13 rigorous numerics for ODE’s joseph galance

Rigorous Numerics for ODE’s
Joseph Galante
September 2008
1 Abstract
We will give an overview of how computers solve ODE’s numerically in a
nonrigorous fashion and examine the sources of error. We will introduce the
tools of interval arithmetic and of Taylor Models to outline a method of how
a rigorous ODE solver can be implemented on a computer system.
2 Introduction
Many (real world) problems have ODE’s describing them that are too com-
plex to estimate by hand and understand in detail. However rigorously ver-
ified information must be known about solutions in order complete proofs.
For example, tracking a solution to see if it connects two regions, or examin-
ing the shape of ball of initial conditions as the flow acts on it. Computers
can be used to quickly give numerical simulations of a particular solution,
however most ODE solvers (for example Runge-Kutta Methods) give that
the computer generated solution is only O(h
n
) where h and n depend on the
method, and the big-O holds unknown derviatives from the ODE in ques-
tion. This is unacceptable for serious proof since we have no assurance that
computer solution and the actual solution agree after some long time accept
for a heurestic ”pick h small and n large”.
2.1 Euler’s Method
It is helpful to have a model method to understand the sources of error a
computer can make. For this we use Euler’s Method. The idea is simple.
1
Suppose we have the initial value problem

˙x = f (x)
x(0) = x

0
(1)
Throughout this paper, we will assume that solutions exist, are unique, and
are defined for all time, and that f is sufficently smooth (either C
∞
or real
analytic). We specify a fixed step size h for the Euler Method. If x(t) is a
solution to the IVP, then we have from Taylor’s Theorem
x(t + h) = x(t) + h
˙
x(t) + O(h
2
) ≈ x(t) + hf (x(t)) (2)
Euler’s Method forgets the remainder and makes the a linear approximation
at each time step to give

t
i
= t
i−1
+ h
x
i
= x
i−1
+ hf (x
i−1
)
(3)
Each step of the Euler Method makes an error of O(h

2
), which for h small
isn’t too bad. The small errors made by disregarding the O(h
2
) causes the
method to track a slightly different solution each timestep. Nearby solutions
usually behave similarly, however after many steps, these small errors can
accumulate and destroy the method’s usefulness by jumping to a solution
which has different b ehavior from the one desired.
Figure 1: Truncation errors can lead to tracking the wrong solution
2
3 Interval Arithmetic
When working on a computer, there is another source of error which must
be accounted for - floating point error. Differential equations usually have
solutions which require real numbers to represent, however a computer is
incapable of representing a general real number. The se t of ”Representable
Numbers” is the set of numbers which our computer can perform computa-
tions with. This is obviously dependent on the computer’s architechture and
software, however we most computers adopt IEEE standards which specify
such representable numbers. We assume that we have adopted such a stan-
dard. For IEEE, the gap inbetween representable numbers is approximately
10
−16
, which is usually denoted called the machine-. When performing com-
putations, all calculations are subject to errors introduced by allowing this
discrepancy - to the computer 3 = 3 +

2
= 3 −


2
.
A well known trick to get around these difficulties is by using so called inter-
val arithmetic. If x is a real number, then on a computer, we can represent
x as the interval [a,b], where a and b are machine representable numbers.
Rules may be developed to handle basic operations. For example if x is
represented by [a,b] and y is represented by [c,d], then to compute x+y, we
compute [a+c,b+d]. We can then perform a ’round’ which ensures that a+c
and b+d are still machine representable. This corresponds roughly to rep-
resenting x+y as [a + c − , b + d + ]. The other operations of subtraction,
multiplication, and division, as well as the concept of ’round’ and repre-
sentable number can be made completely rigorous and are done so in [KM].
As an added benefit, we gain the operations of union and intersection. For
example if we are computing a specific quantity that we know is positive,
then we may take the interval arithmetic calculation and intersect the in-
terval with [0, ∞]. As a down side, we lose the concept of equality. x=y
becomes x-y=0, which says compute x-y, then check if zero is in the interval
[−, ]. Equality on a computer is only good up to the size of machine-.
Another downside from a practical standpoint is that interval operations will
have at least twice as long as conventional computer arithmetic, however we
gain mathematical rigor.
Returning to our problem of rigorous numerics for ODE’s, interval arith-
metic can help to produce a rigorous solver. We must first reformulate the
3
problem as an ’Interval Value Problem’.

