Quiz 4 (Unit V) Study Guide

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Quiz 4 is focused on the material of Unit V, though of course your MATLAB^® knowledge should be cumulative. In preparation for Quiz 4 you should find the study guide/summary below useful.

MATLAB

The main new material is creation, inspection, and manipulation of sparse matrices: MATLAB built-in's spalloc, sparse, issparse, nnz, spy; MATLAB sparse matrix storage; sparse matrix and vector operations; sparse backslash (for sparse Gaussian elimination and back-substitution). Also tic toc for timing loops. And also the MATLAB built-in inv for the evil inverse.

Math and Numerics

Existence and uniqueness of solutions to systems of n equations in n unknowns, in particular the case n = 2 (the 2 x 2 case:two equations in two unknowns): row view; column view; the three possibilities (the solution exists and is unique; the solution exists but is not unique; the solution does not exist); construction of solutions including the general solution in the non-unique case.

Simple spring-mass problems: matrix formulation of the equilibrium problem; the origins of sparsity in "local interactions" between nearest neighbors.

Sparse storage and operation counts for matrix and vector operations, such as the matrix vector product.

Gaussian elimination and back substitution for Au = f: pivots and row operations; construction of U and fhat from A and f; back substitution to obtain u from U and fhat; operation counts for general matrices. Note you should be comfortable both with the general concepts and with "by hand" calculation for small systems (i.e., n = 2 or n = 3).

Gaussian elimination and back-substitution for banded matrices: sparsity and fill-in (in U); storage requirements; operation counts (and hence timings); the special cases of tridiagonal and pentadiagonal matrices.

The Evil Inverse (of A): definition; physical interpretation of the columns of the inverse of A; non-sparsity; relation to solution of Au = f; operation counts and timings (and comparisons with sparse Gaussian elimination and back substitution).

Breakdown of Gaussian elimination: simple examples of deserved and undeserved breakdown; partial pivoting (for undeserved case, and stability more generally). Note you need only understand the very basic concept; you will not be required to replicate the detailed effects of round-off error or perform any partial pivoting exercises.

SPD (symmetric positive definite) matrices A: definition; relation to "energy" (v'*A*v); implications for Gaussian elimination — stability without pivoting.