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Gregory Newton Interpolation Formula: Proof, Examples, FAQs

Newton-Gregory Forward Interpolation Formula is an interpolation method when our data points are evenly spaced. Interpolation is a method in maths used to make educated guesses about values between two points we already know. We can say that the Gregory–Newton forward difference formula involves finite differences that give an approximate value for f(x), where x = x0 + θ.h, and 0 < θ <1. Approximation of f(x) ≈ f0 + θ.Δf0 gives the result of Linear Interpolation.

Here in this article learn about, the Newton-Gregory Interpolation Formula, its Examples, and others in detail.

Table of Content

What is Gregory Newton’s Formula?

Gregory-Newton’s Formula refers to a family of interpolation formulas that approximate a function based on a set of known data points. The formulas are named after mathematicians James Gregory and Isaac Newton, who independently developed similar interpolation techniques. The Gregory Newton’s Formula image is add

Gregory-Newton-Forward-Formula

Gregory Newton Interpolation Formula

Gregory-Newton interpolation formulas are particularly useful when you have a set of discrete data points and you want to estimate the value of the function between these data points. These formulas provide a polynomial function that passes through all given data points, allowing for interpolation within the range of the data.

Gregory Newton Forward Formula

Gregory-Newton Forward Interpolation Formula is a method for interpolating values of a function between given data points. The formula is useful when you have equidistant data points. Given n+1 data points (x0, y0), (x1, y1), …, (xn, yn) with a common difference h between the x-values, the forward interpolation formula for finding the value at a point x in the range (x0, xn) is given by:

f(x) = y0 + (x – x0)Δy0 /h0 + (x – x0)(x – x1)Δ2y0 /2!h2 + (x – x0)(x – x1)(x – x2)Δ3y0 /3!h3 + …

where,

  • h is Common Difference Between x-values
  • f(x) is Interpolated Function value at Point x
  • x0, x1, x2,… are given Data Points (in Ascending Order)
  • y0, y1, y2,… are Corresponding Function Values at Given Data Points
  • Δy0 = y1 – y0
  • Δ2 y0 = y2 – y1

Gregory Newton Backward Formula

Gregory-Newton Backward Interpolation Formula is similar to the forward formula but is used when data points are given in reverse order. The formula is given by:

f(x) = yn + (x – xn)Δyn /h + (x – xn)(x – xn-1)Δ2yn /2!h2 + (x – xn)(x – xn-1)(x – xn-2)Δ3y0 /3!h3 +…

where,

Differences Δyn, Δ2yn, … are Calculated Backward and

  • h is Common Difference Between x-values
  • f(x) is Interpolated Function Value at Point x
  • xn, xn-1, xn-2,… is Given Data Points (in Descending Order)
  • Δyn, Δ2 yn… is Backward Differences
  • Δyn = yn-1  –  yn
  • Δ2yn = yn-2 – yn-1

Applications Of  Gregory Newton Formula

Gregory-Newton Interpolation Formula, finds applications in various fields where there’s a need to approximate values of a function between known data points. Some of its common applications are,

Numerical Analysis: Gregory-Newton formulas are extensively used in numerical analysis for interpolation purposes. They allow mathematicians and engineers to estimate intermediate values of a function based on a set of discrete data points.

Curve Fitting: In experimental sciences and engineering, researchers often collect experimental data points. Gregory-Newton interpolation can be used to fit a curve through these data points, enabling the prediction of values between the measured points.

Data Analysis: In data analysis tasks, especially when dealing with time-series or irregularly sampled data, interpolation is often necessary to fill in missing values or to obtain values at regular intervals. Gregory-Newton formulas provide a way to interpolate these missing values.

Numerical Differentiation and Integration: Gregory-Newton formulas can be used to estimate derivatives and integrals of a function based on discrete data points. By approximating the function with an interpolation polynomial, derivatives and integrals can be computed numerically.

Signal Processing: In digital signal processing, Gregory-Newton interpolation can be used to upsample signals or to perform resampling operations, where new samples are generated between existing ones to change the sampling rate.

Computer Graphics: In computer graphics and image processing, interpolation techniques are commonly used for tasks such as image resizing, texture mapping, and anti-aliasing. Gregory-Newton interpolation can be applied to interpolate pixel values between known points.

Financial Modeling: In finance, Gregory-Newton interpolation can be useful for interpolating yield curves or for estimating the prices of financial instruments between observed data points.

Geographic Information Systems (GIS): In GIS applications, Gregory-Newton interpolation can be used for tasks such as contour mapping, where elevation data is interpolated between known elevation points to generate a continuous surface.

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Examples on Gregory Newton Difference Formula

Few examples on Gregory Newton Interpolation Formula are,

Example 1: For a set of numerical values: (1, 2), (2, 8), (3, 18), and (4, 32). Find y at x = 2.5 using Gregory Newton Forward Formula.

