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ROOT-FINDING Solution to Nonlinear Equation, = 0.
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Bisection Method
This is the simplest and the most robust method for finding a root to an equation. One of the main drawbacks is that we need two initial guesses and which bracket the root. Let = ( ) and = ( ) such that ≤ 0. Clearly, if = 0 then one or both of and must be a root of () = 0 based on the Intermediate Value Theorem. The basic algorithm for the bisection method relies on repeated application of :
Let = ( )/2,
If = () = 0 then = is an exact solution,
elseif < 0 then the root root lies in the interval interval ( , ),
else the root lies in t he interval ( , ).
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Bisection Method
Figure 1: Graphical representation of the bisection method showing two initial guesses ( and bracketting the root). Prepared by: ENGR. ALEXANDER S. CARRASCAL
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False Position (Regula Falsi) Method
This method is similar to the bisection method in that it requires two initial guesses to bracket the root. It is also known as Linear Interpolation Method.
Instead of dividing the region in two, a linear interpolation is used to obtain a new point which is (hopefully, but not necessarily) closer to the root.
The basic algorithm for the linear interpolation method is:
Let = ( ) and = ( ) such that ≤ and let
=
= =
If = () = 0 then = is an exact solution,
else if < 0 then the root lies in the interval ( , ),
else the root lies in t he interval ( , ).
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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False Position Method
Figure 2: Graphical representation of the false position method showing two initial guesses ( and bracketting the root). Prepared by: ENGR. ALEXANDER S. CARRASCAL
Newton-Raphson Method
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Consider the Taylor Series expansion of () about some point = :
() = ( ) ( )′( ) ½( )2 ′′( ) (| |3).
Setting he quadratic and higher terms to zero and solving the linear approximation of () = 0 for x gives
=
Subsequent iterations are defined in a similar manner as
+ =
′
′
Geometrically, + can be interpreted as the value of at which a line, passing through the point ( , ( )) and tangent to the curve () at that point, crosses the axis.
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Newton-Raphson Method
When it works, Newton-Raphson converges much more rapidly than the bisection or linear interpolation.
Figure 3: Graphical representation of the Newton-Raphson method with one initial guesse .
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Newton-Raphson Method However, if ‘() vanishes at an iteration point, or indeed even between the current estimate and the root, then the method will fail to converge.
Figure 4: Divergence of the Newton-Raphson method due to the presence of of a turning point close to the root. Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Secant (Chord) Method
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This method is essentially the same as Newton-Raphson except that the derivative ′() is approximated by a finite difference based on the current and the preceding estimate for the root, i.e.
′ ≈
− −
and this is substituted into the Newton-Raphson algorithm to give
+ =
− −
This formula is identical to that for the Linear Interpolation method. The difference is that instead of replacing one of the two estimates so that the root is always bracketed, the oldest point is always discarded in favour of the new.
This means it is not necessary to have two initial guesses bracketing the root, but on the other hand, convergence is not guaranteed.
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Secant Method
Figure 5: Convergence on the root using the Secant method. Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Secant Method
Figure 6: Divergence from the root using the Secant method. Prepared by: ENGR. ALEXANDER S. CARRASCAL
Direct Iteration
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A simple and often useful method involves rearranging and possibly transforming the function () by ((), ) to obtain () = ((),). The only restriction on ((), ) is that solutions to () = 0 have a one to one relationship with solutions to () = for the roots being sort. The iteration formula for this method is then just
+ = ( ).
The method will converge only if |′| < 1 . The sign of ′ determines whether the convergence (or divergence) is monotonic (positive ′) or oscillatory (negative ′).
Prepared by: ENGR. ALEXANDER S. CARRASCAL
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Direct Iteration Method
Figure 7: Convergence from the root using the Direct Iteration method. Prepared by: ENGR. ALEXANDER S. CARRASCAL
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