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Numerical Analysis

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Numerical Analysis

Computational methods for mathematical problems

152Modules

8Units

1

Error Analysis

Understanding and controlling numerical errors

20 modules

What is Numerical Analysis?INTRO

Why We ApproximateINTRO

Computers and MathematicsINTRO

Floating Point RepresentationLESSON

Floating Point PitfallsEXAMPLE

Floating Point PracticePRACTICE

Machine EpsilonLESSON

Round-off ErrorLESSON

Error Analysis ExamplesEXAMPLE

Error CalculatorACTIVITY

Error Analysis PracticePRACTICE

Truncation ErrorLESSON

Absolute and Relative ErrorLESSON

Error PropagationLESSON

Loss of SignificanceLESSON

Numerical StabilityLESSON

Condition NumbersLESSON

Error Analysis SummarySUMMARY

Error Analysis ReferenceFORMULA SHEET

Error Analysis QuizQUIZ

2

Root Finding

Solving nonlinear equations

22 modules

The Root Finding ProblemINTRO

Where Roots AppearINTRO

Bisection MethodLESSON

Bisection Method ExamplesEXAMPLE

Bisection PracticePRACTICE

Fixed Point IterationLESSON

Newton's MethodLESSON

Newton's Method ExamplesEXAMPLE

Newton's Method PracticePRACTICE

Newton's Method ConvergenceLESSON

Convergence VisualizerACTIVITY

Secant MethodLESSON

Method of False PositionLESSON

Müller's MethodLESSON

Convergence RatesLESSON

Multiple RootsLESSON

Engineering ApplicationsREAL WORLD

Polynomial Root FindingLESSON

Root Finder ToolACTIVITY

Root Finding SummarySUMMARY

Root Finding ReferenceFORMULA SHEET

Root Finding QuizQUIZ

3

Interpolation

Approximating functions from data

22 modules

What is Interpolation?INTRO

Fitting Curves to DataINTRO

Polynomial InterpolationLESSON

Interpolation ToolACTIVITY

Polynomial Interpolation PracticePRACTICE

Lagrange InterpolationLESSON

Lagrange Interpolation ExamplesEXAMPLE

Newton's Divided DifferencesLESSON

Interpolation ErrorLESSON

Runge's PhenomenonLESSON

Chebyshev NodesLESSON

Piecewise Polynomial InterpolationLESSON

Cubic SplinesLESSON

Splines PracticePRACTICE

Natural and Clamped SplinesLESSON

Bézier CurvesLESSON

Computer GraphicsREAL WORLD

Spline Construction ExamplesEXAMPLE

Spline BuilderACTIVITY

Interpolation SummarySUMMARY

Interpolation ReferenceFORMULA SHEET

Interpolation QuizQUIZ

4

Numerical Integration

Approximating definite integrals

22 modules

Numerical IntegrationINTRO

When Integration is HardINTRO

Riemann Sums ReviewLESSON

Trapezoid RuleLESSON

Trapezoid and Simpson ExamplesEXAMPLE

Simpson's RuleLESSON

Newton-Cotes FormulasLESSON

Composite RulesLESSON

Basic Rules PracticePRACTICE

Error in Numerical IntegrationLESSON

Physics ApplicationsREAL WORLD

Numerical Integration CalculatorACTIVITY

Romberg IntegrationLESSON

Gaussian QuadratureLESSON

Gaussian Quadrature ExamplesEXAMPLE

Gaussian Quadrature PracticePRACTICE

Adaptive QuadratureLESSON

Improper IntegralsLESSON

Multidimensional IntegrationLESSON

Numerical Integration SummarySUMMARY

Numerical Integration ReferenceFORMULA SHEET

Numerical Integration QuizQUIZ

5

Numerical Differentiation

Approximating derivatives

17 modules

Numerical DifferentiationINTRO

Challenges of Numerical DifferentiationINTRO

Finite DifferencesLESSON

Finite Difference ExamplesEXAMPLE

Finite Differences PracticePRACTICE

Forward, Backward, Central DifferencesLESSON

Higher-Order FormulasLESSON

Second Derivative ApproximationsLESSON

Error in Numerical DifferentiationLESSON

Differentiation ToolACTIVITY

Richardson ExtrapolationLESSON

Richardson Extrapolation ExamplesEXAMPLE

Richardson Extrapolation PracticePRACTICE

Partial Derivative ApproximationsLESSON

Numerical Differentiation SummarySUMMARY

Numerical Differentiation ReferenceFORMULA SHEET

Numerical Differentiation QuizQUIZ

6

Linear Systems

Solving systems of linear equations

21 modules

Systems of Linear EquationsINTRO

Why Numerical Methods?INTRO

Gaussian EliminationLESSON

Gaussian Elimination ExamplesEXAMPLE

Partial PivotingLESSON

LU DecompositionLESSON

Cholesky DecompositionLESSON

Matrix ConditioningLESSON

Iterative Methods IntroductionLESSON

Iterative Method ExamplesEXAMPLE

Direct Methods PracticePRACTICE

Iterative Methods PracticePRACTICE

Jacobi MethodLESSON

Gauss-Seidel MethodLESSON

Successive Over-RelaxationLESSON

Iterative Method ConvergenceLESSON

Large-Scale SystemsREAL WORLD

Linear System SolverACTIVITY

Linear Systems SummarySUMMARY

Linear Systems ReferenceFORMULA SHEET

Linear Systems QuizQUIZ

7

Ordinary Differential Equations

Numerical methods for ODEs

23 modules

Numerical Solution of ODEsINTRO

Initial Value ProblemsINTRO

Euler's MethodLESSON

Euler's Method ExamplesEXAMPLE

Error in Euler's MethodLESSON

Taylor MethodsLESSON

Euler and Taylor PracticePRACTICE

Runge-Kutta MethodsLESSON

Runge-Kutta ExamplesEXAMPLE

Runge-Kutta PracticePRACTICE

Classical Fourth-Order RKLESSON

Adaptive Step SizeLESSON

Multistep MethodsLESSON

Adams-Bashforth MethodsLESSON

Stiff EquationsLESSON

Systems of ODEsLESSON

Physics and ODEsREAL WORLD

Higher-Order ODEsLESSON

ODE SolverACTIVITY

Phase Portrait PlotterACTIVITY

Numerical ODEs SummarySUMMARY

Numerical ODEs ReferenceFORMULA SHEET

Numerical ODEs QuizQUIZ

8

Course Resources

Comprehensive references and practice materials

5 modules

Course OverviewINTRO

Course SummarySUMMARY

Numerical Analysis Master ReferenceFORMULA SHEET

Method ExplorerACTIVITY

Comprehensive Final ExamQUIZ