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root_sp98

# root_sp98 - CGN 3421 Computer Methods Fall 1997 ROOT...

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CGN 3421 root_sp98.fm page 1 of 12 7/1/98 CGN 3421 Computer Methods Fall 1997 ROOT FINDING We’ll discuss 5 methods. Each has the same goal, and each tries to get there through a series of guesses. What’s different? Each method has devised a different way to get to the next guess (presumably better than the previous guess). 1) Start with some initial guess (or range in which to look) 2) Use some information to make another, closer guess 3) Stop when some definition of error is small enough 1. INCREMENTAL METHOD 1) Start with an initial range, and take even steps within range until a subrange is located. Make this subrange the new range to look in 2) In the new range, take even steps until a new smaller subrange is located. con- tinue this process in an indefinite loop. 3) Stop subdividing when the error is small enough. err=(b-a) OR err=f(b), or f(a) a1 b1 a2 b2 segment into 7 pieces

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CGN 3421 root_sp98.fm page 2 of 12 7/1/98 2. BISECTION METHOD - no special info about function used 1) Define initial range [a b] (possibly the result of a single pass of the incremental method). First guess m=(b+a)/2 2) Find which half the root is in redefine the range b = m and a = a for f(a)*f(m) < 0 OR a = m and b = b for f(a)*f(m) > 0 new guess m=(b+a)/2 3) Stop bisecting when the error is small enough Error - one half the current range err = (b-a)/2 OR Error - the function value at m err = |f(m)| a1 b1 m1 f(m1) b2 m2 f(m2) a2 a3
CGN 3421 root_sp98.fm page 3 of 12 7/1/98 3. FALSE POSITION - Relative height of function at endpoints used 1) Define initial range [a b] (possibly the result of a single pass of the incremental method).

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root_sp98 - CGN 3421 Computer Methods Fall 1997 ROOT...

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