Key takeaways
- Nine units, from types of error to the power method for the largest eigenvalue.
- A comparison table of root-finding methods with their order of convergence.
- Formula tables for trapezoidal, Simpson 1/3 and Simpson 3/8 rules and for Euler, modified Euler and RK4.
- Solved examples on Newton-Raphson, bisection, Gauss-Seidel, Lagrange interpolation, Euler and RK4.
- Original Uncodemy revision notes; check your own university syllabus.
About these handwritten notes
These are original Uncodemy revision notes for Numerical Methods, written in a clean handwritten style so they feel like a topper's notebook rather than a textbook. They are short on purpose: each unit gives the key definitions, formulas, a solved example or two and the points that are easy to forget.
The notes follow the common syllabus that most Indian B.Tech universities use for this subject, including universities that follow the AICTE model curriculum. They are not official notes from any university. Unit names, order and depth differ from one university to another, so check your own university syllabus and mark any topic that is missing before you rely on the notes for an exam. Numerical methods questions are long but predictable, so the notes show the table layout for each method as well as the formula.
What's inside the numerical methods notes
The notes have nine units. One line on each:
Unit 1: Errors
Absolute, relative and percentage error, round-off and truncation error, and significant digits.
Unit 2: Roots of equations f(x) = 0
Bisection, regula falsi, secant, Newton-Raphson and fixed-point iteration, compared by what they need, their order of convergence and whether they always converge, with solved examples for √2 and bisection.
Unit 3: Linear systems
Direct methods (Gauss elimination with partial pivoting, Gauss-Jordan, LU decomposition) and iterative methods (Jacobi, Gauss-Seidel), with a solved Gauss-Seidel example.
Unit 4: Finite differences and interpolation
Forward, backward and shift operators, Newton forward and backward formulas, Lagrange's formula and Newton's divided differences for unequal spacing.
Unit 5: Numerical differentiation
Derivatives from Newton's forward formula and central difference formulas for the first and second derivative.
Unit 6: Numerical integration
Trapezoidal, Simpson 1/3 and Simpson 3/8 rules, with their conditions on n, and a solved example.
Unit 7: Numerical solution of ODEs
Euler, modified Euler (Heun) and fourth order Runge-Kutta, with short notes on the Taylor series, Picard, Milne and Adams-Bashforth methods, plus solved Euler and RK4 steps.
Unit 8: Curve fitting and eigenvalues
Least squares straight line y = a + bx with normal equations, a solved example, and the power method for the largest eigenvalue.
Unit 9: Most asked questions (practice)
Ten practice questions that cover every unit.
How to revise numerical methods with these notes
- Learn the formula and the table. For each method, practise both the formula and the layout of the working table. A neat table earns step marks and avoids errors.
- Keep four decimal places. The notes warn that in RK4 most marks are lost by rounding too early. Write k1, k2, k3 and k4 in a small table with four decimals.
- Know the conditions. Simpson 1/3 needs an even number of intervals and Simpson 3/8 needs a multiple of three. Gauss-Seidel converges reliably for diagonally dominant systems, so rearrange the equations first.
- Practise on a calculator. GATE numericals are calculator-based, so speed matters. Time yourself on Newton-Raphson iterations and Simpson's rule.
- Check with tools. After solving by hand, compare your answer using the Newton-Raphson calculator, Simpson's rule calculator, RK4 calculator or the Newton forward difference table.
- Remember the orders of convergence. Bisection is linear, secant is about 1.618 and Newton-Raphson is quadratic. These come up as one-mark questions.
Exam focus: high-weightage topics
The tip at the end of the notes says GATE often asks Newton-Raphson iterations, trapezoidal and Simpson values, and Euler or RK steps, along with order of convergence facts. University papers add derivations and longer tables.
Topics worth extra time:
- Bisection: four steps for a root of x³ - 2x - 5 = 0.
- Newton-Raphson: a root of x e^x = 1 to three decimals, plus the derivation of the formula and its quadratic convergence.
- Gauss-Seidel for a 3 x 3 diagonally dominant system.
- Interpolation: Newton forward formula from a table, and Lagrange's formula for unequally spaced points.
- Integration: ∫₀¹ 1/(1 + x²) dx by the trapezoidal and Simpson 1/3 rules with h = 0.25, and the estimate of π from it.
- ODEs: one RK4 step for dy/dx = y - x, and a comparison of Euler and modified Euler.
- Errors: truncation and round-off error with examples.
Typical questions
These questions come from the practice set in the PDF. They are practice questions, not a prediction of your exam.
Who these notes are for
- B.Tech students taking numerical methods, often called computer-based numerical and statistical techniques or numerical analysis, in the second or third year.
- GATE aspirants revising numerical methods for engineering mathematics.
- Programming and data science learners. Root finding, iteration, interpolation and least squares are the ideas behind many algorithms, so the notes are a quick refresher before you code them in Python.
Numerical methods also links to other maths subjects. Root finding and interpolation build on calculus, the linear systems unit builds on matrices, and the ODE unit gives a numerical way to check answers from the differential equations notes. Revising them together saves time.
A simple 5-day revision plan
- Day 1: Units 1 and 2. Errors and all root-finding methods, with their orders of convergence.
- Day 2: Unit 3. Gauss elimination, LU, Jacobi and Gauss-Seidel on one 3 x 3 system.
- Day 3: Units 4 and 5. Difference tables, Newton forward and backward, Lagrange and divided differences.
- Day 4: Units 6 to 8. Trapezoidal and Simpson rules, Euler, modified Euler, RK4 and least squares.
- Day 5: Solve the ten practice questions on a calculator with a timer, then revise the formula tables.
Common mistakes to avoid
- Wrong sign check in bisection. The root lies in the half where f changes sign. Check f(a) x f(mid) every step.
- Choosing a bad starting point for Newton-Raphson. If f'(x) is close to zero, the method can jump away. Pick a point near the root, for example from a quick sign check.
- Using Simpson 1/3 with an odd number of intervals. Count the intervals, not the points.
- Mixing up u in interpolation. u = (x - x0)/h for the forward formula and u = (x - xn)/h for the backward formula.
- Using old values in Gauss-Seidel. Use each new value as soon as you find it; that is the difference from Jacobi.
Download Numerical Methods Handwritten Notes PDF
Numerical Methods Handwritten Notes PDF
The PDF carries the same content as this page in a clean, print-ready layout, so you can read it offline and keep it for revision.
7 pages · 250 KB · PDF · Updated 5 Nov 2026