Introduction to Computational Mathematics

Spring 2026

EN.553.385 at Johns Hopkins: floating point, linear systems, least squares, root finding, interpolation and ODEs.
Published

January 20, 2026

Syllabus
(PDF)
Textbook
T. A. Driscoll and R. J. Braun, Fundamentals of Numerical Computation, freely available online. The course follows Chapters 1 to 6.
Class notes
(OneNote, written in class)
Final project
guidelines, project ideas and past projects

About the course

This course offers a broad introduction to numerical computation in the mathematical sciences. Topics include floating point numbers, linear systems, LU factorization, vector and matrix norms, conditioning of linear systems, QR factorizations, the root finding problem, fixed point iterations, Newton’s method in one variable and for nonlinear systems, interpolation, cubic splines, finite differences, numerical integration, initial-value problems for ODEs, Euler’s method, systems of differential equations and Runge-Kutta methods; time permitting, shooting methods for boundary-value problems and topics in numerical optimization.

This is a rigorous mathematics course. The focus is on the mathematical analysis of algorithms: their theoretical foundations, convergence properties, stability and error bounds. Programming assignments, submitted as Jupyter notebooks, are an essential component, but this is not a programming course.

Prerequisites: linear algebra and multivariable calculus at an advanced undergraduate level; some programming experience in Python.

Course outline

The course has six units, each with homework assignments and weekly discussion sections.

  1. Introduction: floating point arithmetic, conditioning and stability
  2. Linear algebra: linear systems, LU factorization, norms and conditioning
  3. Ordinary least squares: normal equations, QR factorization
  4. Root finding: fixed point iteration, Newton’s method, nonlinear systems
  5. Piecewise approximations: interpolation, cubic splines, finite differences, numerical integration
  6. Ordinary differential equations: initial-value problems, Euler’s method, Runge-Kutta methods

The semester ends with a group final project in place of a final exam.

Final project

In place of a final exam, students complete a final project in groups of two or three. Every project balances three elements: learning a method or application beyond the syllabus, analysis that tests its boundaries, and research, a creative modification of the model or algorithm that is then evaluated.

Timeline
Milestone When
In-class group work sessions April 21 and 23
Project proposal End of class on April 23
Presentations May 5 or May 8
Final submission May 8

During the two in-class work sessions I give an overview of the deliverables, answer questions about the requirements, and hand out templates and rubrics for the proposal. Each group develops its proposal, splits up roles and responsibilities, and fills in a check-in sheet summarizing its discussion. The proposal is a concrete project plan with each member’s responsibilities and tasks. All materials are due May 8 regardless of the presentation date, so every group has at least two weeks to complete its project.

Presentation

Slides are assessed on four criteria:

  • Narrative clarity: does the presentation tell a clear, logical story that assumes only the knowledge covered in the course?
  • The “delta”: is the group’s specific modification or research contribution clearly identified and explained?
  • Visual communication: are the slides clean and professional, using effective visualizations rather than walls of text?
  • Technical command: does the group show a solid understanding of its methods, stay within the time limit and answer questions well?

The final presentation is a chance to share findings with the community built over the semester. The criteria keep everyone accountable, but the goal is a low-stress setting: if you have done the work and can explain it clearly, you will do well.

Project archive

The final submission is a zip file containing a Jupyter notebook and everything it needs to run. The notebook is the heart of the project: it is where the technical rigor and the creative “delta” live. It must run from top to bottom in a clean environment (with numpy, scipy, matplotlib and seaborn installed, plus any packages listed in a requirements.txt or environment.yml), use relative paths, and include all of its data files. It is assessed on:

  • Reproducibility: the notebook executes from top to bottom in a clean environment without manual troubleshooting.
  • Literate programming: Markdown cells give a clear narrative explaining the mathematical intuition behind the code.
  • Algorithmic correctness: the methods and modifications are technically sound and mathematically accurate.
  • The “delta”: the creative modification is clearly coded, tested, and compared against a baseline or standard method.
  • Data visualization: plots and tables demonstrate convergence, error analysis or comparative results.
  • Code quality: intuitive variable names, and complex logic broken into readable, well-commented functions.
  • Rigorous analysis: the method’s boundaries are tested, for example its stability, parameter sensitivity or convergence rate.

Project ideas

These are only starting points: the best projects usually come from your own research interests or curiosity, so modify them, combine them or propose something entirely new. Section numbers refer to Driscoll and Braun, Fundamentals of Numerical Computation.

