NE 415 Notes

These draft notes support NE 415 - Introduction to Engineering Analysis.

• Prologue
• Lesson 1 - An Introduction to Python, Spyder, and Jupyter
  • Overview, Objectives, and Key Terms
  • Overview of Python
  • Python Distributions
  • Accessing Python
  • Basic Types
  • On Naming Variables
  • Further Reading
• Lesson 2 - Basic Operators and Common, Built-In Functions
  • Overview, Objectives, and Key Terms
  • Python Operators
  • Exploring Built-In Functions
  • A First Program
  • Further Reading
• Lesson 3 - One-Dimensional Data Processing with NumPy and Matplotlib
  • Overview, Objectives, and Key Terms
  • NumPy and One-Dimensional Arrays
  • Diving into pyplot
  • Data Input and Output via NumPy
  • Further Reading
• Lesson 4 - Two-Dimensional Data Processing with NumPy and Matplotlib
  • Overview, Objectives, and Key Terms
  • Making Two-Dimensional Arrays
  • Visualizing 2-D Arrays
  • Slicing
  • Matrix-Vector and Matrix-Matrix Operations
  • Further Reading
• Lesson 5 - Monte Carlo First
  • Objectives
  • Why Monte Carlo first?
  • Models, simulations, and observables
  • Deterministic and stochastic views
  • A first physical example: attenuation through a slab
  • Random numbers as computational coins
  • Probability is not observed frequency
  • The Monte Carlo pattern
  • A brief historical note
  • A toy source-detector problem
  • Rules for one neutron history
  • Flowchart before code
  • Python implementation
  • Reading the output
  • What changes the model? What changes the run?
  • What we have already used
  • Deterministic and Monte Carlo views of the same problem
  • Looking ahead
• Lesson 6 - Random Variables and Distributions
  • Objectives
  • From computational coins to random variables
  • Discrete and continuous random variables
  • Common distributions
  • The uniform distribution U(0,1)
  • Expectation and basic rules
  • Looking ahead
• Lesson 7 - Sampling Discrete and Continuous Random Variables
  • Objectives
  • Simulation modules and physical parameters
  • Sampling the exponential distribution via inverse transform
  • Empirical distributions and histograms
  • Incorporating exponential flights into the neutron walk
  • Looking ahead
• Lesson 8 - Statistics and Estimators
  • Objectives
  • Revisiting the simulation
  • Modeling the detector response
  • Estimating the parameter \(p\)
  • Sampling distributions and the Central Limit Theorem
  • Example: repeated simulations and a normal overlay
  • Summary and looking forward
• Lesson 9 - Engineering the Simulation
  • Introduction
  • The simulation-engineering workflow
  • Keep notebooks tidy and reproducible
  • Design functions before writing long simulations
  • Unit tests: small checks with known answers
  • Defensive programming: fail early and clearly
  • Random number generators and reproducibility
  • Flowcharts: making the algorithm visible
  • Repeatable runs
  • Documentation habits for student projects
  • Bridge to Lesson 10
  • References and further reading
• Lesson 10 - From Balance Laws to Linear Systems
  • Why linear systems appear in engineering models
  • A fixed-step walk with absorption
  • A boundary-value variant
  • Putting the balances into matrix form
  • Solving first, opening the box later
  • Residuals as a first quality check
  • A second view: source-to-solution response
  • Exercises
  • End-of-lecture summary
• Lesson 11 - Matrix and Vector Operations
  • Why this chapter follows balance laws
  • Vectors as ordered data
  • Dot products as weighted sums
  • Matrix-vector products: the row view
  • Matrix-vector products: the column view
  • The loop pattern behind matrix-vector multiplication
  • Cost of a dense matrix-vector product
  • Matrix-matrix products
  • Array shapes and noncommutativity
  • Transposes
  • Permutation matrices
  • Residuals revisited
  • Exercises
  • End-of-lecture summary
• Lesson 12 - One Matrix, Many Scenarios
  • The repeated-solve pattern
  • A small example with several sources
  • Elimination: removing coupling step by step
  • Augmented matrices for bookkeeping
  • Back substitution
  • From elimination to LU
  • Pivoting
  • Using SciPy for reusable solves
  • Why not compute the inverse?
