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Using complex variables for numerical differentiation was started by Lyness and Moler in 1967. Their algorithm is applicable to higher-order derivatives. A method based on numerical inversion of a complex Laplace transform was developed by Abate and Dubner.

know the derivative (slope) of the solution at the initial condition. The initial slope is simply the right hand side of Equation 1.1. Our first numerical method, known as Euler’s method, will use this initial slope to extrapolate and predict the future. For the case of the function , , the slope at the initial condition is . In Figure 1.2 we show the function and the extrapolation based on the initial condition.

able to come up with methods for approximating the derivatives at these points, and again, this will typically be done using only values that are defined on a lattice. The underlying function itself (which in this cased is the solution of the equation) is unknown. A simple approximation of the first derivative is f0(x) ≈ f(x+h)−f(x) h, (5.1)

We start by introducing the simplest method for numerical differentiation, de- rive its error, and its sensitivity to round-off errors. The procedure used here ...

5 Numerical Differentiation 5.1 Basic Concepts This chapter deals with numerical approximations of derivatives. ... able to come up with methods for approximating the derivatives at these points, and again, this will typically be done using only …

1.1.2 Euler’s method We can use the numerical derivative from the previous section to derive a simple method for approximating the solution to differential equations. When we know the the governingdifferential equation and the start time then we know the derivative (slope) of the solution at the initial condition.

Approximating derivatives is a very important part of any numerical simulation. When it is no longer possible to analytically obtain a value ...

When using numeric differentiation, use at least Central Differences, and if execution time is not a concern or the objective function is such that determining a good static relative step size is hard, Ridders’ method is recommended. Footnotes. 1. Numerical Differentiation. 2 [Press] Numerical Recipes, Section 5.7. 3

Complex step differentiation (CSD) is well known as an efficient numerical differentiation method: f′(x)=Imf(x+ih)h+O(h2),i:=√−1. This method requires the ...

The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods. For example, the first derivative can be calculated by the complex-step derivative formula: This formula can be obtained by Taylor series expansion:

This is another one-sided difference, called a backward difference, approximation to f (a). A third method for approximating the first derivative of f can be ...

In numerical analysis, numerical differentiation describes algorithms for estimating the derivative of a mathematical function or ...

Numerical differentiation is based on the approximation of the function from which the derivative is taken by an interpolation polynomial. All ...

explain the definitions of forward, backward, and center divided methods for numerical differentiation; find approximate values of the first derivative of ...

We refer to a methods as a pth-order method if the truncation error is of the order of O(hp). It is possible to write more accurate formulas than (5.3) for the ...

Numerical Differentiation Formulation of equations for physical problems often involve derivatives (rate-of-change quantities, such as v elocity and acceleration). Numerical solution of such problems involves numerical evaluation of the derivatives. One method for numerically evaluating derivatives is to use Finite DIfferences:

for deriving numerical differentiation methods. In this way you will not only have a number of methods available to you, but you will also be able to develop new methods, tailored to special situations that you may encounter. The basic strategy for deriving numerical differentiation methods is to evalu-

derivative of the curve. • Fit a 2nd order Lagrange interpolating polynomial to each set of 3 adjacent data points: • Does NOT require equally spaced data • Differentiate the Lagrange interpolating polynomial ()xi−1,xi,xi+1 Fit a 2nd order Lagrange interpolating polynomial xi-1 xi X x i+1 x y Known data points Point where derivative is desired