NUMERICAL APPROXIMATION

Lizeth Paola Barrero Riaño

Petroleum Engeeniering

Industrial University of Santander

An approximation is an inexact representation of something that is still close enough to be useful.

It may yield a sufficiently accurate solution while reducing the complexity of the problem

significantly.

Definition

Significant FiguresThe significant figures in a measurement include the certain digits, or digits which the scientist can state are accurate without question, and one uncertain digit, or digit which has some possibility of error.

12.34 ml

12.3 would be the certain digits

The hundredth place (the 4) is the

uncertain digit

It measure

ment could

also be interpreted as

12.34ml ±

0.01ml

Significant figures concept has two important implications in the study of numerical methods.

The numerical methods obtain approximate results. Therefore, must be developed criteria to specify how accurate are the results obtained.

Although certain numbers represent specific number, it cannot be expressed exactly with a finite number of digits.

Significant Figures

Particular situations

All non zero digits are significant. 549 has three significant figures 1.892 has four significant figures

Zeros between non zero digits are significant.

4023 has four significant figures

50014 has five significant figures

Zeros to the left of the first non zero digit are not significant.

0.000034 has only two significant figures. (This is more easily seen if it is written as 3.4x10-5)

0.001111 has four significant figures.

Trailing zeros (the right most zeros) are significant when there is a decimal point in the number. 

400. has three significant figures

2.00 has three significant figures

0.050 has two significant figures

Particular situations

Trailing zeros are not significant in numbers without decimal

points.

470,000 has two significant figures 400 or 4x102 indicates only one significant figure. (To indicate that the trailing zeros are significant a decimal point must be added.

400. has three significant digits and is written as 4.00x102 in scientific notation.)

Exact numbers have an infinite number of significant digits but they are generally not reported.

If you count 2 pencils, then the number of pencils is 2.000...

When adding and subtracting, round the final result to have the same precision (same

number of decimal places) as the least precise initial value,

regardless of the significant figures of

any one term.

98.112 +2.300 100.412

But this value must be rounded to 100.4 (the precision of the least

precise term).

Addition and Subtraction

When multiplying, dividing, or taking

roots, the result should have the same

number of significant figures as the least

precise number in the calculation.

3.69 x 2.3059 8.5088

This value which should be rounded to

8.51 (three significant figures

like 3.69).

Multiplication, Division, and Roots

When calculating the logarithm of a number, retain in the mantissa

(the number to the right of the decimal

point in the logarithm) the same number of significant figures as

there are in the number whose

logarithm is being found.

This result should be rounded to 4.4771

Logarithms and Antilogarithms

But this value should be rounded to 4.5

When calculating the antilogarithm of a

number, the resulting value should have the

same number of significant figures as the mantissa in the

logarithm.

antilog(0.301) = 1.9998, which should be rounded to 2.00

Logarithms and Antilogarithms

antilog(0.301) = 1.9998, which should be rounded to 2.0

Accuracy and Precision

ACCURACY

• In a measurement system, it is the degree of closeness of measurements of a quantity to its actual (true) value.

PRECISION

• In a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.

INACCURACY (also known as bias)

• Is defined as a systematic deviation from the true value.

IMPRECISION (also called uncertainty)

• Refers to the magnitude in the dispersion of values.

Numerical methods must be sufficiently accurate or without bias to satisfy the requirements of a particular engineering problem

Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities. These errors include the following error types:

Truncation errors: Result from the use of approximations as an exact mathematical procedure.

Rounding errors:

Occur when are used numbers which have a limit significant digits to represent accurate numbers.

Numerical Errors

To both types of errors, the relation between the exact or real result and the approximate result is given by:

Numerical Errors

True value = Approximate value + error

Then the numerical error is

Et = True value - Approximate valueWhere Et is used to denote the exact error value.

This definition has the disadvantage of not taking into consideration the magnitude order of the estimated value, so an error of 1ft is much more significant if is measuring a bridge instead of the well depth. To correct this, the error is normalized with respect to the true value, i.e.:

Replacing in the above equation the true error Et is:

True errorRelative fractional true error=

True value

o

True error= 100%

True valuetó

True value-Approximate value= 100%

True valuet

Generally, in so much real applications are unknown the true answer, whereby is used the approximation:

o

Approximate error= 100%

Approximate valuea

There are numerical methods which use iterative method to calculate the results, where do an approximation considering the previous approximation; this process is performed several times or iteratively, hoping for better approaches, therefore the relative percent error is given by:

Actual approach-Previous approach= 100%

Actual approacha

Rules for rounding off non-significant digits:

• Determine how many significant figures the answer should have.

• Look at the next digit to the right of the last significant digit, this number will determine if you will alter the last significant digit.

Rounding

Is the process of elimination

of insignificant

figures.

Method 1

• If this digit is 5 or greater, increase the last significant figure by 1 and drop all non-significant digits. Thus, 2.795 becomes 2.80.

• If this digit is less than 5, then simply drop all the non-significant digits without changing the last significant digit. Thus, 2.794 becomes 2.79.

Method 2

• The zero doesn't really require rounding • 5 is rounded down when the preceding significant digit is even and 5 is rounded up when

the preceding significant digit is odd. Values less than 5 are rounded down and values greater than 5 are rounded up. For example, 2.785 would be rounded down to 2.78 and 2.775 would be rounded up to 2.78.

Rounding

It is the addition of the truncation and rounding errors. The only way to

minimize this type of error is to increase the number of

significant figures.

Total Numerical Error

Taylor's theorem and formula, the Taylor series, is of great

value in the study of numerical methods because it

establishes that any smooth function can be approximated

by a polynomial.

Truncation errors are those that result from using an

approximation rather than an exact mathematical procedure, hence to obtain knowledge of these errors characteristics,

makes use of the series.

Truncation error and Taylor’s

series

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):

o

1i if x f x

Taylor’s series

1 2f x f x

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 ’series expansion and to gain better approach generalizes the series for all functions, as follows:

1 1'i i i i if x f x f x x x

o

11 =

!

nni i i

i

f x x xf x

n

BibliographyCHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2007. 5ª edition.

http://en.wikipedia.org/wiki/Approximation

http://www.mitecnologico.com/Main/ConceptosBasicosMetodosNumericosCifraSignificativaPrecisionExactitudIncertidumbreYSesgo

http://www.ndt-d.org/GeneralResources/SigFigs/SigFigs.htm

http://www.montgomerycollege.edu/Departments/scilcgt/sig-figs.pdf