Interpolation: Polynomials with Newton's Divided Difference and Gregory-Newton Forward, Lecture notes of Commercial Law

Download Interpolation: Polynomials with Newton's Divided Difference and Gregory-Newton Forward and more Lecture notes Commercial Law in PDF only on Docsity!

Nummeth

Interpolation Methods

Interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.

Newton’s Divided Difference Formula

1. The pressure drop that occurs when water flows through an orifice meter is measured using a differential pressure transmitter. The output current is converted to voltage drop by a resistor. The objective is to correlate the flow rate with the voltage drop. The following data were collected from an experiment.

Newton’s Divided Difference Formula F(g/min) (^3 5 6 ) P (mV) (^293 508 585 ) Derive a third order polynomial passing through these data points.

Newton’s Divided Difference Formula To get the nth order interpolating polynomial: p n (x) = f 0

• (x-x 0 )f 01
• (x-x 0 )(x-x 1 )f 012
• (x- x 0 )(x-x 1 )(x-x 2 )f 0123
• …(x-x 0 )(x-x 1

(x-x n- 1 )f 01234…n

Newton’s Divided Difference Formula x f(x) f[x, x]^1 f[x, x, x]^2 f[x,^ x, x, x]^3 3 293

5 508 - 10. 77.00 0. 6 585 - 4.

9 764 Here, x ≡ F and f(x) ≡ P.

Gregory-Newton Forward Interpolation This method uses formula Where f 0 is the first value of the data set, ∆f 0 , ∆ 2 f 0 , ∆ 3 f 0 are the first, second and third forward differences respectively. The variable r is the difference between an unknown point x and a known point x 1 divided by the interval h, that is, 0 0

Gregory-Newton Forward Interpolation Example: Use the Gregory-Newton forward interpolation to compute f(1.03) from the following data. Note: The formula of is valid only when the values are equally spaced with interval.

Gregory-Newton Forward Interpolation Solution: Answer: 1. 0