Lagrange interpolation - closed form coefficients
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I have to find algorithm to easy compute coefficients of Lagrange polynomial.
$$y(x) = sum_{i = 1}^{n} L_i(x) f_itag{1}$$
It is polynomial of Lagrange interpolation. We know that it is of the form:
$$y(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0tag{2}$$
It's easy to observe that $y(0) = a_0$, so we can compute $y(0)$ from formula $(1)$.
We continue:
$$frac{y(x) - a_0}{x} = a_n x^{n-1} + ... + a_2 x + a_1$$
But we cannot compute left side in zero, because we cannot divide by zero. What can we do next?
polynomials interpolation lagrange-interpolation
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
add a comment |
up vote
-1
down vote
favorite
I have to find algorithm to easy compute coefficients of Lagrange polynomial.
$$y(x) = sum_{i = 1}^{n} L_i(x) f_itag{1}$$
It is polynomial of Lagrange interpolation. We know that it is of the form:
$$y(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0tag{2}$$
It's easy to observe that $y(0) = a_0$, so we can compute $y(0)$ from formula $(1)$.
We continue:
$$frac{y(x) - a_0}{x} = a_n x^{n-1} + ... + a_2 x + a_1$$
But we cannot compute left side in zero, because we cannot divide by zero. What can we do next?
polynomials interpolation lagrange-interpolation
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I have to find algorithm to easy compute coefficients of Lagrange polynomial.
$$y(x) = sum_{i = 1}^{n} L_i(x) f_itag{1}$$
It is polynomial of Lagrange interpolation. We know that it is of the form:
$$y(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0tag{2}$$
It's easy to observe that $y(0) = a_0$, so we can compute $y(0)$ from formula $(1)$.
We continue:
$$frac{y(x) - a_0}{x} = a_n x^{n-1} + ... + a_2 x + a_1$$
But we cannot compute left side in zero, because we cannot divide by zero. What can we do next?
polynomials interpolation lagrange-interpolation
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
I have to find algorithm to easy compute coefficients of Lagrange polynomial.
$$y(x) = sum_{i = 1}^{n} L_i(x) f_itag{1}$$
It is polynomial of Lagrange interpolation. We know that it is of the form:
$$y(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0tag{2}$$
It's easy to observe that $y(0) = a_0$, so we can compute $y(0)$ from formula $(1)$.
We continue:
$$frac{y(x) - a_0}{x} = a_n x^{n-1} + ... + a_2 x + a_1$$
But we cannot compute left side in zero, because we cannot divide by zero. What can we do next?
polynomials interpolation lagrange-interpolation
polynomials interpolation lagrange-interpolation
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
edited Nov 20 at 10:08
vrugtehagel
10.7k1549
10.7k1549
asked Dec 6 '17 at 13:20
Margaret
11
asked Dec 6 '17 at 13:20
Margaret
11
asked Dec 6 '17 at 13:20
Margaret
11
11
Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30
add a comment |
Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30
Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30
add a comment |
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Well, that means we can't do that trick.
– vrugtehagel
Nov 20 at 10:10
See en.wikipedia.org/wiki/Vandermonde_matrix#Applications
– lhf
Nov 20 at 10:30