LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle [-1, 1]}

${\displaystyle [-1,1]}$

... is translated to the CAS output ...

Semantic latex: [-1, 1]

Confidence: 0

Mathematica

Translation: [- 1 , 1]

Information

Sub Equations

[- 1 , 1]

Tests

Symbolic

Numeric

SymPy

Translation: [- 1 , 1]

Information

Sub Equations

[- 1 , 1]

Tests

Symbolic

Numeric

Maple

Translation: [- 1 , 1]

Information

Sub Equations

[- 1 , 1]

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$1$

Description

interval

rule

polynomial of degree

common domain of integration

accurate approximation to the integral

quadrature rule

polynomial

other interval

section

Lobatto quadrature of function

low-order quadrature rule

i.e.

Gaussian quadrature rule

integrand

positive weight function

simplest integration problem

general way

integration problem

Legendre polynomial

orthogonal polynomial

Complete translation information:

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Specify your own input

Formula input
Wikitext context	{{Redirect\|Gaussian integration\|the integral of a Gaussian function\|Gaussian integral}} {{more footnotes\|date=September 2018}} [[File:Comparison Gaussquad trapezoidal.svg\|thumb\|upright=2\|alt=Comparison between 2-point Gaussian and trapezoidal quadrature.\|Comparison between 2-point Gaussian and trapezoidal quadrature. The blue line is the polynomial <math display="inline">y(x) = 7x^3 - 8x^2 - 3x + 3</math>, whose integral in {{math\|[−1, 1]}} is {{math\|{{frac\|2\|3}}}}. The [[trapezoidal rule]] returns the integral of the orange dashed line, equal to <math display="inline">y(-1) + y(1) = -10</math>. The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to <math display="inline">y\left(-\sqrt{\frac{1}{3}}\right) + y\left(\sqrt{\frac{1}{3}}\right) = \frac{2}{3}</math>. Such a result is exact, since the green region has the same area as the sum of the red regions.]] In [[numerical analysis]], a '''quadrature rule''' is an approximation of the [[integral\|definite integral]] of a [[function (mathematics)\|function]], usually stated as a [[weighted sum]] of function values at specified points within the domain of integration. (See [[numerical integration]] for more on [[quadrature (mathematics)\|quadrature]] rules.) An ''n''-point '''Gaussian quadrature rule''', named after [[Carl Friedrich Gauss]],<ref>''Methodus nova integralium valores per approximationem inveniendi.'' In: ''Comm. Soc. Sci. Göttingen Math.'' Band 3, 1815, S. 29–76, [http://gallica.bnf.fr/ark:/12148/bpt6k2412190.r=Gauss.langEN Gallica], datiert 1814, auch in Werke, Band 3, 1876, S. 163–196.</ref> is a quadrature rule constructed to yield an exact result for [[polynomial]]s of degree {{math\|2''n'' − 1}} or less by a suitable choice of the nodes {{mvar\|x<sub>i</sub>}} and weights {{mvar\|w<sub>i</sub>}} for {{math\|''i'' {{=}} 1, ..., ''n''}}. The modern formulation using orthogonal polynomials was developed by [[Carl Gustav Jacobi]] 1826.<ref>[[Carl Gustav Jacob Jacobi\|C. G. J. Jacobi]]: ''Ueber Gauß' neue Methode, die Werthe der Integrale näherungsweise zu finden.'' In: ''Journal für Reine und Angewandte Mathematik.'' Band 1, 1826, S. 301–308, [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001&DMDID=DMDLOG_0035 (online)], und Werke, Band 6.</ref> The most common domain of integration for such a rule is taken as {{math\|[−1, 1]}}, so the rule is stated as :<math>\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i),</math> which is exact for polynomials of degree {{math\|2''n'' − 1}} or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if {{math\|''f''(''x'')}} is well-approximated by a polynomial of degree {{math\|2''n'' − 1}} or less on {{math\|[−1, 1]}}. The Gauss-[[Adrien-Marie Legendre\|Legendre]] quadrature rule is not typically used for integrable functions with endpoint [[singularity (math)\|singularities]]. Instead, if the integrand can be written as :<math>f(x) = \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x),\quad \alpha,\beta > -1,</math> where {{math\|''g''(''x'')}} is well-approximated by a low-degree polynomial, then alternative nodes <math>x_i'</math> and weights <math>w_i'</math> will usually give more accurate quadrature rules. These are known as [[Gauss-Jacobi quadrature]] rules, i.e., :<math>\int_{-1}^1 f(x)\,dx = \int_{-1}^1 \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x)\,dx \approx \sum_{i=1}^n w_i' g\left(x_i'\right).