Shyam Agrawal

Welcome to Part 13 and the opening of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Modules 1 to 3 we lived in a perfect world where every formula, from Newton Gregory to Stirling and Bessel, demanded equally spaced arguments. Real data is rarely that polite. When the intervals turn unequal, … Read more

Welcome to Part 12, the final part of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Part 11 we met the Gauss Forward and Backward formulas. They work, but their zig zag paths are clumsy for hand calculation. Mathematicians fixed this by averaging them, and the averages became the two most … Read more

Welcome back to Part 11 of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Part 10 we met the central operators δ\delta and μ\mu. Today we put them to work with the two foundational pillars of central interpolation: Gauss’s Forward and Backward formulas. Both formulas measure the step value u=x−x0hu = … Read more

Welcome to Part 10 and the opening of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Module 2 we predicted values near the beginning and the end of a table. But when the missing value sits exactly in the middle, both Newton Gregory formulas converge slowly. The accurate tools for the … Read more

Welcome to Part 9, the final part of Module 2, updated for the UPSC ISS 2026 to 2027 cycle. The forward and backward formulas need a full difference table. But when a question asks for one missing value from a small dataset, there is a far faster route. The hack: assume the nn known values … Read more

Welcome back to Part 8 of Module 2, updated for the UPSC ISS 2026 to 2027 cycle. In Part 7 we predicted values near the beginning of a table. Today the examiner flips the situation: the target sits near the end of the data. Use the forward formula there and your truncated series carries a … Read more

Welcome to Part 7 and the opening of Module 2, updated for the UPSC ISS 2026 to 2027 cycle. In Part 6 we understood what interpolation is, its assumptions, and the error term. Now it is time for the most famous weapon in your arsenal: the Newton Gregory Forward Interpolation Formula. The formula estimates the … Read more

Welcome to Part 6, the final part of Module 1, updated for the UPSC ISS 2026 to 2027 cycle. With operators, factorial polynomials, and differences of zero behind us, we now step into the heart of Numerical Analysis: Interpolation. Interpolation is the technique of estimating the value of a function for an intermediate value of … Read more

Welcome back to Part 5 of Module 1, updated for the UPSC ISS 2026 to 2027 cycle. You now know operator algebra, separation of symbols, and factorial polynomials. Today we unlock one of the most intimidating yet highest scoring topics in the syllabus: Differences of Zero and Stirling Numbers. The difference of zero, written Δn0m\Delta^n … Read more

Welcome back to Part 4 of Module 1, updated for the UPSC ISS 2026 to 2027 cycle. After operator algebra and separation of symbols, today we tackle a concept that works like an absolute cheat code in Paper 1: the Factorial Representation of a Polynomial. A factorial polynomial, written x(n)x^{(n)}, is the product x(x−h)(x−2h)⋯[x−(n−1)h]x(x-h)(x-2h)\cdots[x-(n-1)h] of … Read more