# Cubic interpolation

• July 31st 2011, 09:14 AM
Bieha
Cubic interpolation
Helle there,

I am currently studying fixed-income market securities as part of a master's degree in Finance, but I seem to have a problem that involves understanding something that seems basic, but since I lack knowledge of any form of Matrix' solving, I suppose the question will do just fine in the Pre-University Math Help forums.

I need to find the interpolated rate R(0,w) with w [v;z] verifieying the three-order polynomial equation: R(0,w) = aw^3 + bw^2 + cw + d

where a, b, c and d sitisfy the system:

R(0,v) = av^3 + bv^2 + cv + d
R(0,x) = ax^3 + bx^2 + cx + d
R(0,y) = ay^3 + by^2 + cy + d
R(0,z) = az^3 + bz^2 + cz + d

I need to solve this cubic polynomial function by solving a matrix:

a
b
c
d

equals

1 1 1 1
8 4 2 1
27 9 3 1
64 16 4 1

by

3%
5%
5,5%
6%

Then what I don't understand, is that it (my book) shows me solve for the vector of constants a, b, c and d by:

x = B^(-1)*A, where x are the constant, B the matrix of maturity terms and A the vector of interest rate (3%, 5%, etc.).

0.0025
-0.0225
0.07
-0.02

I might not even need to know this, but I am currently deeply frustrated by not being able to understand this:)

Who can help me out here?

Regards,

Hans
• August 1st 2011, 03:40 AM
HallsofIvy
Re: Cubic interpolation
Since you have the formula $x= B^{-1}A$ which uses the "inverse" of B, do you know how to find the inverse of a matrix? Or do you know how to do "row reduction" of a matrix?

You might find these interesting:
[&#x202a;Determining Inverse Matrices Using Augmented Matrices&#x202c;&rlm; - YouTube32.3 The Inverse of a Matrix