Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

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Shifted Interpolation Methods for Efficient Resolution of Algebraic and 

Transcendental Equations 

Farah Deeba 𝟏 Najmuddin Ahmad  𝟐∗, Fauzia Shaheen 𝟑 

 1Department of Mathematics & Statistics, Integral University, Lucknow, farah@iul.ac.in. 

 2Department of Mathematics & Statistics, Integral University, Lucknow, 

najmuddinahmad33@gmail.com. 

 3Department of Mathematics & Statistics, Integral University, Lucknow, 

shaheenfauzia97@gmail.com. 

*Corresponding author e-mail: najmuddinahmad33@gmail.com 2 

 

Article History: 

Received:04/09/2025 

Revised:03/10/2025 

Accepted:27/11/2025 

Abstract:  A popular method in mathematics and statistics for 

determining values between any two points is interpolation. For both 

equally and unequally spaced data sets, numerous techniques have been 

employed to derive practical interpolation equations. This work aims to 

develop a difference interpolation formula derived from Gauss's Formula 

and another formula where we substituted 𝜌 for 𝜌 + 4 and reused the 

subscript in Gauss's Forward Formula by four units. Additionally, based on 

the differences, we compared the generated interpolation formula with the 

current interpolation formulae. We discovered from the error analysis that 

the new formula we obtained is significantly more accurate and efficient at 

resolving functional values between the provided data. 

Keywords: Interpolation, Central Difference, Mathematical Norm, 

Gauss’s Formula  

 

Introduction 

A fundamental method in numerical analysis, interpolation is used to estimate values that fall 

between known data sets. Finding new data points within the range of a discrete set of known data 

points is known as interpolation. [5] It is crucial in a variety of domains, including scientific 

computing, computer graphics, signal processing, and geographic information systems (GIS). The 

type of data, the required level of precision, and processing efficiency all influence the interpolation 

method selection. Based on recent research, this study examines a number of widely used 

interpolation techniques, including their theoretical underpinnings, uses, benefits, and drawbacks. 

[10-13] Its primary function is to forecast unknown values for any geographically connected data 

points, like elevation, rainfall, noise level, and so forth. The process of interpolation involves taking 

a discrete data set and using it to create a basic function that passes through the given data points. 

This aids in identifying the data points that lie between the ones that are provided. [3] When 

mailto:farah@iul.ac.in
mailto:najmuddinahmad33@gmail.com
mailto:shaheenfauzia97@gmail.com
mailto:najmuddinahmad33@gmail.com


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calculating a function's value for an intermediate value of the independent function, this procedure is 

always required. The technique of identifying the unknown values that fall between the known data 

points is known as interpolation. Its primary function is to forecast unknown values for any 

geographically connected data points, like elevation, rainfall, noise level, and so forth. 

Interpolation Methods 

There are different types of interpolation methods. They are: 

Nearest Neighbors Method – This method inserts the value of an interpolated point to the value of 

the most adjacent data point. Therefore, this method does not produce any new data points.[6] 

Thin-plate Spline Method – This method consists of smooth surfaces that also extrapolate well.[7] 

It is only for surfaces only. 

Cubic Spline Interpolation Method – This method fits a different cubic polynomial between each 

pair of data points for curves, or between sets of three points for surfaces.[8] 

Linear Interpolation Method – This method applies a distinct linear polynomial between each pair 

of data points for curves, or within the sets of three points for surfaces.[9] 

Shape-Preservation Method – This method is also known as Piecewise Cubic Hermite 

Interpolation (PCHIP). It preserves the monotonicity and the shape of the data. It is for curves only. 

In this paper, we tried to develop a central difference interpolation formula which is obtained from 

Gauss’s Formula and another formula in which we reused the subscript in Gauss’s Forward Formula 

by four units and replacing 𝜌 by 𝜌 + 4 [1]. Also, we compared the obtained interpolation formula 

with the existing interpolation formulae, (Gauss’s, Stirling’s and Bessel’s etc.) based on differences 

and use the concept of mathematical norm to select which method is best for evaluating functional 

values between data. 

