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Chapter 2 of 6

Numerical Analysis

In the UPSC ESE Civil syllabus under Engineering Mathematics · 2 parts

Numerical Methods — Roots of Equations & Linear Systems · Numerical Interpolation, Integration & Solution of ODEs

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📑 Contents (14 sections)

Part 1 of 2

Numerical Methods — Roots of Equations & Linear Systems

Last reviewed 16 Sept 2026 · 7 min read

Errors

Error	Definition
Absolute error	$∣ x_{t r u e} - x_{a pp r o x} ∣$
Relative error	$\frac{∣ x _{t r u e} - x _{a pp r o x} ∣}{∣ x _{t r u e} ∣}$
Percentage error	Relative error × 100
Round-off error	Due to finite number of digits in computation
Truncation error	Due to approximating an infinite process by a finite one (e.g. truncating a Taylor series)
Inherent error	Present in the data/model itself

Significant digits — digits that carry meaning; a number correct to $n$ decimal places has error at most $0.5 \times 1 0^{- n}$ .
Errors propagate: for sums, absolute errors add; for products/quotients, relative errors add (approximately).

Roots of non-linear equations

Intermediate value theorem

If $f$ is continuous on $[a, b]$ and $f (a) f (b) < 0$ , there is at least one root in $(a, b)$ .

1. Bisection method

Choose $a, b$ with $f (a) f (b) < 0$ .
$c = \frac{a + b}{2}$ ; replace the end point whose function value has the same sign as $f (c)$ .
Repeat until the interval is small enough.

Always converges (bracketing), but slowly — linear convergence, error halves each step.
Iterations for accuracy ε: $n \geq \frac{ln ( \frac{b - a}{ε} )}{ln 2}$ .

2. Regula falsi (method of false position)

c = \frac{a f ( b ) - b f ( a )}{f ( b ) - f ( a )}

Keeps a bracket like bisection but uses the chord intercept — usually faster; linear convergence (one end may remain fixed).

3. Secant method

x_{n + 1} = x_{n} - f (x_{n}) \frac{x _{n} - x _{n - 1}}{f ( x _{n} ) - f ( x _{n - 1} )}

No bracketing required; superlinear convergence (order ≈ 1.618); may diverge.

4. Newton–Raphson method

FormulaNewton–Raphson

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

Quadratic convergence (order 2) near a simple root — the number of correct digits roughly doubles each step.
Requires $f^{'} (x)$ ; fails or is slow if $f^{'} (x_{n}) \approx 0$ , near multiple roots (convergence becomes linear), or with poor starting values.
Square root of $N$ : $x_{n + 1} = \frac{1}{2} (x_{n} + \frac{N}{x _{n}})$
Reciprocal of $N$ : $x_{n + 1} = x_{n} (2 - N x_{n})$

5. Fixed-point iteration

Rewrite $f (x) = 0$ as $x = g (x)$ and iterate $x_{n + 1} = g (x_{n})$ .

Converges if $∣ g^{'} (x) ∣ < 1$ near the root (linear convergence with rate $∣ g^{'} (α) ∣$ ).

Order of convergence summary

Method	Order	Bracketing
Bisection	1 (linear)	Yes
Regula falsi	1 (linear)	Yes
Secant	≈ 1.618	No
Newton–Raphson	2 (quadratic)	No
Fixed-point	1 (if $g^{'} (α) \neq = 0$ )	No

Linear systems — direct methods

Gauss elimination

Forward elimination — reduce $[A ∣ B]$ to upper triangular form.
Back substitution.

Partial pivoting — swap rows so the largest absolute element in the column becomes the pivot — reduces round-off errors and avoids division by zero.
Computational effort ≈ $n^{3} /3$ multiplications for large $n$ .

Gauss–Jordan method

Eliminates above and below pivots to obtain the identity matrix — gives the solution directly (more work than Gauss elimination); used to find inverses.

LU decomposition

$A = LU$ , then solve $L Y = B$ (forward substitution) and $U X = Y$ (back substitution) — efficient for many right-hand sides.

Method	Form
Doolittle	$L$ has unit diagonal
Crout	$U$ has unit diagonal
Cholesky	$A = L L^{T}$ for symmetric positive definite matrices (about half the work)

Thomas algorithm

A simplified Gauss elimination for tridiagonal systems (common in finite differences for beams, heat conduction) — work proportional to $n$ .

