Numerical Analysis
Year 2 · 5 sections · 48 concepts · 106 practice questions
What's covered
free1 · Errors and root finding9 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Floating-point numbers and rounding error
- Example 1.2 · Catastrophic cancellation
- Proposition 1.3 · The bisection method
- Theorem 1.4 · Convergence of fixed-point iteration
- Definition 1.5 · Order of convergence
- Theorem 1.6 · Quadratic convergence of Newton's method
- Proposition 1.7 · The secant method
- Check 1.8 · Check: choosing and testing a root-finder
2 · Interpolation10 concepts
- Overview 2 · The plan of this section
- Theorem 2.1 · Existence and uniqueness of the interpolating polynomial
- Definition 2.2 · Divided differences and the Newton form
- Proposition 2.3 · The Newton form interpolates
- Theorem 2.4 · The interpolation error
- Example 2.5 · Runge's phenomenon
- Proposition 2.6 · Chebyshev nodes minimise the node polynomial
- Definition 2.7 · Cubic splines
- Proposition 2.8 · Existence and properties of the natural spline
- Check 2.9 · Check: choosing an interpolation method
3 · Numerical integration8 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Quadrature rules and degree of exactness
- Theorem 3.2 · Trapezium and Simpson rules
- Corollary 3.3 · Composite rules
- Proposition 3.4 · Richardson extrapolation
- Theorem 3.5 · Gaussian quadrature
- Example 3.6 · Two-point Gauss–Legendre rule
- Check 3.7 · Check: choosing and validating a quadrature rule
4 · Linear systems10 concepts
- Overview 4 · The plan of this section
- Theorem 4.1 · LU factorisation
- Proposition 4.2 · Cost of elimination
- Definition 4.3 · Partial pivoting
- Theorem 4.4 · Cholesky factorisation
- Definition 4.5 · Matrix norms and the condition number
- Theorem 4.6 · Sensitivity of linear systems
- Definition 4.7 · Jacobi and Gauss–Seidel iterations
- Theorem 4.8 · Convergence of stationary iterations
- Check 4.9 · Check: solving a linear system numerically
5 · ODE solvers11 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Euler's method
- Definition 5.2 · Local truncation error and order
- Theorem 5.3 · Convergence of Euler's method
- Definition 5.4 · Runge–Kutta methods
- Proposition 5.5 · Orders of Runge–Kutta methods
- Example 5.6 · Adams–Bashforth multistep methods
- Definition 5.7 · Absolute stability
- Proposition 5.8 · Stiffness
- Example 5.9 · Backward Euler on a stiff problem
- Check 5.10 · Check: choosing and testing an ODE solver
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Proposition 1.3 · The bisection method
Bisection: always converges given a sign change; error halves per step (linear, rate ).
Proof
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