Gradient and Newton's Methods

Now we turn to the minimization of a function

of n variables, where

and the partial derivatives of

are accessible.

Steepest Descent or Gradient Method

Let

be a function of

such that

exists for

. The gradient of

, denoted by

, is the vector

(1) .

Illustrations: -

Recall that the gradient vector in (1) points locally in the direction of the greatest rate of increase of . Hence points locally in the direction of greatest decrease . Start at the point and search along the line through in the direction . You will arrive at a point , where a local minimum occurs when the point is constrained to lie on the line . Since partial derivatives are accessible, the minimization process can be executed using either the quadratic or cubic approximation method.

Next we compute

and move in the search direction

. You will come to

, where a local minimum occurs when

is constrained to lie on the line

. Iteration will produce a sequence,

, of points with the property

then

will be a local minimum

Outline of the Gradient Method

Suppose that

has been obtained.

(i) Evaluate the gradient vector

.

(ii) Compute the search direction

.

(iii) Perform a single parameter minimization of

on the interval

, where b is large.
This will produce a value

where a local minimum for

. The relation

shows that this is a minimum for

along the search line

.

(iv) Construct the next point

.

(v) Perform the termination test for minimization, i.e.
are the function values

and

sufficiently close and the distance

small enough ?

Repeat the process.

Comments

Runge-Kutta-Fehlberg Method

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Van Der Pol System

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Newton-Raphson Method

This method is generally used to improve the result obtained by one of the previous methods. Let x o be an approximate root of f(x) = 0 and let x 1 = x o + h be the correct root so that f(x 1 ) = 0. Expanding f(x o + h) by Taylor ’s series, we obtain f(x o ) + hf’(x o ) + h 2 f’’(x o )/2! +-----= 0 Neglecting the second and higher order derivatives, we have f(x o ) + hf’(x o ) = 0 Which gives h = - f(x o )/f’(x o ) A better approximation than x o is therefore given be x 1 , where x 1 = x o – f(x o )/f’(x o ) Successive approximations are given by x 2 , x 3 ,----,x n+1 , where x n+1 = x n – f(x n )/f’(x n ) Which is the Newton-Raphson formula. Example: - Find a real root of the equation x 3 -5x + 3 = 0. Sol n : - Let, f(x) = x 3 -5x + 3 = 0 f’(x) = 3x 2 - 5 Choosing x o = 1 Step-1: f(x o ) = -1 f(x o ) = -2 So, x 1 =1 – ½ = 0.5 Step-2: f(x 1 ) = 0.625 f’(x 1 ) = -4.25 x 2 = 0.5 + 0.625/4.25 = 0.647 S...

Powell's Method

The essence of Powell's method is to add two steps to the process described in the preceding paragraph. The vector represents, in some sense, the average direction moved over the n intermediate steps in an iteration. Thus the point is determined to be the point at which the minimum of the function f occurs along the vector . As before, f is a function of one variable along this vector and the minimization could be accomplished with an application of the golden ratio or Fibonacci searches. Finally, since the vector was such a good direction, it replaces one of the direction vectors for the next iteration. The iteration is then repeated using the new set of direction vectors to generate a sequence of points . In one step of the iteration instead of a zig-zag path the iteration follows a "dog-leg" path. The process is outlined below. Let be an initial guess at the location of the minimum of the function . Let for be the ...

Numerical Analysis

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