</math> Common weights include <math display="inline">\frac{1}{\sqrt{1 - x^2}}</math> ([[Chebyshev–Gauss quadrature\|Chebyshev–Gauss]]) and <math>\sqrt{1 - x^2}</math>. One may also want to integrate over semi-infinite ([[Gauss-Laguerre quadrature]]) and infinite intervals ([[Gauss–Hermite quadrature]]). It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes {{mvar\|x<sub>i</sub>}} are the [[Root of a function\|roots]] of a polynomial belonging to a class of [[orthogonal polynomials]] (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights. == Gauss–Legendre quadrature == <!-- The section "Other forms of Gaussian quadrature" below links to this section --> [[File:Legendrepolynomials6.svg\|thumb\|Graphs of Legendre polynomials (up to {{math\|''n'' {{=}} 5)}}]] For the simplest integration problem stated above, i.e., {{math\|''f''(''x'')}} is well-approximated by polynomials on <math>[-1, 1]</math>, the associated orthogonal polynomials are [[Legendre polynomials]], denoted by {{math\|''P''<sub>''n''</sub>(''x'')}}. With the {{mvar\|n}}-th polynomial normalized to give {{math\|''P''<sub>''n''</sub>(1) {{=}} 1}}, the {{mvar\|i}}-th Gauss node, {{mvar\|x<sub>i</sub>}}, is the {{mvar\|i}}-th root of {{mvar\|P<sub>n</sub>}} and the weights are given by the formula {{Harv\|Abramowitz\|Stegun\|1972\|loc=p. 887}} :<math> w_i = \frac{2}{\left( 1 - x_i^2 \right) \left[P'_n(x_i)\right]^2}.</math> Some low-order quadrature rules are tabulated below (over interval {{math\|[−1, 1]}}, see the section below for other intervals). {\| class="wikitable" style="margin:auto; background:white; text-align:center;" ! Number of points, ''n'' ! colspan="2" \| Points, {{mvar\|x<sub>i</sub>}} ! colspan="2" \| Weights, {{mvar\|w<sub>i</sub>}} \|- \| 1 \| colspan="2" \| 0 \| colspan="2" \| 2 \|- \| 2 \| <math>\pm\frac{1}{\sqrt{3}}</math> \|\| ±0.57735... \| colspan="2" \| 1 \|- \| rowspan="2" \| 3 \| colspan="2" \| 0 \| <math>\frac{8}{9}</math> \|\| 0.888889... \|- \| <math>\pm\sqrt{\frac{3}{5}}</math> \|\| ±0.774597... \| <math>\frac{5}{9}</math> \|\| 0.555556... \|- \| rowspan="2" \| 4 \| <math>\pm\sqrt{\frac{3}{7} - \frac{2}{7}\sqrt{\frac{6}{5}}}</math> \|\| ±0.339981... \| <math>\frac{18 + \sqrt{30}}{36}</math> \|\| 0.652145... \|- \| <math>\pm\sqrt{\frac{3}{7} + \frac{2}{7}\sqrt{\frac{6}{5}}}</math> \|\| ±0.861136... \| <math>\frac{18 - \sqrt{30}}{36}</math> \|\| 0.347855... \|- \| rowspan="3" \| 5 \| colspan="2" \| 0 \| <math>\frac{128}{225}</math> \|\| 0.568889... \|- \| <math>\pm\frac{1}{3}\sqrt{5 - 2\sqrt{\frac{10}{7}}}</math> \|\| ±0.538469... \| <math>\frac{322 + 13\sqrt{70}}{900}</math> \|\| 0.478629... \|- \| <math>\pm\frac{1}{3}\sqrt{5 + 2\sqrt{\frac{10}{7}}}</math> \|\| ±0.90618... \| <math>\frac{322 - 13\sqrt{70}}{900}</math> \|\| 0.236927... \|} == Change of interval == An integral over {{math\|[''a'', ''b'']}} must be changed into an integral over {{math\|[−1, 1]}} before applying the Gaussian quadrature rule. This change of interval can be done in the following way: :<math>\int_a^b f(x)\,dx = \frac{b-a}{2} \int_{-1}^1 f\left(\frac{b-a}{2}\xi + \frac{a+b}{2}\right)\,d\xi.</math> Applying <math>n</math> point Gaussian quadrature <math>(\xi, w)</math> rule then results in the following approximation: :<math>\int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}\xi_i + \frac{a+b}{2}\right).