---
title: "Iterative methods for linear equations"
subject: "Computational Techniques in Civil Engineering"
chapter: 2
key: "semester_8/computational_techniques_in_civil_engineering:iterative-methods-linear-equations"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/iterative-methods-linear-equations
site: BCT Engineering
---

# Iterative methods for linear equations

Computational Techniques in Civil Engineering · Chapter 2: Solutions of linear equations

- Asked in: 2078 Chaitra, 3 marks
- Video: [Jacobi method | Iterative method | Solution of Linear System Equation](https://www.youtube.com/watch?v=jPv5pP1kOco) — Dr.Gajendra Purohit
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/iterative-methods-linear-equations
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-2

## Question

Briefly explain the different iterative methods used for the solution of a given set of equations.

## Short answer

Iterative methods improve an initial guess for large sparse systems. Jacobi uses only previous iteration values; Gauss-Seidel uses latest values immediately, converging twice as fast. Successive over-relaxation (ω between 1 and 2) accelerates convergence. Conjugate gradient minimizes energy along conjugate directions for symmetric positive definite matrices and converges in at most n steps.

## Answer

Iterative methods start from an assumed solution and improve it repeatedly until the change is within tolerance. They are used for large sparse systems $[A]\{x\}=\{b\}$ and need diagonal dominance (or symmetry and positive definiteness) for convergence.

1. **Jacobi method:** every unknown is computed using values of the previous iteration only.
$$
x_i^{(k+1)}=\frac{1}{a_{ii}}\Big(b_i-\sum_{j\ne i}a_{ij}x_j^{(k)}\Big)
$$
2. **Gauss-Seidel method:** newly computed values are used immediately in the same iteration:
$$
x_i^{(k+1)}=\frac{1}{a_{ii}}\Big(b_i-\sum_{j<i}a_{ij}x_j^{(k+1)}-\sum_{j>i}a_{ij}x_j^{(k)}\Big)
$$
It converges about twice as fast as Jacobi.
3. **Successive over-relaxation (SOR):** $x_i^{new}=(1-\omega)x_i^{old}+\omega\,x_i^{GS}$ with $1<\omega<2$ to speed up convergence.
4. **Conjugate gradient method:** for symmetric positive definite matrices; minimises the quadratic energy along conjugate directions and converges in at most $n$ steps.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the Jacobi method each unknown is calculated using...
   A. Newly computed values
   B. Values from the previous iteration only
   C. Gradient of energy
   D. Residual direction
   Answer: B. Jacobi is fully based on old values. (Key point: Jacobi)

2. What does SOR do?
   A. Blends old and Gauss-Seidel values with factor omega (1 to 2) to speed up convergence
   B. Uses conjugate directions
   C. Uses omega < 1 to slow
   D. Uses only old values
   Answer: A. x_new = (1-omega) x_old + omega x_GS. (Key point: SOR)

3. What helps Jacobi/Gauss-Seidel converge?
   A. Matrix being sparse
   B. Matrix being full
   C. Unsymmetry
   D. Diagonal dominance
   Answer: D. Iterative methods need diagonal dominance or symmetry and positive definiteness. (Key point: Convergence)

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