MAT5603

Catalog entry

Prerequisite: MAT1223, MAT2233 (or MAT2253), or consent of instructor.

Contents Rigorous introduction to the fundamental methods of numerical analysis, emphasizing both theoretical foundations and practical applications. Areas of study include the solution of nonlinear equations, interpolation and curve fitting, numerical differentiation and integration, Fourier transforms, and optimization.

Scaffolded Numerical Analysis Curriculum (15 Weeks)

Phase 1: Foundation - Numbers and Errors (Weeks 1-3)

Topics

• Computer arithmetic and floating-point representation
• Error types and propagation
• Condition numbers and stability
• Root-finding for single variables

Student Learning Outcomes (SLOs)

• Explain IEEE 754 floating-point representation and identify representation limitations
• Calculate and distinguish between absolute, relative, forward, and backward errors
• Compute condition numbers and assess problem sensitivity to input perturbations
• Implement and analyze convergence of bisection, fixed-point iteration, and Newton's method
• Select appropriate root-finding methods based on problem characteristics and convergence requirements

Scaffolding

• Establishes error analysis framework used throughout entire course
• Introduces iteration and convergence concepts that extend to matrix methods
• Newton's method foundation for later multivariable extensions

Weekly Content

Week 1: Computer Arithmetic

• IEEE 754 standard, machine epsilon, rounding errors
• Sauer: Chapter 0
• Burden & Faires: Chapter 1.2
• Quarteroni et al.: Chapter 2

Week 2: Error Analysis

• Absolute vs relative error, forward vs backward error analysis
• Condition numbers of problems and algorithms
• Sauer: Chapter 0
• Kincaid & Cheney: Chapter 2

Week 3: Single Variable Root-Finding

• Bisection method, fixed-point iteration, Newton's method, secant method
• Convergence analysis and rates
• Sauer: Chapter 1
• Burden & Faires: Chapter 2
• Quarteroni et al.: Chapter 6

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Phase 2: Linear Foundations (Weeks 4-7)

Topics

• Direct methods for linear systems
• Matrix decompositions (LU, Cholesky)
• Iterative methods for linear systems
• Convergence theory for iterative methods

Student Learning Outcomes (SLOs)

• Perform Gaussian elimination with pivoting and explain stability considerations
• Implement LU decomposition and apply to multiple right-hand sides efficiently
• Estimate condition numbers and predict solution sensitivity
• Implement Jacobi, Gauss-Seidel, and SOR methods
• Analyze convergence of iterative methods using spectral radius
• Choose between direct and iterative approaches based on problem characteristics

Scaffolding

• Extends error analysis to linear algebra settings
• Matrix iteration concepts build on scalar fixed-point theory from Phase 1
• Linear system solving becomes essential tool for subsequent interpolation and optimization phases
• Condition number analysis connects to Phase 1 concepts

Weekly Content

Week 4: Gaussian Elimination and LU Decomposition

• Gaussian elimination, partial pivoting, LU factorization
• Sauer: Chapter 2.1-2.3
• Burden & Faires: Chapter 6.1-6.2
• Quarteroni et al.: Chapter 3

Week 5: Matrix Conditioning and Special Systems

• Condition numbers for matrices, Cholesky decomposition
• Sauer: Chapter 2.4
• Burden & Faires: Chapter 6.6
• Kincaid & Cheney: Chapter 4

Week 6: Iterative Methods - Basic

• Jacobi and Gauss-Seidel methods, matrix splitting
• Sauer: Chapter 2.5
• Burden & Faires: Chapter 7.1-7.2
• Quarteroni et al.: Chapter 4

Week 7: Iterative Methods - Convergence Theory

• Spectral radius, SOR method, convergence analysis
• Sauer: Chapter 2.5
• Burden & Faires: Chapter 7.3
• Quarteroni et al.: Chapter 4

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Phase 3: Interpolation and Approximation (Weeks 8-11)

Topics

• Polynomial interpolation (Lagrange, Newton forms)
• Interpolation error analysis
• Spline interpolation
• Chebyshev interpolation and optimal node placement

Student Learning Outcomes (SLOs)