Part 2 of 2

Numerical Interpolation, Integration & Solution of ODEs

Last reviewed 16 Sept 2026 · 7 min read

Finite differences

For equally spaced values $x_{0}, x_{1} = x_{0} + h, \dots$ with $y_{i} = f (x_{i})$ :

Operator	Definition
Forward difference $Δ$	$Δ y_{i} = y_{i + 1} - y_{i}$
Backward difference $\nabla$	$\nabla y_{i} = y_{i} - y_{i - 1}$
Central difference $δ$	$δ y_{i} = y_{i + 1/2} - y_{i - 1/2}$
Shift operator $E$	$E y_{i} = y_{i + 1}$
Averaging operator $μ$	$μ y_{i} = \frac{1}{2} (y_{i + 1/2} + y_{i - 1/2})$

Relations: $E = 1 + Δ$ ; $\nabla = 1 - E^{- 1}$ ; $δ = E^{1/2} - E^{- 1/2}$ ; $E = e^{h D}$ ( $D = d / d x$ ).

The nth differences of a polynomial of degree n are constant, and higher differences are zero.

Interpolation

Newton's forward difference formula

For $x$ near the beginning of the table, $p = \frac{x - x _{0}}{h}$ :

y = y_{0} + p Δ y_{0} + \frac{p ( p - 1 )}{2 !} Δ^{2} y_{0} + \frac{p ( p - 1 ) ( p - 2 )}{3 !} Δ^{3} y_{0} + \dots

Newton's backward difference formula

For $x$ near the end of the table, $p = \frac{x - x _{n}}{h}$ :

y = y_{n} + p \nabla y_{n} + \frac{p ( p + 1 )}{2 !} \nabla^{2} y_{n} + \dots

(Central difference formulas — Gauss, Stirling, Bessel, Everett — are used near the middle.)

Lagrange's interpolation formula

For unequally spaced points:

P (x) = i = 0 \sum n y_{i} L_{i} (x), L_{i} (x) = j \neq = i \prod \frac{x - x _{j}}{x _{i} - x _{j}}

Newton's divided difference formula

P (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}] + \dots

with $f [x_{0}, x_{1}] = \frac{f ( x _{1} ) - f ( x _{0} )}{x _{1} - x _{0}}$ , etc. — convenient for adding points.

Interpolation error $\frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} \prod (x - x_{i})$ ; high-degree polynomials on equally spaced points may oscillate (Runge phenomenon) — spline interpolation is often preferred.
Extrapolation (outside the data range) is unreliable.

Numerical differentiation

From Newton's forward formula at $x = x_{0}$ :

(\frac{d y}{d x})_{x_{0}} = \frac{1}{h} (Δ y_{0} - \frac{1}{2} Δ^{2} y_{0} + \frac{1}{3} Δ^{3} y_{0} - \dots)

Simple approximations: forward $\frac{y _{1} - y _{0}}{h}$ (error $O (h)$ ); central $\frac{y _{1} - y _{- 1}}{2 h}$ (error $O (h^{2})$ ); second derivative $\frac{y _{1} - 2 y _{0} + y _{- 1}}{h ^{2}}$ .

Numerical integration

FormulaNewton–Cotes rules (h = step, n = number of intervals)

Trapezoidal rule:

\int_{x_{0}}^{x_{n}} y d x \approx \frac{h}{2} [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + \dots + y_{n - 1})]

Error $O (h^{2})$ (global); exact for linear functions.

Simpson's 1/3 rule ( $n$ even):

\int y d x \approx \frac{h}{3} [(y_{0} + y_{n}) + 4 (y_{1} + y_{3} + \dots) + 2 (y_{2} + y_{4} + \dots)]

Error $O (h^{4})$ ; exact for polynomials up to degree 3.

Simpson's 3/8 rule ( $n$ a multiple of 3):

\int y d x \approx \frac{3 h}{8} [(y_{0} + y_{n}) + 3 (y_{1} + y_{2} + y_{4} + y_{5} + \dots) + 2 (y_{3} + y_{6} + \dots)]

Error $O (h^{4})$ .

Weddle's rule ( $n$ multiple of 6): $\frac{3 h}{10} [y_{0} + 5 y_{1} + y_{2} + 6 y_{3} + y_{4} + 5 y_{5} + y_{6}]$ per 6 intervals.

Gaussian quadrature

Chooses optimal points and weights — an $n$ -point Gauss–Legendre rule is exact for polynomials of degree up to $2 n - 1$ .
Two-point rule on $[- 1, 1]$ : $\int_{- 1}^{1} f d x \approx f (- \frac{1}{3}) + f (\frac{1}{3})$ .
For $[a, b]$ , substitute $x = \frac{b - a}{2} t + \frac{b + a}{2}$ .
Widely used in finite element stiffness computations.

Romberg integration

Richardson extrapolation of trapezoidal results with successively halved step sizes: $I = I_{h /2} + \frac{I _{h /2} - I _{h}}{3}$ (gives Simpson accuracy), repeated for higher accuracy.

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