˙x = f (x)
x(0) ∈ I
(4)
where now I is some small interval of initial conditions, x is now made up

of intervals instead of reals, and operations are performed via interval arith-
metic. From a dynamical systems perspective we are seeking to transport a
ball of initial conditions under the flow from the ODE.
We can solve the ’IVP’ using an intervalized Euler Method which uses in-
terval arithmetic. However we can do slightly more. The other source of
error in Euler’s Method is error introduced by truncating off the O(h
2
) term.
Suppose we have information which allows us to bound the error so that
O(h
2
) ∈ E where E is some interval. Then we can make a rigorous solver by
doing

t
i
= t
i−1
+ h
x
i
= x
i−1
+ hf (x
i−1
) + E
(5)
where op erations are carried out via interval arithmetic. As we have included
both the floating point errors, and the truncation errors, then we have pro-
duced a method which rigorously solves the IVP. However in practice this

method is useless. It turns out that (with the exception of a few special
cases) the intervals which contain the true s olution grow very quickly due
to the repeated addition of the interval term E. For example, if E=[-1,1] (a
seemingly reasonable bound on an error term) then after two steps we have
accumulated E+E=[-1,1]+[-1,1]=[-2,2]. In the literature, this is known as
the wrapping effect. Iterating, by the nth step we will have a bound for the
solution at least as bad as [-n,n]. This being completely unacceptable, we
persue an idea which greatly refines interval methods.
4 Taylor Models
Definition: Suppose f(x) is C
n+1
in an open domain D. We define an n-th
order Taylor Model about x
0
∈ D as a pair (P,I) where P is nth order Taylor
Polynomial of f about x
0
, and I is some interval such that for all x ∈ D we
have f(x) ∈ P (x − x
0
) + I.
4
Notice that since f is in C
n+1
, then Taylor’s theorem gives that the size
of I shrinks as n grows. Hence the definition is nothing more than a clever
statement that says that smooth functions behave like their Taylor Poly-
nomial approximations (up to some small error) inside a sufficently small
neighborhood.
Rules for ’Taylor Model Arithmetic’ have been generated in [B M1]. For ex-

ample, suppose T
1
= (P
1
, I
1
) and T
2
= (P
2
, I
2
) are nth order Taylor Models
about x
0
∈ D. Then we have
T
1
+ T
2
= (P
1
+ P
2
, I
1
+ I
2
) (6)
T

1
· T
2
= (P
1
· P
2
− P
h
, B(P
h
) + B(P
1
) · I
2
+ B(P
2
) · I
1
+ I
1
· I
2
(7)
where P
h
is the polynomial made up of all terms of order (n+1) or larger in
P
1
· P

2
, and B(P) is a bound on the polynomial. Notice that since we can
obtain a bound on any polynomial by simply performing interval evaluation
on the domain in which it is defined. Other arithmetic operations, trunca-
tion, and the notion of an antiderviative can also be defined.
One advantage in working with Taylor Models is that bounds for most com-
mon functions are known and can be computed automatically with a com-
puter. Bounds for polynomials, trig, exponential, and logarithmic functions,
as well as operations like 1/x and Sqrt(x) (all referred to a intrinsic functions)
have been computed in [BM1]. All of these quantities are known explicitly
and for the domain [-h,h] have remainders that scale like O(h
n+1
) Since most
complicated expressions (ie the ones we care about) are made from com-
posing these simple known quantities together, then we can get nice Taylor
Models with remainder intervals that scale like O(h
n+1
). This is known as
The Fundamental Theorem of Taylor Model Arithmetic. Suppose
that the function f is described by an nth order Taylor Model (P
f
, I
f
) on its
domain D. Let g be a function which is composed of finitely many elementary
operations and intrinsic functions, and suppose g is defined on the range of
f. Let (P,I) be the Taylor Model which arises by plugging in (P
f
, I
f