Das and Chakrabarty (2016) derived a formular from Lagrange’s interpolation method and this was 

used to obtain a numerical data for total population of India. The work was extended by deriving 

other methods for the same purpose. [2] 

Proposed And Existing Interpolation Formulae [4] 

Given below are the Gauss’s Central-Difference Formulae  

Gauss’s Forward Central-Difference Formula 

𝑦 = 𝑦0 + 𝜌∆𝑦0 + 𝜌(𝜌 − 1)
∆2𝑦−1

2!
 +  𝜌(𝜌 + 1) (𝜌 − 1)

∆3𝑦−1

3!
 +  𝜌(𝜌 − 2) (𝜌 − 1)(𝜌 + 1)

∆4𝑦−2

4!
 + ⋯           

                   (i) 

 

 



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Gauss’s Backward Central-Difference Formula 

𝑦 = 𝑦0 + 𝜌∆𝑦−1 + 𝜌(𝜌 + 1)
∆2𝑦−1

2!
 +  𝜌(𝜌 − 1) (𝜌 + 1)

∆3𝑦−2

3!
 +  𝜌(𝜌 − 2)(𝜌 − 1)(𝜌 + 1)

∆4𝑦−2

4!
 +

 …                 (ii) 

Sterling Interpolation Formula 

𝑦 = 𝑦0 + 𝜌(
∆𝑦0+∆𝑦−1

2
) +

𝜌2

2!
∆2𝑦−1 +

𝜌(𝜌2−1)

3!
(

∆3𝑦0+∆3𝑦−2

2
) +

𝜌2(𝜌2−1)

4!
∆4𝑦−2 + ⋯                  (iii)                    

Bessel’s Interpolation Formula 

𝑦 = 𝑦1 + (𝜌 − 1)∆𝑦0 + 𝜌(𝜌 − 1)
∆2𝑦−1

2!
+ 𝜌(𝜌 − 1)(𝜌 − 2)

∆3𝑦−2

3!
+ 𝜌(𝜌2 − 1)(𝜌 − 2)

∆4𝑦−2

4!
+

𝜌(𝜌2 − 1)(𝜌 − 2)(𝜌 − 3)
∆5𝑦−2

5!
 + …                              (iv) 

𝑦 =  
𝑦0+𝑦1

2
 +  (𝜌 − 

1

2
)∆𝑦0 +

𝜌(𝜌−1)

2!

(∆2𝑦−1+∆2𝑦0)

2
 +  

𝜌(𝜌−
1

2
)(𝜌−1)

3!
∆3𝑦−1 +

 
𝜌(𝜌2−1)(𝜌−2)

4!

(∆4𝑦−2+∆4𝑦−1)

2
 +      

𝜌(𝜌−
1

2
)(𝜌2−1)(𝜌−1)

5!
∆5𝑦−2 + …                            (v) 

Equation (v) is derived as Bessel’s Interpolation Formula. 

Previously Proposed Formula 

𝑦 =  
𝑦−2 + 𝑦0

2
 + [1 −

𝜌

2
{1 +

∆𝑦−1

∆𝑦−2
}] ∆𝑦−2  + 

(𝜌 + 1)

2!
[
(𝜌 + 2)(∆2𝑦−3) + 𝜌∆2𝑦−1

2
]

+ 
(𝜌 + 1)

3!
[
(𝜌 + 2)(𝜌 + 3)∆2𝑦−3 + 𝜌(𝜌 − 1)∆3𝑦−2

2
] + ⋯ 

New Interpolation Formula (NIF) 

To derive the new formula, we retreat the subscript in Gauss’s Forward Interpolation Formula by 

four units and replacing 𝜌 by 𝜌 + 4. 

From equation (i) 

𝑦 = 𝑦0 + 𝜌∆𝑦0 + 𝜌(𝜌 − 1)
∆2𝑦−1

2!
 +  𝜌(𝜌 + 1) (𝜌 − 1)

∆3𝑦−1

3!
 +  𝜌(𝜌 − 2) (𝜌 − 1)(𝜌 + 1)

∆4𝑦−2

4!
 +

⋯                  (vi) 

So, we obtain, 

𝑦 = 𝑦−4 + (𝜌 + 4)∆𝑦−4 + (𝜌 + 4)(𝜌 + 3)
∆2𝑦−5

2!
+ (𝜌 + 4)(𝜌 + 5)(𝜌 + 3)

∆3𝑦−5

3!
+ 



Communications on Applied Nonlinear Analysis 

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  (𝜌 + 4)(𝜌 + 2) (𝜌 + 3)(𝜌 + 5)  
∆4𝑦−6

4!
 + ⋯                        (vii) 

Also, from equation (ii) 

𝑦 = 𝑦0 + 𝜌∆𝑦−1 + 𝜌(𝜌 + 1)
∆2𝑦−1

2!
 +  𝜌(𝜌 − 1) (𝜌 + 1)

∆3𝑦−2

3!
 +  𝜌(𝜌 − 2)(𝜌 − 1)(𝜌 + 1)

