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Steepest Descent Method

Steepest Descent Method. Section 10.4 Burden and Faires. System of Nonlinear Equations. System of Nonlinear Equations. Steepest Descent. Evaluate g at an initial approximation x (0) = (x 1 (0), x 2 (0) ,…,x n (0) ) T

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Steepest Descent Method

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  1. Steepest Descent Method Section 10.4 Burden and Faires Steepest Descent Method

  2. System of Nonlinear Equations Steepest Descent Method

  3. System of Nonlinear Equations Steepest Descent Method

  4. Steepest Descent • Evaluate g at an initial approximation x(0) = (x1(0), x2(0),…,xn(0))T • Determine a direction from x(0) that results in a decrease in the value of g • Move an appropriate amount in this direction and call the new vector x(1) • Repeat steps 1 to 3 with x(0) replaced by x(1) Steepest Descent Method

  5. Questions • How do we determine a direction from x(0) that results in a decrease in the value of g(x)? • What is appropriate amount in this direction? Steepest Descent Method

  6. Definition 10.9 Steepest Descent Method

  7. Minimum Steepest Descent Method

  8. Directional Derivative Steepest Descent Method

  9. Multivariable Calculus When g is differentiable, the direction that produces the maximum value for the directional derivative occurs when vis chosen to be parallelto the gradient of g provided that the gradient is not equal to zero. Steepest Descent Method

  10. Multivariable Calculus Therefore the greatest decrease in the value of g at x is in the direction of –grad(g) Steepest Descent Method

  11. Choosing a We now have the direction, and we need to determine a Steepest Descent Method

  12. Choosing a Minimize Single Variable Function Steepest Descent Method

  13. Approximate Approach Steepest Descent Method

  14. Approximate Approach Steepest Descent Method

  15. Approximate Approach Steepest Descent Method

  16. Finding Minimum Steepest Descent Method

  17. Finding Minimum Steepest Descent Method

  18. Finding Minimum Steepest Descent Method

  19. Finding Minimum Steepest Descent Method

  20. Finding Minimum Steepest Descent Method

  21. Finding Minimum Steepest Descent Method

  22. Finding Minimum Steepest Descent Method

  23. Finding Minimum Steepest Descent Method

  24. Finding Minimum Steepest Descent Method

  25. Example Steepest Descent Method

  26. Example Steepest Descent Method

  27. Example Steepest Descent Method

  28. Example Steepest Descent Method

  29. Example Steepest Descent Method

  30. Example Steepest Descent Method

  31. Steepest Descent Method

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