equations (2.1) and (2.2) of the student's attached article to the matrix formulation in equation (2.4) is provided (in the OTA's attached file). Then an explanation of how the counterpart of that matrix formulation for the case of interpolating monthly data from quarterly data is given (in the "Problem Solution" box), by indicating the specific differences between the two cases.

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.

?

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.