Steepest Descent Method

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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

Section 10.4

Burden and Faires

Steepest Descent Method

System of Nonlinear Equations

Steepest Descent Method

System of Nonlinear Equations

Steepest Descent Method

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
• 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

Steepest Descent Method

Minimum

Steepest Descent Method

Directional Derivative

Steepest Descent Method

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 Calculus

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

Steepest Descent Method

Choosing a

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

Steepest Descent Method

Approximate Approach

Steepest Descent Method

Approximate Approach

Steepest Descent Method

Approximate Approach

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Finding Minimum

Steepest Descent Method

Example

Steepest Descent Method

Example

Steepest Descent Method

Example

Steepest Descent Method

Example

Steepest Descent Method

Example

Steepest Descent Method

Example

Steepest Descent Method