) into g
and evaluating using Taylor Model arithmetic. Then (P,I) is a Taylor Model
for g ◦ f. Furthermore, if the remainder interval I
f
scales like O(h
n+1
), then
so does the remainder interval I.
5
Proof of this theorem, as well as a detailed list of intrinsic functions and
their Taylor Models is found in [BM1]. It basically amounts to induction on
each operation performed by g. This theorem is important since it allows
to think of Taylor Models as data objects which we can move around and
manipulate in a computer without risking losing control over the size of the
remainder bound.
Notice that zeroth-order Taylor Model Arithmetic is simply just interval
arithmetic. Higher order Taylor Models however give us so much more. Con-
sider the function g(x)=x-x. If we feed g the function f(x)=x on [-1,1], and use
interval arithmetic to bound the answer, then we get g(f (x)) ∈ [−2, 2] since
[-1,1]-[-1,1]=[-2,2], which is hardly a tight bound. Now suppose instead we
use Taylor Models to represent f(x) as x+I where x ∈ [−1, 1] and I = [−, ]
(machine precision). Then we get (x+I)-(x+I)=(x-x)+(I-I)=[−2, 2]. This
is quite a dramatic improvement over intervals. Essentially by using a Taylor
Model, we are storing the higher order information about the shape of the
range of f, which can be manipulated and cancellated to gain better bounds.
Due to the this extra storage of information, one of the primary disadvan-
tages to Taylor Models is that they are very slow and require alot of storage
space in comparsion to interval methods. Somewhat efficient methods to do
this have been developed in [BM1]. (A nice trick is that polynomial coef-
ficents below the machine- don’t need to be storaged, they only need to

thrown into the remainder bound.) In practice however, n=5 is usually good
enough in terms of both speed and accuracy, for your average problem.
There is a scripting language called COSY developed by Martin Berz and
Kyoto Makino (currently at MSU) which has Taylor Models as built in ob-
jects that can automatically perform all of the operations describe d above.
Additionally COSY has some more advanced theoretical tricks to obtain the
bounds B(P). It also has a full interval arithmetic package built in and can
convert between Taylor Models, intervals, and reals to the extent that it
makes sense to do so. All intrinsic functions on Taylor Models are built
in and bounds can be instantly obtained by working with them. COSY is
currently avaliable free of charge for academic use (under some restrictions)
at www.cosyinfinity.org. As an added bonus, COSY has been ’verified’ by
other rigorous computer arithmetic packages.[BM1] [BM2] [BM3] [MB1] [M1]
[RMB1]
6
5 Schauder’s Theorem and Verified Integra-
tion
Taylor Models have another operation which can be defined in a natural way,
the operation of antidifferentiation. If (P,I) is a Taylor Model, then we define
the antidifferentation operator
∂
−1
i
(P, I) = (

x
i
0
P
n−1

(x)dx, (B(P − P
n−1
) + I) · B(x
i
)) (8)
where P
n−1
is the (n-1)-th degree truncation of P, and x
i
∈ [a
i
, b
i
]. Notice
this operation is easy to compute since integration of a p olynomial is just
manipulation of its coefficents. The bounds in the remainder term are easily
computed by interval evaluation of the polynomial piece, and interval oper-
ations. This is the key to implementing a rigorous ODE solver.
Recall that every ODE of the form (1) can be written as an integral equation
x(t) = x(0) +

t
0
f(x(s))ds for t ≤ h. We define the Picard operator
A
f
(x)(t) = x(0) +

t
0

f(x(s))ds (9)
A
f
is a map from C
0
([0, h]) onto itself and fixed points of A
f
correspond to
solutions of our IVP. (Note A
f
is continuous because we are assuming that f
is continuous.) We have theorem which gives us the existence of such a fixed
point.
Schauder’s Theorem. Let A be a continuous operator on the Banach Space
X. Let M ⊂ X be compact and convex, and let A(M ) ⊂ M . Then A has a
fixed point in M.
We going to apply this theorem to a subset of X = C
0
([0, h]) which con-
tains all Taylor Models and A=A
f
. This approach was originally done in
[BM4] and we follow it here.
We start by finding large Y ⊂ X which is agreeable to our analysis on Taylor
Models and to which we can apply the Schauder Theorem. Let (P+I) be a
Taylor Model depending on both time and the initial condition x
0
. Then we
7
define the set M

