Courses / Mathematics / Numerical analysis: algorithms, errors and evidence Mathematics · unit version 1 Numerical analysis: algorithms, errors and evidence An undergraduate numerical sequence connecting arithmetic, approximation, solvers and ODE methods with explicit convergence and stability hypotheses. Includes an executable exact-arithmetic adaptive ODE investigation and a sustained project; independent implementation artifacts and proofs require review.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
If f is continuous on [a,b] and f(a) and f(b) have opposite signs, what does this prove—and what does it not prove? A solver needs an existence guarantee distinct from a convergence or uniqueness claim.
Compare reasoning and find a refresher The intermediate value theorem gives at least one zero in (a,b). It does not establish uniqueness or convergence of an arbitrary iteration started nearby.
Review existence and uniqueness arguments
What makes a first-order Taylor approximation useful for predicting a small change? Error estimates and local convergence use the remainder, not only a derivative formula.
Compare reasoning and find a refresher The linear term predicts the leading change, and a remainder estimate quantifies the discrepancy. If the second derivative is bounded by M on the connecting interval, the first-order Taylor remainder has magnitude at most M times the step squared divided by 2.
Review Taylor error bounds
Why can a tiny residual coexist with a large solution error in a linear system? Numerical stability and sensitivity of the underlying problem must be separated.
Compare reasoning and find a refresher For a nonsingular A, error equals A⁻¹ times the residual up to sign. A large inverse norm can magnify a small residual; relative-error bounds also involve the condition number.
Review conditioning
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Numerical analysis: justify the computation and its limits
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Numerical investigation: accepted work and defensible error
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Separate representable numbers, rounding error and exceptional range behavior.
Compute spacing and rounding in a specified toy format. Apply the relative-error model only within its range assumptions. Explain absorption, underflow and the distinction between unit roundoff and spacing at one. 6 practice tasks.
Lesson details and objectives Distinguish absolute and relative sensitivity from errors made by an algorithm.
Derive scalar condition numbers from local perturbations. Interpret absolute versus relative sensitivity near zero. Relate linear-system conditioning to singular directions. 6 practice tasks.
Lesson details and objectives Relate forward error, backward error and conditioning through stable and unstable formulas.
Compute a residual-based backward error. Distinguish intrinsic cancellation from avoidable cancellation. Derive stable reformulations with explicit domains. 6 practice tasks.
Lesson details and objectives Prove bisection convergence and distinguish interval guarantees from residual heuristics.
Maintain a continuous sign-changing bracket. Derive iteration counts with explicit indexing. Diagnose discontinuity and arithmetic stagnation. 6 practice tasks.
Lesson details and objectives Use contraction estimates to prove convergence and computable iteration errors.
Check a self-map and strict contraction. Derive a priori and posteriori error bounds. Distinguish local attraction from global convergence. 6 practice tasks.
Lesson details and objectives Derive Newton iteration, quadratic local error and multiplicity effects.
Compute tangent-based Newton steps. Prove a local quadratic error bound. Diagnose cycles, derivative failure and multiple roots. 6 practice tasks.
Lesson details and objectives Derive secant steps, explain superlinear order and specify a genuinely contracting safeguard.
Compute secant updates and detect a degenerate slope. Explain local superlinear order with its assumptions. Design a bracket-preserving rule with a guaranteed shrink factor. 6 practice tasks.
Lesson details and objectives Construct cardinal polynomials and prove uniqueness and a smooth-function interpolation remainder.
Construct a Lagrange interpolant through distinct nodes. Prove uniqueness using a root count. Derive the remainder formula with its smoothness and distinct-point assumptions. 6 practice tasks.
Lesson details and objectives Derive divided differences, nested evaluation and incremental interpolation.
Construct a divided-difference table. Prove the Newton coefficient recurrence. Evaluate incrementally while distinguishing arithmetic from approximation error. 6 practice tasks.
