Scientific computing¶

Open Frontier

What you will develop¶

Know conditioning, numerical stability, approximation, iterative solution, ODE/PDE discretization and computational verification.

Before you begin¶

Recorded prerequisites:

• Calculus + vector calculus
• Probability

Study and practice¶

Progress by the work you can do, not by a fixed calendar.

Practice sequence 29¶

Spine: Driscoll & Braun, Fundamentals of Numerical Computation

Reading: Ch. 1-3: floating point; linear systems; overdetermined systems

Know: Distinguish model error, conditioning and algorithmic error; solve linear/least-squares problems stably.

Reconstruct: Derive condition number intuition and backward-error concept; explain why normal equations square condition number.

Do: Construct a near-singular calibration problem and compare naive inversion, QR and SVD behavior.

Defend: What does it mean for a numerical answer to be backward stable?

Gate: diagnose whether an error comes from data, model, conditioning or algorithm.

Source: source

Practice sequence 30¶

Spine: Driscoll & Braun, FNC

Reading: Ch. 4 roots/nonlinear equations; Ch. 5 interpolation, finite differences, integration

Know: Implement Newton/secant, interpolation, finite differences and quadrature with convergence/error checks.

Reconstruct: Derive Newton from local linearization; derive finite-difference truncation order with Taylor expansion.

Do: Build a derivative/integral routine that adaptively selects resolution and reports an error estimate.

Defend: Why can higher formal order perform worse in finite precision?

Gate: implement two methods for same quantity, compare convergence and explain discrepancy.

Source: source

Practice sequence 31¶

Spine: Driscoll & Braun, FNC

Reading: Ch. 6 IVPs; Ch. 7 matrix analysis (SVD/eigen); selected Ch. 8 Krylov

Know: Integrate ODEs adaptively; connect eigen/SVD analysis to computational behavior and large systems.

Reconstruct: Derive Euler local error and Runge-Kutta idea; derive power iteration intuition.

Do: Simulate a stiff vs non-stiff dynamical system and document solver failure/success regimes.

Defend: How can a physically stable system still be numerically unstable?

Gate: Closed-book method comparison + one solver implementation from pseudocode.

Source: source

Practice sequence 32¶

Spine: Driscoll & Braun, FNC

Reading: Ch. 10-13: BVPs, diffusion, advection, 2D PDEs

Know: Discretize continuum models and reason about stability, stiffness, convergence and boundary conditions.

Reconstruct: Derive a finite-difference Laplacian and method-of-lines semi-discretization.

Do: Solve 1D heat diffusion and advection; show one unstable discretization and diagnose it.

Defend: What makes a discretization a model in its own right?

Gate: Module gate: turn one PDE into a computational experiment with grid-refinement and conservation checks.

Source: source

Exit gate¶

Closed-book: 120 min coding: conditioning, root solve, integration, ODE solve, eigen/SVD, PDE discretization.

Novel problem: Given a result that changes with mesh/tolerance/precision, diagnose the numerical cause.

Artifact: Build a reproducible solver comparison with convergence plots and conservation/residual checks.

Defend: Explain backward error, conditioning, stability, stiffness and discretization error.

Pass criterion: Pass if learner can separate mathematical/model error from floating-point/algorithm/discretization error.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

1. Floating point: Find an algebraically equivalent expression that is numerically much worse due to cancellation.

2. Condition number: Create Ax=b with large condition number and quantify solution amplification.

3. Root finding: Compare bisection/Newton/secant on a difficult root problem.

4. Interpolation: Demonstrate Runge-style instability or another interpolation pathology.

5. Finite difference: Estimate derivative vs step size and identify truncation vs roundoff regimes.

6. Quadrature: Integrate a sharply varying function adaptively and compare with uniform grid.

7. ODE solver: Compare Euler and adaptive Runge-Kutta on a stiff-ish toy problem.

8. Eigen/SVD: Use power iteration/SVD on a structured matrix and validate residuals.

9. PDE diffusion: Solve heat equation under grid refinement and check stability/convergence.

10. PDE advection: Create an unstable/oscillatory advection discretization and repair it.

Textbooks¶

See the five-book resource page.

Reading options¶

Assigned readings, where available, appear in the practice sequence above. These additional recommendations are not all required reading.

• Numerical Linear Algebra

Continue¶

Connections above reflect recorded prerequisites, not a compulsory calendar.

Integrated practice | Choose a study route