• Construct interpolating polynomials using Lagrange and Newton divided difference forms
• Derive and apply interpolation error bounds
• Explain Runge phenomenon and strategies to avoid it
• Implement natural and clamped cubic splines by solving tridiagonal systems
• Apply Chebyshev nodes to minimize interpolation error
• Select appropriate interpolation methods based on data characteristics and accuracy requirements

Scaffolding

• Uses linear system solving from Phase 2 for spline coefficient determination
• Error analysis from Phase 1 extends to approximation errors
• Establishes approximation theory foundation for later least squares and optimization
• Condition number concepts apply to interpolation matrix analysis

Weekly Content

Week 8: Polynomial Interpolation

• Lagrange interpolation, Newton divided differences
• Sauer: Chapter 3.1-3.2
• Burden & Faires: Chapter 3.1-3.2
• Quarteroni et al.: Chapter 8

Week 9: Interpolation Error Analysis

• Error bounds, Runge phenomenon, Chebyshev nodes
• Sauer: Chapter 3.3
• Burden & Faires: Chapter 3.3
• Quarteroni et al.: Chapter 8

Week 10: Spline Interpolation - Construction

• Piecewise linear interpolation, cubic splines
• Sauer: Chapter 3.4
• Burden & Faires: Chapter 3.5
• Quarteroni et al.: Chapter 8

Week 11: Spline Interpolation - Applications

• Natural, clamped, and periodic boundary conditions
• Implementation and applications
• Sauer: Chapter 3.4
• Burden & Faires: Chapter 3.5
• Kincaid & Cheney: Chapter 6

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Phase 4: Calculus Operations and Data Fitting (Weeks 12-15)

Topics

• Numerical differentiation and Richardson extrapolation
• Numerical integration (Newton-Cotes, composite rules)
• Least squares approximation
• Introduction to nonlinear systems

Student Learning Outcomes (SLOs)

• Derive finite difference formulas and analyze their error behavior
• Apply Richardson extrapolation to improve derivative approximations
• Implement trapezoid, Simpson's, and composite integration rules
• Derive error bounds for numerical integration methods
• Solve linear least squares problems using normal equations and QR decomposition
• Extend Newton's method to nonlinear systems of equations
• Connect optimization concepts to earlier root-finding methods

Scaffolding

• Numerical differentiation uses interpolation polynomials from Phase 3
• Integration connects to interpolation through Newton-Cotes derivation
• Error analysis from Phase 1 applied to differentiation and integration
• Least squares requires linear system solving from Phase 2
• Multivariable Newton's method extends single-variable version from Phase 1
• Sets foundation for optimization methods in advanced courses

Weekly Content

Week 12: Numerical Differentiation

• Finite differences, Richardson extrapolation, error analysis
• Sauer: Chapter 5.1
• Burden & Faires: Chapter 4.1
• Quarteroni et al.: Chapter 10

Week 13: Numerical Integration

• Trapezoid rule, Simpson's rule, composite formulas
• Sauer: Chapter 5.2-5.3
• Burden & Faires: Chapter 4.2-4.3
• Quarteroni et al.: Chapter 10

Week 14: Least Squares and Data Fitting

• Linear least squares, normal equations, QR approach
• Sauer: Chapter 4.1-4.2
• Burden & Faires: Chapter 8.1
• Quarteroni et al.: Chapter 9

Week 15: Nonlinear Systems

• Newton's method for systems, Jacobian matrices
• Sauer: Chapter 4.3
• Burden & Faires: Chapter 10.2
• Quarteroni et al.: Chapter 6

Final Assessment

Final Exam Period: Comprehensive assessment covering error analysis, linear systems, interpolation, and calculus operations with emphasis on method selection and scaffolded problem-solving.

Key Scaffolding Connections

• Error Analysis (Phase 1) threads through everything
• Linear Systems (Phases 2) are used in interpolation, least squares, and nonlinear systems
• Newton's Method evolves: single variable → systems
• Iteration Theory develops: fixed-point → linear systems
• Approximation builds: interpolation → integration → differential equations
• Optimization Connection: Least squares introduces objective function minimization