Gregory Newton Interpolation Formula
Last Updated :
23 Jul, 2025
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.
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 Interpolation FormulaGregory-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 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 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
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.
Read More,
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)
Solution:
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
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.
Similar Reads
Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems,
3 min read
Linear Algebra
MatricesMatrices are key concepts in mathematics, widely used in solving equations and problems in fields like physics and computer science. A matrix is simply a grid of numbers, and a determinant is a value calculated from a square matrix.Example: \begin{bmatrix} 6 & 9 \\ 5 & -4 \\ \end{bmatrix}_{2
3 min read
Row Echelon FormRow Echelon Form (REF) of a matrix simplifies solving systems of linear equations, understanding linear transformations, and working with matrix equations. A matrix is in Row Echelon form if it has the following properties:Zero Rows at the Bottom: If there are any rows that are completely filled wit
4 min read
Eigenvalues and EigenvectorsEigenvalues and eigenvectors are fundamental concepts in linear algebra, used in various applications such as matrix diagonalization, stability analysis and data analysis (e.g., PCA). They are associated with a square matrix and provide insights into its properties.Eigen value and Eigen vectorTable
10 min read
System of Linear EquationsA system of linear equations is a set of two or more linear equations involving the same variables. Each equation represents a straight line or a plane and the solution to the system is the set of values for the variables that satisfy all equations simultaneously.Here is simple example of system of
5 min read
Matrix DiagonalizationMatrix diagonalization is the process of reducing a square matrix into its diagonal form using a similarity transformation. This process is useful because diagonal matrices are easier to work with, especially when raising them to integer powers.Not all matrices are diagonalizable. A matrix is diagon
8 min read
LU DecompositionLU decomposition or factorization of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It is a fundamental technique in linear algebr
6 min read
Finding Inverse of a Square Matrix using Cayley Hamilton Theorem in MATLABMatrix is the set of numbers arranged in rows & columns in order to form a Rectangular array. Here, those numbers are called the entries or elements of that matrix. A Rectangular array of (m*n) numbers in the form of 'm' horizontal lines (rows) & 'n' vertical lines (called columns), is calle
4 min read
Sequence & Series
Calculus
Limits, Continuity and DifferentiabilityLimits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding function behavior and are crucial for solving real-world problems in physics, engineering, and economics.Table of ContentLimitsKey Characteristics of LimitsExample of Limi
10 min read
Cauchy's Mean Value TheoremCauchy's Mean Value theorem provides a relation between the change of two functions over a fixed interval with their derivative. It is a special case of Lagrange Mean Value Theorem. Cauchy's Mean Value theorem is also called the Extended Mean Value Theorem or the Second Mean Value Theorem.According
7 min read
Taylor SeriesA Taylor series represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.Taylor series is a powerful mathematical tool used to approximate complex functions with an infinite sum of terms derived from the function's derivatives at a single poi
8 min read
Inverse functions and composition of functionsInverse Functions - In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. Additionally, this must hold true for every element in the domain co-domain(range) of b. In other words, assuming x and y are constants, if b(x) =
3 min read
Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
8 min read
Application of Derivative - Maxima and MinimaDerivatives have many applications, like finding rate of change, approximation, maxima/minima and tangent. In this section, we focus on their use in finding maxima and minima.Note: If f(x) is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum v
6 min read
Probability & Statistics
Mean, Variance and Standard DeviationMean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various en
10 min read
Conditional ProbabilityConditional probability defines the probability of an event occurring based on a given condition or prior knowledge of another event. It is the likelihood of an event occurring, given that another event has already occurred. In probability, this is denoted as A given B, expressed as P(A | B), indica
12 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Covariance and CorrelationCovariance and correlation are the two key concepts in Statistics that help us analyze the relationship between two variables. Covariance measures how two variables change together, indicating whether they move in the same or opposite directions. Relationship between Independent and dependent variab
6 min read
Practice Questions