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Numerical Analysis and Scientific Computing
Explore errors and equation solving; approximation and quadrature; linear systems and differential equations. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Ordinary differential equations
An ODE relates an unknown function of one variable to its derivatives.
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Errors and equation solving
Floating-point arithmetic
Finite-precision arithmetic introduces rounding and has bounded representable ranges.
Video explanations to be addedConditioning
Conditioning measures sensitivity of a problem's exact solution to perturbations in its data.
Video explanations to be addedNumerical stability
A stable algorithm controls amplification of rounding and data errors relative to the problem.
Video explanations to be addedBisection method
Continuity and opposite endpoint signs allow repeated halving to bracket a root.
Video explanations to be addedNewton's method
A tangent-based iteration approximates a root but convergence depends on the initial point and local hypotheses.
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Approximation and quadrature
Polynomial interpolation
Distinct nodes determine a unique polynomial of bounded degree passing through their data.
Video explanations to be addedSplines
Piecewise polynomials joined with specified smoothness approximate data and curves.
Video explanations to be addedNumerical quadrature
Weighted function samples approximate definite integrals with error depending on smoothness and sampling.
Video explanations to be addedFinite differences
Differences of nearby samples approximate derivatives, balancing truncation and roundoff errors.
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Linear systems and differential equations
LU factorization
With appropriate pivoting, triangular factors reduce linear-system solution to substitution.
Video explanations to be addedIterative linear solvers
Repeated approximations solve large linear systems when the iteration satisfies convergence conditions.
Video explanations to be addedEuler method
Euler's method advances an ODE solution using its current derivative and a step size.
Video explanations to be addedRunge–Kutta methods
Multiple derivative evaluations per step improve ODE approximation accuracy.
Video explanations to be addedStiff equations
Widely separated decay scales can force explicit methods to take very small stable steps.
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