NUMERICAL

APPROXIMATION

The concept of significant

figures has been

developed to designate

formally the reliability of a

numerical value. The

significant figures of a

number are those that can

be used reliably, e.g, used

to determine the accuracy

of the measurement.

The measurement of fluid

volume in the graduated

cylinder is between 38

and 39 cm³. We believe

that it is 38.4 cm or 38.2

cm³. As modified, a move

the last digit is estimated

and therefore

uncertain. The measure of

this volume has three

significant figures.

The concept of significant figures has two important implications for

the study of numerical methods.

The numerical methods obtain approximate

results. Therefore, we must develop criteria to

specify how accurate are the results

obtained. A way to do it is in terms of

significant figures. For example, we can say

that the approach is acceptable to four

significant figures.

1

Although certain quantities such as π or e

represent specific numbers, they can’t to

express exactly with a finite number of digits.

For example:

π = 3.14159265358973238462643... To

infinity.

The computers hold only a finite number of

significant figures, these numbers can never be

represented accurately.

2

1 All nonzero digits are significant:

1,284 g 4 significant figures

1,2 g 2 significant figures

With zeroes, the situation is particularly:

2

Zeroes placed before other digits are

not significant.

0.046 2 significant digits.

4009 kg 4 significant digits.

3Zeroes placed between other digits are

always significant.

4Zeroes placed after other digits

but behind a decimal point are

significant.

7,90 3 significant digits.

8.200 x 103 4 S.F

8.20 x 103 3 S.F

8.2 x 103 2 S.F

5 Zeroes at the end of a number

are significant only if they are

behind a decimal point.

Otherwise, it is impossible to

tell if they are significant. For

example, in the number 8200,

it is not clear if the zeroes are

significant or not. The number

of significant digits in 8200 is at

least two, but could be three or

four. To avoid uncertainty, use

scientific notation to place

significant zeroes behind a

decimal point:

Example:

5.67 J

+ 1.1 J

0.9378 J

--------------------

7.7 J

In math operations, the significant

number its in answer should

equal to the least number of

significant digits in any one of the

numbers being multiplied,

divided etc.

( 3 S.F)

(2 S.F)

(4 S.F)

(2 S.F)

It's called rounding to the process of eliminating non-significant digits

of a number. The rules are the following:

1If the digit removed is greater than 5, the previous

digit increases by one. E.g: 8.236 → 8.24

2If the digit removed is less than 5, the previous digit

is not modified. E.g: 8.231 → 8.23

3

If the digit removed is 5 followed by a different

number than 0, the previous digit increases by one.

E.g: 8.2353→8.24

4

If the digit removed is 5 followed by 0 looks to the

next that follows, if it is odd increase or if it’s pair

remains unchanged. E.g: (1) 8.23503→8.24

(2) 8.23502→8.23

1. Writing more digits in an

answer (intermediate or final)

than justified by the number of

digits in the data.

2. Rounding-off, say, to two

digits in an intermediate answer,

and then writing three digits in

the final answer.

ACCURACY refers to how close is

measured or calculated value to

the true value.

PRECISION refers to how close is

an measured or calculated

individual value with respect to

the others.

THE INACCURACY OR BIAS is

defined as a systematic departure

from the truth.

THE VAGUENESS OR

UNCERTAINTY, refers to the

magnitude of the spread of values.

d

b

c

a

The numerical methods should be sufficiently accurate or no bias

to satisfy the requirements of a particular engineering problem.

Truncation error

Represents the difference between

an exact mathematical formulation

of a problem and the approximation

given by a numerical method.

For the types of errors, the relationship between the exact

or true result and the approximate is given by:

Rounding error

Occurs when the numbers has

a limit of significant figures

which are used to represent

exact numbers.

True value = Approximation + error

True Value - ApproximationRelative Error=

True Value

True Value - Approximation=

True Valuet x 100

• Relative Error

It is the quotient (division) between the absolute error and

the true value. If you multiply by 100 to obtain the true

percentage relative error.

• True or Absolute Error

It is equal to the difference between the true value and

approximate value

or

Approximate Error=

Approximate Valuea x 100

Current approach - Anterior approach=

Current approach a x 100

For numerical methods, the true value will only be known when the

functions can be solved analytically. Thus, in real life to know the true

value early, it is difficult. In these cases, normalizing the error is an

alternative to have the best possible estimate of true value:

Some numerical methods use an iterative method to calculate results. In

such cases, the error is calculated as the difference between the previous

and the current approach. Therefore, the percentage relative error is

given by:

In essence, the Taylor series

provides a means to predict

the value of a function at a

point in terms of the

function value and its

derivatives at another

point. In particular, the

theorem states that any

smooth function can be

approximated with a

polynomial.

Aproximación de la función exponencial

Fuente:

http://upload.wikimedia.org/wikipedia/commo

ns/6/64/Taylorspolynomialexbig.svg

To the Taylor’s series construction makes use of

approximations, what allows us to understand more

about them.

Initially requires a first term which is a zero-order

approximation

(f value at the new point is equal to the value in the

previous point)

If (xi ) is next to (xi+1),then F(xi) soon will be equal

to F(xi+1):

To achieve greater approach adds one more term to

the series; this is an order 1 approximation, which

generates an adjustment for straight lines.

To make the Taylor´s series expansion and to gain

better approach generalizes the series for all functions,

as follows:

Similarly can be added terms and obtain the Taylor series of order n:

(n)

n

n

f ''(a ) f (a )f(x) f(a ) f '(a )(x a ) (x a ) ... (x a ) R

! n!

2

2

x n

(n )

n

a

(x t )R f (t )dt

n!

1

Where Rn term is

included, to notice

the term of n +1 to

infinity. :

“With the Taylor’ series we can estimate the truncation errors”