Solution:

h = (x2 – x1) = 2 – 1 = 1

Δy0 = 8 – 2 = 6

Δ2y0 = 18 – 8 = 10

Apllying Gregory Newton’s Forward Difference Formula,

P(2.5) = 2 + (2.5 – 1)6/1 + (2.5 – 1)(2.5 – 2)10/2

P(2.5) = 2 + (1.5)×6 + (1.5)(0.5×5)

P(2.5) = 2 + 9 + 3.75 = 14.75

Estimated values for y at x = 2.5 is 14.75

Example 2: From the given set of data find f(1.5)

xi

1

2

3

fi

4

9

14

Solution:

xi

fi

Δfi

Δ2fi

1

4

2

9

5

3

14

5

0

Say, a = 1

h = x2 – x1 = 1

Apllying Gregory Newton’s Forward Difference Formula,

f(1.5) = f(1) + (1.5 – 1)/1×5 + (1.5 – 1)(1.5 – 2).0/1

f(1.5) = 4 + 2.5 + 0 = 6.5

Example 3: Given Data: (1, 2), (2, 3), (3, 5). Find the Newton interpolation polynomial.

Construct Divided Difference Table:

x f(x) f[x0, x1] f[x0, x1, x2]
1 2
2 3 1
3 5 1 0

Newton Polynomial:

P(x) = 2 + 1(x-1) + 0(x-1)(x-2)

= 2 + (x-1)

P(x) = x+1

Example 4: Given Data: (1, 1), (2, 4), (3, 9). Find the Newton interpolation polynomial.

Solution:

Construct Divided Difference Table:

x f(x) f[x0, x1] f[x0, x1, x2]
1 1
2 4 3
3 9 5 1

Newton Polynomial:

P(x) = 1 + 3(x-1) + 1(x-1)(x-2)

P(x) = x2

Example 5: Given Data: (0,1), (1,2), (2,4), (3,8). Find the Newton interpolation polynomial.

Solution:

Construct Divided Difference Table:

x f(x) f[x0, x1] f[x0, x1, x2] f[x0, x1, x2, x3]
0 1
1 2 1
2 4 2 1
3 8 4 2 1

Newton Polynomial:

P(x) = 1 + 1(x-0) + 1(x-0)(x-1) + 1(x-0)(x-1)(x-2)

P(x) = x3

Practice Questions on Gregory Newton’s Formula

Problem 1: Given Data: (1,3), (2,7), (4,15). Find the Newton interpolation polynomial.

Problem 2: Given data points (1, 3), (2, 8), (3, 15) with a common difference h = 1, find the value at x = 4 using the Gregory-Newton Forward Formula.

Problem 3: Use Gregory-Newton Forward Formula to estimate the value at x = 6 for the data points (x0, y0) = (1, 2), (x1, y1) = (3, 8), (x2, y2) = (5, 18) with h = 2.

Problem 4: If common difference h is 3, and data points are (0, 1), (3, 10), (6, 25), use Gregory-Newton Forward Formula to find value at x = 9.

Problem 5: Given Data: (2,5), (4,9), (6,17). Find the Newton interpolation polynomial.

Problem 6: Given Data: (1,1), (2,8), (4,64). Find the Newton interpolation polynomial.

Problem 7: Given Data: (0,1), (1,4), (2,9), (3,16). Find the Newton interpolation polynomial.

Problem 8: Given Data: (1,2), (2,6), (3,12), (4,20). Find the Newton interpolation polynomial.

Problem 9: Given Data: (0,0), (1,1), (2,4), (3,9). Find the Newton interpolation polynomial.

Problem 10: Given Data: (0,1), (1,2), (2,4), (3,8). Find the Newton interpolation polynomial.

Gregory Newton Formula FAQs

What is Gregory Newton’s Forward Difference Formula?

Gregory Newton’s forward difference formula is a way used to find value of any function at points where its value is not given but its value at other points is given.

When to use Gregory-Newton Forward Formula, and when Backward Formula?

Gregory-Newton Forward Formula is used when your data points are arranged in ascending order (left to right). Gregory-Newton Backward Formula is used when your data points go in descending order (right to left).

What is Common Difference (h) in Gregory-Newton Formula?

Common difference (h) is the difference between consecutive x-values. It’s essential for calculating differences like Δy, its square and higher powers.

Can Gregory-Newton Formula is Used for Non-Equidistant Data Points?

No, Gregory-Newton Formula is designed for evenly spaced data points. If your data points aren’t equally spaced, it is not used.

Can Gregory-Newton interpolation be used for extrapolation?

Yes, it can be used for extrapolation, but the accuracy of the extrapolation depends on the degree of the polynomial and the distribution of the data points.

What is the role of divided differences in this method?

Divided differences provide a systematic way to compute the coefficients of the Newton polynomial, capturing the function’s behavior based on the given data points.

How can you verify the accuracy of the Newton interpolation polynomial?

Compare the polynomial’s values with the known values of the function at the data points and, if possible, at additional points. You can also use error analysis techniques to estimate the interpolation error.


 


Reffered: https://www.geeksforgeeks.org

Engineering Mathematics

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