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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

Steepest Descent Method

Section 10.4

Burden and Faires

Steepest Descent Method

system of nonlinear equations
System of Nonlinear Equations

Steepest Descent Method

system of nonlinear equations3
System of Nonlinear Equations

Steepest Descent Method

steepest descent
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

questions
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

definition 10 9
Definition 10.9

Steepest Descent Method

minimum
Minimum

Steepest Descent Method

directional derivative
Directional Derivative

Steepest Descent Method

multivariable calculus
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

multivariable calculus10
Multivariable Calculus

Therefore the greatest decrease in the value of g at x is in the direction of –grad(g)

Steepest Descent Method

choosing a
Choosing a

We now have the direction, and we need to determine a

Steepest Descent Method

approximate approach
Approximate Approach

Steepest Descent Method

approximate approach14
Approximate Approach

Steepest Descent Method

approximate approach15
Approximate Approach

Steepest Descent Method

finding minimum
Finding Minimum

Steepest Descent Method

finding minimum17
Finding Minimum

Steepest Descent Method

finding minimum18
Finding Minimum

Steepest Descent Method

finding minimum19
Finding Minimum

Steepest Descent Method

finding minimum20
Finding Minimum

Steepest Descent Method

finding minimum21
Finding Minimum

Steepest Descent Method

finding minimum22
Finding Minimum

Steepest Descent Method

finding minimum23
Finding Minimum

Steepest Descent Method

finding minimum24
Finding Minimum

Steepest Descent Method

example
Example

Steepest Descent Method

example26
Example

Steepest Descent Method

example27
Example

Steepest Descent Method

example28
Example

Steepest Descent Method

example29
Example

Steepest Descent Method

example30
Example

Steepest Descent Method