|
'''Төгсгөлөг ялгавар''' нь ''f''(''x'' + ''b'') − ''f''(''x'' + ''a'') хэлбэрийн математик илэрхийлэл юм. Хэрэв төгсгөлөг ялгавар нь ''b'' − ''a'' гэснээр хуваагдвал, [[ялгаварын ноогдвор]] гарч ирнэ. Төгсгөлөг ялгавараар [[уламжлал]]ыг ойролцоолох нь ялангуяа [[хязгаарын нөхцөлт бодлого]], тэр дундаа [[дифференциал тэгшитгэл]]ийн шийдийг инженер, шинжлэх ухааны хүрээнд [[тоон анализ|тоон]] аргаар бодоход [[төгсгөлөг ялгаварын арга]]-ын гол үүргийг гүйцэтгэдэг.
Гэхдээ, төвийн ялгаварын хамгийн гол асуудал нь хэлбэлзэл бүхий функцын уламжлал тэг байх магадлалтайд оршино. Хэрэв ''f''(''nh'')=1 сондгой n үед, мөн ''f''(''nh'')=2 тэгш n үед гэвэл ''f ' '' (''nh'')=0 болно. Иймэрхүү тохиолдлын функцад төвийн ялгаварыг хэрэглэх үед уламжлал тэг байх болно. Ялангуяа f нь дискрет хэлбэртэй бол иймэрхүү асуудал үүснэ.
==Өндөр эрэмбийн ялгаварууд== <!-- this section is linked to further down in the article -->
In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for {{math|''f ' ''(''x''+''h''/2)}} and {{math|''f ' ''(''x''−''h''/2)}} and applying a central difference formula for the derivative of {{mvar| f '}} at {{mvar|x}}, we obtain the central difference approximation of the second derivative of {{mvar|f}}:
'''2-р эрэмбийн төвийн ялгавар'''
:<math> f''(x) \approx \frac{\delta_h^2[f](x)}{h^2} = \frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . </math>
Similarly we can apply other differencing formulas in a recursive manner.
'''2-р эрэмбийн давших ялгавар'''
:<math> f''(x) \approx \frac{\Delta_h^2[f](x)}{h^2} = \frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . </math>
More generally, the '''{{mvar|n}}-th order forward, backward, and central''' differences are given by, respectively,
'''Давших ялгавар'''
:<math>\Delta^n_h[f](x) =
\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x + (n - i) h),
</math>
or for h=1,
:<math>\Delta^n [f](x)= \sum_{k=0}^n\binom nk(-1)^{n-k}f(x + k)</math>
'''Ухрах'''
:<math>\nabla^n_h[f](x) =
\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),
</math>
'''Төвийн'''
:<math>\delta^n_h[f](x) =
\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).
</math>
These equations are using [[binomial coefficient]]s after the summation sign shown as <math>\ \binom{n}{i}</math>. Each row of [[Pascal's triangle]] provides the coefficient for each value of i.
Note that the central difference will, for odd {{mvar|n}}, have {{mvar|h}} multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of <math>\delta^n[f](x - h/2)</math> and <math>\delta^n[f](x + h/2)</math>.
Forward differences applied to a [[sequence]] are sometimes called the [[binomial transform]] of the sequence, and have a number of interesting combinatorial properties.
Forward differences may be evaluated using the [[Nörlund–Rice integral]]. The integral representation for these types of series is interesting, because the integral can often be evaluated using [[asymptotic expansion]] or [[saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large {{mvar|n}}.
The relationship of these higher-order differences with the respective derivatives is straightforward,
:<math>\frac{d^n f}{d x^n}(x) = \frac{\Delta_h^n[f](x)}{h^n}+O(h) = \frac{\nabla_h^n[f](x)}{h^n}+O(h) = \frac{\delta_h^n[f](x)}{h^n} + O(h^2).</math>
Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order {{mvar|h}}. However, the combination
:<math> \frac{\Delta_h[f](x) - \frac12 \Delta_h^2[f](x)}{h} = - \frac{f(x+2h)-4f(x+h)+3f(x)}{2h} </math>
approximates ''f'''(''x'') up to a term of order {{math|''h''<sup>2</sup>}}. This can be proven by expanding the above expression in [[Taylor series]], or by using the calculus of finite differences, explained below.
