lygin
Newbie level 1
Let the derivative be presented as
\[z^{(k)}(x)=\xi^{(k)}(x,h)+O(h^p)\]
where \[z^{(k)}(x)[/tex:6e3a66056d] is the derivative,\[\xi^{(k)}(x,h)\] is numerical method for calculating the derivative, and \[O(h^p)\] is the error function with order \[p\] which is needed to be increased. And we know that \[p=n+1-k\], where \[n\] is the interpolation polynomial order, \[k\] - the order of the derivative. So there is the task of increasing accuracy order of the algorithm for calculating the derivative based on polynomial interpolation. There are several such methods: the first - to increase the order of the interpolation polynomial (\[n\]), the second - the Runge-Romberg method. After the procedure we will get \[z^{(k)}(x)=\xi^{(k)}(x,h)+O(h^{p+m})\]. Both the first and second methods increase the algorithm and involve more points to calculate the value of the derivative (ie, digital filter buffers which are realized on the two methods will increase identically), both methods in one step of increasing order of accuracy increase the order of the error function at the unit. Ie at first glance, in terms of drawbacks, they are absolutely identical. The question is: how do they differ, and if there are other methods to improve the accuracy order of differentiation, I ask you to list them and give references to the literature. Thanks in advance.
\[z^{(k)}(x)=\xi^{(k)}(x,h)+O(h^p)\]
where \[z^{(k)}(x)[/tex:6e3a66056d] is the derivative,\[\xi^{(k)}(x,h)\] is numerical method for calculating the derivative, and \[O(h^p)\] is the error function with order \[p\] which is needed to be increased. And we know that \[p=n+1-k\], where \[n\] is the interpolation polynomial order, \[k\] - the order of the derivative. So there is the task of increasing accuracy order of the algorithm for calculating the derivative based on polynomial interpolation. There are several such methods: the first - to increase the order of the interpolation polynomial (\[n\]), the second - the Runge-Romberg method. After the procedure we will get \[z^{(k)}(x)=\xi^{(k)}(x,h)+O(h^{p+m})\]. Both the first and second methods increase the algorithm and involve more points to calculate the value of the derivative (ie, digital filter buffers which are realized on the two methods will increase identically), both methods in one step of increasing order of accuracy increase the order of the error function at the unit. Ie at first glance, in terms of drawbacks, they are absolutely identical. The question is: how do they differ, and if there are other methods to improve the accuracy order of differentiation, I ask you to list them and give references to the literature. Thanks in advance.
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