Mixing
Deriving quadratic interpolating function
$begingroup$
I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $iin {0,dots, n}$:
$$p_2(x) = frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + frac{(x-x_{i})(x-x_{i+2})}{(x_{i+1}-x_{i})(x_{i+1}-x_{i+2})}f(x_{i+1})+ frac{(x-x_{i})(x-x_{i+1})}{(x_{i+2}-x_{i})(x_{i+2}-x_{i+1})}f(x_{i+2})$$
For this I take the general polynomial $p(x)=a_2 x^2 + a_1 x + a_0$ and build an augmented Vandermonde system from $n$ polynomials
$$begin{bmatrix}
x_i^2 & x_i & 1 & y_i\
x_{i+1}^2 & x_{i+1} & 1 & y_{i+1}\
x_{i+2}^2 & x_{i+2} & 1 & y_{i+2}
end{bmatrix}
$$
and row-reduce it to get long expressions for $a_2, a_1, a_0$. Then substitute to $p(x)$ to get $p_2(x)$.
I was wondering if this is a correct approach and if there is a shorter and more elegant approach to this?
Also, why is $p_2(x)$ called a piecewise function? And, lastly, why does $n$ have to be even? Doesn't one solve $n/3$ systems like the one above?
numerical-methods computational-mathematics lagrange-interpolation
asked Jan 28 at 0:42
sequencesequence
4,28331437
$endgroup$
add a comment |
$begingroup$
I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $iin {0,dots, n}$:
$$p_2(x) = frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + frac{(x-x_{i})(x-x_{i+2})}{(x_{i+1}-x_{i})(x_{i+1}-x_{i+2})}f(x_{i+1})+ frac{(x-x_{i})(x-x_{i+1})}{(x_{i+2}-x_{i})(x_{i+2}-x_{i+1})}f(x_{i+2})$$
For this I take the general polynomial $p(x)=a_2 x^2 + a_1 x + a_0$ and build an augmented Vandermonde system from $n$ polynomials
$$begin{bmatrix}
x_i^2 & x_i & 1 & y_i\
x_{i+1}^2 & x_{i+1} & 1 & y_{i+1}\
x_{i+2}^2 & x_{i+2} & 1 & y_{i+2}
end{bmatrix}
$$
and row-reduce it to get long expressions for $a_2, a_1, a_0$. Then substitute to $p(x)$ to get $p_2(x)$.
I was wondering if this is a correct approach and if there is a shorter and more elegant approach to this?
Also, why is $p_2(x)$ called a piecewise function? And, lastly, why does $n$ have to be even? Doesn't one solve $n/3$ systems like the one above?
numerical-methods computational-mathematics lagrange-interpolation
asked Jan 28 at 0:42
sequencesequence
4,28331437
$endgroup$
add a comment |
$begingroup$
I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $iin {0,dots, n}$:
$$p_2(x) = frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + frac{(x-x_{i})(x-x_{i+2})}{(x_{i+1}-x_{i})(x_{i+1}-x_{i+2})}f(x_{i+1})+ frac{(x-x_{i})(x-x_{i+1})}{(x_{i+2}-x_{i})(x_{i+2}-x_{i+1})}f(x_{i+2})$$
For this I take the general polynomial $p(x)=a_2 x^2 + a_1 x + a_0$ and build an augmented Vandermonde system from $n$ polynomials
$$begin{bmatrix}
x_i^2 & x_i & 1 & y_i\
x_{i+1}^2 & x_{i+1} & 1 & y_{i+1}\
x_{i+2}^2 & x_{i+2} & 1 & y_{i+2}
end{bmatrix}
$$
and row-reduce it to get long expressions for $a_2, a_1, a_0$. Then substitute to $p(x)$ to get $p_2(x)$.
I was wondering if this is a correct approach and if there is a shorter and more elegant approach to this?
Also, why is $p_2(x)$ called a piecewise function? And, lastly, why does $n$ have to be even? Doesn't one solve $n/3$ systems like the one above?
numerical-methods computational-mathematics lagrange-interpolation
asked Jan 28 at 0:42
sequencesequence
4,28331437
$endgroup$
I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $iin {0,dots, n}$:
$$p_2(x) = frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + frac{(x-x_{i})(x-x_{i+2})}{(x_{i+1}-x_{i})(x_{i+1}-x_{i+2})}f(x_{i+1})+ frac{(x-x_{i})(x-x_{i+1})}{(x_{i+2}-x_{i})(x_{i+2}-x_{i+1})}f(x_{i+2})$$
For this I take the general polynomial $p(x)=a_2 x^2 + a_1 x + a_0$ and build an augmented Vandermonde system from $n$ polynomials
$$begin{bmatrix}
x_i^2 & x_i & 1 & y_i\
x_{i+1}^2 & x_{i+1} & 1 & y_{i+1}\
x_{i+2}^2 & x_{i+2} & 1 & y_{i+2}
end{bmatrix}
$$
and row-reduce it to get long expressions for $a_2, a_1, a_0$. Then substitute to $p(x)$ to get $p_2(x)$.
I was wondering if this is a correct approach and if there is a shorter and more elegant approach to this?
Also, why is $p_2(x)$ called a piecewise function? And, lastly, why does $n$ have to be even? Doesn't one solve $n/3$ systems like the one above?
numerical-methods computational-mathematics lagrange-interpolation
numerical-methods computational-mathematics lagrange-interpolation
asked Jan 28 at 0:42
sequencesequence
4,28331437
asked Jan 28 at 0:42
sequencesequence
4,28331437
asked Jan 28 at 0:42
sequencesequence
4,28331437
asked Jan 28 at 0:42
sequencesequence
4,28331437
asked Jan 28 at 0:42
sequencesequence
4,28331437
4,28331437
add a comment |
add a comment |
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