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618 xr xl and x 618 x x xl r r l step 2 calculate

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Unformatted text preview: uses the golden ratio 0.618 Given: ￿ f (x ) ￿ Interval [a, b ] containing the minimum ￿ Tolerance ￿ > 0 19 / 45 Golden Section Search Algorithm: Step 1 ￿ Let xl = a, xr = b ￿ ￿ Find inner points: xr = xl + .618 ∗ (xr − xl ) and ￿ = x − .618 ∗ (x − x ) xl r r l Step 2 ￿ ￿ Calculate f (x ￿ ) and f (xr ) l ￿ ) < f (x ￿ ): ￿ If f (x r l ￿ ￿ ￿ xl = xl , xr = xr￿ , xr￿ = xl￿ xl￿ = xr − .618 ∗ (xr − xl ) (Note: this is using the new values!) Otherwise: ￿ ￿ xl = xl￿ , xr = xr , xl￿ = xr￿ xr￿ = xl + .618 ∗ (xr − xl ) Step 3 ￿ If xr − xl < ￿ stop and return ￿ Else repeat step 2 xr +xl 2 20 / 45 Golden Section Example ￿ ￿ ￿ f (x ) = 4x 2 + 6x − 7 a = −10, b = 10 ￿ = .001 21 / 45 Golden Section Example Start with: ￿ xl = −10 ￿ xr = 10 ￿ x ￿ = 10 − .618 ∗ (20) l ￿ ￿ xr = −10 + .618 ∗ (20) 22 / 45 Golden Section Example 23 / 45 Golden Section Example 24 / 45 Golden Section Example 25 / 45 Golden Section Example 26 / 45 Golden Section Example 27 / 45 Golden Section Example 28 / 45 Golden Section Example 29 / 45 Steepest Gradient Descent Now assume we have a continuously differentiable f : Rn ...
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This note was uploaded on 12/10/2013 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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