(P,I)
so that M
(P,I)
⊂ X = C
0
([0, t
0
]) and for x ∈ M
(P,I)
we
have
x(0) = x
0
(10)
x(t) ∈ P + I∀t ∈ [0, h]∀x
0
(11)
|x(t

) − x(t

)| ≤ k|t

− t

|∀t

, t

∈ [0, h]∀x

0
(12)
The last condition is a Lipschitz condition for existence/uniqueness of solu-
tions to the ODE. We take k to be some Lipschitz bound on the function k.
Define Y as
Y =

(P,I)
M
(P,I)
(13)
So Y will contain all Taylor Models. Now if M ⊂ Y and if x
1
, x
2
∈ M , then
ax
1
+(1−a)x
2
∈ M ∀a ∈ [0, 1] since (ax
1
+(1−a)x
2
)(0) = x
0
, ax
1
+(1−a)x
2

is also Lipschitz with constant k, and ax
1
+ (1 − a)x
2
will be in the same
Taylor Models as x
1
and x
2
(due to FTTMA). Hence M is convex. Some
point-set topology and an application of Ascoli-Arzela Theorem shows that
M is compact. Finally note that A maps Y into self since (A
f
(x))(0) = x
0
.
A
f
(x) is continuous due to the integral and Lipschitz continuous with con-
stant k because f is bounded by k. Lastly, since A is made up of intrinsic
functions, then FTTMA gives that A maps Taylor Models to Taylor Models,
ie Y into Y.
To apply Schauder’s Theorem, we must then find a Taylor Model (P,I) so
that A(P + I) ⊂ P + I. Then the fixed point, ie solution of the ODE will be
contained in the Taylor Model. Notice that if I is small, then we will have
succeeded in closely modeling the solution with the polynomial part. Find-
ing such a Taylor Model is relatively easy computationally. Start with the
zero polynomial, and repeatedly iterate it through A, each time disregarding
terms of order (n+1) or higher.
Claim. After (n+1) steps, this will produce an nth degree polynomial invari-

ant under A.
Proof. We will show that after k applications of A to the zero polynomial,
all terms of degree (k-1) will be fixed. Since the Taylor Model is of degree n,
8
then applying A (n+1)-times will produce the result.
Let P = A
k
(0). Then it suffices to show that A(P + O(t
k
)) = P + O(t
k
). We
proceed by induction on k.
Basis: k=1. P = A(0) = x
0
.
A(x
0
+ O(t)) = x
0
+

t
0
f(x
0
+ O(τ ))dτ = x
0
+ O(t) (14)
since all terms in the integral will pick up at least a factor of t after the

integration. Now assume the result holds true for k. We will show it holds
for (k+1). Let Q = A(P ) = A
k+1
(0). By the inductive hypothesis, Q =
R+S +O(t
k+1
) where R is the degree k-1 polynomial such that P = R+O(t
k
)
(hence R is fixed under iterates of A), and S is the polynomial composed of
the degree k terms in Q.
A(Q) =A(R + S + O(t
k+1
)) = x
0
+

t
0
f(R + S + O(τ
k+1
))dτ (15)
= x
0
+

t
0
f(R + S) + f


(R + S) · O(τ
k+1
) +
f

(R + S)
2
O(τ
k+1
)
2
+ dτ
(16)
= A(R + O(t
k
)) +

t
0
O(τ
k+1
)(stuff)dτ (17)
= R + O(t
k
) + O(t
k+2
) (18)
= R + (kth order terms) + O(t
k+1
) (19)