</math> == Other forms == The integration problem can be expressed in a slightly more general way by introducing a positive [[weight function]] {{mvar\|ω}} into the integrand, and allowing an interval other than {{math\|[−1, 1]}}. That is, the problem is to calculate :<math> \int_a^b \omega(x)\,f(x)\,dx </math> for some choices of {{mvar\|a}}, {{mvar\|b}}, and {{mvar\|ω}}. For {{math\|''a'' {{=}} −1}}, {{math\|''b'' {{=}} 1}}, and {{math\|ω(''x'') {{=}} 1}}, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for [[Abramowitz and Stegun]] (A & S). {\| class="wikitable" style="margin:auto; background:white; text-align:center;" ! Interval ! {{math\|''ω''(''x'')}} ! Orthogonal polynomials ! A & S ! For more information, see ... \|- \| {{math\|[−1, 1]}} \|\| {{math\|1}} \|\| [[Legendre polynomials]] \|\| 25.4.29 \|\| {{section link\|\|Gauss–Legendre quadrature}} \|- \| {{math\|(−1, 1)}} \|\| <math>\left(1 - x\right)^\alpha \left(1 + x\right)^\beta,\quad \alpha, \beta > -1</math> \|\| [[Jacobi polynomials]] \|\| 25.4.33 ({{math\|''β'' {{=}} 0}}) \|\| [[Gauss–Jacobi quadrature]] \|- \| {{math\|(−1, 1)}} \|\| <math>\frac{1}{\sqrt{1 - x^2}}</math> \|\| [[Chebyshev polynomials]] (first kind) \|\| 25.4.38 \|\| [[Chebyshev–Gauss quadrature]] \|- \| {{math\|[−1, 1]}} \|\| <math>\sqrt{1 - x^2}</math> \|\| Chebyshev polynomials (second kind) \|\| 25.4.40 \|\| [[Chebyshev–Gauss quadrature]] \|- \| {{math\|[0, ∞)}} \|\| <math> e^{-x}\, </math> \|\| [[Laguerre polynomials]] \|\| 25.4.45 \|\| [[Gauss–Laguerre quadrature]] \|- \| {{math\|[0, ∞)}} \|\| <math> x^\alpha e^{-x},\quad \alpha>-1 </math> \|\| Generalized [[Laguerre polynomials]] \|\| \|\| [[Gauss–Laguerre quadrature]] \|- \| {{math\|(−∞, ∞)}} \|\| <math> e^{-x^2} </math> \|\| [[Hermite polynomials]] \|\| 25.4.46 \|\| [[Gauss–Hermite quadrature]] \|} === Fundamental theorem === Let {{mvar\|p<sub>n</sub>}} be a nontrivial polynomial of degree {{mvar\|n}} such that :<math>\int_a^b \omega(x) \, x^k p_n(x) \, dx = 0, \quad \text{for all } k = 0, 1, \ldots, n - 1.</math> If we pick the {{mvar\|n}} nodes {{mvar\|x<sub>i</sub>}} to be the zeros of {{mvar\|p<sub>n</sub>}}, then there exist {{mvar\|n}} weights {{mvar\|w<sub>i</sub>}} which make the Gauss-quadrature computed integral exact for all polynomials {{math\|''h''(''x'')}} of degree {{math\|2''n'' − 1}} or less. Furthermore, all these nodes {{mvar\|x<sub>i</sub>}} will lie in the open interval {{math\|(''a'', ''b'')}} {{harv\|Stoer\|Bulirsch\|2002\|pp=172–175}}. The polynomial {{mvar\|p<sub>n</sub>}} is said to be an orthogonal polynomial of degree {{mvar\|n}} associated to the weight function {{math\|''ω''(''x'')}}. It is unique up to a constant normalization factor. The idea underlying the proof is that, because of its sufficiently low degree, {{math\|''h''(''x'')}} can be divided by <math>p_n(x)</math> to produce a quotient {{math\|''q''(''x'')}} of degree strictly lower than {{mvar\|n}}, and a remainder {{math\|''r''(''x'')}} of still lower degree, so that both will be orthogonal to <math>p_n(x)</math>, by the defining property of <math>p_n(x)</math>. Thus :<math> \int_a^b \omega(x)\,h(x)\,dx = \int_a^b \omega(x)\,r(x)\,dx. </math> Because of the choice of nodes {{mvar\|x<sub>i</sub>}}, the corresponding relation :<math>\sum_{i=1}^n w_i h(x_i) = \sum_{i=1}^n w_i r(x_i)</math> holds also. The exactness of the computed integral for <math>h(x)</math> then follows from corresponding exactness for polynomials of degree only {{mvar\|n}} or less (as is <math>r(x)</math>). ==== General formula for the weights ==== The weights can be expressed as {{NumBlk\|:\|<math>w_{i} = \frac{a_{n}}{a_{n-1}}\frac{\int_{a}^{b}\omega(x)p_{n-1}\left(x\right)^{2}dx}{p'_{n}(x_{i})p_{n-1}(x_{i})}</math>\|{{EquationRef\|1}}}} where <math>a_{k}</math> is the coefficient of <math>x^{k}</math> in <math>p_{k}(x)</math>. To prove this, note that using [[Lagrange interpolation]] one can express {{math\|''r''(''x'')}} in terms of <math>r(x_{i})</math> as :<math>r(x) = \sum_{i=1}^{n}r(x_{i})\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}</math> because {{math\|''r''(''x'')}} has degree less than {{mvar\|n}} and is thus fixed by the values it attains at {{mvar\|n}} different points. Multiplying both sides by {{math\|''ω''(''x'')}} and integrating from {{mvar\|a}} to {{mvar\|b}} yields :<math>\int_{a}^{b}\omega(x)r(x)dx= \sum_{i=1}^{n}r(x_{i})\int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx</math> The weights {{mvar\|w<sub>i</sub>}} are thus given by :<math>w_{i} = \int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx</math> This integral expression for <math>w_{i}</math> can be expressed in terms of the orthogonal polynomials <math>p_{n}(x)</math> and <math>p_{n-1}(x)</math> as follows. We can write :<math>\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\left(x-x_{j}\right) = \frac{\prod_{1\leq j\leq n} \left(x - x_{j}\right)}{x-x_{i}} = \frac{p_{n}(x)}{a_{n}\left(x-x_{i}\right)}</math> where <math>a_{n}</math> is the coefficient of <math>x^n</math> in <math>p_{n}(x)</math>. Taking the limit of {{mvar\|x}} to <math>x_{i}</math> yields using L'Hôpital's rule :<math>\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\left(x_{i}-x_{j}\right) = \frac{p'_{n}(x_{i})}{a_{n}}</math> We can thus write the integral expression for the weights as {{NumBlk\|:\|<math>w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</math>\|{{EquationRef\|2}}}} In the integrand, writing :<math>\frac{1}{x-x_i} = \frac{1 - \left(\frac{x}{x_i}\right)^{k}}{x - x_i} + \left(\frac{x}{x_i}\right)^{k} \frac{1}{x - x_i}</math> yields :<math>\int_a^b\omega(x)\frac{x^kp_n(x)}{x-x_i}dx= x_i^k\int_{a}^{b}\omega(x)\frac{p_n(x)}{x-x_i}dx</math> provided <math>k \leq n</math>, because :<math>\frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}}</math> is a polynomial of degree {{math\|k − 1}} which is then orthogonal to <math>p_{n}(x)</math>. So, if {{math\|''q''(''x'')}} is a polynomial of at most nth degree we have :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{1}{q(x_{i})}\int_{a}^{b}\omega(x)\frac{q(x)p_{n}(x)}{x-x_{i}}dx </math> We can evaluate the integral on the right hand side for <math>q(x) = p_{n-1}(x)</math> as follows. Because <math>\frac{p_{n}(x)}{x-x_{i}}</math> is a polynomial of degree {{math\|n − 1}}, we have :<math>\frac{p_{n}(x)}{x-x_{i}} = a_{n}x^{n-1} + s(x)</math> where {{math\|''s''(''x'')}} is a polynomial of degree <math>n - 2</math>. Since {{math\|''s''(''x'')}} is orthogonal to <math>p_{n-1}(x)</math> we have :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{p_{n-1}(x_{i})}\int_{a}^{b}\omega(x)p_{n-1}(x)x^{n-1}dx </math> We can then write :<math>x^{n-1} = \left(x^{n-1} - \frac{p_{n-1}(x)}{a_{n-1}}\right) + \frac{p_{n-1}(x)}{a_{n-1}}</math> The term in the brackets is a polynomial of degree <math>n - 2</math>, which is therefore orthogonal to <math>p_{n-1}(x)</math>. The integral can thus be written as :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{a_{n-1}p_{n-1}(x_{i})}\int_{a}^{b}\omega(x)p_{n-1}(x)^{2}dx </math> According to equation ({{EquationNote\|2}}), the weights are obtained by dividing this by <math>p'_{n}(x_{i})</math> and that yields the expression in equation ({{EquationNote\|1}}). <math>w_{i}</math> can also be expressed in terms of the orthogonal polynomials <math>p_{n}(x)</math> and now <math>p_{n+1}(x)</math>. In the 3-term recurrence relation <math>p_{n+1}(x_{i}) = (a)p_{n}(x_{i}) + (b)p_{n-1}(x_{i})</math> the term with <math>p_{n}(x_{i})</math> vanishes, so <math>p_{n-1}(x_{i})</math> in Eq. (1) can be replaced by <math display="inline">\frac{1}{b}p_{n+1}\left(x_i\right)</math>. ====Proof that the weights are positive==== Consider the following polynomial of degree <math>2n - 2</math> :<math>f(x) = \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}}\frac{\left(x - x_j\right)^2}{\left(x_i - x_j\right)^2}</math> where, as above, the {{mvar\|x<sub>j</sub>}} are the roots of the polynomial <math>p_{n}(x)</math>. Clearly <math>f(x_j) = \delta_{ij}</math>. Since the degree of <math>f(x)</math> is less than <math>2n - 1</math>, the Gaussian quadrature formula involving the weights and nodes obtained from <math>p_{n}(x)</math> applies. Since <math>f(x_{j}) = 0</math> for j not equal to i, we have :<math>\int_{a}^{b}\omega(x)f(x)dx=\sum_{j=1}^{N}w_{j}f(x_{j}) = \sum_{j=1}^{N} \delta_{ij} w_j = w_{i} > 0.