Constructive Mathematics. QS 2
1. Let f : [a0 , b0 ] R be continuous with f (a0 )f (b0 ) < 0. A method for the approximation
of (a0 , b0 ) such that f () = 0 is defined which is identical to the Bisection Method
except that instead of testing the midpoint
Constructive Mathematics. QS 3
1. Suppose f C 4 in a interval containing the root, and that Newtons method gives a
sequence of iterates xk , k = 0, 1, 2, . . . which converge to . Show that Newtons method
is at least quadratically convergent to if f 0 ()
Constructive Mathematics. QS 4
1. Let p(z) = an z n + an1 z n1 + . . . + a1 z + a0 be a polynomial with real coefficients. If
u, w R are given and
p(z) = (z 2 uz w)s(z) + c1 z + c0
where
s(z) = cn z n2 + cn1 z n3 + . . . + c3 z + c2 ,
derive a nested mult
Constructive Mathematics
Developed by Andy Wathen,
minor edits by Raphael Hauser
April 21, 2015
Introduction
Much of Mathematics is abstract. However a surprisingly large number of algorithms - that is procedures which one can carry out in practice or in
MODULE 11a
Topics: The vector norm of a matrix
Let k k denote a norm on Rm and Rn . Typically, we think of kxk = kxk = maxi |xi |,
but it can be any norm.
We define the vector norm of a matrix A by
kAk = max kAxk.
kxk=1
We say that the vector norm kAk is