Numerical Methods — Roots of Equations & Linear Systems — Engineering Mathematics Notes

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 $n$ .

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Numerical Methods — Roots of Equations & Linear Systems