Lecture7

Lecture7 - Engineering Analysis ENG 3420 Fall 2009 Dan C....

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1 Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
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2 Lecture 7 Lecture 7 ± Last time: ² Roundoff and truncation errors ² More on Matlab ± Today: ² Approximations ² Finding the roots of the equation f(x)=0 ² Structured programming ² File creation and file access ² Relational operators ± Next Time ² Open methods for finding the roots of the equation f(x) = 0 Note: No office hours on Thursday. The TA will come to the class and answer questions about Project1.
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3 The Taylor Series fx i + 1 () = i + f ' x i h + f '' x i ( ) 2! h 2 + f (3) x i 3! h 3 + L + f ( n ) x i ( ) n ! h n + R n
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4 Truncation Error ± In general, the n th order Taylor series expansion will be exact for an n th order polynomial. ± In other cases, the remainder term R n is of the order of h n+1 , meaning: ² The more terms are used, the smaller the error, and ² The smaller the spacing, the smaller the error for a given number of terms.
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5 Numerical Differentiation ± The first order Taylor series can be used to calculate approximations to derivatives: ² Given: ² Then: ± This is termed a “forward” difference because it utilizes data at i and i +1 to estimate the derivative. f ( x i + 1 ) = f ( x i ) + f ' ( x i ) h + O ( h 2 ) f ' ( x i ) = f ( x i + 1 ) f ( x i ) h + O ( h )
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6 ± There are also backward difference and centered difference approximations, depending on the points used: ± Forward: ± Backward: ± Centered: f ' ( x i ) = Differentiation (cont’d) f ( x i + 1 ) f ( x i ) h + O ( h ) f ' ( x i ) = f ( x i ) f ( x i 1 ) h + O ( h ) f ' ( x i ) = f ( x i + 1 ) f ( x i 1 ) 2 h + O ( h 2 )
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7 Total Numerical Error ± The total numerical error is the summation of the truncation and roundoff errors. ± The truncation error generally increases as the step size increases, while the roundoff error decreases as the step size increases - this leads to a point of diminishing returns for step size.
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8 Other Errors ± Blunders - errors caused by malfunctions of the computer or human imperfection. ± Model errors - errors resulting from incomplete mathematical models. ± Data uncertainty - errors resulting from the accuracy and/or precision of the data.
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9 Finding the roots of the equation f(x)=0 ± Graphical method Æ plot of the function and observe where it crosses the x -axis ± Bracketing methods Æ making two initial guesses that “bracket” the root - that is, are on either side of the root ² Bisection Æ divide the interval in half ² False position Æ connect the endpoints of the interval with a straight line and determine the location of the intercept of the x-axis.
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10 Graphical Methods ± A simple method for obtaining the estimate of the root of the equation f(x) =0 is to make a plot of the function and observe where it crosses the x -axis. ± Graphing the function can also indicate where roots may be and where some root-finding methods may fail: a) Same sign, no roots b) Different sign, one root c) Same sign, two roots d) Different sign, three roots
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11 Bracketing Methods ± Bracketing methods are based on making two initial guesses that “bracket” the root - that is, are on either side of the root.
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Lecture7 - Engineering Analysis ENG 3420 Fall 2009 Dan C....

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