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# Chapter 1

## Mathematical Preliminaries and Error Analysis

Department of Mathematics
University of California, Berkeley

## Limits and Continuity

Definition
A function f defined on a set X of real numbers has the limit L at
x0 , written limxx0 f (x) = L, if, given any real number > 0,
there exists a real number > 0 such that
|f (x) L| < ,

whenever

## x X and 0 < |x x0 | < .

Definition
Let f be a function defined on a set X of real numbers and
x0 X. Then f is continuous at x0 if
lim f (x) = f (x0 ).

xx0

## The function f is continuous on the set X if it is continuous at

each number in X.

Limits of Sequences
Definition
Let {xn }
n=1 be an infinite sequence of real of complex numbers.
The sequence {xn }
n=1 has the limit x is, for any > 0, there
exists a positive integer N () such that |xn x| < , whenever
n > N (). The notation
lim xn = x, or xn x as n ,

## means that the sequence {xn }

n=1 converges to x.
Theorem
If f is a function defined on a set X of real numbers and x0 X,
then the following statements are equivalent:
1
2

f is continuous at x0 ;
If the sequence {xn }
n=1 in X converges to x0 , then
limn f (xn ) = f (x0 ).

Derivatives

Definition
Let f be a functions defined in an open interval containing x0 .
The function f is differentiable at x0 if
f 0 (x0 ) = lim

xx0

f (x) f (x0 )
x x0

## exists. The number f 0 (x0 ) is called the derivative of f at x0 . A

function that has a derivative at each number in a set X is
differentiable on X.
Theorem
If the function f is differentiable at x0 , then f is continuous at x0 .

Derivative Theorems
Theorem Rolles Theorem
Suppose f C[a, b] and f is differentiable on (a, b). If
f (a) = f (b), then a number c in (a, b) exists with f 0 (c) = 0.
Theorem Mean Value Theorem
If f C[a, b] and f is differentiable on (a, b), then a number c in
(a, b) exists with
f 0 (c) =

f (b) f (a)
.
ba

## Theorem Extreme Value Theorem

If f C[a, b], then c1 , c2 [a, b] exist with f (c1 ) f (x) f (c2 ),
for all x [a, b]. In addition, if f is differentiable on (a, b), then
the numbers c1 and c2 occur either at the endpoints of [a, b] or
where f 0 is zero.

Integrals

Definition
The Riemann integral of the function f on the interval [a, b] is the
following limit, provided it exists:
Z

f (x) dx =
a

lim

max xi 0

n
X

f (zi )xi ,

i=1

## where the numbers x0 , x1 , . . . , xn satisfy

a = x0 x1 xn = b, and where xi = xi xi1 , for each
i = 1, 2, . . . , n, and zi is arbitrarily chosen in the interval [xi1 , xi ].

Integrals

## Theorem Weighted Mean Value Theorem for Integrals

Suppose f C[a, b], the Riemann integral of g exists on [a, b], and
g(x) does not change sign on [a, b]. Then there exists a number c
in (a, b) with
Z

Z
f (x)g(x) dx = f (c)

g(x) dx.
a

Generalizations

## Theorem Generalized Rolles Theorem

Suppose f C[a, b] is n times differentiable on (a, b). If f (x) is
zero at the n + 1 distinct numbers x0 , . . . , xn in [a, b], then a
number c in (a, b) exists with f (n) (c) = 0.
Theorem Intermediate Value Theorem
If f C[a, b] and K is any number between f (a) and f (b), then
there exists a number c in (a, b) for which f (c) = K.

Taylor Polynomials
Theorem Taylors Theorem
Suppose f C n [a, b], that f (n+1) exists on [a, b], and x0 [a, b].
For every x [a, b], there exists a number (x) between x0 and x
with f (x) = Pn (x) + Rn (x), where
f 00 (x0 )
(x x0 )2 +
2!
n
X
f (k) (x0 )
f (n) (x0 )
n
+ +
(x x0 ) =
(x x0 )k
n!
k!

k=0

and
Rn (x) =

f (n+1) ((x))
(x x0 )n+1 .
(n + 1)!

## Long real (double precision) format

Default data type in MATLAB, double in C
Base 2, 1 sign bit, 11 exponent bits, 52 significand bits:
x

xxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Represented number:

(1)s 2c1023 (1 + f )

## Decimal Floating-Point Numbers

Base-10 Floating-Point
For simplicity we study k-digit decimal machine numbers:
0.d1 d2 . . . dk 10n ,

1 d1 9,

0 di 9

## Any positive number within the range can be written:

y = 0.d1 d2 dk dk+1 dk+2 . . . 10n
Two ways to represent y with k digits:
Chopping: Chop off after k digits:
f l(y) = 0.d1 d2 . . . dk 10n
Rounding: Add 5 10n(k+1) and chop:
f l(y) = 0.1 2 . . . k 10n

## Errors and Significant Digits

Definition
If p is an approximation to p, the absolute error is |p p |, and
the relative error is |p p |/|p|, provided that p 6= 0.
Definition
The number p is said to approximate p to t significant digits (or
figures) if t is the largest nonnegative integer for which
|p p |
5 10t .
|p|

## Floating Point Operations

Finite-Digit Arithmetic
Machine addition, subtraction, multiplication, and division:
x
+ y = f l(f l(x) + f l(y)),

x
y = f l(f l(x) f l(y))

x
y = f l(f l(x) f l(y)),

x
y = f l(f l(x) f l(y))

## Round input, perform exact arithmetic, round the result

Cancelation
Common problem: Subtraction of nearly equal numbers:
f l(x) = 0.d1 d2 . . . dp p+1 p+2 . . . k 10n
f l(y) = 0.d1 d2 . . . dp p+1 p+2 . . . k 10n
gives fewer digits of significance:
f l(f l(x) f l(y)) = 0.p+1 p+2 . . . k 10np

## Error Growth and Stability

Definition
Suppose E0 > 0 is an initial error, and En is the error after n
operations.
En CnE0 : linear growth of error
En C n E0 : exponential growth of error
Stability
Stable algorithm: Small changes in the initial data produce
small changes in the final result
Unstable or conditionally stable algorithm: Large errors in
final result for all or some initial data with small errors

## Rate of Convergence (Sequences)

Definition

Suppose {n }
n=1 is a sequence converging to zero, and {n }n=1
converges to a number . If a positive constant K exists with
|n | K|n |,

for large n,

## then we say that {n }

n=1 converges to with rate of convergence
O(n ), indicated by n = + O(n ).
Polynomial rate of convergence
Normally we will use
n =

1
,
np

n = + O(1/np ).

## Rate of Convergence (Functions)

Definition
Support that limh0 G(h) = 0 and limh0 F (h) = L. If a positive
constant K exists with
|F (h) L| K|G(h)|,

## then we write F (h) = L + O(G(h)).

Polynomial rate of convergence
Normally we will use
G(h) = hp ,
and look for the largest value p > 0 such that
F (h) = L + O(hp ).
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