If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.
===Дурын хэмжээтэй цөмүүд===
Using a little linear algebra, one can fairly easily construct approximations, which sample an arbitrary number of points to the left and a (possibly different) number of points to the right of the center point, for any order of derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points, around the center point, well approximates the Taylor expansion of the desired derivative.
This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side.
The details are outlined in these [http://commons.wikimedia.org/wiki/File:FDnotes.djvu notes].
===Шинж чанар===
* For all positive ''k'' and ''n''
:<math>\Delta^n_{kh} (f, x) = \sum\limits_{i_1=0}^{k-1} \sum\limits_{i_2=0}^{k-1} \cdots \sum\limits_{i_n=0}^{k-1} \Delta^n_h (f, x+i_1h+i_2h+\cdots+i_nh).</math>
* [[Leibniz rule (generalized product rule)|Leibniz rule]]:
:<math>\Delta^n_h (fg, x) = \sum\limits_{k=0}^n \binom{n}{k} \Delta^k_h (f, x) \Delta^{n-k}_h(g, x+kh).</math>
==Төгсгөлөг ялгаварын аргууд==
An important application of finite differences is in [[numerical analysis]], especially in [[numerical partial differential equations|numerical differential equations]], which aim at the numerical solution of [[ordinary differential equation|ordinary]] and [[partial differential equation]]s respectively. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called [[finite difference method]]s.
Common applications of the finite difference method are in computational science and engineering disciplines, such as [[thermal engineering]], [[fluid mechanics]], etc.
==Ньютоны цуваа==
The '''[[Newton polynomial|Newton series]]''' consists of the terms of the '''Newton forward difference equation''', named after [[Isaac Newton]]; in essence, it is the '''Newton interpolation formula''', first published in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' in 1687,<ref>Newton, Isaac, (1687). [http://books.google.com/books?id=KaAIAAAAIAAJ&dq=sir%20isaac%20newton%20principia%20mathematica&as_brr=1&pg=PA466#v=onepage&q&f=false ''Principia'', Book III, Lemma V, Case 1]</ref> namely the discrete analog of the continuum Taylor expansion,
{{Equation box 1
|indent =::
|equation = <math>f(x)=\sum_{k=0}^\infty\frac{\Delta^k [f](a)}{k!} ~(x-a)_k
= \sum_{k=0}^\infty {x-a \choose k}~ \Delta^k [f](a) ~,
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
which holds for any [[polynomial]] function ''f'' and for most (but not all) [[analytic function]]s. Here, the expression
:<math>{x \choose k} = \frac{(x)_k}{k!}</math>
is the [[binomial coefficient]], and
:<math>(x)_k=x(x-1)(x-2)\cdots(x-k+1)</math>
is the "[[falling factorial]]" or "lower factorial", while the [[empty product]] (''x'')<sub>0</sub> is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of ''x'', ''h'' = 1 of the generalization below.
Note also the formal correspondence of this result to [[Taylor's theorem]]. Historically, this, as well as the [[Chu–Vandermonde identity]],
:<math>(x+y)_n=\sum_{k=0}^n {n \choose k} (x)_{n-k} ~(y)_k ~,</math>
(following from it, and corresponding to the [[binomial theorem]]), are included in the observations which matured to the system of the [[umbral calculus]].
