CAAM 353 Computational Numerical Analysis Spring 2013

general information reading and lectures assignments

Notes are posted in PDF. A free PDF viewer is available for most computer systems from clicking on the the icon shown below.

• Week 1 (1/5-1/8): Introduction to Scientific Computing .
  • Chapter 1 in Michael T. Heath : Scientific Computing .
  • Introduction: Motivating scientific computing examples.
  • Approximation in Scientific Computation: Read SS 1.1, 1.2.
  • Introduction to Computer Arithmetic: Read SS 1.3.1 - 1.3.2 .
  • For decimal to IEEE conversion see decimal_2_binary
  • Suggested reading for more detail: Michael Overton Numerical Computing with IEEE Floating Point Arithmetic, SIAM, 2001.
  
  
• Week 2 (1/15): More on Computer Arithmetic. (1/17): Solving f(x) = 0.
  • Chapters 1 and 5 in Michael T. Heath : Scientific Computing .
  • Floating point arithmetic, round off, cancellation: Read SS 1.3.4 - 1.3.9.
  • Nonlinear Equations in One Dimenson: Read SS 5.5.
    
  • Matlab:
    demo_fl_1.m, demo_fl_2.m, sfl.m, sflop.m.
  • Additional reading:
    • SUN Microsystems Numerical Computation Guide
    • What every computer scientist should know about floating-point arithmetic by D. Goldberg.
    • Christopher Essex, Matt Davison, and Christian Schulzky. Numerical Monsters, SIGSAM Bull., Vol 34, No. 4, 2000, pages 16-32
    • Some disasters attributable to bad numerical computing, compiled by Douglas Arnold.
    • Collection of software bugs, compiled by Thomas Huckle.
  
  
• Week 3 (1/21-1/25): Solving systems of linear equations.
  • Chapters 5 and 2 in Michael T. Heath : Scientific Computing .
  • Review of basic linear algebra concepts and operations: Read SS 2.1.
  • Matrix-Vector formulation of linear systems: Read SS 2.2.
  • Triangular linear systems, LU Decompositon, Pivoting: Read SS 2.4.1-2.4.5
  
  
• Week 4 (1/28-2/1): Solving Systems of Linear Equations
  • Chapter 2 in Michael T. Heath : Scientific Computing .
  • Implementing LU Decompositon and Solving Ax = b : Read SS 2.4.6,2.4.7.
  • Solving Slightly Modified Systems, Re-Using LU: Read SS 2.4.9.
  • Systems with special structure: Read SS 2.5.
  • Matrix and Vector Norms: Read SS 2.3.1.
  
  
• Week 5 (2/5): Understanding and Assessing Accuracy in Ax = b Solution. (2/7): Systems of Nonlinear Equations
  • Chapter 2,5 in Michael T. Heath : Scientific Computing .
  • Sensitivity to data and numerical error: Condition Number, Read SS 2.3.3.
  • Backward Error Analysis: Assessing accuracy from residual and Condition Number, Read SS 2.3.4
  • Solving f(x) = 0 : Read SS 5.1-5.3.
  • Convergence Rates and Stopping Criteria: Read SS 5.4.
  • Systems of Nonlinear Equations: Read SS 5.6.1-5.6.2.
  
  
• Week 6 (2/12): Newton's method in higher dimensions (2/14): Linear Least Squares Problems
  • Chapter 5,3 in Michael T. Heath : Scientific Computing .
  • Safeguarded Newton's method: Read SS 5.6.2.
  • Secant Updating and Newton-like methods: Read SS 5.6.3, 5.6.4
  • Least Squares Problems: Read SS 3.1.
  • The Normal Equations, existence, uniqueness: Read SS 3.2.1, 3.4.1, 3.4.2
  
  
• Week 7 (2/19, 2/21): Linear Least Squares Problems
  • Chapter 3 in Michael T. Heath : Scientific Computing .
  • Orthgonality and Orthogonal Transformation Methods: Read SS 3.2.2, 3.4.5, 3.5.1-3.5.3
  • Rank Deficiency: Read SS 3.5.4.
  • The Singular Value Decomposition(SVD): Read SS 3.6.
  • Low Rank Approximation: Read SS 3.6.1 and Class Notes.
  
  
• Week 8 (2/28, 3/1): Spring break
  
  
• Week 9 (3/5): Regularization for Least Squares (3/7) Eigenvalue Problems
  • Chapter 3,4 in Michael T. Heath : Scientific Computing .
  • Sensitivity and Conditioning: Read SS 3.3.
  • Regularization: Class Notes.
  • The eigenvalue problem: Read SS 4.1,4.2.
  • Power Method and Inverse Iteration:Read SS 4.5.1,4.5.2
  
  
• Week 10 (3/12): QR for Eigenvalues, (3/15): Polynomial Interpolation and Splines
  • Chapter 4,7 in Michael T. Heath : Scientific Computing .
  • The QR Iteration: Read SS 4.5.6.
  • Polynomial Interpolation: Read SS 7.3
  • Piecewise Polynomial Interpolation: Read SS 7.4
  
  
• Week 11 (3/18-3/22): Numerical Integration
  • Chapter 8 in Michael T. Heath : Scientific Computing .
  • Problem Statement: Read SS 8.1, 8.2.
  • Quadrature: Read SS 8.3.1,8.3.2.
  • Gaussian Quadrature: Read SS 8.3.3.
  • Orthogonal Polynomials and Gauss Quadrature: Read Orthogonal_Polynomials.
  • Composite and Adaptive Quadrature: Read 8.3.5, 8.3.6
  
  
• Week 12 (3/29): Numerical Differentiation
  • Chapter 8 in Michael T. Heath : Scientific Computing .
  • Finite Differences: Read SS 8.6.1.
  • Automatic Differentiation: Read SS 8.6.2.
  
  
• Week 12 (3/28-29): Midterm Recess