L23 Numerical Methods part 3

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Presentation transcript:

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 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

H22

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

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

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

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

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!

Golden section Figure 10.9 Graphic of a section partition.

Descent Algorithm? Descent is guaranteed!

Steepest descent algorithm How does it work?

“Modified” Steepest-Descent Algorithm

Ex 10.4 Use Solver to find α*

EX 10.4 ||c||=0 Done!

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

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