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Numerical methods wiki map

Each card shows which pages a guide, derivation or exercise points to. The counter tells how many pages reference it.

Most referenced pages

Taylor and the truncation error14 Heun's method12 Newton-Raphson method12 Euler's method11 Closed Newton-Cotes: trapezoid, Simpson and more10 Runge-Kutta method (RK4)10 Lagrange interpolation9 Finite differences: the first derivative8 Newton for nonlinear systems8 Newton interpolation and divided differences7 Adams-Bashforth methods7 Convergence order and efficiency7

Foundations

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Errors in numerical computing

Significant figures and round-off Taylor and the truncation error Exercise: error of the sin x ≈ x approximation Exercise: iterative error with e^0.5

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Significant figures and round-off

Errors in numerical computing Taylor and the truncation error

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Taylor and the truncation error

Errors in numerical computing Derivation: finite differences from Taylor Exercise: Taylor of cos(x) up to order 3 Finite differences: the first derivative

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Derivation: finite differences from Taylor

Taylor and the truncation error

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Exercise: error of the sin x ≈ x approximation

Errors in numerical computing

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Exercise: iterative error with e^0.5

Errors in numerical computing Taylor and the truncation error

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Exercise: Taylor of cos(x) up to order 3

Taylor and the truncation error

Interpolation

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Interpolation: idea, existence and error

Newton interpolation and divided differences Lagrange interpolation Hermite interpolation Cubic splines

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Newton interpolation and divided differences

Interpolation: idea, existence and error Lagrange interpolation Hermite interpolation Derivation: Newton coefficients via divided differences Exercise: Newton with population data Finite differences: the first derivative

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Lagrange interpolation

Interpolation: idea, existence and error Newton interpolation and divided differences Hermite interpolation Derivation: Lagrange basis functions Exercise: Lagrange with population data

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Hermite interpolation

Lagrange interpolation Newton interpolation and divided differences Exercise: Hermite with the Bessel function

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Cubic splines

Interpolation: idea, existence and error Newton interpolation and divided differences Lagrange interpolation Linear systems: error, residual and conditioning

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Derivation: Newton coefficients via divided differences

Newton interpolation and divided differences

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Derivation: Lagrange basis functions

Lagrange interpolation

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Exercise: Hermite with the Bessel function

Hermite interpolation

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Exercise: Newton with population data

Newton interpolation and divided differences Exercise: Lagrange with population data

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Exercise: Lagrange with population data

Lagrange interpolation Exercise: Newton with population data

Integration

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Numerical quadrature from Lagrange

Lagrange interpolation Taylor and the truncation error Closed Newton-Cotes: trapezoid, Simpson and more Open Newton-Cotes and midpoint Gaussian quadrature

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Closed Newton-Cotes: trapezoid, Simpson and more

Numerical quadrature from Lagrange Derivation: trapezoid rule Derivation: composite trapezoid Derivation: Simpson 1/3 Exercise: trapezoid and Simpson on a smooth integral

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Open Newton-Cotes and midpoint

Numerical quadrature from Lagrange Derivation: simple midpoint Derivation: composite midpoint Exercise: trapezoid and midpoint Closed Newton-Cotes: trapezoid, Simpson and more

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Gaussian quadrature

Numerical quadrature from Lagrange Derivation: two-point Gauss-Legendre Exercise: Gauss-Legendre on a general interval Exercise: Gauss-Chebyshev and an error bound

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Numerical multiple integration

Closed Newton-Cotes: trapezoid, Simpson and more Gaussian quadrature Exercise: double integral with Gauss-Legendre Exercise: double integral of a surface

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Derivation: trapezoid rule

Numerical quadrature from Lagrange Closed Newton-Cotes: trapezoid, Simpson and more Exercise: trapezoid and Simpson on a smooth integral

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Derivation: composite trapezoid

Derivation: trapezoid rule Closed Newton-Cotes: trapezoid, Simpson and more Exercise: trapezoid and midpoint