We have used the Taylor Series expansion of f in terms of its argument x,
and the inductive hypothesis. Our claim will be complete if the (kth order
terms)=S. Notice that since deg(S)=k we have

t
0
f(R + S)dτ =

t
0
f(R) + f

(R)S +
f

(R)
2
S
2
+ dτ (20)
=

t
0
f(R)dτ +

t
0
R · (stuff)dτ (21)
=


t
0
f(R)dτ + O(t
k+1
) (22)
9
ie the (kth order terms) are actually independent of S since all the terms in
S get integrated and land in the O(t
k+1
). Now
A(P ) = A(R + (P − R)) (23)
= x
0
+

t
0
f(R + (P − R ))dτ (24)
= x
0
+

t
0
f(R) + f

(R)(P − R) + dτ (25)
= A(R) +


t
0
(P − R)(stuff)dτ (26)
= A(R) + O(t
k+1
) (27)
since deg(P-R)=k. So we have A(Q) = A(R) + O(t
k+1
) and A(R) = A(P ) +
O(t
k+1
) = A(A
k
(0)) + O(t
k+1
) = Q + O(t
k+1
), so we must have that A(Q) =
Q + O(t
k+1
) = R + S + O(t
k+1
) which implies that (kth order terms)=S.
Using this algorithm, we can generate a polynomial invariant under A. To
complete the application of Schauder’s Theorem we must find a Taylor Model
which is invariant under A. Let P be the A invariant nth degree polynomial.
We desire an interval I so that A(P + I) ⊂ P + I, ie A(P + I) − P ⊂ I We
have A(P + I) = x
0
+


t
0
f(P + I)dτ . By FTTMA f(P+I) will be a Taylor
Model. We can decompose as f (P + I) = Q + R +
ˆ
I where Q is all terms of
degree (n-1) or less, R is all degree n terms, and
ˆ
I is the remainder. Since
A(P ) = P + O(t
n+1
) and since deg(R)=n it will integrate to an (n+1) order
term, and then we must have that P = x
0
+

t
0
Qdτ. Thus the other terms
will contribute only to the remainder. ie
A(P + I) − P ⊂

t
0
R +
ˆ
Idτ (28)
We want to better understand this relation since
ˆ

I depends upon I. We
10
actually have
A(P + I) = x
0
+

t
0
f(P + I)dτ (29)
= x
0
+

t
0
f(P ) + f

(P )I +
f

(P )
2
I
2
+ dτ (30)
= x
0
+


t
0
Q + R + f

(P )I +
f

(P )
2
I
2
+ dτ (31)
= P +

t
0
R + f

(P )I +
f

(P )
2
I
2
+ dτ (32)
(33)
Note that we can have the ’ ’ will converge due to class of functions we are
working in. Now suppose the interval I ⊂ [−d, d] where d ≤ 1. Then we
would have

A(P + I) − P ⊂

t
0
R + f

(P )I +
f

(P )
2
I
2
+ dτ (34)
⊂

t
0
R + I · (f

(P ) +
f

(P )
2
+ )dτ (35)
(36)
Notice that the Taylor expansion of f(P+1) about the point P gives
f(P + 1) = f(P ) + f


(P ) +
f

(P )
2
+ (37)
where the sum will again terminate if we use f(P+1)=f(P+1+[0,0]) as a
Taylor Model. Hence we have
A(P + I) − P ⊂

