Numerical methods and dynamical systems
Approximate roots and integrals with explicit error checks, analyze discrete updates, and distinguish numerical instability from modeled continuous behavior.
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Recommended preparation and ways to catch up
- Evaluate functions, manipulate a linear equation and read signed graphs.
- Derivatives and integrals are explained in the calculus course; direct entry includes a local recap.
- Guided calculations require no programming; independent implementation is an optional deeper deliverable.
Find a foundation course · Open contextual refreshers. Each investigation includes a short recap; preparation never locks access.
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Your units and lessons
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Approximate with a checkable bound
Preserve root brackets and compare quadrature rules against known exact values.
Trap a root before approximating it
What can a shrinking interval prove about an approximate answer?
Approximate accumulation without hiding its error
How do left rectangles, midpoint rectangles and trapezoids differ?
Read a density as probability per unit
Connect continuous probability, area, Gaussian scale and numerical integration.
Understand repeated updates
Find fixed points and classify contraction, oscillation and perturbation growth.
A repeated rule can settle, alternate or grow
What determines whether errors around a fixed point shrink?
Approximate continuous dynamics
Relate a differential equation to time steps and test stability separately from accuracy and model validity.
A decaying equation can have a growing approximation
When does a time step invent behavior absent from the differential equation?
Connect dynamics estimates and constraints
Step a feedback system and separate numerical accuracy from stability and measurement assumptions.
Build a numerical result you can audit
Prepare a reproducible numerical investigation combining an approximation method and a time-evolution model. Declare tolerance, workload and stopping rules, compare with an exact or independently checked case, and demonstrate a failure caused by an invalid setup or coarse step. Use at least two mechanisms; a smooth-looking plot is not validation.
Build from lesson evidence, compare alternatives, write an artifact, and tackle unfamiliar numerical cases. Use the rubric with a teacher, peer or colleague to assess the quality of your reasoning.
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