Lesson details and objectives Analyze cardinal-weight amplification, Runge behavior and Chebyshev node geometry.
Bound data amplification using cardinal weights. Explain why equispaced high-degree interpolation can fail. Derive the Chebyshev node-product bound and distinguish it from a universal convergence claim. 6 practice tasks.
Lesson details and objectives Derive natural cubic spline equations and explain boundary conditions and uniqueness.
Count and impose cubic spline constraints. Derive the tridiagonal equations for nodal second derivatives. Explain boundary-condition choices and error limitations. 6 practice tasks.
Lesson details and objectives Derive QR least squares and distinguish residual minimization from accurate coefficient recovery.
Reduce full-rank least squares to a triangular QR solve. Explain orthogonal residuals and uniqueness. Compare normal equations with QR under finite precision. 6 practice tasks.
Lesson details and objectives Derive composite rules, polynomial exactness and derivative-based error bounds.
Compute midpoint, trapezoid and Simpson approximations. Derive second-order composite bounds from local errors. State Simpson smoothness and mesh hypotheses. 6 practice tasks.
Lesson details and objectives Derive Richardson error estimates and explain adaptive allocation and failure detection.
Derive the Simpson refinement divisor 15. Allocate local tolerance with a global sum bound. Construct counterexamples to trusting sampled error estimates blindly. 8 practice tasks.
Lesson details and objectives Derive pivoted LU, explain growth and connect residuals with componentwise or normwise errors.
Perform elimination with row pivoting. Explain multiplier control and element growth. Interpret residual-based backward error with explicit perturbation conventions. 6 practice tasks.
Lesson details and objectives Derive Jacobi and Gauss–Seidel and prove convergence under spectral or norm conditions.
Construct stationary iteration matrices. Prove a sufficient norm contraction and characterize spectral convergence. Compare Jacobi and Gauss–Seidel updates and stopping evidence. 6 practice tasks.
Lesson details and objectives Analyze power and shifted inverse iteration, residuals and spectral gaps.
Derive power convergence under a strict dominant-eigenvalue hypothesis. Explain how a shift changes inverse-iteration dominance. Use a residual certificate with the correct operator assumptions. 6 practice tasks.
Lesson details and objectives Derive explicit Euler and separate one-step defect from accumulated global error.
Compute Euler steps for a stated initial-value problem. Derive the O(h²) one-step defect. Explain why global error is generally O(h) under stability assumptions. 6 practice tasks.
Lesson details and objectives Derive second-order stages and the classical RK4 update while distinguishing order from stability.
Compute explicit midpoint and Heun steps. Verify second-order cancellation through Taylor expansion. State RK4 stages and test-problem stability polynomial without overclaiming a general proof. 6 practice tasks.
Lesson details and objectives Derive stability regions and distinguish stability restrictions from accuracy requirements.
Compute amplification factors for explicit and implicit methods. Derive Euler stability conditions on real and complex test modes. Explain stiffness and the limits of A-stability. 6 practice tasks.
Lesson details and objectives Prove a one-step error bound and interpret adaptive error control with clear limitations.
Derive a discrete Grönwall-style global error estimate. Distinguish method order from estimator order in step selection. Specify acceptance, rejection and resource limits for adaptive integration. 6 practice tasks.
Lesson details and objectives Combine method choice, independent references, error budgets and implementation evidence.
Select methods from assumptions and failure modes. Construct refinement and residual evidence without overstating it. Write a reproducible error and resource budget with independent checks. 6 practice tasks.
Lesson details and objectives Execute and audit a bounded adaptive Euler step-doubling trace.
Execute and audit a bounded adaptive Euler step-doubling trace. Preserve accepted state on rejection and count all derivative work. 6 practice tasks.
Lesson details and objectives Prove and accumulate an exact global error bound over accepted half steps.
Prove and accumulate an exact global error bound over accepted half steps. Distinguish exact certificates, approximate references and incomplete solves. 6 practice tasks.
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