The most common way for time differentiation is Euler differentiation.
We can see that this way of differentiation on t has an error of order . A better way to achieve better accuracy is the leap-frog differentiation
The compensation for the high order accuracy of leap-frog differentiation is that it needs level values from two different time step( t and ) to get values at the third time step( ). At any time in the evolution of the system variables, we need to save two contiguous time step values, instead of one for the Euler differentiation. For small programs this extra demand poses no difficulty, but for large programs we have to balance the accuracy and resources to obtain the optimal result.
For the first-order space differentiation, there are three basic types. We will discuss these types for the one-dimensional case.
Forward differentiation:
Backward differentiation:
Centered differentiation:
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We can see that centered differentiation has better accuracy, and for this reason it is most commonly used in first-order space differentiation.
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In some circumstances, like those depicted below, one may wish to smooth the derivatives to get a more realistic picture of what is happening. That is the case with the analysis of projectile motion.
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Click here to see example
The following example shows how derivatives are taken of List data.
Click here to see example
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Background.
Numerical differentiation formulas formulas can be derived by first constructing the
Lagrange interpolating polynomial through three points, differentiating the
Lagrange polynomial, and finally evaluating at the desired point. In this module the truncation error will be investigated, but round off error from computer arithmetic using computer numbers will be studied in another module.
Theorem (Three point rule for derivative, based on three points is
).
The central difference formula for the first and the remainder term is
,
.
Together they make the equation bound is
, and the truncation error where for :
. This gives rise to the Big "O" notation for the error term
.
Computer Programs Numerical Differentiation Numerical Differentiation
Project I.
Investigate the numerical differentiation formula and truncation error bound
where . The truncation error is investigated. The round off error from computer arithmetic using computer numbers will be studied in another module.
Enter the three point formula for numerical differentiation.
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Aside.
From a mathematical standpoint, we expect that the limit of the difference quotient is the derivative. Such is the case, check it out.
Example 1.
Consider the function derivative
. Find the formula for the third
, it will be used in our explorations for the remainder term and the truncation error bound. Graph . Find the bound . Look at it's graph and estimate the value , be sure to take the absolute value if necessary.
Solution 1.
, using Example 2 (a).
Compute numerical approximations for the derivative step sizes , include the details.
2 (b).
Compute numerical approximations for the derivatives
, using step sizes
2 (c).
Plot the numerical approximation it with the graph of
Solution 2 (a).
Solution 2 (b).
Solution 2 (c).
over the interval .
over the interval
.
. Compare
Example 3.
Plot the absolute error
, and estimate the maximum absolute error over the interval.
over the interval
3 (a).
Compute the error bound and observe that over .
3 (b).
Since the function f[x] and its derivative is well known, and we have the graph for
, we can observe that the maximum error on the given interval occurs at x=0. Thus we can do better that "theory", we see that
over .
Solution 3.
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Example 4.
of
Investigate the behavior of
then the error bound is reduced by
Solution 4.
. If the step size is reduced by a factor
. This is the behavior.
Example 9.
Given derivative
Solution 9.
, find numerical approximations to the
, using two points and the forward difference formula.
Example 9.
Given derivative
, find numerical approximations to the
, using two points and the forward difference formula.
Solution 9.
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Animation 1. ( Numeirical Dfferentiation Numerical Differentiation ).
(c) John H. Mathews 2004
Example 10.
Given derivative
Solution 10.
, find numerical approximations to the
, using two points and the backward difference formula.
Example 10.
Given derivative
, find numerical approximations to the
, using two points and the backward difference formula.
Solution 10.
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