Terrain Modeling and Elevation Analysis
Interpolate scattered elevation data using splines to create terrain models. Compute derived quantities (slope, aspect, curvature) using finite differences on the interpolated surface, and study accuracy versus data density.
Connection to the course (textbook sections): 5.3 Cubic splines, 5.4–5.5 Finite differences
Newton’s Fractals and Basins of Attraction
Apply Newton’s method to polynomials and transcendental functions in the complex plane to generate fractal basin boundaries. Study how basin structure changes with polynomial degree, explore modified Newton methods with relaxation parameters, and analyze fractal dimension of boundaries.
Connection to the course (textbook sections): 4.3 Newton’s method, 4.5 Newton for systems
Chemical Equilibrium Concentrations
Solve systems of nonlinear equations for equilibrium concentrations in chemical reactions. Use Newton’s method, study convergence dependence on initial guesses, and extend to parameter studies varying temperature or pressure.
Connection to the course (textbook sections): 4.5 Newton for nonlinear systems
Mechanical Oscillators: Pendulums and Springs
Model single and coupled oscillators (pendulum, spring-mass systems) as ODE systems. Extend to nonlinear oscillators (large-angle pendulum), compare numerical solutions with linearized approximations, and study energy conservation.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4 Runge-Kutta methods
Mandelbrot and Julia Sets Visualization
Generate high-resolution visualizations of Mandelbrot and Julia sets using iterative complex mappings. Implement zoom functionality with arbitrary precision, create color schemes based on escape time and orbit behavior, and explore the connection between Mandelbrot set parameters and Julia set shapes.
Connection to the course (textbook sections): 4.1 Fixed-point iteration, 4.3 Convergence analysis
Heat Distribution in a Rod
Discretize the steady-state heat equation on a 1D rod to obtain a linear system. Solve for temperature distribution with various boundary conditions, study how mesh refinement affects accuracy and matrix conditioning.
Connection to the course (textbook sections): 2.3–2.4 Linear systems and LU factorization, 2.8 Conditioning
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Adaptive Integration for Challenging Functions
Implement adaptive quadrature with automatic error estimation and subdivision. Extend to handling integrands with singularities, sharp peaks, or oscillations. Compare adaptive strategy efficiency with fixed high-order methods on a diverse test suite.
Connection to the course (textbook sections): 5.7 Adaptive integration
Orbit Determination from Observations
Estimate orbital parameters from position observations using nonlinear least squares. Study how observation timing and quantity affect parameter accuracy, compare Gauss-Newton with Levenberg-Marquardt, and analyze residual patterns.
Connection to the course (textbook sections): 4.7 Nonlinear least squares
Curve Fitting for Kinetics and Growth Models
Fit kinetic models (Michaelis-Menten, enzyme kinetics) and growth models (logistic, Gompertz) to experimental data using Gauss-Newton and Levenberg-Marquardt. Analyze parameter confidence intervals and compare convergence behavior across methods.
Connection to the course (textbook sections): 4.7 Nonlinear least squares
Audio Signal Fitting and Compression
Fit audio signals with sums of sinusoids using least squares. Study how the number of frequencies affects fit quality and conditioning, compare time-domain versus frequency-domain approaches, and analyze compression tradeoffs.
Connection to the course (textbook sections): 3.1–3.4 Least squares methods
Balancing Chemical Equations
Formulate chemical equation balancing as a linear system and solve for stoichiometric coefficients. Extend to complex reaction networks, study underdetermined systems with free parameters, and analyze solution uniqueness.
Connection to the course (textbook sections): 2.3–2.4 Linear systems and LU factorization
Iterative Method Convergence Visualization
Visualize convergence behavior of iterative methods (Jacobi, Gauss-Seidel, SOR, conjugate gradient) for linear systems. Create animated visualizations showing error evolution in 2D/3D, generate convergence rate heatmaps as parameters vary, and study spectral radius effects on iteration paths.
Connection to the course (textbook sections): 2.5 Iterative methods, 2.7–2.8 Norms and conditioning
Electrical Network Analysis
Model resistor networks as linear systems using Kirchhoff’s laws. Solve for node voltages and branch currents, study how network topology affects matrix structure and conditioning, and analyze sensitivity to component tolerances.
Connection to the course (textbook sections): 2.3–2.4 Linear systems and LU factorization, 2.7–2.8 Norms and conditioning
Mass-Spring System Dynamics
Model coupled mass-spring systems as linear systems for equilibrium positions. Solve for displacements under applied forces, study how spring constants and topology affect conditioning, and extend to analyzing vibrational modes.
Connection to the course (textbook sections): 2.3–2.4 Linear systems and LU factorization, 2.7–2.8 Norms and conditioning