</math> Since both <math>\omega(x)</math> and <math>f(x)</math> are non-negative functions, it follows that <math>w_{i} > 0</math>. === Computation of Gaussian quadrature rules === There are many algorithms for computing the nodes {{mvar\|x<sub>i</sub>}} and weights {{mvar\|w<sub>i</sub>}} of Gaussian quadrature rules. The most popular are the Golub-Welsch algorithm requiring {{math\|''O''(''n''<sup>2</sup>)}} operations, Newton's method for solving <math>p_n(x) = 0</math> using the [[Orthogonal polynomials#Recurrence relation\|three-term recurrence]] for evaluation requiring {{math\|''O''(''n''<sup>2</sup>)}} operations, and asymptotic formulas for large ''n'' requiring {{math\|''O''(''n'')}} operations. ==== Recurrence relation ==== Orthogonal polynomials <math>p_r</math> with <math>(p_r, p_s) = 0</math> for <math>r \ne s</math> for a scalar product <math>(\, \,)</math>, degree <math>(p_r) = r</math> and leading coefficient one (i.e. [[monic polynomial\|monic]] orthogonal polynomials) satisfy the recurrence relation :<math>p_{r+1}(x) = (x - a_{r,r})p_r(x) - a_{r,r-1}p_{r-1}(x)\cdots - a_{r,0}p_0(x)</math> and scalar product defined :<math>(f(x),g(x))=\int_a^b\omega(x)f(x)g(x)dx</math> for <math>r = 0, 1, \ldots, n - 1</math> where {{mvar\|n}} is the maximal degree which can be taken to be infinity, and where <math display="inline">a_{r,s} = \frac{\left(xp_r, p_s\right)}{\left(p_s, p_s\right)}</math>. First of all, the polynomials defined by the recurrence relation starting with <math>p_0(x) = 1</math> have leading coefficient one and correct degree. Given the starting point by <math>p_0</math>, the orthogonality of <math>p_r</math> can be shown by induction. For <math>r = s = 0</math> one has :<math>(p_1,p_0)=(x-a_{0,0})(p_0,p_0)=(xp_0,p_0)-a_{0,0}(p_0,p_0)=(xp_0,p_0)-(xp_0,p_0)=0.</math> Now if <math>p_0, p_1, \ldots, p_r</math> are orthogonal, then also <math>p_{r+1}</math>, because in :<math>(p_{r+1}, p_s) = (xp_r, p_s) - a_{r,r}(p_r, p_s) - a_{r,r-1}(p_{r-1}, p_s)\cdots - a_{r,0}(p_0, p_s)</math> all scalar products vanish except for the first one and the one where <math>p_s</math> meets the same orthogonal polynomial. Therefore, :<math>(p_{r+1},p_s)=(xp_r,p_s)-a_{r,s}(p_s,p_s)=(xp_r,p_s)-(xp_r,p_s)=0.</math> However, if the scalar product satisfies <math>(xf, g) = (f,xg)</math> (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For <math>s < r - 1, xp_s</math> is a polynomial of degree less than or equal to {{math\|''r'' − 1}}. On the other hand, <math>p_r</math> is orthogonal to every polynomial of degree less than or equal to {{math\|''r'' − 1}}. Therefore, one has <math>(xp_r, p_s) = (p_r, xp_s) = 0</math> and <math>a_{r,s} = 0</math> for {{math\|''s'' < ''r'' − 1}}. The recurrence relation then simplifies to :<math>p_{r+1}(x)=(x-a_{r,r})p_r(x)-a_{r,r-1}p_{r-1}(x)</math> or :<math>p_{r+1}(x)=(x-a_r)p_r(x)-b_rp_{r-1}(x)</math> (with the convention <math>p_{-1}(x) \equiv 0</math>) where :<math>a_r:=\frac{(xp_r,p_r)}{(p_r,p_r)},\qquad b_r:=\frac{(xp_r,p_{r-1})}{(p_{r-1},p_{r-1})}=\frac{(p_r,p_r)}{(p_{r-1},p_{r-1})}</math> (the last because of <math>(xp_r, p_{r-1}) = (p_r, xp_{r-1}) = (p_r, p_r)</math>, since <math>xp_{r-1}</math> differs from <math>p_r</math> by a degree less than {{mvar\|r}}). ==== The Golub-Welsch algorithm ==== The three-term recurrence relation can be written in matrix form <math>J\tilde{P} = x\tilde{P} - p_n(x) \times \mathbf{e}_n</math> where <math>\tilde{P} = \begin{bmatrix} p_0(x) & p_1(x) & \ldots & p_{n-1}(x) \end{bmatrix}^\mathsf{T}</math>, <math>\mathbf{e}_n</math> is the <math>n</math>th standard basis vector, i.e., <math>\mathbf{e}_n = \begin{bmatrix} 0 & \ldots & 0 & 1 \end{bmatrix}^\mathsf{T}</math>, and {{mvar\|J}} is the so-called Jacobi matrix: :<math>\mathbf{J}=\begin{pmatrix} a_0 & 1 & 0 & \ldots & \ldots & \ldots \\ b_1 & a_1 & 1 & 0 & \ldots & \ldots \\ 0 & b_2 & a_2 & 1 & 0 & \ldots \\ 0 & \ldots & \ldots & \ldots & \ldots & 0 \\ \ldots & \ldots & 0 & b_{n-2} & a_{n-2} & 1 \\ \ldots & \ldots & \ldots & 0 & b_{n-1} & a_{n-1} \end{pmatrix}</math> The zeros <math>x_j</math> of the polynomials up to degree {{mvar\|n}}, which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this [[tridiagonal matrix]]. This procedure is known as ''Golub–Welsch algorithm''. For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix <math>\mathcal{J}</math> with elements :<math>\begin{align} \mathcal{J}_{i,i} = J_{i,i} &= a_{i-1} && i=1,\ldots,n \\ \mathcal{J}_{i-1,i} = \mathcal{J}_{i,i-1} = \sqrt{J_{i,i-1}J_{i-1,i}} &= \sqrt{b_{i-1}} && i=2,\ldots,n. \end{align}</math> {{math\|'''J'''}} and <math>\mathcal{J}</math> are [[similar matrices]] and therefore have the same eigenvalues (the nodes). The weights can be computed from the corresponding eigenvectors: If <math>\phi^{(j)}</math> is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated to the eigenvalue {{mvar\|x<sub>j</sub>}}, the corresponding weight can be computed from the first component of this eigenvector, namely: :<math>w_j=\mu_0 \left(\phi_1^{(j)}\right)^2</math> where <math>\mu_0</math> is the integral of the weight function :<math>\mu_0=\int_a^b \omega(x) dx.