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 x = \frac{y-\mu}{\sqrt{2} \sigma} \Leftrightarrow y = \sqrt{2} \sigma x + \mu}

${\displaystyle x={\frac {y-\mu }{{\sqrt {2}}\sigma }}\Leftrightarrow y={\sqrt {2}}\sigma x+\mu }$

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

Semantic latex: x = \frac{y-\mu}{\sqrt{2} \sigma} \Leftrightarrow y = \sqrt{2} \sigma x + \mu

Confidence: 0

Mathematica

Translation:

Information

Symbol info

(LaTeX -> Mathematica) No translation possible for given token: Unknown relation. Cannot translate: \Leftrightarrow [\Leftrightarrow]

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Unknown relation. Cannot translate: \Leftrightarrow [\Leftrightarrow]

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) No translation possible for given token: Unknown relation. Cannot translate: \Leftrightarrow [\Leftrightarrow]

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$y$

Description

integration by substitution

integral

expectation

variable

Hermite polynomial

Complete translation information:

{
  "id" : "FORMULA_e12b2c8ae811a7a0fa36df107f98f258",
  "formula" : "x = \\frac{y-\\mu}{\\sqrt{2} \\sigma} \\Leftrightarrow y = \\sqrt{2} \\sigma x + \\mu",
  "semanticFormula" : "x = \\frac{y-\\mu}{\\sqrt{2} \\sigma} \\Leftrightarrow y = \\sqrt{2} \\sigma x + \\mu",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) No translation possible for given token: Unknown relation. Cannot translate: \\Leftrightarrow [\\Leftrightarrow]"
        }
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Unknown relation. Cannot translate: \\Leftrightarrow [\\Leftrightarrow]"
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Unknown relation. Cannot translate: \\Leftrightarrow [\\Leftrightarrow]"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 1,
    "word" : 28
  } ],
  "includes" : [ "y" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "integration by substitution",
    "score" : 0.7125985104912714
  }, {
    "definition" : "integral",
    "score" : 0.6859086196238077
  }, {
    "definition" : "expectation",
    "score" : 0.5988174995334326
  }, {
    "definition" : "variable",
    "score" : 0.5988174995334326
  }, {
    "definition" : "Hermite polynomial",
    "score" : 0.5500952380952381
  } ]
}

Specify your own input

Formula input
Wikitext context	{{short description\|Form of Gaussian quadrature}} [[File:Gauss-Hermite quadrature weights.svg\|thumb\|Weights versus ''x<sub>i</sub>'' for four choices of ''n'']] In [[numerical analysis]], '''Gauss–Hermite quadrature''' is a form of [[Gaussian quadrature]] for approximating the value of integrals of the following kind: :<math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx.</math> In this case :<math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</math> where ''n'' is the number of sample points used. The ''x''<sub>''i''</sub> are the roots of the physicists' version of the [[Hermite polynomial]] ''H''<sub>''n''</sub>(''x'') (''i'' = 1,2,...,''n''), and the associated weights ''w''<sub>''i''</sub> are given by <ref name="AS">[[Abramowitz and Stegun\|Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'']], 10th printing with corrections (1972), Dover, {{ISBN\|978-0-486-61272-0}}. Equation 25.4.46.</ref> :<math>w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.</math> ==Example with change of variable== Consider a function ''h(y)'', where the variable ''y'' is [[Normal distribution\|Normally distributed]]: <math> y \sim \mathcal{N}(\mu,\sigma^2)</math>. The [[Expected value\|expectation]] of ''h'' corresponds to the following integral: <math>E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(y-\mu)^2}{2\sigma^2} \right) h(y) dy</math> As this doesn't exactly correspond to the Hermite polynomial, we need to change variables: <math>x = \frac{y-\mu}{\sqrt{2} \sigma} \Leftrightarrow y = \sqrt{2} \sigma x + \mu</math> Coupled with the [[integration by substitution]], we obtain: <math>E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sqrt{\pi}} \exp(-x^2) h(\sqrt{2} \sigma x + \mu) dx</math> leading to: <math>E[h(y)] \approx \frac{1}{\sqrt{\pi}} \sum_{i=1}^n w_i h(\sqrt{2} \sigma x_i + \mu)</math> ==References== <references/> * {{dlmf\| id = 3.5.E28\| title=Quadrature: Gauss–Hermite Formula}} * {{cite journal\|first1=T. S. \| last1=Shao \| first2=T. C. \| last2=Chen \| first3= R. M. \| last3=Frank \|title=Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials \|year=1964 \|journal=Math. Comp. \| mr=0166397 \| volume=18 \| number=88 \| pages=598–616 \|doi=10.1090/S0025-5718-1964-0166397-1\| doi-access=free }} * {{cite journal\|first1=N. M. \| last1=Steen \| first2=G. D. \| last2=Byrne \|first3=E. M. \| last3=Gelbard \| title = Gaussian quadratures for the integrals <math>\textstyle\int_0^\infty e^{-x^2} f(x) dx</math> and <math>\textstyle\int_0^b e^{-x^2} f(x) dx</math> \|journal = Math. Comp. \| year=1969 \|volume=23 \| number=107 \| pages=661–671 \| mr=0247744 \| doi=10.1090/S0025-5718-1969-0247744-3 \| doi-access=free }} * {{cite journal\|first1= B. \| last1=Shizgal \| title = A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems \|journal=J. Comput. Phys. \| volume=41 \| pages=309–328 \| year=1981 \|doi=10.1016/0021-9991(81)90099-1 }} ==External links== * For tables of Gauss-Hermite abscissae and weights up to order ''n'' = 32 see http://www.efunda.com/math/num_integration/findgausshermite.cfm. * [http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html Generalized Gauss–Hermite quadrature], [[free software]] in C++, Fortran, and Matlab {{DEFAULTSORT:Gauss-Hermite quadrature}} [[Category:Numerical integration (quadrature)]]
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LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

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