To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the [[Fibonacci sequence]] {{mvar|f}} = 2, 2, 4, ... One can find a [[polynomial]] that reproduces these values, by first computing a difference table, and then substituting the differences which correspond to ''x''<sub>0</sub> (underlined) into the formula as follows,
:<math>
\begin{matrix}
\begin{array}{|c||c|c|c|}
\hline
x & f=\Delta^0 & \Delta^1 & \Delta^2 \\
\hline
1&\underline{2}& & \\
& &\underline{0}& \\
2&2& &\underline{2} \\
& &2& \\
3&4& & \\
\hline
\end{array}
&
\quad \begin{align}
f(x) & =\Delta^0 \cdot 1 +\Delta^1 \cdot \dfrac{(x-x_0)_1}{1!} + \Delta^2 \cdot \dfrac{(x-x_0)_2}{2!} \quad (x_0=1)\\
\\
& =2 \cdot 1 + 0 \cdot \dfrac{x-1}{1} + 2 \cdot \dfrac{(x-1)(x-2)}{2} \\
\\
& =2 + (x-1)(x-2) \\
\end{align}
\end{matrix}
</math>
For the case of nonuniform steps in the values of ''x'', Newton computes the [[divided differences]],
:<math>\Delta _{j,0}=y_j,\quad \quad \Delta _{j,k}=\frac{\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_j}\quad \ni \quad \left\{ k>0,\ \ j\le \max \left( j \right)-k \right\},\quad \quad \Delta 0_k=\Delta _{0,k}</math>
the series of products,
:<math>{P_0}=1,\quad \quad P_{k+1}=P_k\cdot \left( \xi -x_k \right) ~,</math>
and the resulting polynomial is the [[scalar product]], <math>f(\xi ) = \Delta 0 \cdot P\left( \xi \right)</math> .<ref>[[Robert D. Richtmyer|Richtmeyer, D.]] and Morton, K.W., (1967). ''Difference Methods for Initial Value Problems'', 2nd ed., Wiley, New York.</ref>
In analysis with [[p-adic number]]s, [[Mahler's theorem]] states that the assumption that ''f'' is a polynomial function can be weakened all the way to the assumption that ''f'' is merely continuous.
[[Carlson's theorem]] provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.
The Newton series, together with the [[Stirling series]] and the [[Selberg class|Selberg series]], is a special case of the general [[difference series]], all of which are defined in terms of suitably scaled forward differences.
In a compressed and slightly more general form and equidistant nodes the formula reads
:<math>f(x)=\sum_{k=0}{\frac{x-a}h \choose k} \sum_{j=0}^k (-1)^{k-j}{k\choose j}f(a+j h).</math>
==Төгсгөлөг ялгаварын тооцоолол==
The forward difference can be considered as a difference [[Operator (mathematics)|operator]],<ref>[[George Boole|Boole, George]], (1872). ''A Treatise On The Calculus of Finite Differences'', 2nd ed., Macmillan and Company. [https://archive.org/details/cu31924031240934 On line]. Also, [Dover edition 1960]</ref><ref>Jordan, Charles, (1939/1965). "Calculus of Finite Differences", Chelsea Publishing. On-line: [http://books.google.com/books?hl=en&lr=&id=3RfZOsDAyQsC&oi=fnd&pg=PA1&ots=AqSuAgOKs3&sig=fzPpAdvnzp7sG6PorqIe5qFjD2Q#v=onepage]</ref> which maps the function {{mvar|''f''}} to {{math|Δ<sub>''h''</sub>[''f'' ]}}. This operator amounts to
::<math>\Delta_h = T_h-I, \,</math>
where {{math|''T''<sub>''h''</sub>}} is the [[shift operator]] with step ''h'', defined by {{math|''T''<sub>''h''</sub>[''f'' ](''x'') {{=}} ''f''(''x''+''h'')}}, and {{mvar|''I''}} is the [[identity operator]].
The finite difference of higher orders can be defined in recursive manner as {{math| Δ<sub>''h''</sub><sup>''n''</sup> ≡ Δ<sub>''h''</sub> (Δ<sub>''h''</sub><sup>''n''−1</sup>}}). Another equivalent definition is {{math| Δ<sub>''h''</sub><sup>''n''</sup> {{=}} [''T''<sub>''h''</sub> −''I'']<sup>''n''</sup>}}.
