C3 THEORY OF COMPUTATIONAL DYNAMICS
EXAMPLE SHEET 4
Stars indicate level of difficulty.
1.
Let M be the matrix M = Id + E, where E is the matrix all of whose entries are 0, except for the ijth,
which is -lij. Thus
M
=
1 L
L 0
0 1
M


M O
M



−
O
M
l
ij




L 1
0
Verify that multiplying a matrix A on the left by M subtracts lij × jth row of A from the ith row of A.
2.
Let M be as in Q1. Show that
(2Id - M)M
=
Id
and hence that
M-1
3.
=
1 L
L 0
0 1
M


M M O



l
O
M
ij




L 1
0
Suppose that
L
=
1
L
0
1

 M −l j +1, j O

M


−l ij

M

 0 −l

nj
L 0
M






O M
L 1
By writing L as a product of matrices of the form of M in Q1, and using Q2 or otherwise, show
that
(2Id - L)L
and hence that
=
Id
C3 Exercise Sheet 4
L -1
4.
1 L
0
1

 M l j +1, j O

M


l ij

M

0 l

nj
=
2
L 0
M






O M
L 1
Suppose that the n×n matrix A is invertible and let B be the matrix of cofactors of A, thus (-1)i+jBij
is the determinant of the (n-1)×(n-1) matrix obtained from A by removing the ith row and j t h
column. By considering the usual expansion of det A using the ith row, or otherwise, show that
Bi1Ai1 + Bi2Ai2 + … + BinAin
=
det A
Now consider the expansion of det C using the ith row, where C is the matrix obtained from A by
replacing the ith row of A by the jth row of A use this to show that
Bi1Aj1 + Bi2Aj2 + … + BinAjn
=
0
for any i ≠ j. Hence derive Cramer’s rule
A-1
1
†
det A B
=
where † denotes transpose, so that (B†)ij = Bji.
5.
Estimate the number of multiplications and divisions required to compute A-1 by Cramer’s rule.
Compare this to the number needed for the LR-decomposition method.
6.
If A is diagonalisable, that is there exists and invertible matrix P such that P -1AP is a diagonal
matrix show that
det A
=
λ1λ2 … λn
where λ1,λ2, …, λn are the eigenvalues of A.
7.
C3 May 1994, Q4 i).
8.
C3 May 1994, Q4 ii).
9.
C3 May 1995, Q4 i).
10. C3 May 1995, Q4 ii).
11. Suppose that multiple shooting is used to find a period q orbit of f : Rn → Rn, but that the initial
guess (x 1, x2, …, xq)∈Rqnis such that all the xi∈Rn are equal. By considering the structure of the
Jacobian of the error function show that the Newton update is of the form (δ, δ, …, δ), where
δ∈Rn is the Newton step for finding a fixed point of f with initial guess x = x1 = x2 = … = xq. Hence
conclude that we shall have x1 = x2 = … = xq at every step of the multiple shooting algorithm. Can
this algorithm converge to an orbit with minimal period q with such an initial guess.