Lect29 - Numerical Solution of Equations Solutions to the...

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92.131 Lecture29 1 of 20 Ronald Brent © 2010 All rights reserved. Numerical Solution of Equations Solutions to the problem 0 ) ( = x f are called roots. 1) Finding critical points of functions. 2) Finding the x -intercepts of graphs of functions Rarely can solutions be found analytically. Newton’s method is a numerical method that works well in computing extremely accurate approximations to the actual solutions.
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92.131 Lecture29 2 of 20 Ronald Brent © 2010 All rights reserved. Consider the problem of find the root of 2 ) ( 3 = x x x f . The graph of 2 ) ( 3 = x x x f is shown below. x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 y -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 The solution of this problem certainly seems to be near 5 . 1 = x . By zeroing on this point we get:
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92.131 Lecture29 3 of 20 Ronald Brent © 2010 All rights reserved. x 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 y -1.00 -0.50 0.00 0.50 1.00 This graph suggest the root to be near 1.525. Repeating the zoom gives:
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92.131 Lecture29 4 of 20 Ronald Brent © 2010 All rights reserved. x 1.510 1.512 1.514 1.516 1.518 1.520 1.522 1.524 1.526 1.528 1.530 y -0.05 -0.04 -0.03 -0.02 -0.01 -0.00 0.01 0.02 0.03 0.04 0.05 This graph suggests the root to be about 1.5215. Let’s do this one more time.
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92.131 Lecture29 5 of 20 Ronald Brent © 2010 All rights reserved. x 1.5210 1.5212 1.5214 1.5216 1.5218 1.5220 y -0.002 -0.001 0.000 0.001 0.002 If our desire is accuracy within 0.00001, then a good result would be 1.521375. But what if we need more accuracy? Say to 50 10 . This method can breakdown.
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92.131 Lecture29 6 of 20 Ronald Brent © 2010 All rights reserved. Going back to our original picture: Let’s draw the tangent line at 2 = x Since 2 ) ( 3 = x x x f , 1 3 ) ( 2 = x x f . 4 ) 2 ( = f , 11 ) 2 ( = f , and so the tangent line is 18 11 = x y . x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 y -2 -1 0 1 2 3 4 Notice how the x -intercept of the tangent line approaches the root of the original function.
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect29 - Numerical Solution of Equations Solutions to the...

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