  • Checking many solutions at once
  • Exercises
  • End-of-lecture summary
• Lesson 13 - Symbolic Computation with SymPy
  • Why symbolic computation belongs here
  • Symbols and expressions
  • Equations are not assignments
  • A two-cell symbolic balance model
  • Matrix methods in SymPy
  • Substitution and numerical evaluation
  • Simplify, expand, factor, and collect
  • Turning symbolic expressions into numerical functions
  • Symbolic versus numerical workflows
  • Exercises
  • End-of-lecture summary
• Lesson 14 - Norms, Sparse Matrices, and Iterative Methods
  • The basic question
  • Vector norms
  • Residual versus error
  • Why sparsity appears
  • Sparse matrices in SciPy
  • Direct solves and fill-in
  • Never form the inverse
  • From physics to fixed-point iteration
  • Jacobi and Gauss–Seidel as corrected residual updates
  • Convergence as error multiplication
  • Why this leads to subspaces
  • Exercises
  • End-of-lecture summary
• Lesson 15 - Orthogonality, Gram-Schmidt, and Projection
  • From operations to geometry
  • Orthogonality
  • Projection onto a line
  • Projection onto a subspace
  • Least squares as projection
  • Gram–Schmidt
  • Why orthonormal bases are computationally useful
  • Connection to Krylov ideas
  • Connection to eigenvectors
  • Exercises
  • End-of-lecture summary
• Lesson 16 - One Eigenpair from Iteration
  • Starting from Richardson iteration
  • Why repeated multiplication reveals a preferred direction
  • Power iteration
  • What the magnitude tells us
  • Smallest eigenvalue, inverse iteration, and shifted inverse iteration
  • Applications of the dominant eigenpair
  • Exercises
  • End-of-lecture summary
• Lesson 17 - The Whole Spectrum at Once
  • From one eigenpair to the full spectrum
  • Low dimensions: what can be done analytically
  • The full spectral picture
  • Zero eigenvalues and invertibility
  • Computing eigenpairs in practice
  • What about the eigenvectors?
  • Exercises
  • End-of-lecture summary
• Lesson 18 - Eigenvalue Applications
  • What eigenvalues add to the story
  • Discrete-time growth and decay
  • Stability of an iterative method
  • A transport-motivated mode calculation
  • Continuous-time systems
  • Coupled variables move as modes
  • A simplified two-group reactor-physics eigenproblem
  • Subcritical, critical, and supercritical interpretation
  • Full-spectrum versus dominant-mode thinking
  • Communicating an eigenvalue result
  • Exercises
  • End-of-lecture summary
• Lesson 19 - Numerical Differentiation
  • Overview, Objectives, and Key Terms
  • If, in the beginning, \(\Delta\) didn’t go to 0…
  • Back to Order
  • When Smaller Is Not Always Better
  • Approximating the Second Derivative
  • From Derivatives to Gradients
  • Optional Extension: Higher-Order Stencils
  • Further Reading
• Lesson 20 - Numerical Integration
  • Overview, Objectives, and Key Terms
  • Riemann Sums in Practice
  • Beyond the One-Side Sums
  • Library Tools for Sampled Data
  • What About Accuracy?
  • Using quad for Callable Functions
  • Multiple Integrals and Vector-Valued Integrands
  • Numerical Probability Accounting
  • Further Reading
• Lesson 21 - Nonlinear Equations
  • Overview, Objectives, and Key Terms
  • Nonlinear Equations
  • Determining Roots Numerically
  • Derivative Information from Lesson 19
  • Solving Systems of Nonlinear Equations Numerically
  • Further Reading
• Lesson 22 - Unconstrained Optimization
  • Overview, Objectives, and Key Terms
  • Some Nomenclature
  • Finding Extrema
  • The Connection To Nonlinear Functions
  • Solving Unconstrained Optimization Problems
  • Gradients Make Optimization a Root-Finding Problem
  • Preview: Constrained Optimization Comes Next
  • Further Reading
• Lesson 23 - Constrained Optimization
  • Overview, Objectives, and Key Terms
  • Start With the minimize Convention
  • Where linprog Deviates
  • Picking the Right SciPy Tool
  • Case Study 1: When Should We Count an Activated Sample?
  • Boundaries, Active Constraints, and a Tiny Bit of Theory
  • Example 2: Minimum-Cost Shielding With a Nonlinear Dose Response
  • Example 3: A Linear Attenuation-Budget Model for linprog
  • A Practical Decision Rule
  • Reading Solver Output
  • Further Reading
• Lesson 24 - Modeling Data with Least-Squares Fits
  • Overview, Objectives, and Key Terms
  • Noisy Data and \(y = at + b\)
  • What “Best” Means
  • Beyond Linear Models
  • Beyond Polynomial Models
  • Further Reading
• Lesson 25 - Modeling Data with Interpolation
  • Overview, Objectives, and Key Terms
  • The Teeter-Totter: Two Points, Two Weights
  • The 1-D Tool: np.interp
  • Smooth 1-D Interpolation: CubicSpline
  • Regular Grids: Repeat the Same Idea
  • Unstructured Points: Barycentric Weights
  • Tool Choice
  • Further Reading
• Lesson 26 - Advection, Diffusion, and Reaction Models
  • Overview, Objectives, and Key Terms
  • The L10 Picture Still Works
  • Count Beans Before Writing Derivatives
  • The Integral Balance
  • Mechanisms Are Easier When Units Lead
  • From Integral Balance to Continuum Model
  • One Parent Balance, Several Child Models
  • Named-State Balances: U-238 to Pu-239 as a Linear Chain
  • Geometry Changes the Weights, Not the Bookkeeping
  • Finite Differences Enforce the Same Balance
  • A Steady Diffusion-Reaction Row
  • What To Carry Forward
• Lesson 27 - Solving ODEs Symbolically
  • Overview, Objectives, and Key Terms
  • From L26 to L27
  • Symbols Are Not Functions
  • A Reusable Symbolic Scaffold
  • Example 1: Pure Reaction with a Constant Source
  • Example 2: Steady Advection-Reaction in a Flow
  • Ansatz: Guess a Shape, Reduce to Algebra
  • PDE Example: Separation of Variables
  • Boundary Conditions Select the Allowed Modes
  • What SymPy Is Really Buying Us
• Lesson 28 - Solving IVPs with Euler's Method
  • Overview, Objectives, and Key Terms
  • Differential Equations
  • Where Does the Standard First-Order Linear IVP Come From?
