Numerical approximation
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1) An approximation is an inexact representation of something that is still close enough to be useful as it may yield an accurate solution while reducing complexity. 2) Significant figures include certain digits that are accurate and one uncertain digit that has some error possibility. 3) When adding and subtracting numbers, the final result should be rounded to the same precision as the least precise initial value. When multiplying, dividing, or taking roots the result should have the same number of significant figures as the least precise number.
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This document defines error in numerical methods and discusses different types of errors. It defines error as the difference between the true value and approximate value. The main types of errors are absolute error, relative error, and percentage error. Absolute error is the difference between the true and approximate values. Relative error is the error divided by the true value. Percentage error is the relative error expressed as a percentage. An example is provided to illustrate these different error types.
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1701173877972_Lecture # 2.pptx
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lecture01.ppt
This document provides an overview of the topics that will be covered in the Numerical Methods course, including an introduction to numerical methods, Taylor series, solution techniques for nonlinear and linear equations, curve fitting, interpolation, numerical integration and differentiation, and solving ordinary and partial differential equations. The course will also cover number representation, floating point representation, rounding and truncation errors, and accuracy and precision in numerical calculations. The document outlines the basic needs for numerical methods, accuracy and practical computation, and summarizes the key concepts and methods that will be discussed over the duration of the course.
Statistical analysis in analytical chemistry
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NUMERICA METHODS 1 final touch summary for test 1
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
Numerical approximation
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numerical analysis
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Lecture8 Applied Econometrics and Economic Modeling
This document discusses regression analysis and issues that can arise, such as multicollinearity and outliers. It provides examples using data on employee salaries from a bank to examine potential gender discrimination. Key findings include:
1) When regressing salary on foot length variables for both left and right feet, multicollinearity causes unreliable coefficient estimates.
2) Potential outliers are identified in the salary and residual vs fitted value plots, but removing them does not significantly change conclusions about gender discrimination.
3) Regression in blocks, adding variables if they pass a partial F-test, finds job grade affects salary more than education level.
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Lecture8 Applied Econometrics and Economic Modeling
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Numerical approximation
- 1. NUMERICAL APPROXIMATION Lizeth Paola Barrero Riaño PetroleumEngeeniering Industrial University of Santander
- 2. Definition 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.
- 3. SignificantFigures The 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.
- 7. 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
- 8. 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
- 9. Logarithms and Antilogarithms 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 But this value should be rounded to 4.5
- 10. Logarithms and Antilogarithms 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 antilog(0.301) = 1.9998, which should be rounded to 2.0
- 11. Accuracy and Precision Numerical methods must be sufficiently accurate or without bias to satisfy the requirements of a particular engineering problem
- 13. Numerical Errors
- 14. Numerical Errors To both types of errors, the relation between the exact or real result and the approximate result is given by: True value = Approximate value + error Then the numerical error is Et = True value - Approximate value Where Etis used to denote the exact error value.
- 15. o 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.: ó Replacing in the above equation the true error Etis:
- 16. o Generally, in so much real applications are unknown the true answer, whereby is used the approximation: 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:
- 17. Rounding Is the process of elimination of insignificant figures.
- 18. Rounding
- 19. Total Numerical Error 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.
- 20. Bibliography CHAPRA, 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