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Student Learning Objectives

• SLO#1: Quantify, by theoretical arguments, how the size of a matrix, the physical units of the matrix entries, and the scaling thereof impact the overall scaling and practical solvability of a problem.
  • Assessment#1: Homework, Theoretical
  • Activity: Lecture

• SLO#2: Implement in software a code which (1) computes the Singular Value Decomposition; (2) geometrically represents the resulting matrices, and (3) correctly identifies the resulting rows or columns in both the matrix- and geometric representations of the SVD.
  • Assessment#2: Homework, Software
  • Activity: Lecture

• SLO#3: Implement, in software, the reduced QR-factorization for an arbitrary sized matrix, using the Classical Gram-Schmidt orthogonalization; then demonstrate the limitations of the algorithm by constructing an example matrix for which it fails; explain the limitation by applying appropriate theoretical arguments.
  • Assessment#3: Homework, Software & Theoretical
  • Activity: Lecture

• SLO#4: Implement, in software, the modified Gram-Schmidt algorithm, for an arbitrary sized matrix; identify appropriate examples demonstrating that this mathematically equivalent algorithm is more numerically stable than the Classical Gram-Schmidt algorithm; apply the theory to construct examples where the results of the two algorithms deviate.
  • Assessment#4: Homework, Software & Theoretical
  • Activity: Lecture

• SLO#5: Quantify, numerically or theoretically, the probability distributions of the eigenvalues of randomly constructed dense large scale matrices, and interpret the impact on the condition numbers of matrices.
  • Assessment#5: Homework
  • Activity: Lecture