t
0
R + I · (f (P + 1) − f(P ))dτ (38)
⊂ B(

h
0
Rdτ ) + IB(

h
0
(f(P + 1) − f(P ))dτ ) (39)
(40)
We can now solve for I to get
B(

h
0
Rdτ )
1 − B(


h
0
(f(P + 1) − f(P ))dτ )
⊂ I. (41)
11
Since we are choosing I, we choose it to the the left hand side, and we get our
desired inclusion. However there is a catch. We have made the assumption
that I ⊂ [−d, d] where d ≤ 1. There is no reason why apriopi, this must
be true. However we have one more variable which we can control. Now
we can control t. Suppose t ∈ [−h, h]. Clearly for h=0, then left hand side
evaluates to zero, which is to say that to model the initial condition will have
no error. Also the expression is continuous as a function of h, except at the
point where the denominator is zero, ie the point where the bounds become
(−∞, ∞). By the intermediate value theorem, there is some h so that we
can have d ≤ 1. And we are done.
Notice that the remainder interval I scales with h. T his makes sense since it
is easier to model the flow for a short period of time, than for a longer period.
What is hidden is exactly how well it scales. The term R which is the nth
degree pieces of f(P+I) will behave like O(h
n+1
) so decreasing h (the time we
are modeling the ODE for), or increasing n will result in a dramatic increase
in accuracy. This is precisely what was s aid of the non-rigorous integrators
in the introduction. The difference is that we have explicit formulas for the
error now. We also can carry them with us through the integration. Our
algorithm is now
Input: f, initial conditions (as intervals), h
1) Make invariant polynomial to model flow and compute remainder interval
for time up to t=h

2) Evaluate at the polynomial t=h and add the remainder interval. Use this
as a new initial condition and goto 1.
As a result, we will get a rigorous integration thanks to the Schauder The-
orem. Notice that we have used all of the derviatives that were originally
left out of the Euler Method. Where are they? They are absorbed into the
remainder interval. And due to the FTTMA, we know this explicitly. Fur-
thermore, unlike the Euler Method, we actually have a model for the entire
flow, not just its value at some test points.
Remark: The ab ove results carry through without difficulty into multidi-
mensional system case. The only detail to note is the the remainder interval
must be the same in e ach dimension. This can avoided with some more
lengthy expressions for the remainder interval error, however we do not per-
12
sue this.
It is instructive to work an example. Consider the simple linear ODE

˙x = λx
x(0) = x
0
(42)
We will consider a degree n=3 Taylor Model and model up to time t=h.
Iteration of the operator A gives the polynomial
P (t) = x
0
· (1 + λt +
(λt)
2
2
+
(λt)

3
6
) (43)
And the remainder interval is now
B(

h
0
x
0
(λτ )
3
6
dτ)
1 − B(

h
0
(λ · (P + 1) − λ(P ))dτ )
(44)
Which gives
d =
x
0
λ
3
h
4
24(1 − λh)
(45)

Notice that for λ > 0 we have to worry about the denominator going to
zero, which would force us to take h small to avoid having d > 1. This
phenomemon is known as stiffness and is present in other non-rigorous ODE
solvers. Dynamically this just says it is hard to model the flow exactly for
long periods when it is growing exponentially.
Notice as well that we have not really used exactly what x
0
is, except that
we require it to be at least an interval since we are working on a computer. A
more fruitful approach is to take x
0
itself to be a Taylor Model. In this way,
if we want to push a larger box of initial conditions with the flow, then we
only need to represent the box as represent range of Taylor Model, and use
the Taylor Model as our initial condition. All of the analysis goes through
exactly as above, with our P as a Taylor Model in both initial conditions
and time. In fact we can obtain substantial savings since we can get some
of the beneficial cancellation properties described in section 4 since we will
be carrying an even smaller remainder interval. The downside is increased
computation time. Additionally, pushing a box too large will require pro-
hibitively short time intervals since our remainder interval e stimate must
apply to all points inside the box. However there are methods avaliable to
deal with exactly this problem.
13
6 Shrink Wrapping
We have succeeded in creating a rigorous integrator. Moreover (depending on
the ODE) it works for reasonable timescales. However, as above with our in-
terval integrator, we have the same troubles that interval remainders start to
accumulate and ruin the entire result. By using Taylor Models for the initial
conditions, we have partially cancelled this wrapping effect by using objects