The difference operator {{math|Δ<sub>''h''</sub>}} is a [[linear operator]] and it satisfies a special [[Leibniz rule (generalized product rule)|Leibniz rule]] indicated above,
{{math|Δ<sub>''h''</sub>(''f''(''x'')''g''(''x'')) {{=}} (Δ<sub>''h''</sub>''f''(''x'')) ''g''(''x''+''h'') + ''f''(''x'') (Δ<sub>''h''</sub>''g''(''x''))}}. Similar statements hold for the backward and central differences.
Formally applying the [[Taylor series]] with respect to ''h'', yields the formula
:<math> \Delta_h = hD + \frac{1}{2} h^2D^2 + \frac{1}{3!} h^3D^3 + \cdots = \mathrm{e}^{hD} - I ~, </math>
where ''D'' denotes the continuum derivative operator, mapping ''f'' to its derivative ''f'''. The expansion is valid when both sides act on [[analytic function]]s, for sufficiently small ''h''. Thus, {{math|''T''<sub>''h''</sub>{{=}}e<sup>''hD''</sup>}}, and formally inverting the exponential yields
:<math> hD = \log(1+\Delta_h) = \Delta_h - \tfrac{1}{2} \Delta_h^2 + \tfrac{1}{3} \Delta_h^3 + \cdots. \, </math>
This formula holds in the sense that both operators give the same result when applied to a polynomial.
Even for analytic functions, the series on the right is not guaranteed to converge; it may be an [[asymptotic series]]. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to {{math|''f’''(''x'')}} mentioned at the end of the [[#Higher-order differences|section ''Higher-order differences'']].
The analogous formulas for the backward and central difference operators are
:<math> hD = -\log(1-\nabla_h) \quad\text{and}\quad hD = 2 \, \operatorname{arsinh}(\tfrac12\delta_h). </math>
The calculus of finite differences is related to the [[umbral calculus]] of combinatorics. This remarkably systematic correspondence is due to the identity of the [[commutators]] of the umbral quantities to their continuum analogs ({{math|''h''→0}} limits),
{{Equation box 1
|indent =::
|equation =
<math> \Bigl[ \frac{\Delta_h}{h} ~,~ x\, T^{-1}_h \Bigr] = [ D ~,~ x ] = I ~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
A large number of formal differential relations of standard calculus involving
functions {{math|''f''(''x'')}} thus ''map systematically to umbral finite-difference analogs'' involving {{math|''f''(''xT''<sub>h</sub><sup>−1</sup>)}}.
For instance, the umbral analog of a monomial ''x''<sup>n</sup> is a generalization of the above falling factorial ([[Pochhammer k-symbol]]),
:<math>~(x)_n\equiv (xT_h^{-1})^n=x (x-h) (x-2h) \cdots (x-(n-1)h)</math> ,
so that
::<math>\frac{\Delta_h}{h} ~(x)_n=n ~(x)_{n-1} ~,</math>
hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function ''f''(''x'') in such symbols), and so on.
For example, the umbral sine is
:<math>\sin (x\,T_h^{-1}) = x -\frac{(x)_3}{3!} + \frac{(x)_5}{5!} - \frac{(x)_7}{7!} + \cdots .</math>
As in the continuum limit, the eigenfunction of {{math| Δ<sub>''h''</sub> /''h''}} also happens ''to be an exponential'',
::<math>\frac{\Delta_h}{h}~(1+\lambda h)^{x/h} =\frac{\Delta_h}{h} ~e^{\ln (1+\lambda h) ~x/h}= \lambda ~e^{\ln (1+\lambda h) ~x/h} ~,</math>
and hence ''Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully'', i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials.<ref>{{cite journal |last =Zachos|first =C.| authorlink =Cosmas Zachos| year =2008| title =Umbral Deformations on Discrete Space-Time | journal =International Journal of Modern Physics A| volume =23 | issue=13| pages =2005–2014 | doi = 10.1142/S0217751X08040548 }}</ref> This umbral exponential thus amounts to the exponential [[generating function]] of the [[Pochhammer symbol]]s.
