Best Interpolation Method for Adding New Data Points UPSC ISS 2020 Paper-1 | Problem-61 | RitwikMath

Learn which interpolation method is most efficient when new data points are added in this important Numerical Analysis PYQ discussion. In this video, we compare Lagrange Interpolation, Newton Forward Interpolation, and Newton’s Divided Difference Formula to determine which method avoids recomputing the entire interpolation polynomial.

Perfect for students preparing for UPSC Statistics Optional, ISS, CSIR NET, GATE, IIT JAM, and other mathematics/statistics competitive exams.

📌 Problem Discussed:
Which interpolation method is best when an additional interpolation point is inserted later without recalculating the entire polynomial?

📌 Key Comparisons:

### ❌ Lagrange Interpolation
In Lagrange interpolation:
\[
P(x)=\sum_{i=0}^{n} y_iL_i(x)
\]

If a new point is added, all basis polynomials:
\[
L_i(x)
\]
change completely.

Hence, the whole polynomial must be recomputed.

---

### ❌ Newton Forward Interpolation
Newton Forward Interpolation depends on:
- Forward difference tables
- Equally spaced points

Adding new points may require extensive recomputation of the difference table.

---

### ✅ Newton’s Divided Difference Formula
Newton’s Divided Difference interpolation has the form:
\[
P_n(x)=f(x_0)+(x-x_0)f[x_0,x_1]
+(x-x_0)(x-x_1)f[x_0,x_1,x_2]+\cdots
\]

When a new point is added:
- Previous computations remain unchanged
- Only the new divided difference term is added

📌 Final Conclusion:
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📌 Concepts Explained:
- Lagrange Interpolation
- Newton Forward Interpolation
- Newton Divided Difference Formula
- Updating Interpolation Polynomials
- Numerical Analysis PYQ
- Computational Efficiency

🎯 Useful For:
- UPSC Statistics Optional
- ISS Exam
- CSIR NET Mathematical Sciences
- GATE Mathematics
- IIT JAM
- Engineering Mathematics
- Applied Mathematics Students

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