Solving Mathematical Equations Using Numerical Analysis Methods Bisection Method, Fixed Point Iteration, Newton

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Solving Mathematical Equations Using Numerical
Analysis MethodsBisection Method, Fixed Point
Iteration, Newtons MethodPrepared byParag
JainMohamed ToureDowling College, Oakdale,
NYFor Research Topics in Computer Science---The
Computer Science Lyceum---Spring 2002
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CONTENTS

• Notice
• Abstract
• Introduction
• Bisection Method
• Fixed-Point Iteration Method
• Newtons Method
• Conclusion
• Acknowledgements
• References

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NOTICE

• This project and the contents herein are provided
  as a convenience to the user. The contents of
  this project are provided on "as is" and "as
  available" basis. We have done this work in good
  faith. This material is purely for educational
  purpose and the user of this document should have
  no commercial intention. We are not responsible
  for any damage this material might cause.
• NO WARRANTY OF ANY KIND IS MADE IN RELATION TO
  THE AVAILABILITY, ACCURACY, RELIABILITY OR
  CONTENT OF THIS MATERIAL.
• Suggestions related to how to improve this
  material are welcome. Email us at
  Fausto09_at_yahoo.com or Jain9164_at_dowling.edu

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ABSTRACT

• In our daily life we encounter several problems
  that need to be analyzed and solved. In
  Mathematics, it is usually impossible to solve to
  exactitude real world problems. It is therefore
  useful to have an idea about what the solution to
  the problem would look like.
• We, have been working in Numerical Analysis since
  long, and the challenge involved motivated us to
  implement a nice GUI for the user to help him
  solve problems using certain methods of
  approximation such as
• Bisection Method
• Fixed Point Iteration Method
• Newtons Method

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INTRODUCTION

• Our program aims at finding the roots of
  mathematical equations utilizing the three
  methods Bisection Method, Fixed Point Iteration
  Method, and Newtons Method.
• The idea is to use a library of equations from
  which the user chooses one, selects the method
  that he/she wants to utilize for solving that
  equation, defines an interval or initial guess
  (as the case might be), and defines the accuracy.
• The program then finds the solution to the
  equation to the defined accuracy (say 110-10)
  and to achieve that performs iterations of the
  chosen method.
• The program displays the roots for the equation
  and the number of iterations performed.

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INTRODUCTION CONTD

• A slightly different approach is used while
  solving an equation using Fixed Point Iteration
  and Newtons Method. This approach is called
  Sampling.
• We utilize the Bisection Method to determine
  those regions in the interval where the function
  has a root, and we look for the root in these
  semi-intervals only using Fixed Point Iteration
  or Newtons Method.
• This expedites the root finding process and
  reduces the probability of failure of these two
  methods.
• The failure happens when we start looking for
  roots in an interval that is not continuous, has
  no roots, function not self-mapped (in
  Fixed-Point Iteration Method), or the initial
  guess leads to a critical point (in Newtons
  Method).

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BISECTION METHOD

• The Bisection Method gives a proof of the
  Intermediate Value Theorem and provides a
  practical method to find roots of equations at
  the same time.
• Recalling the Intermediate Value Theorem, we have
  that if f(x) be a continuous function on the
  interval a,b and if d belongs to f(a),f(b),
  then there is a point c in a,b such that f(c)
  d.
• By replacing f(x) by f(x) d, we may assume
  that d 0 it then suffices to obtain the
  following version as a rule for the Bisection
  Method - let f(x) be a continuous function on the
  interval a,b. If f(a) and f(b) have opposite
  signs, then there is a point c in a,b such that
  f(c) 0.

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BISECTION METHOD CONTD

• We illustrate Bisection Method by considering the
  following polynomial p(x) x7 9x5 - 13x - 17
• Note that p(0)-17 and p(2)373. Therefore, since
  p(x) is a continuous function (i.e., its graph
  has no breaks), we know that there must be a
  root, say r, in the interval (0,2) by the
  Intermediate Value Theorem.
• To shrink the interval in on r, we now evaluate p
  at the midpoint of (0,2) which is 1. p(1)-20.
  Now we see that r must actually lie in the
  interval (1,2) since p switches signs from
  negative to positive as x ranges from 1 to 2. So
  we have reduced the interval under consideration
  from (0,2) to (1,2). We have cut the length of
  our interval in half, or bisected it.
• We look next at the midpoint of (1, 2), namely
  1.5, and p(1.5)48.929. Thus, r must be in the
  interval (1, 1.5). We continue this procedure
  until a desired accuracy has been achieved. The
  table summarizes the results of iterating this
  technique several times.
• A plot of these values follows a general pattern
  shown in the graph.
• This is how our applet works.

