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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?

special-functions approximation nonlinear-optimization interpolation

share|cite|improve this question

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

$endgroup$

add a comment | 

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?

special-functions approximation nonlinear-optimization interpolation

share|cite|improve this question

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

$endgroup$

add a comment | 

$begingroup$

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?

special-functions approximation nonlinear-optimization interpolation

share|cite|improve this question

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

$endgroup$

share|cite|improve this question

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

share|cite|improve this question

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

edited 15 hours ago

J. M. is not a mathematician

61.3k5152290

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518

asked 18 hours ago

Tal-BotvinnikTal-Botvinnik

1,482518