Boundary Value Problems via Shooting Methods
Solve two-point BVPs (e.g., steady-state heat conduction, beam deflection) using single and multiple shooting. Combine ODE integration with Newton’s method for rootfinding, and study sensitivity of shooting to initial guesses and problem stiffness.
Connection to the course (textbook sections): 6.3–6.4 ODE methods, 4.3–4.5 Newton’s method
Intersection of Curves and Surfaces
Find intersection points of implicit curves and surfaces using Newton’s method for nonlinear systems. Study convergence behavior, handle multiple intersections, and extend to ray-surface intersection for simple rendering applications.
Connection to the course (textbook sections): 4.5 Newton for nonlinear systems
Polynomial Interpolation for Data Reconstruction
Implement Newton’s divided differences and barycentric Lagrange interpolation to reconstruct missing data points (e.g., sensor readings, historical records). Compare stability, demonstrate Runge phenomenon, and study convergence with Chebyshev nodes.
Connection to the course (textbook sections): 2.1 Polynomial interpolation, 2.8 Conditioning, 5.1 Interpolation
Phase Portrait and Vector Field Visualization
Create phase portraits for 2D dynamical systems showing trajectories, nullclines, and equilibria. Implement interactive vector field plotting with color-coded magnitudes, visualize limit cycles and separatrices, and study bifurcations as parameters change.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4 Runge-Kutta methods
Image Deblurring via Linear Systems
Model image blur as a linear system and solve for the original image using direct methods. Study how blur kernel size affects conditioning, compare solution methods, and analyze regularization approaches for noisy images.
Connection to the course (textbook sections): 2.3–2.4 Linear systems and LU factorization, 2.8 Conditioning
Camera Calibration from Images
Estimate camera parameters (focal length, position, orientation) from known calibration points using linear least squares, then refine with nonlinear least squares. Study how point configuration affects conditioning and parameter accuracy.
Connection to the course (textbook sections): 3.1–3.3 Least squares methods, 4.7 Nonlinear least squares
Predator-Prey and Ecological Models
Model ecological dynamics (Lotka-Volterra, competing species, SIR epidemics) as ODE systems. Solve with RK methods, compare accuracy, and study qualitative behavior (oscillations, equilibria, bifurcations) as parameters vary.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4 Runge-Kutta methods
Numerical Integration: Algorithms and Applications
Implement composite quadrature rules (trapezoid, Simpson, Gauss-Legendre) and compare on standard test integrals. Extend to computing quantities of interest (arc length, surface area, centers of mass) and analyzing efficiency versus accuracy tradeoffs.
Connection to the course (textbook sections): 5.6 Numerical integration
Chaotic Dynamics: Lorenz System and Double Pendulum
Simulate chaotic systems using RK methods. Extend to quantifying sensitivity to initial conditions, studying how numerical error affects long-time behavior, and comparing adaptive versus fixed-step methods for chaotic trajectories.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4–6.5 Runge-Kutta methods
GPS Positioning via Least Squares
Estimate position from satellite range measurements using linearized least squares and iterate via Gauss-Newton. Study how satellite geometry affects accuracy (geometric dilution of precision) and analyze sensitivity to measurement noise.
Connection to the course (textbook sections): 3.1–3.3 Least squares methods, 4.7 Nonlinear least squares
Cubic Splines for Curve and Surface Modeling
Use cubic splines to interpolate parametric curves (2D paths, fonts, animations). Extend to tensor-product spline surfaces for terrain or shape modeling. Study how parameterization and knot placement affect curve/surface quality.
Connection to the course (textbook sections): 5.3 Cubic splines
Projectile Motion and Trajectory Optimization
Model projectile motion with realistic drag models as ODE systems. Extend to solving targeting problems using shooting methods (combining ODE integration with rootfinding) to determine launch parameters for desired endpoints.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4 Runge-Kutta methods, 4.3 Newton’s method
Heat Equation Animation and PDE Visualization
Solve time-dependent heat equation using method of lines (spatial discretization + ODE solver). Create animated visualizations of temperature evolution with various boundary conditions and initial states. Study stability, accuracy, and visual phenomena like diffusion smoothing.
Connection to the course (textbook sections): 6.3–6.4 ODE systems and methods, 5.4 Finite differences
Data Linearization and Nonlinear Model Fitting
Fit nonlinear models (exponential, power law, logistic) to data using linearization techniques and compare with direct nonlinear least squares (Gauss-Newton). Analyze when linearization introduces bias and compare parameter accuracy.
Connection to the course (textbook sections): 3.1–3.3 Least squares methods, 4.7 Nonlinear least squares
