PERTEMUAN 3
Numerical differentiation
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
achieve better accuracy is the leap-frog differentiation
. A better way to
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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Some improved formulas exist for the higher derivatives
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
where
for
, and the truncation error
. 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
. Find the formula for the third
derivative
, 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.
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
over the interval
it with the graph of
over the interval
.
Solution 2 (a).
Solution 2 (b).
Solution 2 (c).
, using
. Compare
Example 3. Plot the absolute error
over the interval
, and estimate the maximum absolute error 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. Investigate the behavior of
of
then the error bound is reduced by
Solution 4.
. If the step size is reduced by a factor
. This is the
behavior.
Various Scenarios and Animations for Numerical Differentiation.
Example 9. Given
, find numerical approximations to the
derivative
, using two points and the forward difference formula.
Solution 9.
Example 9. Given
, find numerical approximations to the
derivative
, 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
, find numerical approximations to the
derivative
, using two points and the backward difference formula.
Solution 10.
Example 10. Given
, find numerical approximations to the
derivative
, using two points and the backward difference formula.
Solution 10.
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Terima kasih
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