MATINF 4170/9170 - Lecture 9 - 15/3-2017Last time: Chapter 4• Weekly problem 3.2• Knot insertion• Blossoms

Today: Chapter 4 • Recap: Blossoms• Knot insertion• Sign changes

Previously, for

In particular

The B-spline coefficients are functions of the knots

by the blossom

Spline coefficients for polynomials

Proof: use

Inserting one knot

Example

Inserting several knots

Simple knot insertion matrices

• two-banded (n+1)xn matrix• nonzero elements• rows sum to one (row stochastic)

General knot insertion matrices

Sign changes

Descartes rule of signsRecall: z is a zero of f with multiplicity r>0 if

Spline sign changes

Spline sign changes

Spline sign changes