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Derivation: simple midpoint

Open Newton-Cotes and midpoint Derivation: composite midpoint Exercise: trapezoid and midpoint

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Derivation: composite midpoint

Derivation: simple midpoint Open Newton-Cotes and midpoint Exercise: trapezoid and midpoint

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Derivation: Simpson 1/3

Numerical quadrature from Lagrange Closed Newton-Cotes: trapezoid, Simpson and more Exercise: trapezoid and Simpson on a smooth integral

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Derivation: two-point Gauss-Legendre

Gaussian quadrature Exercise: Gauss-Legendre on a general interval

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Exercise: trapezoid and Simpson on a smooth integral

Closed Newton-Cotes: trapezoid, Simpson and more Derivation: composite trapezoid Derivation: Simpson 1/3

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Exercise: trapezoid and midpoint

Open Newton-Cotes and midpoint Closed Newton-Cotes: trapezoid, Simpson and more Derivation: simple midpoint Derivation: composite midpoint Derivation: composite trapezoid Exercise: trapezoid and Simpson on a smooth integral

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Exercise: work from tabulated data

Closed Newton-Cotes: trapezoid, Simpson and more Open Newton-Cotes and midpoint Exercise: trapezoid and Simpson on a smooth integral Exercise: trapezoid and midpoint

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Exercise: Gauss-Legendre on a general interval

Gaussian quadrature Derivation: two-point Gauss-Legendre Exercise: double integral with Gauss-Legendre

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Exercise: Gauss-Chebyshev and an error bound

Gaussian quadrature Exercise: Gauss-Legendre on a general interval

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Exercise: double integral with Gauss-Legendre

Numerical multiple integration Gaussian quadrature Exercise: double integral of a surface

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Exercise: double integral of a surface

Numerical multiple integration Closed Newton-Cotes: trapezoid, Simpson and more Exercise: double integral with Gauss-Legendre

ODEs

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Initial value problems

Taylor and the truncation error Euler's method Heun's method Runge-Kutta method (RK4)Convergence, consistency and order Adams-Bashforth methods Exercise: the SIR model with Euler, Heun and RK4

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Euler's method

Initial value problems Taylor and the truncation error Derivation: Euler's method and its order Derivation: implicit Euler Heun's method Convergence, consistency and order Exercise: Euler and RK4 by hand on a first-order ODE Exercise: stability of explicit and implicit Euler Exercise: numerical order estimation (Euler, Heun, RK4)Newton-Raphson method

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Heun's method

Euler's method Derivation: Heun's method Runge-Kutta method (RK4)Predictor-corrector methods Exercise: numerical order estimation (Euler, Heun, RK4)Derivation: trapezoid rule

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Runge-Kutta method (RK4)

Heun's method Derivation: fourth-order Runge-Kutta Adams-Bashforth methods Exercise: numerical order estimation (Euler, Heun, RK4)Exercise: the SIR model with Euler, Heun and RK4 Exercise: second-order ODE as a system, by hand Derivation: Simpson 1/3 Initial value problems

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Convergence, consistency and order

Euler's method Errors in numerical computing Heun's method Runge-Kutta method (RK4)Exercise: numerical order estimation (Euler, Heun, RK4)Exercise: stability of explicit and implicit Euler

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Adams-Bashforth methods

Euler's method Lagrange interpolation Adams-Moulton methods Predictor-corrector methods Derivation: two-step Adams-Bashforth (AB2)Exercise: AB2 and order estimation Runge-Kutta method (RK4)Initial value problems Heun's method

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Adams-Moulton methods

Adams-Bashforth methods Predictor-corrector methods Derivation: one-step Adams-Moulton (AM2)Newton-Raphson method Stiff problems and stability

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Predictor-corrector methods

Adams-Bashforth methods Adams-Moulton methods Heun's method Exercise: explicit versus predictor-corrector Euler's method

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Stiff problems and stability

Adams-Moulton methods Predictor-corrector methods Exercise: stability of explicit and implicit Euler

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Derivation: Euler's method and its order

