Picard Iteration

The term "Picard iteration" occurs two places in undergraduate mathematics. In numerical analysis it is used when discussing fixed point iteration for finding a numerical approximation to the equation

. In differential equations, Picard iteration is a constructive procedure for establishing the existence of a solution to a differential equation

that passes through the point

The first type of Picard iteration uses computations to generate a "sequence of numbers" which converges to a solution. We will not present this application, but mention that it involves the traditional role of computer software as a "number cruncher."

The goal of this article is to illustrate the second application of Picard iteration; i. e. how to use a computer to efficiently generate a "sequence of functions" which converges to a solution. We will see that computer software can perform the more sophisticated task of "symbol cruncher

Most differential equations texts give a proof for the existence and uniqueness of the solution to a first order differential equation. Then exercises are given for performing the laborious details involved in the method of successive approximations. The concept seems straightforward, just repeated integration, but students get bogged down with the details. Now computers can do all the drudgery and we can get a better grasp on how the process works.

Theorem 1 (Existence Theorem):
If both

are continuous on the rectangle

and

, then there exists a unique solution to the initial value problem (I.V.P.)

(1)

for all values of x in some (smaller) interval

contained in

Picard's Method for D.E.'s

The method of successive approximations uses the equivalent integral equation for (1) and an iterative method for constructing approximations to the solution. This is a traditional way to prove (1) and appears in most all differential equations textbooks. It is attributed to the French mathematician Charles Emile Picard (1856-1941).

Theorem 2 (Successive Approximations - Picard Iteration):

The solution to the I.V.P in (1) is found by constructing recursively a sequence of functions

, and
(2)
.

Then the solution to (1) is given by the limit:

(3) .

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Runge-Kutta-Fehlberg Method

One way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice using step sizes h and and compare answers at the mesh points corresponding to the larger step size. But this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the agreement is not good enough. The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve this problem. It has a procedure to determine if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each Runge-Kutta-Fehlberg step requires the use of the following six values: Then an approximation to the solution of the I.V.P.

Van Der Pol System

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting . It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by If , the equation reduces to the equation of simple harmonic motion The van der Pol equation is , where is a constant. When the equation reduces to , and has the familiar solution . Usually the term in equation (1) should be regarded as friction or resistance, and this is the case when the coefficient is positive. However, if the coefficient is negative then we have the case of "negative resistance." In the age of "vacuum tube" radios, the " tetrode vacuum tube " (cathode, grid, plate), was used for a power amplifie

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

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

Numerical Analysis

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