Teaching · Course design
This is one way to turn the book into a semester. It is not a mandate. The useful part is the sequence: understand the problem, expose the algorithm, run it, and then change something important enough that the result might surprise us.
A traditional numerical analysis course can spend much more time on proofs than CMNA does, and there is nothing wrong with that. In that course, this map works best as the computational spine running alongside the analytical material. In a computational methods or scientific computing course, the balance can move farther toward implementation and experiment.
Either way, I would resist the temptation to treat the laboratories as demonstrations we perform after the lecture. They are more useful when students have to make a prediction first. The computer should settle an argument, not start one.
Reading: Chapter 1: Numerical Analysis, efficiency, data types, elementary problems
Start with a problem that has an exact mathematical answer and ask why we might still want an approximation. Then compare two implementations that compute the same thing differently.
Assignment seed: Compare naive and improved implementations for summation or polynomial evaluation. Record when the answers begin to separate.
Reading: Chapter 2: Accuracy, precision, floating point, roundoff, loss of significance, stability
Make floating-point arithmetic visibly disagree with real arithmetic. Do not rescue the example too quickly.
Laboratory: Finite differences
Assignment seed: Find a calculation that becomes worse when a numerical parameter is made smaller. Explain the mechanism.
Reading: Chapter 3.1–3.2: Vectors, matrices, row operations, Gaussian elimination, tridiagonal systems
Work one elimination far enough by hand that the row operations are concrete, then read the R implementation as the same sequence.
Assignment seed: Solve one system by row reduction, then verify it with R. Explain any difference between the algorithm and the production function.
Reading: Chapter 3.3–3.5: LU, Cholesky, Jacobi, Gauss-Seidel, least squares
Put Jacobi and Gauss-Seidel on the same system and ask students to predict which residual falls faster before running either method.
Laboratory: Jacobi vs. Gauss-Seidel
Assignment seed: Modify a well-behaved system until one iterative method converges poorly or fails to converge. Diagnose the matrix, not the symptoms.
Reading: Chapter 4: Polynomial, piecewise, spline, Bézier, multidimensional interpolation
Use the same data for a global polynomial, piecewise linear interpolation, and a spline. Move one point and watch which parts of each model move with it.
Laboratory: Interpolation comparison
Assignment seed: Construct a data set for which the global polynomial behaves badly between otherwise ordinary points.
Reading: Chapter 5.1: Finite differences and second derivatives
Begin with a secant line and shrink h. Ask when the secant should become the tangent, then let floating point complicate the answer.
Laboratory: Finite differences
Assignment seed: For a function with a known derivative, sweep h across several orders of magnitude and explain the error curve.
Reading: Chapter 5.2: Multipanel rules and Newton-Cotes error
Draw the geometry of midpoint, trapezoid, and Simpson rules before comparing their numerical results.
Laboratory: Simpson's rule
Assignment seed: Choose an integrand where increasing the panel count matters a great deal, and another where it hardly matters at all.
Reading: Chapter 5.3–5.4: Gaussian quadrature, adaptive integration, Romberg, Monte Carlo
Put Gaussian quadrature and Monte Carlo next to each other. One carefully chooses points; the other chooses them randomly. Both are trying to spend function evaluations intelligently.
Laboratory: Gaussian quadrature · Monte Carlo
Assignment seed: Use the same smooth integrand with Simpson, Gauss-Legendre, and Monte Carlo. Compare error against the number of function evaluations.
Reading: Chapter 6.1: Bisection, Newton, secant
Ask what information each method requires before discussing speed. A method that cannot legally start does not get credit for theoretical convergence.
Laboratory: Root-finder comparison
Assignment seed: Find one function where Newton is dramatically faster than bisection and another where the starting value makes Newton embarrassing.
Reading: Chapter 6.2: Golden-section search and gradient descent
Compare interval reduction with slope following. They solve related problems but ask the function for different information.
Laboratory: Golden-section search
Assignment seed: Use a multimodal function to show why an interval or starting point can decide which extremum we find.
Reading: Chapter 6.3–6.4: Multidimensional gradient descent, hill climbing, simulated annealing, applications
Put gradient descent and simulated annealing on the same landscape. One follows local slope; the other is sometimes permitted to make a worse move.
Laboratory: Gradient descent · Simulated annealing
Assignment seed: Change only the starting point or random seed and explain why the final answers differ.
Reading: Chapter 7.1: Euler, Runge-Kutta, linear multistep methods
Advance one Euler step by hand, then unpack the four slopes in RK4. The difference between the methods should be visible before it is numerical.
Laboratory: Euler, midpoint, and RK4
Assignment seed: Hold the differential equation fixed and vary h. Compare how the three methods respond.
Reading: Chapter 7.2: Systems of ODEs and boundary value problems
Use a coupled system to show that the numerical method has not fundamentally changed; only the state being advanced has.
Laboratory: R Workbench
Assignment seed: Implement or adapt a small system example and inspect how an error in one component propagates through the rest of the state.
Reading: Chapter 7.3–7.4: Heat equation, wave equation, applications
Run a stable discretization first. Then violate the stability condition on purpose. Students should see the numerical solution fail before we ask them to describe why.
Laboratory: Heat equation · Wave equation
Assignment seed: Construct one stable and one unstable discretization for the same initial condition. Explain the difference in terms of the numerical scheme.
Compressing the course
For an accelerated course, I would combine Weeks 1 and 2, combine the two linear algebra weeks, combine differentiation with Newton-Cotes integration, combine root finding with one-dimensional optimization, and treat the final two differential-equation weeks as a single unit. Interpolation deserves its own week because the distinction between global and local models is too useful to rush.
The thing I would not compress is the prediction step. If time is short, remove an algorithm before removing the opportunity to ask what an algorithm should do. Coverage is easy to measure and remarkably easy to overvalue.
