Class09 - Class meeting #9 Wednesday, Sept 22nd GEEN 1300...

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1 GEEN 1300 Introduction to Engineering Computing Class meeting #9 Wednesday, Sept 22 nd solving nonlinear equations live solution using bisection spherical tank problem Spreadsheet Problem Solving circular calculations and the iterative solver unstable iteration and how it works 1 TABLE function application Homework #4, due Wednesday Section Test 1, next Monday, Sept 27th, 7–9pm, HUMN 1B50 and RAMY C250 Coverage: Class #1 through #8 today, Labs 1-5 Text, pages 1-274 Section Test 1, next Monday, Sept 27th, 7–9pm, HUMN 1B50 and RAMY C250 Coverage: Class #1 through #8 today, Labs 1-5 Text, pages 1-274 REMINDER Text, pages 1 274 HUMN 1B50 (cap’y 284) Sections 101, 102, 103, 104, 105, 106 (139 students) RAMY C250 (cap’y 204) Sections 107, 109, 110, 111, 112 (109 students) 2 Bring only calculator (spare batteries), two pencils and eraser
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2 Issue 2 : What to do if the calculation blows up For example, with our original equation,  1 x sin x solve for the x in the sin(x) term, 1  1 x sin x   This is an alternate form of the same equation.   x gx Try the formula copy technique on this form. 3 Note: initial estimate of 0.5 won’t work, but 1.1 (near the solution) will Replace the initial estimate of 1.1 with the pointer formula, =B3, Press F9 three times, and . . . the calculation blows up! So, when you are setting up a circular calculation, you 4 always have to rearrange your equation in the form. You do this by solving for one of the x’s in the equation. If your formulation blows up, you can usually find one that will converge by solving for one of the other x’s in the equation. In other words, we were just lucky that our first formulation converged to the right answer!  
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3 What’s behind this convergence and blow-up business? A typical graph of   gxv sx will reveal the problem. the initial estimate, x 1 , is plugged into g(x) to yield g(x 1 ) Note: the solution is where the g(x) curve intersects the 45-degree line, shown as a dashed line g(x)  1 g x 2 g x g(x 1 ) becomes x 2 , the next estimate the pattern repeats: x 2 is plugged into g(x) to yield g(x 2 ), which becomes x 5 x 1 x 2 x 3 x 3 evidently, the circular calculation is proceeding toward the solution, indicated by the circle This is a similar scenario, but with a different g(x) curve. Notice that the process is diverging away from the solution – in other words, it’s blowing up! g(x) 1 x 1 g x 2 x 3 x 2 g x Now, observe what’s different between the two examples. The slope of the g(x) line in the first example is less than the slope 6 x of the 45-degree line, less than 1. The slope in this example is > 1 The condition required for the solution to converge is that the absolute value of the slope (or derivative) of g(x) at or near the solution is less than one.
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4 For most engineering problems, multiple solutions are just a nuisance, since only one of the solutions will make physical sense. That may not be the case in abstract math problems.
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Class09 - Class meeting #9 Wednesday, Sept 22nd GEEN 1300...

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