Measuring Errors Major All Engineering Majors Authors Autar
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Measuring Errors ﺧﻄﺎﻫﺎی ﻣﺤﺎﺳﺒﺎﺗی Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Presented by: ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for Amol Undergraduates Students 15/09/2020 http: //numericalmet hods. eng. usf. edu 1
Measuring Errors ﺧﻄﺎﻫﺎی ﻣﺤﺎﺳﺒﺎﺗی http: //numericalmethods. eng. usf. edu 2 http: //numericalmet hods. eng. usf. edu 15/09/2020
True Error Defined as the difference between the true value in a calculation and the approximate value found using a numerical method etc. . ﺧﻄﺎ = ﻣﻘﺪﺍﺭ ﻭﺍﻗﻌی – ﻣﻘﺪﺍﺭ ﻣﺤﺎﺳﺒﻪ ﺷﺪﻩ True Error = True Value – Approximate Value n 4 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example (cont. ) ﺍﺩﺍﻣﻪ Solution: ﺟﻮﺍﺏ a) For 6 and ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example (cont. ). ﻣﻘﺪﺍﺭ ﻭﺍﻗﻌی ﺍﺯ ﻣﻘﺪﺍﺭ ﺧﻮﺩ ﺗﺎﺑﻊ ﺑﻪ ﺩﺳﺖ ﻣی آیﺪ Solution: b) The exact value of can be found by using our knowledge of differential calculus. So the true value of is : ﺧﻄﺎ True error is calculated as True Value – Approximate Value 7 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Relative True Error ﺧﻄﺎی ﻧﺴﺒی n Defined as the ratio between the true error, and the true value. Relative True Error ( 8 )= True Error True Value ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example—Relative True Error ﺧﻄﺎی ﻧﺴﺒی ﻭﺍﻗﻌی Following from the previous example for true error, find the relative true error for at with From the previous example, Relative True Error is defined as as a percentage, 9 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Approximate Error ﺧﻄﺎی ﺗﻘﺮیﺒی n n What can be done if true values are not known or are very difficult to obtain? Approximate error is defined as the difference between the present approximation and the previous approximation. Approximate Error ( ) = Present Approximation – Previous Approximation ﺧﻄﺎی ﺗﻘﺮیﺒی ﺍﺯ ﺗﻔﺎﻭﺕ ﺧﻄﺎﻫﺎی ﺟﺎﺭی ﻭ ﻗﺒﻠی ﺑﺪﺳﺖ ﻣی ، ﺍگﺮ ﻣﻘﺎﺩیﺮﻭﺍﻗﻌی ﻣﺘﻐیﺮ ﻣﻌﻠﻮﻡ ﻧﺒﺎﺷﺪ . آیﺪ 10 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example—Approximate Error For at find the following, a) using b) using c) approximate error for the value of Solution: a) For and 11 for part b) ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example (cont. ) Solution: (cont. ) b) For 12 and ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example (cont. ) Solution: (cont. ) c) So the approximate error, is Present Approximation – Previous Approximation 13 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Relative Approximate Error n Defined as the ratio between the approximate error and the present approximation. Relative Approximate Error ( ) = 14 Approximate Error Present Approximation ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example—Relative Approximate Error ﺧﻄﺎی ﻧﺴﺒی ﺗﻘﺮیﺒی For at , find the relative approximate error using values from and Solution: From Example 3, the approximate value of using and using Present Approximation – Previous Approximation 15 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Example (cont. ) Solution: (cont. ) Approximate Error Present Approximation as a percentage, Absolute relative approximate errors may also need to be calculated, ﺧﻄﺎی ﻣﻄﻠﻖ 16 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺧﻄﺎی ﻣﻄﻠﻖ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻌیﺎﺭ ﺗﻮﻗﻒ ﺑﺮﻧﺎﻣﻪ How is Absolute Relative Error used as a stopping criterion? If where is a pre-specified tolerance, then no further iterations are necessary and the process is stopped. کﻤﺘﺮ ﺍﺯ ﻣیﺰﺍﻥ ﻣﻄﻠﻮﺏ ، ﺗﻮﻗﻒ ﺑﺮﻧﺎﻣﻪ ﻭﻗﺘی کﻪ ﺧﻄﺎ . گﺮﺩیﺪ If at least m significant digits are required to be correct in the final answer, then ﺗﻮﻗﻒ ﺑﺮﻧﺎﻣﻪ ﺑﺎ چﻨﺪ ﺭﻗﻢ ﺑﺎ ﻣﻌﻨی 17 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Table of Values For 18 at with varying step size, 0. 3 10. 263 N/A 0 0. 15 9. 8800 0. 038765% 3 0. 10 9. 7558 0. 012731% 3 0. 01 9. 5378 0. 024953% 3 0. 001 9. 5164 0. 002248% 4 ﺩکﺘﺮ ﺍﺑﻮﺍﻟﻔﻀﻞ ﺭﻧﺠﺒﺮ ﻧﻮﻋی آﻤﻞ - آﻤﻞ 89 -90 -2 lmethods. eng. usf. edu 15/09/2020 http: //numerica
Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/measuring _errors. html 19 http: //numericalmet hods. eng. usf. edu 15/09/2020
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