</math> See, for instance, {{harv\|Gil\|Segura\|Temme\|2007}} for further details. === Error estimates === The error of a Gaussian quadrature rule can be stated as follows {{harv\|Stoer\|Bulirsch\|2002\|loc=Thm 3.6.24}}. For an integrand which has {{math\|2''n''}} continuous derivatives, :<math> \int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i) = \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n, p_n) </math> for some {{mvar\|ξ}} in {{math\|(''a'', ''b'')}}, where {{mvar\|p<sub>n</sub>}} is the monic (i.e. the leading coefficient is {{math\|1}}) orthogonal polynomial of degree {{mvar\|n}} and where :<math> (f,g) = \int_a^b \omega(x) f(x) g(x) \, dx.</math> In the important special case of {{math\|''ω''(''x'') {{=}} 1}}, we have the error estimate {{Harv\|Kahaner\|Moler\|Nash\|1989\|loc=§5.2}} :<math> \frac{\left(b - a\right)^{2n+1} \left(n!\right)^4}{(2n + 1)\left[\left(2n\right)!\right]^3} f^{(2n)} (\xi), \qquad a < \xi < b.</math> Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the order {{math\|2''n''}} derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss–Kronrod quadrature rules can be useful. === Gauss–Kronrod rules === {{main\|Gauss–Kronrod quadrature formula}} If the interval {{math\|[''a'', ''b'']}} is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers), and thus the integrand must be evaluated at every point. ''Gauss–Kronrod rules'' are extensions of Gauss quadrature rules generated by adding {{math\|''n'' + 1}} points to an {{mvar\|n}}-point rule in such a way that the resulting rule is of order {{math\|2''n'' + 1}}. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension is often used as an estimate of the approximation error. === Gauss–Lobatto rules === Also known as '''Lobatto quadrature''' {{Harv\|Abramowitz\|Stegun\|1972\|loc=p. 888}}, named after Dutch mathematician [[Rehuel Lobatto]]. It is similar to Gaussian quadrature with the following differences: # The integration points include the end points of the integration interval. # It is accurate for polynomials up to degree {{math\|2''n'' – 3}}, where {{mvar\|n}} is the number of integration points {{harv\|Quarteroni\|Sacco\|Saleri\|2000}}. Lobatto quadrature of function {{math\|''f''(''x'')}} on interval {{math\|[−1, 1]}}: :<math>\int_{-1}^1 {f(x) \, dx} = \frac {2} {n(n-1)}[f(1) + f(-1)] + \sum_{i = 2}^{n-1} {w_i f(x_i)} + R_n.</math> Abscissas: {{mvar\|x<sub>i</sub>}} is the <math>(i - 1)</math>st zero of <math>P'_{n-1}(x)</math>, here <math>P_m(x)</math>denotes the standard Legendre polynomial of m-th degree and the dash denotes the derivative. Weights: :<math>w_i = \frac{2}{n(n - 1)\left[P_{n-1}\left(x_i\right)\right]^2}, \qquad x_i \ne \pm 1.</math> Remainder: :<math>R_n = \frac{-n\left(n - 1\right)^3 2^{2n-1} \left[\left(n - 2\right)!\right]^4}{(2n-1) \left[\left(2n - 2\right)!\right]^3} f^{(2n-2)}(\xi), \qquad -1 < \xi < 1.</math> Some of the weights are: {\| class="wikitable" style="margin:auto; background:white; text-align:center;" ! Number of points, ''n'' ! Points, {{mvar\|x<sub>i</sub>}} ! Weights, {{mvar\|w<sub>i</sub>}} \|- \| rowspan="2" \| <math>3</math> \| <math>0</math> \|\| <math>\frac{4}{3}</math> \|- \| <math>\pm 1</math> \|\| <math>\frac{1}{3}</math> \|- \| rowspan="2" \| <math>4</math> \| <math>\pm \sqrt{\frac{1}{5}}</math> \|\| <math>\frac{5}{6}</math> \|- \| <math>\pm 1</math> \|\| <math>\frac{1}{6}</math> \|- \| rowspan="3" \| <math>5</math> \| <math>0</math> \|\| <math>\frac{32}{45}</math> \|- \| <math>\pm\sqrt{\frac{3}{7}}</math> \|\| <math>\frac{49}{90}</math> \|- \| <math>\pm 1</math> \|\| <math>\frac{1}{10}</math> \|- \| rowspan="3" \| <math>6</math> \| <math>\pm\sqrt{\frac{1}{3}-\frac{2\sqrt{7}}{21}}</math> \|\| <math>\frac{14+\sqrt{7}}{30}</math> \|- \| <math>\pm\sqrt{\frac{1}{3} + \frac{2\sqrt{7}}{21}}</math> \|\| <math>\frac{14 - \sqrt{7}}{30}</math> \|- \| <math>\pm 1</math> \|\| <math>\frac{1}{15}</math> \|- \| rowspan="4" \| <math>7</math> \| <math>0</math> \|\| <math>\frac{256}{525}</math> \|- \| <math>\pm\sqrt{\frac{5}{11}-\frac{2}{11}\sqrt{\frac{5}{3}}}</math> \|\| <math>\frac{124 + 7\sqrt{15}}{350}</math> \|- \| <math>\pm\sqrt{\frac{5}{11} + \frac{2}{11}\sqrt{\frac{5}{3}}}</math> \|\| <math>\frac{124 - 7\sqrt{15}}{350}</math> \|- \| <math>\pm 1</math> \|\| <math>\frac{1}{21}</math> \|} An adaptive variant of this algorithm with 2 interior nodes<ref>{{cite journal \|title=Adaptive Quadrature - Revisited\|last1=Gander \|first1=Walter \|last2=Gautschi \|first2=Walter \|journal=BIT Numerical Mathematics \|date=2000 \|volume=40 \|issue=1 \|pages=84–101 \|doi=10.1023/A:1022318402393\|url=https://www.inf.ethz.ch/personal/gander/}}</ref> is found in [[GNU Octave]] and [[MATLAB]] as <code>quadl</code> and <code>integrate</code>.<ref>{{cite web \|title=Numerical integration - MATLAB integral \|url=https://www.mathworks.com/help/matlab/ref/integral.html}}</ref><ref>{{cite web \|title=Functions of One Variable (GNU Octave) \|url=https://octave.org/doc/v4.2.2/Functions-of-One-Variable.html#XREFquadl \|accessdate=28 September 2018}}</ref> == References == * [http://www.sscm.kg.ac.rs/jsscm/downloads/Vol11No1/Vol11No1_02.pdf Implementing an Accurate Generalized Gaussian Quadrature Solution to Find the Elastic Field in a Homogeneous Anisotropic Media] * {{AS ref\| 25.4, Integration }} {{cite journal\|first1=Donald G. \|last1=Anderson\|title=Gaussian quadrature formulae for <math>\int_0^1 -\ln(x)f(x) dx</math> \|year=1965 \| volume=19 \| number=91 \| pages=477–481 \| journal=Math. Comp. \| doi=10.1090/s0025-5718-1965-0178569-1\| doi-access=free }} {{citation\| title = Calculation of Gauss Quadrature Rules \| journal = Mathematics of Computation \|first=Gene H. \|last=Golub \|authorlink=Gene Golub\| first2= John H. \|last2= Welsch\|volume= 23\| issue= 106 \|year= 1969\|pages=221–230\| jstor = 2004418\| doi = 10.1090/S0025-5718-69-99647-1\|doi-access= free}} {{cite news\|first1=Walter \| last1=Gautschi\|title=Construction of Gauss–Christoffel Quadrature Formulas\|journal= Math. Comp. \| year=1968 \| volume=22 \| issue= 102 \| pages=251–270\|doi=10.1090/S0025-5718-1968-0228171-0 \| mr=0228171}} {{cite news\|first1=Walter \| last1=Gautschi\|title=On the construction of Gaussian quadrature rules from modified moments \|journal= Math. Comp. \| year=1970 \| volume=24 \| pages=245–260\|doi=10.1090/S0025-5718-1970-0285117-6 \| mr=0285177}} {{cite news\|first1=R.\| last1=Piessens \| title=Gaussian quadrature formulas for the numerical integration of Bromwich's integral and the inversion of the laplace transform\|year=1971 \| volume=5 \| number=1 \| journal= J. Eng. Math. \| pages=1–9\|doi=10.1007/BF01535429\|bibcode=1971JEnMa...5....1P}} {{cite news\|first1=Bernard \| last1=Danloy \| title=Numerical construction of Gaussian quadrature formulas for <math>\int_0^1 (-\log x) x^\alpha f(x) dx</math> and <math>\int_0^\infty E_m(x) f(x) dx</math> \| journal = Math. Comp. \| year=1973\|volume=27 \| number=124 \| pages=861–869 \| doi= 10.1090/S0025-5718-1973-0331730-X \|mr=0331730}} {{citation \| last1=Kahaner \| first1=David \| last2=Moler \| first2=Cleve \| author2-link=Cleve Moler \| last3=Nash \| first3=Stephen \| title=Numerical Methods and Software \| year=1989 \| publisher=[[Prentice-Hall]] \| isbn=978-0-13-627258-8 \| url-access=registration \| url=https://archive.org/details/numericalmethods0000kaha }} {{cite journal \| first1=Robin P. \| last1=Sagar\|title=A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals \|year=1991 \| journal=Comput. Phys. Commun. \| volume=66 \| pages=271–275 \|number=2–3\|doi=10.1016/0010-4655(91)90076-W \|bibcode = 1991CoPhC..66..271S }} {{cite journal \| first1=E. \| last1=Yakimiw\| title=Accurate computation of weights in classical Gauss-Christoffel quadrature rules \|year=1996 \| journal=J. Comput. Phys. \| volume=129 \| issue=2\| pages=406–430\| bibcode=1996JCoPh.129..406Y \|doi=10.1006/jcph.1996.0258}} {{citation\| first1=Dirk P. \|last1=Laurie\|title=Accurate recovery of recursion coefficients from Gaussian