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L23 Numerical Methods part 3

L23 Numerical Methods part 3. Project Homework Review Steepest Descent Algorithm Summary Test 4 results. H22 ans. optimum solution __0.444___ min value __0.0494__ interval of uncertainty__0.889__ number of fcn evals __6____. H22 cont’d.

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L23 Numerical Methods part 3

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  1. L23 Numerical Methods part 3 • Project • Homework • Review • Steepest Descent Algorithm • Summary • Test 4 results

  2. H22 ans optimum solution __0.444___ min value __0.0494__ interval of uncertainty__0.889__ number of fcn evals __6____

  3. H22 cont’d For iterations # 2 on….The interval is reduced to 0.888/1.333 =67% …. For the cost of 2 function evaluations. If we create a measure of efficiency

  4. H22

  5. H22 optimum solution __.472____ min value __.0124______ interval of uncertainty___0.764_____ number of fcn evals ___5_____ For iterations # 2 on….The interval is reduced to 61.8% of interval, I for the cost of only 1 function evaluation. If we create a measure of efficiency Golden Section Best

  6. Search algorithm? 1. Find a direction, then 2. Find best step size for alpha 3. Repeat steps 1 and 2 ‘til “done”

  7. Unimodal functions in “locale” monotonic decreasing then monotonic increasing monotonic increasing then monotonic decreasing Figure 10.4 Unimodal function f().

  8. Review: Step Size Methods • “Analytical” Search direction = (-) gradient, (i.e. line search) Findf’(α)=0, f’’(α)≥0 • Region Elimination (“interval reducing”) Equal interval Alternate equal interval Golden Section • Others Newton-Raphson Successive quadratic Interpolation

  9. Successive Alternate Equal Interval Assume bounding phase has found Min can be on either side of Point values… not a line But for sure its not in this region!

  10. Golden section Figure 10.9 Graphic of a section partition.

  11. Descent Algorithm? Descent is guaranteed!

  12. Steepest descent algorithm How does it work?

  13. “Modified” Steepest-Descent Algorithm

  14. Ex 10.4 Use Solver to find α*

  15. EX 10.4 ||c||=0 Done!

  16. H22 Prob 10.52 Let’s use SteepDescentTemplate.xls to set up 10.52 and solve.

  17. Summary • Step size methods: analytical, region elimin. • Golden Section is very efficient • Algorithms include stopping criteria (||c||,∆f ) • Steepest descent algorithm Convergence is assured Lots of Fcn evals (in line search) Each iteration is independent of previous moves (i.e. totally “local” ) Successive iterations slow down.. may stall

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