which absorb and cancel part of the some of the error. This resulted in much
smaller remainders to carry around. However these will still accumulate over
a long integration, or for a complicated ODE. We use a technique called
’Shrink Wrapping’ outlined in [BM5] which works by attempting to absorb a
large remainder bound into the polynomial part of a Taylor Model. In doing
so, the error can then be manipulated (and hopefully cancelled) along with
polynomial parts. We outline only the ideas and refer to [BM5] for the proof.
(The proofs simply rely on taking norms and measuring distances to images
of boxes.)
Suppose (P,I) is a v-dimensional Taylor Model of order n about the point
x
0
in the domain B = [−1, 1]
v
. (Note we can always rescale variables so
that this holds.) Let C be the constant terms of the Taylor Model and M be
the linear terms. We apply the operator L(·) = M
−1
(· − C) to P+I to get
L(P + I) = Id + S +
ˆ
I ( where Id is the identity). We define the following
constants.
ˆ
I ⊂ [−d, d]
v
(46)
s ≤ |S
i
(x)|∀x ∈ B, 1 ≤ i ≤ v (47)

t ≤ |
∂S
i
(x)
∂x
i
|∀x ∈ B, 1 ≤ i, j ≤ v (48)
We define the Shrink Wrap Factor q was
q = 1 +
d
(1 − (v − 1)t)(1 − s)
(49)
We have the following theorem from [BM5].
(Berz and Makino) Shrink Wrapping. Let Q = Id+S the Taylor Model
described above, and suppose R = [−d, d]
v
+

x∈B
Q(x) is the range of the
Taylor Model over the domain B. Let q be the Shrink Wrap Factor defined
14
above. Then R ⊂

x∈B
(qQ)(x). Hence multiplying the polynomial part by q
allows us to set the interval remainder bound to be zero.
Shrink Wrapping is incredibly useful since it allows us to absorb the in-
terval remainders generated after each integration step into the polynomial
part where it can more easily manipulated. Doing so allows the rigorous

integrator to work for a much longer period, since after each step, we shrink
wrap the new initial conditions s o that going into the next step, there is no
interval remainder. However Shrink Wrapping has its limits. Generally we
hope that the shrink wrap factor q is very close to one. However, if it is too
large (say q ≈ 1.1) for too many steps of integration, then the range of our
Taylor Model grows rapidly and we quickly have a useless integrator.
There are two ways to combat this. One way is to linearize the Taylor
Model Q, ie truncate to a linear polynomial, bound the nonlinear terms, and
add them to the remainder. We can then Shrink Wrap the linearized Taylor
Model. This has the effect of setting the factor t to zero, and increasing d.
If t was originally large, then 1-(v-1)t is close to zero, and q would be huge.
By linearizing we lose this. In most cases the factor d will grow, but not too
badly, and we have an improvement.
The other approach to improving Shrink Wrapping is known as Blunting.
Notice that to apply the Shrink Wrap Theorem, we needed to apply the
operator L to our Taylor Model. If M is a ’nice’ matrix, ie it has a small
condition number, then applying M
−1
won’t cause M
−1
I to grow too much.
Unfortunately this is not always the case. A numerical trick, is to change
the operator L to something nicer to work with numerically. Let
ˆ
L(·) =
(B + M)
−1
(· + Bx − C). Then applying to our Taylor Model, we get
ˆ
L(P + I) = Id + (B + M )

−1
S + (B + M )
−1
I. We can then apply the
Shrink Wrap, and when done, apply
ˆ
L
−1
. If B is a matrix so that (B+M)
has a smaller condition number than M, then inverting will produce much
less numerical error, and a much smaller (B + M )
−1
I to work with in shrink
wrapping. In [BM5] there are detailed ways to produce such a matrix B.
Generally speaking, it works to use B=M+Ev where E is an orthonormal-
ization of M, and v is some vector we pick.
15
7 Conclusion
We have constructed a rigorous integrator which accounts for both the error
in truncation of Taylor Series for the solution to an ODE and floating point
error. The method allows us to rigorously push boxes of solutions around
under the flow. We have numerical methods available which allows us to do
this for reasonably long intevals of time. Taylor Methods are still a niche in
the field of numerical analysis, however new methods to improve integrators
are being developed, for example in [BM6], [BM7], [N2]. As computer speed
and storage capacity increases, runtimes for code decrease, and it becomes
easier to run these verified methods. In one hundred years from now, might
mathematicians lo ok back onto the dark ages of numerics when things were
done without verification and wonder why?
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18