∆4𝑦−2

4!
 +

 …                 (viii) 

So, we obtain, 

𝑦 = 𝑦−4 + (𝜌 + 4)∆𝑦−5 + (𝜌 + 4)(𝜌 + 5)
∆2𝑦−5

2!
+ (𝜌 + 4)(𝜌 + 3)(𝜌 + 5)

∆3𝑦−6

3!
+ (𝜌 + 4)(𝜌 +

2)(𝜌 + 3)(𝜌 + 5)
∆4𝑦−6

4!
 +  …               (ix) 

Taking the mean of equations (viii) and (ix) we get the New Interpolation Formula 

𝑦 = 𝑦−4  + (𝜌 + 4) (
∆𝑦−4+∆𝑦−5

2
) + 

(𝜌+4)2

2!
∆2𝑦−5  + 

(𝜌+4)((𝜌+4)2−1)

3!
(

∆3𝑦−4+∆3𝑦−6

2
) +

 
(𝜌+4)2((𝜌+4)2−1)

4!
∆4𝑦−6  +  …                                   (x) 

Numerical Examples 

To evaluate the assess of our proposed interpolation formula, we examined several examples in 

comparison with established methods. 

Example 1: 

Consider the algebraic function 𝑦 = 3𝑡2 + 2𝑡 + 1 for some equidistantly spaced value of t.  

Here we take 𝑡 = 6, ℎ = 1, 𝑡0 = 2 then 𝜌 =
𝑡−𝑡0

ℎ
 =

6−2

1
= 4. 

Actual value of 𝑦(6)  =  121. 

Table 1: Difference Table for Example 1. 

t y Δy Δ²y Δ³y Δ⁴y 

-2 9 -7    

-1 2 -1 6   

0 1 5 6 0  

1 6 11 6 0 0 

2 17 17 6 0  

3 34 23 6   

4 57 29    

5 86     

 



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Table 2: Comparison of Different Interpolation Techniques along with Error. 

Sr. No. Methods Approximate Value 

of y at 𝒕 = 𝟔 

Error % Error 

1.  Gauss’s Forward 121 0 0 

2.  Gauss’s Backward 121 0 0 

3.  Stirling’s Method 121 0 0 

4.  Bessel’s Method 121 0 0 

5.  Previously proposed 121 0 0 

6.  NIF 121 0 0 

 

Example 2: 

Consider the algebraic function 𝑦 = 4𝑡2 + 3𝑡 + 2 for some equidistantly spaced value of t.  

Here we take 𝑡 = 4.5 𝑡0 = 0, ℎ = 1, and 𝜌 =  
𝑡−𝑡0

ℎ
 =

4.5−0

1
 =  4.5. 

Actual value of  𝑦(4.5)  =  96.5. 

Table 3: Difference Table for Example 2 

  t y Δy Δ²y Δ³y Δ⁴y 

-5 87 -33    

-4 54 -25 8   

-3 29 -17 8 0  

-2 12 -9 8 0 0 

-1 3 -1 8 0 0 

0 2 7 8 0 0 

1 9 15 8 0  

2 24 23 8   

3 47 31    

4 78     

Table 4: Comparison of Different Interpolation Techniques along with Error. 

Sr. No. Methods Approximate Value 

of y at 𝒕 = 4.5  

Error % Error 

1.  Gauss’s Forward 96.5 0 0 

2.  Gauss’s Backward 96.5 0 0 

3.  Stirling’s Method 96.5 0 0 



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4.  Bessel’s Method 96.5 0 0 

5.  Previously proposed 96.5 0 0 

6.  NIF 96.5 0 0 

 

Example 3:  

Consider the transcendental function 𝑦 = 𝑐𝑜𝑠𝜃 for some equidistantly spaced. Here we have to 

calculate value of y at 𝜃=520. 

Actual value of   𝑦(520) =  0.615661. 

Table 5: Difference Table for Example 3. 

θ (°) y Δy Δ²y Δ³y Δ⁴y Δ⁵y Δ⁶y 

0 1.000000 –0.01519 –0.02992 +0.001367 +0.000874 –0.001115 –0.001071 

10 0.984807 –0.04511 –0.02855 +0.001989 –0.000241 –0.002186 +0.000752 

20 0.939693 –0.07366 –0.02631 +0.001748 –0.002427 –0.001434 +0.003978 

30 0.866025 –0.09998 –0.02381 –0.000679 –0.003861 +0.002544  

40 0.766044 –0.12325 –0.01953 –0.004540 –0.001317   

50 0.642788 –0.14278 –0.01574 –0.005857    

60 0.500000 –0.15798 -0.010392     

70 0.342020 –0.16837      

80 0.173648       

 

Table 6: Comparison of Different Interpolation Techniques along with Error. 

