# Newton-Raphson iteration

## Newton-Raphson iteration

(algorithm)
An iterative algorithm for solving equations. Given an equation,

f x = 0

and an initial approximation, x(0), a better approximation is given by:

x(i+1) = x(i) - f(x(i)) / f'(x(i))

where f'(x) is the first derivative of f, df/dx.

Newton-Raphson iteration is an example of an anytime algorithm in that each approximation is no worse than the previous one.
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The solution procedure for the multi-grid finite element method is that the algebraic equations formed by the finite element method are in turn smoothed by the interpolation from the coarse grids to the fine grids and the restriction from the fine grids to the coarse grids, which is the finite Newton-Raphson iteration.
The Newton-Raphson iteration scheme for solving algebraic equations is derived by the first-order Taylor expansion and gives a recurrence formula for the approximate solution of an equation.
Recall that all these terms have to be calculated, added up to give the sine and cosine values, then used in the Newton-Raphson iteration formula to get our next estimate for [pi].
A new registration algorithm based on Newton-Raphson iteration is proposed to align images with rigid body transformation.
The system of nonlinear melt flow equations is solved via a Newton-Raphson iteration that facilitates efficient computation of the design sensitivities.
For linear problems, we c an solve the matrix equation directly from the start with Gaussian elimination using partial pivoting, while for non-linear problems, we use a modified Newton-Raphson iteration.
The most common approach is to use a division-free Newton-Raphson iteration to get an approximation to the reciprocal of the denominator (division) or the reciprocal square root, and then multiply by the numerator (division) or input argument (square root).
FET circuits), a standard Newton-Raphson iteration may be used as an update mechanism even though no starting point information is available (i.
An algorithm, employing third-order Schroder iteration supported by Newton-Raphson iteration, is provided for computing x when a, P(a, x), and Q(a, x) are given.
This study presents an incremental finite element formulation with an equilibrium check by the Newton-Raphson iteration (14) for 2-dimensional axisymmetric deformation using conical elements, or triangular elements.
However, the final set of equations become nonlinear, and Newton-Raphson iteration is required to solve the system of equations for power-law fluids.

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