Error estimate in the approximation of Incomplete Beta FunctionInterpolation errorlinear interpolation error estimate for non-smooth functionPolynomial InterpolationOptimal way to find derivative - numericallyInterpolation of polynomialsUpper bound for the error magnitudeComparing a function and its estimatenewton's interpolation error for non-differentiable functionLinear Interpolation Error estimationfind interpolation polynomial for $f(x) = x^4+3x^2$ of degree $le3$ , such that $max_xin[-1,1]|f(x)-p(x)|$ is minimal

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Error estimate in the approximation of Incomplete Beta Function


Interpolation errorlinear interpolation error estimate for non-smooth functionPolynomial InterpolationOptimal way to find derivative - numericallyInterpolation of polynomialsUpper bound for the error magnitudeComparing a function and its estimatenewton's interpolation error for non-differentiable functionLinear Interpolation Error estimationfind interpolation polynomial for $f(x) = x^4+3x^2$ of degree $le3$ , such that $max_xin[-1,1]|f(x)-p(x)|$ is minimal













0












$begingroup$


In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by



$$f_a,b(x):=1-(b+1)binoma+b+1aint_0^xt^a(1-t)^b,dt$$



for fixed integers $a,b$. If I approximate this function by linear interpolation with only four points ($0,x_0,x_1,1$), I can prove the desired property in my other function $u$, so according to the plots, this interpolation should be enough:





Figure: Plot of $f_5,9(x)$. Note that the interpolation points are such that $f_5,9(x_0)=1-varepsilon$, $f_5,9(x_1)=varepsilon$ for $varepsilon=0.03$.



I now want to estimate the error of the interpolation, in each interval. It is easy to estimate the error in the extreme intervals, because of how the points were chosen. But I am having trouble to estimate the error in the middle interval, which (see this for instance) is given by



begineqnarray
&&frac(x_1-x_0)^28max_[x_0,x_1]|f_a,b''(x)|,\
&=&(b+1)binoma+b+1afrac(x_1-x_0)^28max_[x_0,x_1]x^a-1(1-x)^b-1|a(1-x)-bx|,
endeqnarray



where, as before, $f_a,b(x_0)=1-varepsilon$, $f_a,b(x_1)=varepsilon$.
Any clever ideas to show that this error is bounded by a constant?










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by



    $$f_a,b(x):=1-(b+1)binoma+b+1aint_0^xt^a(1-t)^b,dt$$



    for fixed integers $a,b$. If I approximate this function by linear interpolation with only four points ($0,x_0,x_1,1$), I can prove the desired property in my other function $u$, so according to the plots, this interpolation should be enough:





    Figure: Plot of $f_5,9(x)$. Note that the interpolation points are such that $f_5,9(x_0)=1-varepsilon$, $f_5,9(x_1)=varepsilon$ for $varepsilon=0.03$.



    I now want to estimate the error of the interpolation, in each interval. It is easy to estimate the error in the extreme intervals, because of how the points were chosen. But I am having trouble to estimate the error in the middle interval, which (see this for instance) is given by



    begineqnarray
    &&frac(x_1-x_0)^28max_[x_0,x_1]|f_a,b''(x)|,\
    &=&(b+1)binoma+b+1afrac(x_1-x_0)^28max_[x_0,x_1]x^a-1(1-x)^b-1|a(1-x)-bx|,
    endeqnarray



    where, as before, $f_a,b(x_0)=1-varepsilon$, $f_a,b(x_1)=varepsilon$.
    Any clever ideas to show that this error is bounded by a constant?










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by



      $$f_a,b(x):=1-(b+1)binoma+b+1aint_0^xt^a(1-t)^b,dt$$



      for fixed integers $a,b$. If I approximate this function by linear interpolation with only four points ($0,x_0,x_1,1$), I can prove the desired property in my other function $u$, so according to the plots, this interpolation should be enough:





      Figure: Plot of $f_5,9(x)$. Note that the interpolation points are such that $f_5,9(x_0)=1-varepsilon$, $f_5,9(x_1)=varepsilon$ for $varepsilon=0.03$.



      I now want to estimate the error of the interpolation, in each interval. It is easy to estimate the error in the extreme intervals, because of how the points were chosen. But I am having trouble to estimate the error in the middle interval, which (see this for instance) is given by



      begineqnarray
      &&frac(x_1-x_0)^28max_[x_0,x_1]|f_a,b''(x)|,\
      &=&(b+1)binoma+b+1afrac(x_1-x_0)^28max_[x_0,x_1]x^a-1(1-x)^b-1|a(1-x)-bx|,
      endeqnarray



      where, as before, $f_a,b(x_0)=1-varepsilon$, $f_a,b(x_1)=varepsilon$.
      Any clever ideas to show that this error is bounded by a constant?










      share|cite|improve this question











      $endgroup$




      In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by



      $$f_a,b(x):=1-(b+1)binoma+b+1aint_0^xt^a(1-t)^b,dt$$



      for fixed integers $a,b$. If I approximate this function by linear interpolation with only four points ($0,x_0,x_1,1$), I can prove the desired property in my other function $u$, so according to the plots, this interpolation should be enough:





      Figure: Plot of $f_5,9(x)$. Note that the interpolation points are such that $f_5,9(x_0)=1-varepsilon$, $f_5,9(x_1)=varepsilon$ for $varepsilon=0.03$.



      I now want to estimate the error of the interpolation, in each interval. It is easy to estimate the error in the extreme intervals, because of how the points were chosen. But I am having trouble to estimate the error in the middle interval, which (see this for instance) is given by



      begineqnarray
      &&frac(x_1-x_0)^28max_[x_0,x_1]|f_a,b''(x)|,\
      &=&(b+1)binoma+b+1afrac(x_1-x_0)^28max_[x_0,x_1]x^a-1(1-x)^b-1|a(1-x)-bx|,
      endeqnarray



      where, as before, $f_a,b(x_0)=1-varepsilon$, $f_a,b(x_1)=varepsilon$.
      Any clever ideas to show that this error is bounded by a constant?







      special-functions approximation nonlinear-optimization interpolation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 15 hours ago









      J. M. is not a mathematician

      61.3k5152290




      61.3k5152290










      asked 18 hours ago









      Tal-BotvinnikTal-Botvinnik

      1,482518




      1,482518




















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