Multivariate Regression and Feature Analysis
Fit multivariate linear models to datasets with multiple predictors. Study conditioning of design matrices with correlated features, compare normal equations versus QR, and analyze feature importance through coefficient stability.
Connection to the course (textbook sections): 3.1–3.4 Least squares methods
Interest Rates and Financial Equations
Solve nonlinear equations arising in finance: internal rate of return, bond yield to maturity, and option implied volatility. Compare rootfinding methods and study conditioning of these problems near economically meaningful solutions.
Connection to the course (textbook sections): 4.1–4.4 Rootfinding methods
Adaptive ODE Solvers: Error Control and Efficiency
Implement adaptive RK methods using embedded pairs (e.g., RK4(5)) with step size control. Extend to comparing PI/PID controllers for step selection and testing on problems with shocks, rapid transients, or near-discontinuities.
Connection to the course (textbook sections): 6.5 Adaptive Runge-Kutta
Font Design and Glyph Interpolation
Design letter shapes using cubic splines with control over endpoint derivatives. Study how boundary conditions affect letterform aesthetics, implement smooth connections between strokes, and create interpolations between font weights.
Connection to the course (textbook sections): 5.3 Cubic splines
Bézier Curves and Animation Paths
Implement Bézier curves using de Casteljau’s algorithm for computer graphics and animation. Create interactive curve editors with control point manipulation, generate smooth animation paths with velocity control, and compare with spline interpolation for path planning.
Connection to the course (textbook sections): 5.1–5.2 Interpolation and approximation, 5.3 Cubic splines
Motion Capture and Trajectory Smoothing
Fit smooth curves to noisy motion capture data using least squares with polynomial or spline basis functions. Compare fitting approaches, study regularization to enforce smoothness, and analyze residual patterns.
Connection to the course (textbook sections): 3.1–3.4 Least squares methods, 5.3 Cubic splines
Numerical Integration for Probability and Statistics
Apply numerical integration to compute probabilities, expected values, and moments for distributions without closed-form CDFs. Extend to comparing quadrature efficiency for different distribution shapes and tail behaviors.
Connection to the course (textbook sections): 5.6–5.7 Numerical integration
Work and Energy Calculations
Compute work done by variable forces along paths using numerical integration. Extend to computing potential energy functions, comparing quadrature methods for smooth versus discontinuous force profiles, and analyzing error propagation to energy estimates.
Connection to the course (textbook sections): 5.6–5.7 Numerical integration
Orbital Mechanics and Satellite Motion
Simulate two-body orbital motion using RK methods. Extend to studying energy and angular momentum conservation, comparing long-time accuracy of different methods, and modeling orbital maneuvers or perturbations.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4–6.5 Runge-Kutta methods
Fourier Series and Signal Reconstruction Visualization
Visualize how Fourier series approximate periodic functions by animating partial sums with increasing terms. Create interactive visualizations showing frequency spectrum, demonstrate Gibbs phenomenon at discontinuities, and apply to audio waveform analysis and image compression.
Connection to the course (textbook sections): 3.1–3.4 Least squares (Fourier coefficients as projections), 5.1 Approximation theory
Chemical Reaction Kinetics
Model chemical reaction networks as ODE systems using mass-action kinetics. Extend to systems with disparate time scales (fast/slow reactions), comparing explicit and implicit methods, and studying steady-state approximations.
Connection to the course (textbook sections): 6.3 Systems of ODEs, 6.4 Runge-Kutta methods

Student projects

Spring 2026
  • Response transformations: when to linearize and when to optimize nonlinearly
  • Evaluating numerical ODE solvers in a one-compartment pharmacokinetic model
  • Modeling a tennis serve with air resistance using ODEs
  • Least-squares sinusoidal compression for audio denoising
  • A comparative study of root-finding methods for implied volatility under the Black-Scholes model
  • Lucy-Richardson image deblurring: convergence of the accelerated method and a modified variant
  • Simulating Formula 1 laps
  • Image reconstruction using interpolation methods
  • Camera calibration via reprojection error minimization
  • Adams-Bashforth methods for drag ODEs in parachuting
  • Chaos and computation: estimating Lyapunov exponents in the Lorenz system
  • An exponentially-fitted Simpson’s rule for present value estimation
  • RSSI-based localization using the multivariate Newton’s method
  • Support vector machines and the Pegasos algorithm
  • Numerical integration in several variables and on spheres
  • Game map generation using Perlin noise
  • Anchored local volatility for index options: a Crank-Nicolson PDE approach to out-of-sample forecasting
  • Balancing chemical equations