Derivation of the NewtonRaphson Method

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Derivation of the Newton-Raphson Method
1. Recall Taylor series expansion.
where h xi1 - xi
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Derivation of the Newton-Raphson Method
2. Truncate the series after the first
derivative term.
3. At the intersection of the x-axis, f(xi1)
0.
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Derivation of the Newton-Raphson Method
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Derivation of the Newton-Raphson Method
4. Manipulate the equation to solve for xi1,
which yields the final form of the equation.
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Derivation of the Newton-Raphson Method
5. Iterations can be terminated when a specified
tolerance (Es) has been achieved.
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Newton-Raphson Method applied to the Green-Ampt
Equation
1. Green-Ampt Equation.
2. Based on the equation, F(t) is a function of
itself, therefore an iterative process is
required to solve.
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Newton-Raphson Method applied to the Green-Ampt
Equation
3. Let x F(t).
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Newton-Raphson Method applied to the Green-Ampt
Equation
4. Let g(x) represent a function based on the
variable x.
5. Manipulate the equation so that x is on one
side of the equal sign.
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Newton-Raphson Method applied to the Green-Ampt
Equation
6. Let g(x) represent the derivative of g(x)
with respect to x.
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Newton-Raphson Method applied to the Green-Ampt
Equation
7. Apply Newton-Raphson Method along with the
stopping criterion.
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Newton-Raphson Method applied to the Green-Ampt
Equation
8. Use F(t) Kt as an initial guess of xi.
9. Perform iterative process until acceptable
tolerance has been reached.
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Mannings Equation
1. Mannings Equation.
2. For a rectangular channel.
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Mannings Equation
3. Therefore,
4. Rearranging
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Mannings Equation
5. Perform Newton-Raphson Method until
acceptable tolerance has been reached, and an
accurate solution is obtained.