Sr. No Methods Approximate Value 

of y at  𝜃 = 520 

Error % Error 

1.  Gauss’s Forward 0.615661475 0.000609 0.09891807 

2.  Gauss’s Backward 0.615858611 0.000197 0.03209737 

3.  Stirling’s Method 0.615661475 0.00000047 0.00007715 

4.  Bessel’s Method 0.61556142 0.00009958 0.01617448 

5.  Previously proposed 0.61556153 0.00009947 0.01615662 



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6.  NIF 0.615053 0.000608 0.09875565 

 

 
 

Figure 1 - Graphical Representation for Problem 

Example 4:  

Consider the exponential function 𝑦 = 𝑒𝑡 for some equidistantly spaced values of t. In this problem 

we have to find out the value of y at 𝑡 = 0.74. 

Actual value of 𝑦(0.74) = 2.09532. 

Table 7: Difference Table for Example 4. 

t y Δy Δ²y Δ³y Δ⁴y 

0.64 1.896481 0.019060 0.000192 0.000002 -0.00000001 

0.65 1.915541 0.019252 0.000193 0.000002 0.00000004 

0.66 1.934792 0.019445 0.000195 0.000002 0.00000002 

0.67 1.954237 0.019640 0.000197 0.000002 0.00000000 

0.68 1.973878 0.019838 0.000199 0.000002 0.00000004 

0.69 1.993716 0.020037 0.000201 0.000002 0.00000002 

0.70 2.013753 0.020239 0.000203 0.000002 0.00000000 

0.71 2.033991 0.020442 0.000205 0.000002 0.00000006 

0.72 2.054433 0.020647 0.000208 0.000002 -0.00000003 

0.73 2.075081 0.020855 0.000210 0.000002  

0.74 2.095936 0.021065 0.000212  — 

0.75 2.117000 0.021276 — — — 

0.76 2.138276 — — — — 

 



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Table 8: Comparison of Different Interpolation Techniques along with Error. 

Sr. No. Methods Approximate Value of 

y at  𝑡 = 0.74 

Error % Error 

1.  Gauss’s Forward 2.09532 0 0 

2.  Gauss’s Backward 2.09532 0 0 

3.  Stirling’s Method 2.09532 0 0 

4.  Bessel’s Method 2.07422 0.0211 1.00700 

5.  Previously proposed 2.017533 0.077787 3.7124162 

6.  NIF 2.09585 0.00053 0.025294 

Table 9: Analysis of all Problems. 

Examples Gauss’s 

Forward 

Formula 

 

Gauss’s 

Backward 

Formula 

 

Stirling’s 

Formula 

Bessel’s 

formula 

Previously 

Proposed 

Formula 

New 

Interpol

ation 

formula 

True 

Value 

Example 1 121 121 121 121 121 121 121 

Example 2 96.5 96.5 96.5 96.5 96.5 96.5 96.5 

Example 3 0.6156615 0.61585867 0.6156615 0.6156615 0.6155615 0.615053 0.6156615 

Example 4 2.09532 2.09532 2.09532 2.07422 2.017533 2.09585 2.09532 

 

Conclusion 

This paper presents a new interpolation formula and compares it with existing interpolation models 

using the concept of mathematical norms. The proposed model, given in Equation (viii), is designed 

for cases where the point to be interpolated lies near the center of the given data. It was derived by 

modifying Gauss’s Forward Interpolation Formula specifically, by shifting the subscript back by four 

units and replacing the variable p with p+4. This modified expression was then averaged with 

Gauss’s Backward Interpolation Formula to produce the new model. The performance of this 

proposed formula was evaluated and compared with established methods, including Gauss’s Forward 

Interpolation, Gauss’s Backward Interpolation, Bessel’s Interpolation and Stirling’s Interpolation. 

From all the tables above, we have shown that how different techniques are applied on different 

problems along with error. So as a result, we have found that the proposed interpolation technique 



Communications on Applied Nonlinear Analysis 

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NIF shows better results in comparison to previously defined interpolation techniques. So, the NIF 

stands as a robust and reliable alternative for central interpolation tasks in numerical analysis. 

Acknowledgement: Manuscript Communication Number (MCN): IU/R&D/2025-MCN0003888 

office of research and development Integral University, Lucknow. 

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