Error bound of interpolating polynomial for equally spaced points
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I am using the following error formula for polynomial interpolation
$frac{f^{(n+1)}(xi)}{(n+1)!}$$prod_{j=0}^n x-x_j$
where n is the degree of the interpolating polynomial and j+1 is the number of data points. I am trying to arrive at a general formula for an upper bound where data points are equidistant in an interval $[x_0 x_n]$ I am considering the product part of the error formula. For this I set,
$x_n=x_0+nh$ where is h the distance between successive data points. I have
$(x-x_0)(x-x_1)(x-x_2)...(x-x_n)$ (1)
At this point I introduce a new quantity $alpha$ which I saw in the book Numerical methods for Engineers by Chapra and Canale:
$alpha=frac{x-x_0}{h}$
Plugging this into (1) I get
$alpha h (alpha h-h)(alpha h-2h)...(alpha h-nh)$
and I get
$h^{(n+1)}alpha (alpha-1)(alpha-2)..(alpha-n)$
Now, I want to maximize the absolute value of $alpha(alpha-1)(alpha-2)..(alpha-n)$ in order to establish an upper bound for the associated error. How can I achieve that?
interpolation
asked Nov 24 at 15:41
Ali Kıral
63
add a comment |
up vote
0
down vote
favorite
I am using the following error formula for polynomial interpolation
$frac{f^{(n+1)}(xi)}{(n+1)!}$$prod_{j=0}^n x-x_j$
where n is the degree of the interpolating polynomial and j+1 is the number of data points. I am trying to arrive at a general formula for an upper bound where data points are equidistant in an interval $[x_0 x_n]$ I am considering the product part of the error formula. For this I set,
$x_n=x_0+nh$ where is h the distance between successive data points. I have
$(x-x_0)(x-x_1)(x-x_2)...(x-x_n)$ (1)
At this point I introduce a new quantity $alpha$ which I saw in the book Numerical methods for Engineers by Chapra and Canale:
$alpha=frac{x-x_0}{h}$
Plugging this into (1) I get
$alpha h (alpha h-h)(alpha h-2h)...(alpha h-nh)$
and I get
$h^{(n+1)}alpha (alpha-1)(alpha-2)..(alpha-n)$
Now, I want to maximize the absolute value of $alpha(alpha-1)(alpha-2)..(alpha-n)$ in order to establish an upper bound for the associated error. How can I achieve that?
interpolation
asked Nov 24 at 15:41
Ali Kıral
63
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am using the following error formula for polynomial interpolation
$frac{f^{(n+1)}(xi)}{(n+1)!}$$prod_{j=0}^n x-x_j$
where n is the degree of the interpolating polynomial and j+1 is the number of data points. I am trying to arrive at a general formula for an upper bound where data points are equidistant in an interval $[x_0 x_n]$ I am considering the product part of the error formula. For this I set,
$x_n=x_0+nh$ where is h the distance between successive data points. I have
$(x-x_0)(x-x_1)(x-x_2)...(x-x_n)$ (1)
At this point I introduce a new quantity $alpha$ which I saw in the book Numerical methods for Engineers by Chapra and Canale:
$alpha=frac{x-x_0}{h}$
Plugging this into (1) I get
$alpha h (alpha h-h)(alpha h-2h)...(alpha h-nh)$
and I get
$h^{(n+1)}alpha (alpha-1)(alpha-2)..(alpha-n)$
Now, I want to maximize the absolute value of $alpha(alpha-1)(alpha-2)..(alpha-n)$ in order to establish an upper bound for the associated error. How can I achieve that?
interpolation
asked Nov 24 at 15:41
Ali Kıral
63
I am using the following error formula for polynomial interpolation
$frac{f^{(n+1)}(xi)}{(n+1)!}$$prod_{j=0}^n x-x_j$
where n is the degree of the interpolating polynomial and j+1 is the number of data points. I am trying to arrive at a general formula for an upper bound where data points are equidistant in an interval $[x_0 x_n]$ I am considering the product part of the error formula. For this I set,
$x_n=x_0+nh$ where is h the distance between successive data points. I have
$(x-x_0)(x-x_1)(x-x_2)...(x-x_n)$ (1)
At this point I introduce a new quantity $alpha$ which I saw in the book Numerical methods for Engineers by Chapra and Canale:
$alpha=frac{x-x_0}{h}$
Plugging this into (1) I get
$alpha h (alpha h-h)(alpha h-2h)...(alpha h-nh)$
and I get
$h^{(n+1)}alpha (alpha-1)(alpha-2)..(alpha-n)$
Now, I want to maximize the absolute value of $alpha(alpha-1)(alpha-2)..(alpha-n)$ in order to establish an upper bound for the associated error. How can I achieve that?
interpolation
interpolation
asked Nov 24 at 15:41
Ali Kıral
63
asked Nov 24 at 15:41
Ali Kıral
63
asked Nov 24 at 15:41
Ali Kıral
63
asked Nov 24 at 15:41
Ali Kıral
63
asked Nov 24 at 15:41
Ali Kıral
63
63
add a comment |
add a comment |
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