22 Questions
What is the part of logarithm after the decimal point?
Mantissa
The result obtained from a numerical operation is always exact.
False
What is the difference between the true value and the approximation of that value?
Error
The numerical result obtained from a numerical method is always ______________________.
approximate
Match the following types of errors with their definitions:
Truncation Error = Error due to approximation of an infinite series by a finite number of terms Round-off Error = Error due to approximation of a number by a nearby number with fewer digits Absolute Error = Error that is the difference between the true value and the approximation of that value Relative Error = Error that is the ratio of the absolute error to the true value
What is the ratio of the absolute error to the true value?
Relative Error
Infinite series errors are a type of truncation error.
True
What is the purpose of learning about errors in numerical computation?
To understand errors occurring in numerical computation.
What is the term for the difference between the true value and the approximation of that value?
Error
Truncation error occurs when a finite number of terms is used to approximate an infinite series.
True
What is the formula to find the relative error?
|(approximate value - true value) / true value|
Rounding off a number to a certain number of decimal digits results in a ______________ error.
round-off
Match the following types of errors with their definitions:
Absolute Error = The difference between the true value and the approximation of that value Relative Error = The ratio of the absolute error to the true value Truncation Error = The error that occurs when a finite number of terms is used to approximate an infinite series Round-off Error = The error that occurs when a number is rounded off to a certain number of decimal digits
If x = 0.0083465, find the relative error if x is rounded off to three decimal digits.
0.0185%
What is the definition of an error in numerical computation?
The difference between the actual value and the approximate value
Truncation errors occur when digits are included in the number.
False
What is the main reason for errors in computation when using normalized floating-point representation?
Some digits have to be shortened to fit the number in normalized float-point form, resulting in errors.
The infinite series expansion for sin(x) is terminated after a finite number of terms, resulting in a type of error known as _______________________.
truncation error
Match the following types of errors with their definitions:
Truncation Error = a) Excluding digits from a number Round-off Error = b) Representing a number with a limited number of digits Absolute Error = c) The difference between the actual value and the approximate value Relative Error = d) The ratio of the absolute error to the actual value
What is the result of terminating an infinite series expansion, such as sin(x), after a finite number of terms?
Truncation error
What is the main reason for errors in computation when using infinite series expansions, such as sin(x) or cos(x)?
The infinite series is terminated after a finite number of terms, resulting in errors due to omitted terms.
In a normalized floating-point representation, the mantissa can accommodate only a limited number of digits, resulting in a type of error known as _______________________.
round-off error
Study Notes
Errors in Numerical Computation
- Error is defined as the difference between the actual value and the value obtained from the numerical operation.
- Error = actual value – approximate value
- Error = x - xa, where x is the actual value and xa is the approximation.
Errors in Computation
- Computational errors occur due to:
- Normalized floating-point representation
- Truncation of infinite series expansion
- Inefficient algorithm
Truncation and Round-off Errors
- Truncation error occurs when some digits are excluded from the number.
- Truncation error can occur in two situations:
- When numbers are represented in normalized floating-point form.
- The mantissa can accommodate only a few digits.
Truncation Error Examples
- In a hypothetical computer, 0.0356879 would take the form 0.3568E-1 in normalized floating-point representation.
- The mantissa is the part of the logarithm after the decimal point.
Learning Objectives
- Students will learn and understand errors occurring in numerical computation.
- Errors occur due to inefficient algorithms, normalized floating-point representation, and truncation of infinite series expansion.
Absolute and Relative Errors
- Error is the difference between the true value and the approximation of that value.
- Modulus is the positive value of any problem.
Assignment and Activity
- Calculate approximate values, absolute errors, relative errors, and relative percentage errors.
- Find the relative error in computation of x + y.
- Calculate the maximum static error in a thermometer.
- Calculate the maximum error possible in a calculator.
This quiz covers the concepts of errors in numerical computation, including the definition and types of errors such as truncation and round-off errors.
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