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.

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.

1. a) Consider the problem of cubic polynomial interpolation
p(xi) = yi, I = 0,1,23
with deg(p) ≤ 3 and x0, x1, x2, x3 distinct. Convert the problem of finding p(x) to another problem involving the solution of a system of linear equations.
b) Express the system from (a) in the form Ax = b, i

The row and column indices in the nxn Fourier matrix A run from 0 to n-1, and the i,j entry is E^ij, where E^ij = e^(2*PI*i/n). This matrix solves the following interpolation problem: Given complex numbers b_0, ... b_(n-1), find a complex polynomial f(t) = c_0 + c_1 + ... + c_(n-1) t^(n-1) such that f(E^v) = b_v.
(i) Explain

2. Use Eq. (3.10) to construct interpolating polynomials of degrees one, two, and three for the following data. Approximate the specified value using each of the polynomials.
a. f(0.43) if f(0)=1,f(.25)=1.64872 f(.5)=2.71828,f(.75)=4.48169
b. f(0) if f(-.5)=1.93750,f(-.25)=1.33203,f(.25)=.800781,f(.5)=.687500

This solution provides the learner with an understanding of TOWS matrix. In particular, this solution describes the value of the TOWS Matrix in strategy formulation. Further included in this solution is a discussion on the use of TOWS matrix and the potential for generating strategic alternatives.

Please explain the steps on how to solve the problem below.
The accompanying table lists the area of a circle corresponding to several values of the diameter. Find the area corresponding to a diameter of 15 in. by linear interpolation.
Diameter x | 0 | 4 | 8 | 12 | 16 | 20

1.Expand the following function into Maclaurin Series (see attached file) using properties of the power series.
2. The Lagrange interpolation polynomial may be compactly written as is a shape function. Sketch the shape function in a graphic form.
3. Write a forward and backward difference Newton's interpolationformulas b

I am having problems trying to figure this out. My company is considering whether to invest in a project with an initial cost $538,000 that will provide annual net cash inflows of $82,500 for 10 years. I need to use the interpolation to estimate the internal rate of return.

I am submitting a complex set of data points that I would like someone to use, employing LaGrange Interpolation (or any other process that might be more appropriate), to give me another detailed example to use as I try to learn the process and employ it to other problems.
The points are:
(0,214), (0.11,2022), (0.65,1131), (1