Thus, for instance, the [[Dirac delta function]] maps to its umbral correspondent, the [[Sinc function|cardinal sine function]],
:<math>\delta (x) \mapsto \frac{\sin \bigl[ \frac{\pi}{2}(1+x/h) \bigr]}{ \pi (x+h) }~,</math>
and so forth.<ref>{{cite doi|10.3389/fphy.2013.00015|noedit}}</ref> [[Difference equation]]s can often be solved with techniques very similar to those for solving [[differential equation]]s.
The inverse operator of the forward difference operator, so then the umbral integral, is the [[indefinite sum]] or antidifference operator.
==Төгсгөлөг ялгаварын операторын тооцоолох дүрмүүд==
Analogous to [[Derivative#Rules|rules for finding the derivative]], we have:
* '''Constant rule''': If ''c'' is a [[Constant (mathematics)|constant]], then
:<math>\Delta c = 0{\,}</math>
* '''[[Linearity of differentiation|Linearity]]''': if ''a'' and ''b'' are [[Constant (mathematics)|constants]],
:<math>\Delta (a f + b g) = a \,\Delta f + b \,\Delta g</math>
All of the above rules apply equally well to any difference operator, including <math>\nabla</math> as to <math>\Delta</math>.
* '''[[Product rule]]''':
:<math> \Delta (f g) = f \,\Delta g + g \,\Delta f + \Delta f \,\Delta g </math>
:<math> \nabla (f g) = f \,\nabla g + g \,\nabla f - \nabla f \,\nabla g </math>
* '''[[Quotient rule]]''':
:<math>\nabla \left( \frac{f}{g} \right) = \frac{1}{g} \det \begin{bmatrix} \nabla f & \nabla g \\ f & g \end{bmatrix}
\left( \det {\begin{bmatrix} g & \nabla g \\ 1 & 1 \end{bmatrix}}\right)^{-1} </math>
::or
:<math>\nabla\left( \frac{f}{g} \right)= \frac {g \,\nabla f - f \,\nabla g}{g \cdot (g - \nabla g)}</math>
:<math>\Delta\left( \frac{f}{g} \right)= \frac {g \,\Delta f - f \,\Delta g}{g \cdot (g + \Delta g)}</math>
* '''Summation rules''':
:<math>\sum_{n=a}^{b} \Delta f(n) = f(b+1)-f(a)</math>
:<math>\sum_{n=a}^{b} \nabla f(n) = f(b)-f(a-1)</math>
<br /><ref>{{cite book|last=Levy|first=H.|author2=Lessman, F.|title=Finite Difference Equations|year=1992|publisher=Dover|isbn=0-486-67260-3}}</ref><ref>Ames, W. F., (1977). ''Numerical Methods for Partial Differential Equations'', Section 1.6. Academic Press, New York. ISBN 0-12-056760-1.</ref><ref>[[Francis B. Hildebrand|Hildebrand, F. B.]], (1968). ''Finite-Difference Equations and Simulations'', Section 2.2, Prentice-Hall, Englewood Cliffs, New Jersey.</ref><ref>{{Cite journal
| first1 = Philippe | last1 = Flajolet
| authorlink2 = Robert Sedgewick (computer scientist) | first2 = Robert | last2 = Sedgewick
| url = http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz
| title = Mellin transforms and asymptotics: Finite differences and Rice's integrals
| journal=Theoretical Computer Science
| volume = 144 | issue = 1–2 | year = 1995 | pages = 101–124
| doi = 10.1016/0304-3975(94)00281-M
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
==Ерөнхийлөл буюу нэгтгэл==
*A '''generalized finite difference''' is usually defined as
:<math>\Delta_h^\mu[f](x) = \sum_{k=0}^N \mu_k f(x+kh),</math>
where <math>\mu = (\mu_0,\ldots,\mu_N)</math> is its coefficients vector. An '''infinite difference''' is a further generalization, where the finite sum above is replaced by an [[Series (mathematics)|infinite series]]. Another way of generalization is making coefficients <math>\mu_k</math> depend on point <math>x</math> : <math>\mu_k=\mu_k(x)</math>, thus considering '''weighted finite difference'''. Also one may make step <math>h</math> depend on point <math>x</math> : <math>h=h(x)</math>. Such generalizations are useful for constructing different [[modulus of continuity]].
