Derivative of a function in a particular point

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george veropoulos 2021-7-16

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评论： george veropoulos 2021-7-19

采纳的回答： george veropoulos

I have an external function y= function fa(x)

y=sin(x./pi)

end

i want in the main program to find the derivative of fa in numerical point

thank you

George

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george veropoulos 2021-7-16

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format  long 
x = 2 ;
h =  sqrt(eps) ;
f1=(sin(x + h)./(x+h) - sin(x)./x) / h % -0.400000
% -0.435397773981094
 f2=cos(x)./x-sin(x)./x^2 
 
 % -0.435397774979992

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Walter Roberson 2021-7-16

在 MATLAB Online 中打开

format  long 
x = 2 ;
h =  sqrt(eps(x)) ;
f1=(sin(x + h)./(x+h) - sin(x)./x) / h
f1 = 
  -0.435397778831590
 f2=cos(x)./x-sin(x)./x^2 
f2 = 
  -0.435397774979992

george veropoulos 2021-7-16

yes sqrt(eps(x) )

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Jan 2021-7-16

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https://ww2.mathworks.cn/matlabcentral/answers/880038-derivative-of-a-function-in-a-particular-point#answer_748208

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Use one of the quotients of differences to get a numerical approximation of the derivative:

x  = 1.2345;
h  = sqrt(eps);
dy_right = (fa(x + h) - fa(x)) / (h)
dy_left  = (fa(x) - fa(x - h)) / (h)
dy_both  = (fa(x + h) - fa(x - h)) / (2 * h)
function y = fa(x)
y = sin(x./pi) 
end

@george veropoulos: The numerical analysis for the optimal choice of h is still worth to write a PhD thesis. As said before, the 2nd derivative of the function matters. To estimate this, you need a further small variation. Because this is expensive, if the function to be evaluated is huge, some heuristics are useful. This can be an important part of the processing time if you optimize an expensive function. A related field is the optimal choice of temproal and spatial steps sizes in finite element problems, e.g. the simulation of the earth clima.

The rule of thumb: If the function and the argument are about 1 and do not explode nearby, sqrt(eps) is a fair choice between the cancellation and discretization error. The "fairness" can be checkd by testing 10*sqrt(eps) and 0.1*sqrt(eps): If they reply the same derivative, you can assume to be on the right side. For a professional simulation, the explanation must sound more seriously. ;-)

george veropoulos 2021-7-19