LaTeX to CAS translator
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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 n}
... is translated to the CAS output ...
Semantic latex: n
Confidence: 0
Mathematica
Translation: n
Information
Sub Equations
- n
Free variables
- n
Tests
Symbolic
Numeric
SymPy
Translation: n
Information
Sub Equations
- n
Free variables
- n
Tests
Symbolic
Numeric
Maple
Translation: n
Information
Sub Equations
- n
Free variables
- n
Tests
Symbolic
Numeric
Dependency Graph Information
Is part of
Description
- degree
- weight
- orthogonal polynomial of degree
- node
- point
- polynomial of degree
- quadrature rule
- different point
- maximal degree
- nontrivial polynomial of degree
- number of integration point
- rule
- point Gaussian quadrature rule
- exact result for polynomial
- Carl Friedrich Gauss
- suitable choice of the node
- common domain of integration
- polynomial
- yield
- operation
- coefficient
- i.e.
- order
- derivative
- remainder
- Orthogonal polynomial
- integrand
- term
- term of the orthogonal polynomial
- zero
- scalar product
- integral
- formula
- 3-term recurrence relation
- error estimate
- point rule
- relation
- Gauss-Jacobi quadrature rule
- root
- Gaussian quadrature
- asymptotic formula
- continuous derivative
- Eq
- idea
- integral expression for the weight
- L'Hôpital 's rule
- limit
- Lobatto quadrature of function
- low degree
- lower degree
- Mathematica source code
- proof
- property
- quotient
- root of the polynomial
- similar matrix
- st
- way
- accurate approximation to the integral
- Legendre polynomial
- Abscissas
- choice of node
- eigenvalue of this tridiagonal matrix
- equation
- expression in equation
- Golub-Welsch algorithm
- high precision
- i.e. monic
- important special case
- infinity
- matrix form
- node for the Gaussian quadrature
- recurrence relation
- side
- so-called Jacobi matrix
- standard basis vector
- term in the bracket
- three-term recurrence relation
- value
- weight function
- simplest integration problem
- Gauss node
- approximation
- dash
- decimal place
- exactness
- exactness for polynomial
- gauss-quadrature
- Gaussian quadrature formula
- integral expression
- interval
- Legendre-Gaussian quadrature weight
- Newton 's method
- note
- nth degree
- standard Legendre polynomial of m-th degree
- three-term recurrence for evaluation
- abscissa
- element
- extension of Gauss quadrature rule
- practice
- right hand side
- same eigenvalue
- symmetric tridiagonal matrix
- Gauss -- Kronrod rule
- Lagrange interpolation one
- Stoer
- actual error
- Bulirsch remark that this error estimate
Complete translation information:
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"Maple" : {
"translation" : "n",
"translationInformation" : {
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"freeVariables" : [ "n" ]
},
"numericResults" : {
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"numberOfTests" : 0,
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"wasAborted" : false,
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}
}
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"includes" : [ ],
"isPartOf" : [ "2n - 1", "i = 1, ..., n", "\\int_{-1}^1 f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)", "\\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)", "P_{n}(x)", "P_{n}(1) = 1", "P_{n}", "w_i = \\frac{2}{\\left( 1 - x_i^2 \\right) \\left[P'_n(x_i)\\right]^2}", "\\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)", "p_{n}", "\\int_a^b \\omega(x) \\, x^k p_n(x) \\, dx = 0, \\quad \\text{for all } k = 0, 1, \\ldots, n - 1", "p_n(x)", "\\sum_{i=1}^n w_i h(x_i) = \\sum_{i=1}^n w_i r(x_i)", "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})}", "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}}", "\\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", "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", "p_{n}(x)", "p_{n-1}(x)", "\\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)}", "a_{n}", "x^n", "\\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}}", "w_{i} = \\frac{1}{p'_{n}(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx", "\\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", "k \\leq n", "\\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", "q(x) = p_{n-1}(x)", "\\frac{p_{n}(x)}{x-x_{i}}", "n - 1", "\\frac{p_{n}(x)}{x-x_{i}} = a_{n}x^{n-1} + s(x)", "n - 2", "\\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", "x^{n-1} = \\left(x^{n-1} - \\frac{p_{n-1}(x)}{a_{n-1}}\\right) + \\frac{p_{n-1}(x)}{a_{n-1}}", "\\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", "p'_{n}(x_{i})", "p_{n+1}(x)", "p_{n+1}(x_{i}) = (a)p_{n}(x_{i}) + (b)p_{n-1}(x_{i})", "p_{n}(x_{i})", "p_{n-1}(x_{i})", "\\frac{1}{b}p_{n+1}\\left(x_i\\right)", "2n - 2", "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}", "O(n^{2})", "p_n(x) = 0", "O(n)", "r = 0, 1, \\ldots, n - 1", "J\\tilde{P} = x\\tilde{P} - p_n(x) \\times \\mathbf{e}_n", "\\tilde{P} = \\begin{bmatrix} p_0(x) & p_1(x) & \\ldots & p_{n-1}(x) \\end{bmatrix}^\\mathsf{T}", "\\mathbf{e}_n", "\\mathbf{e}_n = \\begin{bmatrix} 0 & \\ldots & 0 & 1 \\end{bmatrix}^\\mathsf{T}", "\\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}", "\\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}", "2n", "\\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)", "\\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", "n + 1", "2n + 1", "2n - 3", "\\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", "P'_{n-1}(x)", "w_i = \\frac{2}{n(n - 1)\\left[P_{n-1}\\left(x_i\\right)\\right]^2}, \\qquad x_i \\ne \\pm 1", "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", "n=2", "n=64" ],
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}