quadrature formulas \|year=1999 \| volume=112 \| number=1–2 \| pages=165–180\|journal=J. Comput. Appl. Math. \| doi=10.1016/S0377-0427(99)00228-9 \| doi-access=free }} * {{ cite journal\| first1=Dirk P. \| last1=Laurie\|title=Computation of Gauss-type quadrature formulas \|year=2001 \| pages=201–217 \| volume=127 \| number=1–2 \| journal=J. Comput. Appl. Math. \| doi=10.1016/S0377-0427(00)00506-9 \|bibcode = 2001JCoAM.127..201L \| doi-access=free }} {{cite journal \| first1=Cordian \|last1=Riener\|last2=Schweighofer \| first2=Markus\|title=Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions \|year=2018 \| journal=Journal of Complexity \| volume=45 \| pages=22–54 \|doi=10.1016/j.jco.2017.10.002 \|arxiv=1607.08404 }} {{citation \| last1=Stoer \| first1=Josef \| last2=Bulirsch \| first2=Roland \| year=2002 \| title=Introduction to Numerical Analysis \| edition=3rd \| publisher=[[Springer-Verlag\|Springer]] \| isbn=978-0-387-95452-3<!-- 0-387-95452-X --> }}. {{dlmf\| title=§3.5(v): Gauss Quadrature \| id=3.5.v \| last=Temme \| first=Nico M.}} {{Citation\|last1=Press\|first1=WH\|last2=Teukolsky\|first2=SA\|last3=Vetterling\|first3=WT\|last4=Flannery\|first4=BP\|year=2007\|title=Numerical Recipes: The Art of Scientific Computing\|edition=3rd\|publisher=Cambridge University Press\| location=New York\|isbn=978-0-521-88068-8\|chapter=Section 4.6. Gaussian Quadratures and Orthogonal Polynomials\|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=179}} * {{citation\| last1=Gil \| first1=Amparo \| last2=Segura \| first2=Javier \|last3=Temme \| first3=Nico M. \| chapter=§5.3: Gauss quadrature\| title=Numerical Methods for Special Functions \| year=2007 \| publisher=SIAM \| isbn=978-0-89871-634-4 }} * {{cite book \| last1 = Quarteroni \| first1 = Alfio \| first2 = Riccardo \| last2 = Sacco \| first3 = Fausto \| last3 = Saleri \| title = Numerical Mathematics \| location = New York \| publisher = [[Springer-Verlag]] \| pages = [https://archive.org/details/numericalmathema00quar_579/page/n439 422], 425 \| date = 2000 \| isbn = 0-387-98959-5 \| title-link = Numerical Mathematics }} * Walter Gautschi: "A Software Repository for Gaussian Quadratures and Christoffel Functions", SIAM, {{ISBN\|978-1-611976-34-2}} (2020). ;Specific <references /> == External links == * {{springer\|title=Gauss quadrature formula\|id=p/g043510}} * [http://www.alglib.net/integral/gq/ ALGLIB] contains a collection of algorithms for numerical integration (in C# / C++ / Delphi / Visual Basic / etc.) * [https://www.gnu.org/software/gsl/ GNU Scientific Library] — includes [[C (programming language)\|C]] version of [[QUADPACK]] algorithms (see also [[GNU Scientific Library]]) [http://ans.hsh.no/home/skk/Publications/Lobatto/PRIMUS_KHATTRI.pdf From Lobatto Quadrature to the Euler constant e] [http://numericalmethods.eng.usf.edu/topics/gauss_quadrature.html Gaussian Quadrature Rule of Integration – Notes, PPT, Matlab, Mathematica, Maple, Mathcad] at ''Holistic Numerical Methods Institute'' * {{MathWorld\|id=Legendre-GaussQuadrature\|title=Legendre-Gauss Quadrature}} * [http://demonstrations.wolfram.com/GaussianQuadrature/ Gaussian Quadrature] by Chris Maes and Anton Antonov, [[Wolfram Demonstrations Project]]. * [https://pomax.github.io/bezierinfo/legendre-gauss.html Tabulated weights and abscissae with Mathematica source code], high precision (16 and 256 decimal places) Legendre-Gaussian quadrature weights and abscissas, for ''n''=2 through ''n''=64, with Mathematica source code. * [https://web.archive.org/web/20121122202131/http://people.sc.fsu.edu/~jburkardt/math_src/arbitrary_weight_rule/arbitrary_weight_rule.html Mathematica source code distributed under the GNU LGPL] for abscissas and weights generation for arbitrary weighting functions W(x), integration domains and precisions. * [http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/gauss.html Gaussian Quadrature in Boost.Math, for arbitrary precision and approximation order] * [http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/gauss_kronrod.html Gauss-Kronrod Quadrature in Boost.Math] [[Category:Numerical integration (quadrature)]]
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