ME 142 Engineering Computation I Computer Precision & Round-Off Error.

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Presentation transcript:

ME 142 Engineering Computation I Computer Precision & Round-Off Error

Key Concepts Numerical Error Significant Figures Accuracy vs. Precision Converting between Decimal & Binary

Numerical Error

What are the 2 major forms of Numerical Error?

Numerical Error  Round-off Error Is due to the fact that computers can represent only quantities with a finite number of digits  Truncation Error Is introduced because numerical methods employ approximations to represent exact mathematical operations and quantities

Significant Figures

What do we mean by Significant Figures?

Significant Figures  What do we mean by Significant Figures? Significant digits of a number are those that can be used with confidence  What is typical in Engineering? Three or four significant digits is common in the final answer  What is an example of rounding off too soon? Keep all available significant figures on intermediate calculations  What is an example of keeping too many decimal places?  How do we control significant figures in Excel?

Accuracy vs. Precision

What is the difference between Accuracy & Precision?

Accuracy vs. Precision  What is the difference between Accuracy and Precision? Accuracy refers to how closely a computed or measured value agrees with the true value. Precision refers to how closely individual computed or measured values agree with each other.  Are both Accuracy and Precision necessary?

Increasing Accuracy Increasing Precision Inaccurate & ImpreciseAccurate & Imprecise Inaccurate & Precise Accurate & Precise

Computer Representation of Numbers

How do computers represent numbers?

Computer Representation of Numbers  How do computers represent numbers? Binary digits or bits Groupings of bits into bytes and words

Computer Representation of Numbers  Do most engineering calculations use floating-point or integer numbers?  Why?  What is scientific notation?  What level of precision is used in Excel?  Is that acceptable for most engineering work?  What are some common ways that significant digits may be lost in computer calculations?

Base x 10 0 = 5 0 x 10 1 = 00 4 x 10 2 = x 10 3 = x 10 4 =

Base x 2 0 = 1 0 x 2 1 = 00 1 x 2 2 = x 2 3 = x 2 4 = = 1 = 0 = 4 = 8 = 16 29

Example Problem  Make a table comparing decimal, binary, and hexidecimal numbers:  Perform the following conversions: Binary to decimal Binary to hexidecimal Decimal to binary  Use Excel to perform the previous conversions:

Review Questions

Review Question Significant Figures Rounding to 3 or 4 significant digits on intermediate calculations is typically an acceptable practice. A.True B.False

Review Question Accuracy vs. Precision Which of the following graphics is an example of precision without accuracy: AB CD

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