Can higher order partial derivatives be derived or approximated from lower order partial derivatives?
There is no specific equation to state the partial derivatives, but they can be measured empirically.
Can higher order partial derivatives be derived from lower order partial derivatives, like 3rd (4th) order from 2nd (3rd) order? And how long can you continue this approximation of an order from the preceding order?
Empirically measuring the higher order partial derivatives is computationally too expensive in this case.
To obtain an exact mathematical solution is necessary and sufficient.
1. Firstly, it is necessary to establish what quality of movement is being studied: translational, parabolic, elliptic, hyperbolic, stochastic.
2. Secondly - determine: how much data you want to use:
- so, for a stochastic movement one point is enough;
- for translational motion, two points are sufficient;
- for parabolic, elliptical and hyperbolic movements, three points are sufficient.
3. Thirdly, the generalized finite difference method is used to equation the model of established motion. The method gives a function of the dependence of the amplitude on a given interval from frequency. Each new frequency is a new derivative. Choose the right one for the direct task. Select the amplitude and find the frequency - for the inverse problem.
To obtain physical information from an exact mathematical solution, it is necessary and sufficient:
1. Choose the general quality of movement - stochastic.
2. Use the function of determinate probability.
3. Find the desired derivative and its value. You solve the direct or inverse problem by initially determining the total, kinetic and potential energies of the process at the measurement point.
A classical tool to do what you are asking are divided differences. A survey of this tecnhique can be found here
http://emis.ams.org/journals/SAT/papers/2/2.pdf
but the topic is covered in all classical numerical analysis textbooks. In general, even if you start from data obtained by the sampling of a smooth function, each time you iterate the procedure you lose one order of convergence in the approximation. If your data is not guaranteed to be the evaluation of something smooth, things can go wrong quite fast, it is easy to build examples by taking some smooth function and adding noise.
Dear Brian Barnard ,
Yes, higher-order partial derivatives can be derived up to any order when enough data is given. Difference tables can do the job by direct computations.
Suitable simple programs can do the calculations within a fraction of one second.
Best regards
Higher-order partial derivatives can be derived from lower-order partial derivatives, Also, you can do it using difference tables with enough data.
See some numerical analysis textbooks.
Rasheed B. Al-Salih
Are you trying to paraphrase the previous answers?
To obtain an exact mathematical solution is necessary and sufficient.
1. Firstly, it is necessary to establish what quality of movement is being studied: translational, parabolic, elliptic, hyperbolic, stochastic.
2. Secondly - determine: how much data you want to use:
- so, for a stochastic movement one point is enough;
- for translational motion, two points are sufficient;
- for parabolic, elliptical and hyperbolic movements, three points are sufficient.
3. Thirdly, the generalized finite difference method is used to equation the model of established motion. The method gives a function of the dependence of the amplitude on a given interval from frequency. Each new frequency is a new derivative. Choose the right one for the direct task. Select the amplitude and find the frequency - for the inverse problem.
To obtain physical information from an exact mathematical solution, it is necessary and sufficient:
1. Choose the general quality of movement - stochastic.
2. Use the function of determinate probability.
3. Find the desired derivative and its value. You solve the direct or inverse problem by initially determining the total, kinetic and potential energies of the process at the measurement point.
In principle, partial derivatives can be approximated by finite differences. But there are some caveats: The accuracy of the approximation depends on the grid of points where the function is evaluated. Furthermore, in higher dimensions or in the presence of curved lines, surfaces etc , one may have rather complicated expressions for the difference type approximation,
and thus, the numerical effort may be rather large.
It's not necessary to use finite differences. It suffices to expand the function in some suitable basis and express the derivatives of the functions of the basis in terms of the same functions-and this is known.
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