Roots of Equations

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Roots of Equations

• Bracketing Methods

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Root

• We are given f(x), a function of x, and we want
  to find a such that
• f(a) 0
• a is called the root of the equation f(x) 0, or
  the zero of the function f(x)

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Example Interest Rate

• Suppose you want to buy an electronic appliance
  from a shop and you can either pay an amount of
  12,000 or have a monthly payment of 1,065 for 12
  months. What is the corresponding interest rate?

We know the payment formulae is
A is the monthly payment P is the loan amount x
is the interest rate per period of time n is the
loan period
To find the yearly interest rate, x, you have to
find the zero of
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Finding Roots Graphically

• Not accurate
• However, graphical view can provide useful info
  about a function.
• Multiple roots?
• Continuous?
• etc.

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• The following root finding methods will be
  introduced
• A. Bracketing Methods
• A.1. Bisection Method
• A.2. Regula Falsi (False-position Method)
• B. Open Methods
• B.1. Fixed Point Iteration
• B.2. Newton Raphson's Method
• B.3. Secant Method

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Bracketing Methods

• Theorem If a function f(x) is continuous in the
  interval a, b and f(a)f(b) lt 0, then the
  equation f(x) 0 has at least one real root in
  the interval (a, b).

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• Usually
• f(a)f(b) gt 0 implies zero or even number of roots
  
• figure (a) and (c)
• f(a)f(b) lt 0 implies odd number of roots
• figure (b) and (d)

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• Exceptional Cases
• Multiple roots
• Roots that overlap at one point.
• e.g. f(x) (x-1)(x-1)(x-2) has a multiple root
  at x1.
• Functions that discontinue within the interval

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Algorithm for bracketing methods

• Step 1 Choose two points xl and xu such that
  f(xl)f(xu) lt 0
• Step 2 Estimate the root xr (note xl lt xr lt
  xu)
• Step 3 Determine which subinterval the root
  lies
• if f(xl)f(xr) lt 0 // the root lies in the
  lower subinterval
• set xu to xr and goto step 2
• if f(xl)f(xr) gt 0 // the root lies in the
  upper subinterval
• set xl to xr and goto step 2
• if f(xl)f(xr) 0
• xr is the root

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How to select xr in step 2?

• Bisection Method
• Guess without considering the characteristics of
  f(x) in (xl, xu)
• False Position Method (Regula Falsi)
• Use "average slope" to predict the location of
  the root

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A.1. Bisection Method

• Each guess reduce the search interval by half

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Bisection Method Example
Find the root of f(x) 0 with an approximated
error below 0.5. (True root a14.7802)
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Example (continue)
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Error Bounds

• The true root, a, must lie between xl and xu.

xl
xu
xr
Let xr(n) denotes xr in the nth iteration
After the 1st iteration, the solution, xr(1),
should be within an accuracy of
xl(1)
xu(1)
xr(1)
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Error Bounds
Suppose the root lies in the lower subinterval.
xl(2)
xu(1)
xu(2)
xr(2)
After the 2nd iteration, the solution, xr(2),
should be within an accuracy of
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Error Bounds
In general, after the nth iteration, the
solution, xr(n), should be within an accuracy of
If we want to achieve an absolute error of no
more than Ea
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Implementation Issues

• The condition f(xl)f(xr) 0 (in step 3) is
  difficult to achieve due to errors.
• We should repeat until xr is close enough to the
  root, but we don't know what the root is!
• One possible solution is to estimate the error as

and repeat until ea lt es (acceptable error)
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Bisection Method (as C function)

• // xl, xu Lower and upper bound of the interval
• // es Acceptable relative percentage error
• // iter_max Maximum of iterations
• double Bisect(double xl, double xu, double es,
• int iter_max)
• double xr // Est. root
• double xr_old // Est. root in the previous
  step
• double ea // Est. error
• int iter 0 // Keep track of of iterations
• xr xl // Initialize xr in order to
• // calculating "ea". Can also be
  "xu".
• do
• iter
• xr_old xr

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xr (xl xu) / 2 // Estimate root
if (xr ! 0) ea fabs((xr xr_old) /
xr) 100 test f(xl) f(xr) if
(test lt 0) xu xr else if (test gt
0) xl xr else ea 0
while (ea gt es iter lt iter_max) return
xr
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Additional Implementation Issues

• Function call is a relatively slow operation.
• In the previous example, function f() is called
  twice in each iteration. Is it necessary?
• We only need to update one of the bounds (see
  step 3 in the algorithm for the bracketing
  method).

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Revised Bisection Method (as C function)

• double Bisect(double xl, double xu, double es,
• int iter_max)
• double xr // Est. Root
• double xr_old // Est. root in the previous
  step
• double ea // Est. error
• int iter 0 // Keep track of of iterations
• double fl, fr // Save values of f(xl) and
  f(xr)
• xr xl // Initialize xr in order to
• // calculating "ea". Can also be
  "xu".
• fl f(xl)
• do
• iter
• xr_old xr
• xr (xl xu) / 2 // Estimate root
• fr f(xr)

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if (xr ! 0) ea fabs((xr xr_old)
/ xr) 100 test fl fr if (test lt
0) xu xr else if (test gt 0)
xl xr fl fr else
ea 0 while (ea gt es iter lt
iter_max) return xr
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Additional Implementation Issues

• Testing if f(xl)f(xr) is positive or negative
  directly could result in underflow or overflow.
• How can we address this problem?
• We can test if both the values have the same sign

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Comments on Bisection Method

• The method is guaranteed to converge.
• However, the convergence is slow as we gain only
  one binary digit in accuracy in each iteration.

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A.2. Regula Falsi Method

• Also known as the false-position method, or
  linear interpolation method.
• Unlike the bisection method which divides the
  search interval by half, regula falsi
  interpolates f(xu) and f(xl) by a straight line
  and the intersection of this line with the x-axis
  is used as the new search position.
• The slope of the line connecting f(xu) and f(xl)
  represents the "average slope" (i.e., the value
  of f'(x)) of the points in xl, xu .

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False-position vs Bisection

• False position in general performs better than
  bisection method.
• Exceptional Cases
• (Usually) When the deviation of f'(x) is high and
  the end points of the interval are selected
  poorly.
• For example,

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Bisection Method (Converge quicker)
False-position Method
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Summary

• Bracketing Methods
• f(x) has the be continuous in the interval xl,
  xu and f(xl)f(xu) lt 0
• Always converge
• Usually slower than open methods
• Bisection Method
• Slow but guarantee the best worst-case convergent
  rate.
• False-position method
• In general performs better than bisection method
  (with some exceptions).