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FIXED POINT ITERATION METHOD

• This method is based on a different principle
  than that of bisection. In this case, one seeks
  the roots of the equation x g(x).
• By definition, a fixed point for a function g(x)
  is the number x that verifies g(x) x. Note that
  if one defines f(x) x - g(x) 0 one is reduced
  to seeking the zero of the function f(x) 0. One
  gets an idea of the zeros location by using the
  Bisection Method through Sampling as discussed
  earlier.
• This method works only if g(x) is continuous and
  self-mapped.
• We assume that we are seeking the zeros of f(x)
  0. This problem can be transformed to the fixed
  point g(x) x form in several ways
• g(x) f(x) x x
• g(x) h(x)f(x) x x (h(x) ! 0 in the
  interval of x we are considering the solution to
  lie in.)
• When solving the equation x g(x) we start with
  an initial guess and perform iterations with
  subsequent values of g(x) for x.
• Example

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NEWTONS METHOD

• Newtons Method is a very efficient way to
  approximate the roots of equations of the type
  f(x) 0, where f is a given function.
• Suppose that you start at a point x0 you think is
  close to the root. The tangent line at the point
  x0 is pretty close to the curve. Hence, the
  intersection point of the tangent line and the
  x-axis should be closer to the root.
• Since the slope of the tangent line is f(x0),
  that point of intersection is x1 x0
  f(x0)/f(x0). So x1 is a better approximation
  than x0
• If the result is not up to the expected
  approximation then the same iteration is used
  with x1 as the initial guess to generate x2

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NEWTONS METHOD CONTD

• As you have noticed, Newtons Method is based on
  iteration and uses f(x) in addition to f(x).
• It also gives a fixed-point function g(x) given
  f(x) as follow
• xn xn-1 - f(xn-1)/f(xn-1) where n1, 2, 3,
• x0 is the starting point
• xn g(xn-1)
• The previous algorithm is used to generate a
  sequence of points xn?n0 that converges to x,
  where x is the true, exact root of f.
• For a lot of functions, this iterative method
  works very well. But not for all functions. For
  instance, it doesn't work for any function that
  doesn't have a root. How could it? And it doesn't
  always work for some functions that do have
  roots, for instance, the cube root function.
• Lets examine the following graphs

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NEWTONS METHOD CONTD

• How can one implement such a nice algorithm
  bounded by chances of failure?
• The final approach that we have adopted is to use
  an already defined method that confirmed the
  existence of a root.
• The idea is to use the Bisection Method in order
  to sample the values of the given function, and
  determine whether or not there is a change in
  sign. If yes, we use Newtons Method with the
  initial guess at a point chosen in that interval
  that generated the change in sign.
• This ensures that there is a root, and this
  process will accelerate Newtons Method.

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NEWTONS METHOD CONTD

• However, there might be more than one solution to
  the problem. The next approach will be to divide
  the initial function by (x-previous root), and
  regenerate the process previously described.
• If the solution goes to the same root as the
  previous, then that root has an order of
  multiplicity.
• Repeat the process until there is no more root
  (i.e. f(x) constant or an equation with
  imaginary roots).
• It is easy to show that Newtons Method fails
  infinitely many times depending on the point
  chosen as the initial guess.
• The sampling actually avoids those fatalities
  (i.e. it reduces the probability of failure by
  choosing the initial point as close as possible
  to a critical point of the function).
• If this algorithm works, it is one of the most
  efficient in approximation theory.

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CONCLUSION

• The project demonstrated how we could find roots
  of equations using suitable techniques.
• The heuristic that we implement in the program
  for Fixed-Point Iteration and Newtons Method
  makes these methods faster and workable even in
  situations in which they would have otherwise
  failed.
• Having a predefined library of equations limits
  the applicability of the program so we are still
  working on the implementation of a parser in our
  project.
• We are still in a process of exploring how we
  could make our root-finding methods better, more
  user-friendly, and more efficient. It is an
  unending quest.

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ACKNOWLEDGEMENTS

• We acknowledge the support and encouragement
  that we got from Professor Herbert J. Bernstein
  throughout this semester. He was a pathfinder to
  us and his guidance was really invaluable.
• Also we acknowledge Professor Raymond Grinnell
  for the masterminded mathematical help that we
  got from him.
• Thanks to all our colleagues and friends for
  their camaraderie.

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REFERENCES

• Richard L. Burden J. Douglas Faires, Numerical
  Analysis (Boston, MA PWS Publishing Company,
  1993).
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  tml
• http//members.tripod.com/showing/VisualMath.html
  
• http//www.csulb.edu/wziemer/FixedPoint/FixedPoin
  t.html
• http//www.dgp.toronto.edu/people/JamesStewart/app
  lets/roots/roots.html
• http//delphi.about.com/library/bluc/blrtlmain.htm
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• http//donald.phast.umass.edu/kicons/greek.html
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