Taylor and the truncation error Euler's method Derivation: implicit Euler Finite differences: the first derivative Interpolation: idea, existence and error Heun's method Runge-Kutta method (RK4)Adams-Bashforth methods

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Derivation: implicit Euler

Derivation: Euler's method and its order Euler's method Stiff problems and stability Finite differences: the first derivative Newton-Raphson method

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Derivation: Heun's method

Derivation: Euler's method and its order Taylor and the truncation error Heun's method Derivation: trapezoid rule Derivation: one-step Adams-Moulton (AM2)

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Derivation: fourth-order Runge-Kutta

Derivation: Simpson 1/3 Derivation: Heun's method Runge-Kutta method (RK4)

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Derivation: two-step Adams-Bashforth (AB2)

Lagrange interpolation Euler's method Adams-Bashforth methods Derivation: one-step Adams-Moulton (AM2)Derivation: Lagrange basis functions Interpolation: idea, existence and error Heun's method Exercise: AB2 and order estimation

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Derivation: one-step Adams-Moulton (AM2)

Derivation: two-step Adams-Bashforth (AB2)Adams-Moulton methods Derivation: trapezoid rule

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Exercise: Euler and RK4 by hand on a first-order ODE

Euler's method Runge-Kutta method (RK4)Exercise: second-order ODE as a system, by hand

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Exercise: second-order ODE as a system, by hand

Initial value problems Runge-Kutta method (RK4)Exercise: Euler and RK4 by hand on a first-order ODE Exercise: the SIR model with Euler, Heun and RK4

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Exercise: numerical order estimation (Euler, Heun, RK4)

Convergence, consistency and order Euler's method Heun's method Runge-Kutta method (RK4)Exercise: AB2 and order estimation

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Exercise: stability of explicit and implicit Euler

Euler's method Convergence, consistency and order Stiff problems and stability Newton-Raphson method

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Exercise: the SIR model with Euler, Heun and RK4

Initial value problems Runge-Kutta method (RK4)Exercise: second-order ODE as a system, by hand

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Exercise: AB2 and order estimation

Adams-Bashforth methods Exercise: numerical order estimation (Euler, Heun, RK4)Heun's method Derivation: two-step Adams-Bashforth (AB2)

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Exercise: explicit versus predictor-corrector

Predictor-corrector methods Exercise: AB2 and order estimation Adams-Moulton methods

Linear systems

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Linear systems: error, residual and conditioning

Jacobi method Gauss-Seidel method Convergence and spectral radius Fixed-point iteration

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Jacobi method

Linear systems: error, residual and conditioning Gauss-Seidel method Derivation: Jacobi iteration matrix Exercise: Jacobi and Gauss-Seidel compared

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Gauss-Seidel method

Jacobi method Convergence and spectral radius Exercise: Jacobi and Gauss-Seidel compared

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Convergence and spectral radius

Jacobi method Gauss-Seidel method Exercise: convergence by spectral radius

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Over-relaxation methods (SOR)

Gauss-Seidel method Convergence and spectral radius

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Derivation: Jacobi iteration matrix

Jacobi method

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Exercise: Jacobi and Gauss-Seidel compared

Jacobi method Gauss-Seidel method

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Exercise: convergence by spectral radius

Convergence and spectral radius

Nonlinear equations

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Nonlinear equations: the problem and iterative methods

Bisection method Fixed-point iteration Newton-Raphson method Derivative-free methods: secant and Steffensen Convergence order and efficiency High-order methods: Halley, Traub, Ostrowski and Jarratt Errors in numerical computing Exercise: numerical comparison of iterative methods

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Bisection method

Nonlinear equations: the problem and iterative methods Derivation: bisection error bound Newton-Raphson method Exercise: bisection by hand

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Fixed-point iteration

Nonlinear equations: the problem and iterative methods Derivation: fixed-point convergence and order Newton-Raphson method Convergence order and efficiency

