Desired interpolation in MATLAB

[firestarter7]

Homework Statement

My questions concern interpolation and a method to assess if the interpolations I have made could be refined in regards of ill condition numbers etc.

How can I make this interpolation better? I have read about interpolation techniques in Matlab but I don’t know which one to use…how can i assess the ill condition of such interpolation? Is the Lagrange polynomial interpolation the best one to use for this purpose?
Can anybody help or show me a way to assess the best method for this to minimize the interpolation errors?

Homework Equations

The Attempt at a Solution

I had the following dataset...

I have the following x and y values:
X1: [243.6, 199.8, 166.8, 133.2, 16.4]
Y1: [1, 1.078,1.1184,1.1648, 1.238]

X2: [286.6, 253.2, 209.4, 183.6, 56.6]
Y2: [1, 1.06,1.1452,1.182,1.3]

X3: [333.6,310.2,259.8,240,113.6]
Y3: [1.0512,1.166,1.212, 1.348,1.3]
.
.
X8: [...]
X8: [...]

Using Excel I got the following (trend lines):

y = -0.000003623485 x2 - 0.000104061124 x + 1.240802644498
y = -0.000003689281 x2 - 0.000050097492 x + 1.314941155615
y = -0.000005138283 x2 + 0.000694149114 x + 1.335972900207
.
.
.

Using a constant called a and jotting down

a = -3.62E-06, -3.69E-06, -5.14E-06...a8
b = -0.000104061, -5.00975E-05, 0.000694149...b8
c = 1.240802644, 1.314941156, 1.3359729...c8

and plotting all the a values I obtained the following higher order polynomial…

y = -0.021888102619 x6 + 0.081260616942 x5 - 0.121696999709 x4 + 0.093940463998 x3 - 0.039420446179 x2 + 0.008521813480 x - 0.000744464046

and for b
y = 18.562953566900 x6 - 68.758956799423 x5 + 102.790440490807 x4 - 79.186370651594 x3 + 33.144419473323 x2 - 7.144313735101 x + 0.619166739240

and for c

y = -4,074.658731371160 x6 + 15,081.496519386700 x5 - 22,571.220590122000 x4 + 17,426.445712900200 x3 - 7,311.335903832610 x2 + 1,580.188986737920 x - 136.160481054492

Can anyone help me with this?

 

[lippyka]

The best way to assess the ill condition of an interpolation is to use a numerical analysis technique such as the Richardson Extrapolation method. This method uses a sequence of increasingly accurate approximations to estimate the true value of a function, and then compares the differences between them to determine the quality of the interpolation. Additionally, using a numerical optimization technique such as a least-squares fit can be used to fit a polynomial to the data points, which will help reduce the ill condition of the interpolation. Finally, the Lagrange polynomial interpolation is a common and reliable method for interpolating data points. This method is generally considered to be more accurate than other methods, and is the preferred method for many applications.