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| In [[numerical analysis]], '''Hermite interpolation''', named after [[Charles Hermite]], is a method of [[interpolation|interpolating data points]] as a [[polynomial function]]. The generated '''[[Hermite polynomial]]''' is closely related to the [[Newton polynomial]], in that both are derived from the calculation of [[divided differences]].
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| Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first ''m'' derivatives. This means that ''n''(''m'' + 1) values
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| :<math>
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| \begin{matrix}
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| (x_0, y_0), &(x_1, y_1), &\ldots, &(x_{n-1}, y_{n-1}), \\
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| (x_0, y_0'), &(x_1, y_1'), &\ldots, &(x_{n-1}, y_{n-1}'), \\
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| \vdots & \vdots & &\vdots \\
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| (x_0, y_0^{(m)}), &(x_1, y_1^{(m)}), &\ldots, &(x_{n-1}, y_{n-1}^{(m)})
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| \end{matrix}
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| </math>
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| must be known, rather than just the first ''n'' values required for Newton interpolation. The resulting polynomial may have degree at most ''n''(''m'' + 1) − 1, whereas the Newton polynomial has maximum degree ''n'' − 1. (In the general case, there is no need for ''m'' to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial may have degree ''N'' − 1, with ''N'' the number of data points.)
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| == Usage ==
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| === Simple case ===
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| When using divided differences to calculate the Hermite polynomial of a function ''f'', the first step is to copy each point ''m'' times. (Here we will consider the simplest case <math>m = 1</math> for all points.) Therefore, given <math>n + 1</math> data points <math>x_0, x_1, x_2, \ldots, x_n</math>, and values <math>f(x_0), f(x_1), \ldots, f(x_n)</math> and <math>f'(x_0), f'(x_1), \ldots, f'(x_n)</math> for a function <math>f</math> that we want to interpolate, we create a new dataset
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| :<math>z_0, z_1, \ldots, z_{2n+1}</math> | |
| such that
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| :<math>z_{2i}=z_{2i+1}=x_i.</math>
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| Now, we create a [[Divided differences|divided differences table]] for the points <math>z_0, z_1, \ldots, z_{2n+1}</math>. However, for some divided differences,
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| :<math>z_i = z_{i + 1}\implies f[z_i, z_{i+1}] = \frac{f(z_{i+1})-f(z_{i})}{z_{i+1}-z_{i}} = \frac{0}{0}</math>
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| which is undefined!
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| In this case, we replace the divided difference by <math>f'(z_i)</math>. All others are calculated normally.
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| === General case ===
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| In the general case, suppose a given point <math>x_i</math> has ''k'' derivatives. Then the dataset <math>z_0, z_1, \ldots, z_{N}</math> contains ''k'' identical copies of <math>x_i</math>. When creating the table, [[divided differences]] of <math>j = 2, 3, \ldots, k</math> identical values will be calculated as
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| :<math>\frac{f^{(j)}(x_i)}{j!}.</math>
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| For example,
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| :<math>f[x_i, x_i, x_i]=\frac{f''(x_i)}{2}</math> | |
| :<math>f[x_i, x_i, x_i, x_i]=\frac{f^{(3)}(x_i)}{6}</math>
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| etc.
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| === Example ===
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| Consider the function <math>f(x) = x^8 + 1</math>. Evaluating the function and its first two derivatives at <math>x \in \{-1, 0, 1\}</math>, we obtain the following data:
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| :{| class="wikitable" style="text-align: center; padding: 1em;"
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| |-
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| | ''x'' || ''ƒ''(''x'') || ''ƒ''<nowiki>'</nowiki>(''x'') || ''ƒ''<nowiki>''</nowiki>(''x'')
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| |-
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| | −1 || 2 || −8 || 56
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| |-
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| | 0 || 1 || 0 || 0
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| |-
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| | 1 || 2 || 8 || 56
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| |}
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| Since we have two derivatives to work with, we construct the set <math>\{z_i\} = \{-1, -1, -1, 0, 0, 0, 1, 1, 1\}</math>. Our divided difference table is then:
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| :<math>
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| \begin{matrix}
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| z_0 = -1 & f[z_0] = 2 & & & & & & & & \\
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| & & \frac{f'(z_0)}{1} = -8 & & & & & & & \\
