Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 09

1. Section 3.5: The cubic spline with not-a-knot condition gives rise to a linear system Ax = b where A is an (n + 1) × (n + 1) matrix and x = (c_{0}, c_{1}, ⋯, c_{n})^{T}. Eliminating c_{0} and c_{n} results in a new equivalent linear system Ãx̃ = b̃ where x̃ = (c_{1}, ⋯, c_{n−1})^{T} and Ã can be made symmetric. Carry out the details for the case h_{0} = ⋯ = h_{n} = h and show that Ã is symmetric and positive definite (hence the original A is non-singular).

2. Section 4.1: Problems 19, 24, 26, 28, 29.