Page 321 - Numerical Analysis Using MATLAB and Excel

Summary

•The Gregory−Newton Backward Interpolation method uses the formula

)
(
(
(
rr + 1 ) 2 rr + 1 r + 2 ) 3
fx() = f + rΔf + ------------------Δ f + -----------------------------------Δ f + …
0 – 1 2! – 2 3! – 3
2 3
where is the first value of the data set, Δf – 1 , Δ f – 2 , and Δ f – 3 are the first, second and third
f
0
backward differences, and
( x – x )
1
r = -------------------
h
This formula is valid only when the values x x x … x, 0 1 , 2 , , n are equally spaced with interval . h
It is used to interpolate values near the end of the data set, that is, the larger values of . Back-
x
ward interpolation is an expression to indicate that we use the differences in a backward
sequence, that is, the last entries on the columns where the differences appear.
• If the increments in values are small, we can use the Excel VLOOKUP function to perform
x
interpolation.

• We can perform interpolation to verify our results with the MATLAB functions
interp1(x,y,x ), interp1(x,y,x ,’method’) where method allows us to specify nearest (nearest
i
i
neighbor interpolation), linear (linear interpolation, the default interpolation), spline (cubic
spline interpolation which does also extrapolation), cubic (cubic interpolation which requires
equidistant values of ), and interp2(x,y,z,x ,y ) which is similar to interp1(x,y,x ) but per-
x
i
i
i
forms two dimensional interpolation;

Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−43
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