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| z_1 = -1 & f[z_1] = 2 & & \frac{f''(z_1)}{2} = 28 & & & & & & \\
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| & & \frac{f'(z_1)}{1} = -8 & & f[z_3,z_2,z_1,z_0] = -21 & & & & & \\
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| z_2 = -1 & f[z_2] = 2 & & f[z_3,z_2,z_1] = 7 & & 15 & & & & \\
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| & & f[z_3,z_2] = -1 & & f[z_4,z_3,z_2,z_1] = -6 & & -10 & & & \\
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| z_3 = 0 & f[z_3] = 1 & & f[z_4,z_3,z_2] = 1 & & 5 & & 4 & & \\
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| & & \frac{f'(z_3)}{1} = 0 & & f[z_5,z_4,z_3,z_2] = -1 & & -2 & & -1 & \\
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| z_4 = 0 & f[z_4] = 1 & & \frac{f''(z_4)}{2} = 0 & & 1 & & 2 & & 1 \\
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| & & \frac{f'(z_4)}{1} = 0 & & f[z_6,z_5,z_4,z_3] = 1 & & 2 & & 1 & \\
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| z_5 = 0 & f[z_5] = 1 & & f[z_6,z_5,z_4] = 1 & & 5 & & 4 & & \\
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| & & f[z_6,z_5] = 1 & & f[z_7,z_6,z_5,z_4] = 6 & & 10 & & & \\
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| z_6 = 1 & f[z_6] = 2 & & f[z_7,z_6,z_5] = 7 & & 15 & & & & \\
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| & & \frac{f'(z_7)}{1} = 8 & & f[z_8,z_7,z_6,z_5] = 21 & & & & & \\
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| z_7 = 1 & f[z_7] = 2 & & \frac{f''(z_7)}{2} = 28 & & & & & & \\
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| & & \frac{f'(z_8)}{1} = 8 & & & & & & & \\
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| z_8 = 1 & f[z_8] = 2 & & & & & & & & \\
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| \end{matrix}
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| </math> | |
| and the generated polynomial is
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| :<math>
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| \begin{align}
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| P(x) &= 2 - 8(x+1) + 28(x+1) ^2 - 21 (x+1)^3 + 15x(x+1)^3 - 10x^2(x+1)^3 \\
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| &\quad{} + 4x^3(x+1)^3 -1x^3(x+1)^3(x-1)+x^3(x+1)^3(x-1)^2 \\
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| &=2 - 8 + 28 - 21 - 8x + 56x - 63x + 15x + 28x^2 - 63x^2 + 45x^2 - 10x^2 - 21x^3 \\
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| &\quad {}+ 45x^3 - 30x^3 + 4x^3 + x^3 + x^3 + 15x^4 - 30x^4 + 12x^4 + 2x^4 + x^4 \\
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| &\quad {}- 10x^5 + 12x^5 - 2x^5 + 4x^5 - 2x^5 - 2x^5 - x^6 + x^6 - x^7 + x^7 + x^8 \\
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| &= x^8 + 1.
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| \end{align}
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| </math>
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| by taking the coefficients from the diagonal of the divided difference table, and multiplying the ''k''th coefficient by <math>\prod_{i=0}^{k-1} (x - z_i)</math>, as we would when generating a Newton polynomial.
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| ==Error==
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| Call the calculated polynomial ''H'' and original function ''f''. Evaluating a point <math>x \in [x_0, x_n]</math>, the error function is
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| :<math>f(x) - H(x) = \frac{f^{(K)}(c)}{K!}\prod_{i}(x - x_i)^{k_i}</math>
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| where ''c'' is an unknown within the range <math>[x_0, x_N]</math>, ''K'' is the total number of data-points plus one, and <math>k_i</math> is the number of derivatives known at each <math>x_i</math> plus one.
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| ==See also==
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| *[[Cubic Hermite spline]]
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| *[[Newton series]], also known as [[finite differences]]
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| *[[Neville's schema]]
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| *[[Polynomial interpolation]]
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| *[[Lagrange polynomial|Lagrange form]] of the interpolation polynomial
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| *[[Bernstein polynomial|Bernstein form]] of the interpolation polynomial
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| *[[Chinese remainder theorem#Applications | Chinese remainder theorem - Applications]]
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| ==References==
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| * {{ cite book|last1=Burden|first1=Richard L.|first2= J. Douglas |last2=Faires|title=Numerical Analysis|publisher= Belmont: Brooks/Cole|year= 2004}}
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| * {{Citation |last=Spitzbart |first=A. |title=A Generalization of Hermite's Interpolation Formula |journal=[[American Mathematical Monthly]] |volume=67 |issue=1 |pages=42–46 |date=January 1960 |jstor=2308924 |doi= }}
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| ==External links==
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| *[http://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html Hermites Interpolating Polynomial] at Mathworld
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| [[Category:Interpolation]]
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| [[Category:Finite differences]]
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| [[Category:Factorial and binomial topics]]
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