Numerical Computation

The route

Lessons are in the order that makes each one easier than the last. Each unit ends with a short quiz.

1. 0/4

   Unit 1

   Errors in Numerical Computation

   Your calculator is a polite liar: every number it gives you is slightly wrong, and the skill is knowing how wrong.
   1. Introduction to Numerical AnalysisFree Exact answers are a luxury, and your calculator has been politely rounding them for you all along. 4 min
   3. Absolute, Relative and Percentage Errors Measuring how wrong you are takes two numbers, absolute and relative, and the second one usually matters more. 4 min
   4. Significant Figures, Decimal Places, Rounding and Chopping Digits that matter: "correct to 2 decimal places" is a precise promise, and here is exactly what it means. 4 min
   5. Sources and Types of Errors Five places a mistake can sneak into a calculation, and one tiny error that cost 28 lives. 10 min
   6. Unit 1 quiz6 questions on this unit 8 min
2. 0/3

   Unit 2 Full plan

   Bracketing Methods for Roots of Nonlinear Equations

   Trap the root, then squeeze it: Bisection and False Position never lose a root once they have caught it.
   1. Location of Roots and the Intermediate Value Theorem The sign-change trap: if a curve starts below the axis and ends above it, it must cross somewhere in between. 5 min
   2. The Bisection Method Halve it and keep the half with the sign change: seven halvings turn a hundred possibilities into one. 10 min
   3. The False Position (Regula Falsi) Method Bisection ignores how big the numbers are — false position lets them vote on where the root is hiding. 11 min
   4. Unit 2 quiz8 questions on this unit 10 min
3. 0/4

   Unit 3 Full plan

   Open Methods for Roots of Nonlinear Equations

   Ride the tangent home: Newton-Raphson and the Secant method are faster than any bracket, if you trust them.
   1. The Newton-Raphson Method One guess and a slope: ride the tangent line down to the axis, and watch the correct digits double each step. 11 min
   2. Limitations of the Newton-Raphson Method When Newton goes wrong, a flat tangent can fling you miles away. Know the traps before you start. 3 min
   3. The Secant Method and Fixed-Point Iteration Newton without calculus: no derivative needed, because two points and a straight line do almost as well. 6 min
   4. Comparison of Root-Finding Methods Choosing your weapon: four root-finders in your bag, and three quick questions that pick the right one. 5 min
   5. Unit 3 quiz7 questions on this unit 9 min
4. 0/5

   Unit 4 Full plan

   Solution of Systems of Linear Equations

   Untangling three equations at once: Gaussian elimination, row echelon form and Gauss-Jordan give one tidy machine for every simultaneous system.
   1. Augmented Matrices and Elementary Row Operations From equations to a grid of numbers: strip away the letters, and three simple row moves do all the work. 4 min
   2. Gaussian Elimination with Back Substitution Make zeros, build a staircase, then walk back up it one unknown at a time. 7 min
   3. Row Echelon Form and the Row Echelon Method That staircase shape has a precise definition — learn three rules and you'll spot row echelon form at a glance. 4 min
   4. Gauss-Jordan Elimination Don't stop at the staircase — keep clearing until the solution reads itself straight off the last column. 5 min
   5. LU Decomposition and Iterative Methods (Jacobi and Gauss-Seidel) Beyond the basics: what computers do when the system has a million unknowns instead of three. 8 min
   6. Unit 4 quiz7 questions on this unit 9 min
5. 0/4

   Unit 5 Full plan

   Matrix Operations, Determinants and Eigenvalues

   Multiplying matrices, checking them with determinants, squeezing them into diagonal form, and finding the directions a matrix respects.
   1. Matrix Multiplication and Its Properties Rows meet columns: one row-times-column rule builds every entry, and the order you multiply in matters. 7 min
   2. Determinants and Cofactor Expansion A one-number health check that says whether a matrix can be undone, and checks your row operations for free. 3 min
   3. Diagonal Matrices and Reduction to Diagonal Form The dream shape: the friendliest matrix there is, and how to squeeze any square matrix into it. 6 min
   4. Eigenvalues, Eigenvectors and Diagonalisation Stretch dough in one direction and some lines never turn — those are the directions a matrix secretly respects. 8 min
   5. Unit 5 quiz7 questions on this unit 9 min

Practise and remember