Root Finding Methods in Numerical Analysis: Bisection, Secant, and Newton's Method, Lecture notes of Matlab skills

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The roots of equations

Root Finding Methods

1. Graphical Method

2. Bisection Method (Binary search)

3. Secant Method

4. Modified Secant

5. Successive approximation

6. Newton-Raphson method

Graphical Method

• To obtain the root of the equation f(x) = 0, make a plot of the function and observe where it crosses the x-axis.
• This gives an approximation of the root.
• This method is limited because of the lack of preceision. Example: Use the graphical approach to determine the drag coefficient C needede for a parachutist of mass m = 68.1 kg to have a velocity 40 m/s after free falling for time t= 10 s. Solution: We can find the value of C by detrmining the root of the previous equation

Example: Root finding

• We have derived a model to predict the parachutist’s velocity as a function of time:
• Suppose we want to determine the drag coefficient for a parachutist of a given mass to attain a prescribed velocity in certian time period.
• There is no way to rearrange the above equation to isolate the variable C in one side, the parameter is implicit
• The solution can be achieved by finding the value of C that makes the following function equals to zero

Graphical Method – cont.

• To check the validity of the obtained result, subsitiute back into the equation:
• The difference is small and close to zero. The fall speed with this value is: Which is very close to the value of v = 40 m/s

Bracketing method

• A root, r, of function f occurs when f(r) = 0.
• For example:
  • f(x) = x 2 - 2x – 3

has two roots at r = - 1 and r = 3.

• f(-1) = 1 + 2 – 3 = 0
• f(3) = 9 – 6 – 3 = 0
• We can also look at f in its factored form.

f(x) = x

2

• 2 x – 3 = ( x + 1)( x – 3)

İllustration of different cases the root may

occur – cont.

• Figure ( b) depicts the case where a single root is

bracketed by negative and positive values of f ( x ).

• Figure ( d) , where f ( xl ) and f ( xu ) are also on

opposite sides of the x- axis, shows three roots

occurring within the interval.

• In general, if f ( xl ) and f ( xu ) have opposite signs,

there are an odd number of roots in the interval.

• As indicated by Fig. ( a) and (c), if f ( xl ) and f ( xu )

have the same sign, there are either no roots or

an even number of roots between the values.

İllustration of different cases the root may

occur – cont.

some exceptions to the general cases: (a) Multiple root that occurs when the function is tangential to the x- axis. For this case, although the end points are of opposite signs, there are an even number of axis intersections for the interval. (b) Discontinuous function where end points of opposite sign bracket an even number of roots. Special strategies are required for determining the roots for these cases. 11

• General solution exists for equations such as ax 2 + bx + c = 0
• The quadratic formula provides a quick answer to all quadratic equations.
• However, no exact general solution (formula) exists for equations with exponents greater than 4.
• Transcendental equations: involving geometric functions ( sin, cos ), log, exp. These equations cannot be reduced to solution of a polynomial. Root Finding – Solving equations

Examples

Bisection Method

• Based on the fact that the function will change signs as it passes thru the root.
• Suppose we know a function has a root between a and b. (… and the function is continuous, … and there is only one root) _f(a)f(b) <_* 0
• Once we have a root bracketed , we simply evaluate the mid-point and halve the interval.
• The root can be located in a smaller interval until reasonable accuracy is achieved

Bisection Method

• c=(a+b)/ 2 a c b f(a)> f(b)< f(c)> New positive values replace old positive values

Bisection method

Bisection method – cont.

• If stop the iterations, else, repeat

the above