Numerical & Statistical methods (2140706) Darshan Ins

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Slide1

Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology

Unit-1

Computer Arithmetic

2140706

– Numerical & Statistical MethodsSlide2

Errors

An error is defined as the

difference

between the actual value and the approximate value obtained from the experimental observation or from numerical computation. Error ctual valueApproximate value Or

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Types of Errors

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Absolute error

If

is the true value of a quantity and

is its approximate value, then Absolute error is denoted by .

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Relative error

The relative error is defined by

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Percentage error

The percentage error is

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Sources of Errors

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Inherent error

The errors which are

already Present in the statement

of problem before its solution is obtained.Such are either due to the given data being approximated or due to limitation of mathematical measurements.Numerical and statistical method (2140706) Darshan Institute of engineering & Technology8Slide9

Rounding-off error

Rounding errors arise from the process of rounding off the numbers during the computation.

There are

numbers with large number of digits.i.e., .This process of dropping unwanted digits is called rounding off. Numerical and statistical method (2140706) Darshan Institute of engineering & Technology9Slide10

Truncation error

Truncation

errors are caused by using approximate results or on

replacing an infinite process by a finite one.If we are using a decimal computer having a fixed word length of 4 digits, rounding off of 13.658 gives 13.66 whereas truncation gives 13.65i.e. if

(say) is replaced by

(

say), then the truncation error is

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Significant Figures

The digits used to express a number are called significant numbers.

All nonzero digits are considered as significant, e.g., 9345, 123.9 have four significant figures.

All zeros between two nonzero digits are significant, e.g., 10011, 120.03 have 5 significant figures.Leading zeros are not significant, e.g., 0.0012, 0.13 have two significant figures.Numerical and statistical method (2140706) Darshan Institute of engineering & Technology11Slide12

Accuracy and Precision

(June-2016)

The

concept of accuracy and precision are closely related to significant digits. They are related as follows:Accuracy refers to how closely a computed or measured value agree with true value.Precision refers to how closely individually computed or measured value agree with each other. Numerical and statistical method (2140706) Darshan Institute of engineering & Technology12Slide13

Accuracy and Precision

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High accuracy High precision

Low accuracy High precision

High accuracy Low

precision

Low accuracy Low precisionSlide14

Example-3

Find error and relative error in the following cases:

Solution:

Absolute error:

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Relative error:

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Example-4

Find the relative error is the computation of

for

and having absolute error Solution:

R

elative error in

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Relative error in y

Relative error in

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Example-5

Find (

i

) Absolute error (ii)Relative error (iii)percentage error, If is approximated to four significant digits.Solution:Here, Absolute Error

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Relative error:

Percentage error:

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Example-9

The solution of a problem is given as

It is known that the absolute error in the solution is less than

Find the interval within which the exact value must lie.Solution:Here,

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Example-12

Given the solution of a problem

with relative error in the solution at most

. Find, to four decimal digits, the range of values within which the exact value of the solution must lie.Solution:

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Solution is,

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Mathematical Modelling

Mathematical Modelling is the method of translating the problems from real life systems into conformable and manageable mathematical expressions whose analytical consideration determines an insight and orientation for solving a problem and provides us with a technique for better development of the system.

Mathematical models are used in various fields including natural sciences, engineering and social sciences.

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Steps of problem solving:

1. Data Analysis:

In this phase problem is analyzed and required data is collected for modelling.

2. Designing of Mathematical model:In this phase, the structure of the solution like objective of the model, bounds of the system, performance measures, etc. is defined.Numerical and statistical method (2140706) Darshan Institute of engineering & Technology23Slide24

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3. Computer simulation and post processing or graphic result:

By inputting required data, we get the result in form of data or graph using mathematical model by computer simulation software.4. Validation/Verification:During validation phase, Mathematical model’s result is verified.5. Implementation: We can implement model of problem in real world.Slide25

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This is the whole view of whole process leading from a problem to its solution by scientific computation.Slide26

Important Results

Consider

be any number expressed as

. Where

are decimal digits.

Result

1 : Absolute Error because of Truncation

If

_ais

the approximate value of x after truncation to k digits , then

Result

2 : Relative Error because of Truncation

If

is the approximate value of

after truncation to k digits, then

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Result 3 : Absolute Error because of Rounding-Off

If

is the approximate value of x after rounding-off to k digits, then

Result 4 : Relative Error Because Of Rounding – OffIf is the approximate value of x after rounding – off to k digits, then

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