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Newton-Raphson method

Fixed-point iteration Taylor and the truncation error Derivation: Newton-Raphson and its quadratic order Derivative-free methods: secant and Steffensen High-order methods: Halley, Traub, Ostrowski and Jarratt Exercise: Newton on x=cos²x Newton for nonlinear systems Bisection method Adams-Moulton methods

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Derivative-free methods: secant and Steffensen

Newton-Raphson method Newton interpolation and divided differences Exercise: the secant by hand Newton for nonlinear systems

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Convergence order and efficiency

Fixed-point iteration High-order methods: Halley, Traub, Ostrowski and Jarratt Exercise: numerical comparison of iterative methods Convergence, consistency and order Derivation: Newton-Raphson and its quadratic order Derivation: fixed-point convergence and order Exercise: Newton on x=cos²x

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High-order methods: Halley, Traub, Ostrowski and Jarratt

Newton-Raphson method Convergence order and efficiency Exercise: numerical comparison of iterative methods Derivation: Newton-Raphson and its quadratic order Heun's method Derivation: trapezoid rule Derivation: Simpson 1/3

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Derivation: Newton-Raphson and its quadratic order

Taylor and the truncation error Newton-Raphson method Derivation: fixed-point convergence and order Derivation: Euler's method and its order High-order methods: Halley, Traub, Ostrowski and Jarratt

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Derivation: bisection error bound

Bisection method

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Derivation: fixed-point convergence and order

Fixed-point iteration Taylor and the truncation error Derivation: Newton-Raphson and its quadratic order

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Exercise: bisection by hand

Bisection method Derivation: bisection error bound Exercise: Newton on x=cos²x

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Exercise: Newton on x=cos²x

Newton-Raphson method Convergence order and efficiency Exercise: bisection by hand

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Exercise: the secant by hand

Derivative-free methods: secant and Steffensen Exercise: Newton on x=cos²x

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Exercise: numerical comparison of iterative methods

High-order methods: Halley, Traub, Ostrowski and Jarratt Convergence order and efficiency

Nonlinear systems

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Systems of nonlinear equations

Nonlinear equations: the problem and iterative methods Convergence order and efficiency Newton for nonlinear systems Cost and efficiency in dimension n High order in systems: Traub, Golden Ratio, NA, Jarratt and RN Linear systems: error, residual and conditioning Derivation: fixed-point convergence and order

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Newton for nonlinear systems

Newton-Raphson method Systems of nonlinear equations Derivation: Newton for systems by linearization Linear systems: error, residual and conditioning High order in systems: Traub, Golden Ratio, NA, Jarratt and RN Exercise: Newton for a system, by hand Exercise: Newton on a 2×2 system with iteration table Cost and efficiency in dimension n High-order methods: Halley, Traub, Ostrowski and Jarratt

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Cost and efficiency in dimension n

Newton for nonlinear systems Convergence order and efficiency High order in systems: Traub, Golden Ratio, NA, Jarratt and RN Exercise: numerical comparison on systems

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High order in systems: Traub, Golden Ratio, NA, Jarratt and RN

Newton for nonlinear systems Cost and efficiency in dimension n High-order methods: Halley, Traub, Ostrowski and Jarratt Exercise: numerical comparison on systems

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Derivation: Newton for systems by linearization

Derivation: Newton-Raphson and its quadratic order Newton for nonlinear systems Exercise: Newton on a 2×2 system with iteration table

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Exercise: Newton for a system, by hand

Newton for nonlinear systems Exercise: Newton on a 2×2 system with iteration table

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Exercise: Newton on a 2×2 system with iteration table

Newton for nonlinear systems Derivation: Newton for systems by linearization Exercise: numerical comparison on systems

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Exercise: numerical comparison on systems

High order in systems: Traub, Golden Ratio, NA, Jarratt and RN Cost and efficiency in dimension n Exercise: numerical comparison of iterative methods

Numerical methods wiki map

Most referenced pages

Foundations

Interpolation

Differentiation

Integration

ODEs

Linear systems

Nonlinear equations

Nonlinear systems