Approximate formula for $f''(x_0)$ of a smooth function $f(x)$ using an interpolating polynomial
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Given three points $x_{-1}$, $x_{0} = x_{-1} + h_0$, and $x_{1} = x_{0} + h_1$ where $h_1 neq h_0$ use the second degree interpolating polynomial in Newton form, differentiated twice to obtain
$$f''(x_0) approx 2f[x_{-1},x_0,x_1]$$
Show that this approximation is, in general, first order accurate.
My attempt so far has been to use the polynomial interpolation error formula
$$f(x) = p_2(x) + f[x_{-1},x_0,x_1,x](x-x_{-1})(x-x_0)(x-x_1)$$
differentiated twice and evaluated at $x_0$. So far, I have not seen anything that would give me a first order result. Any solutions/hints would be greatly appreciated.
numerical-methods
asked Nov 16 at 15:21
geo17
877
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up vote
0
down vote
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Given three points $x_{-1}$, $x_{0} = x_{-1} + h_0$, and $x_{1} = x_{0} + h_1$ where $h_1 neq h_0$ use the second degree interpolating polynomial in Newton form, differentiated twice to obtain
$$f''(x_0) approx 2f[x_{-1},x_0,x_1]$$
Show that this approximation is, in general, first order accurate.
My attempt so far has been to use the polynomial interpolation error formula
$$f(x) = p_2(x) + f[x_{-1},x_0,x_1,x](x-x_{-1})(x-x_0)(x-x_1)$$
differentiated twice and evaluated at $x_0$. So far, I have not seen anything that would give me a first order result. Any solutions/hints would be greatly appreciated.
numerical-methods
asked Nov 16 at 15:21
geo17
877
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given three points $x_{-1}$, $x_{0} = x_{-1} + h_0$, and $x_{1} = x_{0} + h_1$ where $h_1 neq h_0$ use the second degree interpolating polynomial in Newton form, differentiated twice to obtain
$$f''(x_0) approx 2f[x_{-1},x_0,x_1]$$
Show that this approximation is, in general, first order accurate.
My attempt so far has been to use the polynomial interpolation error formula
$$f(x) = p_2(x) + f[x_{-1},x_0,x_1,x](x-x_{-1})(x-x_0)(x-x_1)$$
differentiated twice and evaluated at $x_0$. So far, I have not seen anything that would give me a first order result. Any solutions/hints would be greatly appreciated.
numerical-methods
asked Nov 16 at 15:21
geo17
877
Given three points $x_{-1}$, $x_{0} = x_{-1} + h_0$, and $x_{1} = x_{0} + h_1$ where $h_1 neq h_0$ use the second degree interpolating polynomial in Newton form, differentiated twice to obtain
$$f''(x_0) approx 2f[x_{-1},x_0,x_1]$$
Show that this approximation is, in general, first order accurate.
My attempt so far has been to use the polynomial interpolation error formula
$$f(x) = p_2(x) + f[x_{-1},x_0,x_1,x](x-x_{-1})(x-x_0)(x-x_1)$$
differentiated twice and evaluated at $x_0$. So far, I have not seen anything that would give me a first order result. Any solutions/hints would be greatly appreciated.
numerical-methods
numerical-methods
asked Nov 16 at 15:21
geo17
877
asked Nov 16 at 15:21
geo17
877
asked Nov 16 at 15:21
geo17
877
asked Nov 16 at 15:21
geo17
877
asked Nov 16 at 15:21
geo17
877
877
add a comment |
add a comment |
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