Exactness Error Bounds UPSC ISS: The Quadrature and ODE Tables

Q: Are Simpson's 1/3 and 3/8 rules equally exact?

Yes. Both cap at degree 3 with local O(h5)O(h^5) and global O(h4)O(h^4) errors. They differ only in the interval constraint, even versus multiples of 3.

Q: When does the exactness bypass fail?

When the integrand is not a polynomial, such as sin⁡x\sin x or exe^x, or when its degree exceeds the rule's ceiling. In those cases the formula must genuinely be applied.

Welcome back to Part 31 of Module 8, updated for the UPSC ISS 2026 to 2027 cycle. Part 30 gave you the formulas, which tell you how to solve a problem. Today’s tables tell you something more valuable: whether you need the formula at all. With calculators banned, examiners embed cheat codes into their questions through exactness degrees and error bounds, and knowing these properties by heart converts ten minute fractional grinds into two second ticks.

Picture the classic setup: a question asks for the error when $\int_0^5 x^3\,dx$ is computed by Simpson’s 1/3 rule. The average aspirant builds a table and divides fractions for ten minutes. You read the integrand, see degree 3, recall that Simpson’s 1/3 is exact up to degree 3, and tick zero. Print both tables below and stick them above your desk.

This post is Part 31 of the Last Minute Revision Guide (Module 8), inside our UPSC ISS Numerical Analysis Complete Guide. Roadmap at the UPSC ISS hub.

Table of Contents

Master Table 1: Numerical Integration (Quadrature)

Watch the distinction between the assumed curve degree, the geometry used to build the rule, and the degree of exactness, the magical ceiling below which the error completely vanishes.

Rule	Interval Constraint	Assumed Curve Degree	Degree of Exactness	Local Error	Global Error
Trapezoidal Rule	Any integer	1 (straight line)	Exactly 1	$O(h^3)$ O(h3)	$O(h^2)$ O(h2)
Simpson’s 1/3 Rule	Multiple of 2 (even)	2 (parabola)	Exactly 3	$O(h^5)$ O(h5)	$O(h^4)$ O(h4)
Simpson’s 3/8 Rule	Multiple of 3	3 (cubic)	Exactly 3	$O(h^5)$ O(h5)	$O(h^4)$ O(h4)
Weddle’s Rule	Multiple of 6	6	Exactly 5	$O(h^7)$ O(h7)	$O(h^6)$ O(h6)

Simpson’s 1/3 assumes a degree 2 curve yet is exact for degree 3. Did you remember this exactness trick? Tell us below.

Simpson's 1/3 assumes a degree 2 curve yet is exact for degree 3. Did you remember this exactness trick? Tell us below.x

Key quadrature takeaways. First, the global truncation error is always one degree lower than the local error, because the number of strips is proportional to $\tfrac{1}{h}$ . Second, the Weddle trap: multiples of 6 intervals mean a minimum of 7 consecutive data points, $y_0$ through $y_6$ . Third, Weddle’s rule is the precision champion of the syllabus, with a global error of only $O(h^6)$ and exactness up to degree 5.

Master Table 2: ODE Solvers

The Runge Kutta family from Module 7, with the error limits the examiners test most heavily.

Method	Equivalent RK Order	Local Error	Global Error
Euler’s Method	First order (RK1)	$O(h^2)$ O(h2)	$O(h)$ O(h)
Modified Euler’s Method	Second order (RK2)	$O(h^3)$ O(h3)	$O(h^2)$ O(h2)
Runge Kutta Fourth Order	Fourth order (RK4)	$O(h^5)$ O(h5)	$O(h^4)$ O(h4)

Key ODE takeaways. First, the $h^5$ trap: RK4 matches the Taylor series up to $h^4$ , so its local error starts at the very next power, $O(h^5)$ , the single most repeated theory question of the module. Second, the predictor corrector upgrade tightens Euler’s drifting global $O(h)$ to the Modified Euler’s $O(h^2)$ . Third, the integration bridge: when $\dfrac{dy}{dx} = f(x, y)$ is independent of $y$ , RK4 collapses exactly into Simpson’s 1/3 Rule.

Which error bound do you keep mixing up in mocks? Drop a comment so we can help.

Which error bound do you keep mixing up in mocks? Drop a comment so we can help.x

StatChakravyuh Pro Tips

Read the prefix before the word error. Both $O(h^5)$ and $O(h^4)$ will sit among the RK4 options. “Local” points to the higher power $h^5$ ; “global”, “total”, or “accumulated” points to the lower power $h^4$ .
The bypass drill. Asked for $\int_0^2 (x^3 + 2x)\,dx$ by Simpson’s 3/8 rule, do not touch the formula. Degree 3 is within the exactness ceiling of 3, so plain calculus finishes it: $\left[\tfrac{x^4}{4} + x^2\right]_0^2 = 4 + 4 = 8$ .
The difference framing. Given an exact integral $A$ A and its numerical value $B$ B, the quantity $\left|A – B\right|$ is simply the error. For a degree 2 integrand under Simpson’s 1/3 rule, that error is instantly zero.

Frequently Asked Questions

Why is the global error always one power below the local error?

The number of steps covering a fixed range is proportional to $\tfrac{1}{h}$ , so summing the per step local errors multiplies by $\tfrac{1}{h}$ and drops one power of $h$ .

Are Simpson’s 1/3 and 3/8 rules equally exact?

Yes. Both cap at degree 3 with local $O(h^5)$ and global $O(h^4)$ errors. They differ only in the interval constraint, even versus multiples of 3.

When does the exactness bypass fail?

When the integrand is not a polynomial, such as $\sin x$ or $e^x$ , or when its degree exceeds the rule’s ceiling. In those cases the formula must genuinely be applied.

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Coming up in Part 32: the traps that cost ranks. The top five silly mistakes of Numerical Analysis and exactly how to avoid each one.

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Part 31: Exactness and Error Bounds for UPSC ISS: The Two Master Tables