  • The General Problem \(y' = f(y(t), t)\)
  • There Once was a Man Named Euler
  • Chopping Up Time
  • A Look at Errors and Stability (Optional)
  • Further Reading
• Lesson 29 - Solving Systems of IVPs
  • Overview, Objectives, and Key Terms
  • A System Is One IVP With More Bookkeeping
  • From One Spring Equation to Two First-Order Equations
  • Euler Methods for a Linear System
  • solve_ivp: The General-Purpose IVP Tool
  • A Fast/Slow Balance System
  • What To Check Every Time
  • Further Reading
• Lesson 30 - Solving BVPs Numerically
  • Overview, Objectives, and Key Terms
  • Chopping Up the BVP
  • A Simple Example
  • Eigenvalue Problems
  • Comment on FDM and Alternatives
  • Further Reading
• Appendix A01 - Algorithms, Flowcharts, and Pseudocode
  • Overview, Objectives, and Key Terms
  • Algorithms
  • Sequence by Example: Computing Grades
  • Selection
  • Iteration
  • Practice Problems
  • Further Reading
• Appendix A02 - Conditional Statements and the Structure of Python Code
  • Overview, Objectives, and Key Terms
  • The Simplest if
  • The else Clause
  • The elif Clause
  • Other Tidbits
  • Further Reading
• Appendix A03 - More on Conditionals
  • Overview, Objectives, and Key Terms
  • Motivating Nested Conditionals - Life Insurance
  • But Do We Need Them?
  • Debugging
  • Further Reading
• Appendix A04 - Loops in Python
  • Overview, Objectives, and Key Terms
  • A Motivating Problem
  • The break Clause
  • Numerical Convergence
  • Further Reading
• Appendix A05 - More on Loops
  • Overview, Objectives, and Key Terms
  • The for Loop
  • A Pythonic Quirk
  • Nested for loops
  • One Last Tidbit: continue
  • Further Reading
• Appendix A06 - Pythonic Containers
  • Overview, Objectives, and Key Terms
  • What is a sequential type?
  • The list Type
  • Mutability
  • The tuple Type
  • The dict Type
  • Further Reading
• Appendix A07 - Processing Text Data
  • Overview, Objectives, and Key Terms
  • Reading Text Files
  • Parsing Strings
  • Putting It All Together: Processing data.txt
  • Writing to File
  • Further Reading
• Appendix A08 - Basics of Functions
  • Overview, Objectives, and Key Terms
  • What is a Function?
  • Functions with Multiple Return Values
  • Functions with Multiple Arguments
  • Function Documentation
  • Functions and Flowcharts
  • Further Reading
• Appendix A09 - More on Functions
  • Overview, Objectives, and Key Terms
  • *args and **kwargs
  • Anonymous Functions
  • Recursive Functions
  • Further Reading
• Appendix A10 - Recursive Functions
  • Overview, Objectives, and Key Terms
• Appendix A11 - Algorithm Complexity and Numerical Experiments
  • Overview, Objectives, and Key Terms
• Appendix A12 - Searching
  • Overview, Objectives, and Key Terms
  • The Basic, Linear Search
  • A Bit About Order
  • Binary Search
  • A Comment on Built-In Functions
  • Further Reading
• Appendix A13 - Sorting
  • Overview, Objectives, and Key Terms
  • Sorting is Hard Work
  • Quantifying that Hard Work
  • Better Sorting for Bigger Data
  • Built-in Sorting
  • Further Reading
• Appendix A14 - Modules and Packages
  • Overview, Objectives, and Key Terms
  • Creating Modules
  • Running Modules as Scripts
  • Using Modules on the Moon
  • Packages
  • Further Reading
• Appendix A15 - Unit Tests and Defensive Programming
  • Overview, Objectives, and Key Terms
  • Be on the Defensive
  • Testing, 1-2-3
  • Further Reading