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Let xl a and xr b 2 let xm 3 repeat x l x r 2 31

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Unformatted text preview: th of the search interval |x − x | < ￿ 9 / 45 Bisection Method Given: ￿ f (x ) ￿ Interval [a, b ] containing the minimum ￿ Tolerance ￿ > 0 Algorithm: 1. Let xl = a and xr = b 2. Let xm = 3. Repeat: x l +x r 2 3.1 If f ￿ (xm ) = 0 or |xl − xr | < ￿ return x ∗ = xm 3.2 Otherwise, if f ￿ (xl )f ￿ (xm ) > 0 set xl = xm ; else set xr = xm 3.3 Set xm = xl +xr 2 10 / 45 Bisection Method Example ￿ ￿ ￿ f (x ) = 4x 2 + 6x − 7 a = −10, b = 10 ￿ = .001 11 / 45 Bisection Method Example 12 / 45 Bisection Method Example 13 / 45 Bisection Method Example 14 / 45 Bisection Method Example 15 / 45 Bisection Method Example 16 / 45 Bisection Method Example 17 / 45 Bisection Method Example 18 / 45 Golden Section Search Still want to minimize f : R → R, unimodal over an interval [a, b ], but without requiring that f be continuously differentiable. Golden section search only uses f (x ), does not require finding f ￿ (x ). Called “golden section search” because it...
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