*The generalized difference can be seen as the polynomial rings <math>R[T_h]</math> . It leads to difference algebras.
*Difference operator generalizes to [[Möbius inversion]] over a [[partially ordered set]].
*As a convolution operator: Via the formalism of [[incidence algebra]]s, difference operators and other Möbius inversion can be represented by [[convolution]] with a function on the poset, called the [[Möbius function]] μ; for the difference operator, μ is the sequence (1, −1, 0, 0, 0, ...).
==Зарим өөрчлөлтүүд дахь төгсгөлөг ялгавар==
Finite differences can be considered in more than one variable. They are analogous to [[partial derivative]]s in several variables.
Some partial derivative approximations are (using central step method):
:<math> f_{x}(x,y) \approx \frac{f(x+h ,y) - f(x-h,y)}{2h} \ </math>
:<math> f_{y}(x,y) \approx \frac{f(x,y+k ) - f(x,y-k)}{2k} \ </math>
:<math> f_{xx}(x,y) \approx \frac{f(x+h ,y) - 2 f(x,y) + f(x-h,y)}{h^2} \ </math>
:<math> f_{yy}(x,y) \approx \frac{f(x,y+k) - 2 f(x,y) + f(x,y-k)}{k^2} \ </math>
:<math> f_{xy}(x,y) \approx \frac{f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)}{4hk} ~. </math>
Alternatively, for applications in which the computation of {{mvar|f}} is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is
:<math> f_{xy}(x,y) \approx \frac{f(x+h, y+k) - f(x+h, y) - f(x, y+k) + 2 f(x,y) - f(x-h, y) - f(x, y-k) + f(x-h, y-k)}{2hk} ~,</math>
since the only values to be computed which are not already needed for the previous four equations are {{math|''f''(''x''+''h'', ''y''+''k'')}} and {{math|''f''(''x''−''h'', ''y''−''k'')}}.
==Мөн үзэх==
{{columns-list|3|
* [[Carlson's theorem]]
* [[Divided differences]]
* [[Finite difference coefficients]]
* [[Finite difference method]]
* [[Five-point stencil]]
* [[Gilbreath's conjecture]]
* [[Lagrange polynomial]]
*[[Central differencing scheme]]
* [[Upwind differencing scheme for convection]]
* [[Modulus of continuity]]
* [[Newton polynomial]]
* [[Nörlund–Rice integral]]
* [[Numerical differentiation]]
* [[Sheffer sequence]]
* [[Summation by parts]]
* [[Table of Newtonian series]]
* [[Taylor series]]
* [[Time scale calculus]]
* [[Umbral calculus]]
}}
== Лавлахууд ==
<references/>
== Гадаад линк ==
* {{springer|title=Finite-difference calculus|id=p/f040230}}
* [http://reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html#c:4 Table of useful finite difference formula generated using [[Mathematica]] ]
* [http://www.stanford.edu/~dgleich/publications/finite-calculus.pdf Finite Calculus: A Tutorial for Solving Nasty Sums]
* [http://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/ Discrete Second Derivative from Unevenly Spaced Points]
*[https://whatinsider.com/isaac-newton-biography-discoveries-laws-and-death/ The Universal Law of Gravitation]
{{DEFAULTSORT:Төгсгөлөг ялгавар}}
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