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 w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.}

${\displaystyle w_{i}={\frac {2^{n-1}n!{\sqrt {\pi }}}{n^{2}[H_{n-1}(x_{i})]^{2}}}.}$

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

Semantic latex: w_i = \frac{2^{n-1} n! \sqrt{\cpi}}{n^2 [\HermitepolyH{n-1}@{x_i}]^2}

Confidence: 0.86627249984448

Mathematica

Translation: Subscript[w, i] == Divide[(2)^(n[-]*1)* (n)![Sqrt[Pi]],((n[HermiteH[n[-]*1, Subscript[x, i]]])^(2))^(2)]

Information

Sub Equations

Subscript[w, i] = Divide[(2)^(n[-]*1)* (n)![Sqrt[Pi]],((n[HermiteH[n[-]*1, Subscript[x, i]]])^(2))^(2)]

Free variables

Subscript[w, i]

Subscript[x, i]

i

Symbol info

You use a typical letter for a constant [the imaginary unit == the principal square root of -1].

We keep it like it is! But you should know that Mathematica uses I for this constant. If you want to translate it as a constant, use the corresponding DLMF macro \iunit

Pi was translated to: Pi

Hermite polynomial; Example: \HermitepolyH{n}@{x}

Will be translated to: HermiteH[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Mathematica: https://reference.wolfram.com/language/ref/HermiteH.html

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \HermitepolyH [\HermitepolyH]

Tests

Symbolic

Numeric

Maple

Translation: w[i] = ((2)^(n(-)*1)* factorial(n)(sqrt(Pi)))/(((n(HermiteH(n(-)*1, x[i])))^(2))^(2))

Information

Sub Equations

w[i] = ((2)^(n(-)*1)* factorial(n)(sqrt(Pi)))/(((n(HermiteH(n(-)*1, x[i])))^(2))^(2))

Free variables

i

w[i]

x[i]

Symbol info

You use a typical letter for a constant [the imaginary unit == the principal square root of -1].

We keep it like it is! But you should know that Maple uses I for this constant. If you want to translate it as a constant, use the corresponding DLMF macro \iunit

Pi was translated to: Pi

Hermite polynomial; Example: \HermitepolyH{n}@{x}

Will be translated to: HermiteH($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=HermiteH

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$x_{i}$

$w_{i}$

$n$

Description

associated weight

root of the physicists ' version

Hermite polynomial

Complete translation information:

{
  "id" : "FORMULA_d63515a7330191b7725d61ec4cbb99b8",
  "formula" : "w_i = \\frac {2^{n-1} n! \\sqrt{\\pi}} {n^2[H_{n-1}(x_i)]^2}",
  "semanticFormula" : "w_i = \\frac{2^{n-1} n! \\sqrt{\\cpi}}{n^2 [\\HermitepolyH{n-1}@{x_i}]^2}",
  "confidence" : 0.8662724998444776,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[w, i] == Divide[(2)^(n[-]*1)* (n)![Sqrt[Pi]],((n[HermiteH[n[-]*1, Subscript[x, i]]])^(2))^(2)]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[w, i] = Divide[(2)^(n[-]*1)* (n)![Sqrt[Pi]],((n[HermiteH[n[-]*1, Subscript[x, i]]])^(2))^(2)]" ],
        "freeVariables" : [ "Subscript[w, i]", "Subscript[x, i]", "i" ],
        "tokenTranslations" : {
          "i" : "You use a typical letter for a constant [the imaginary unit == the principal square root of -1].\nWe keep it like it is! But you should know that Mathematica uses I for this constant.\nIf you want to translate it as a constant, use the corresponding DLMF macro \\iunit\n",
          "\\cpi" : "Pi was translated to: Pi",
          "\\HermitepolyH" : "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/18.3#T1.t1.r13\nMathematica:  https://reference.wolfram.com/language/ref/HermiteH.html",
          "n" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\HermitepolyH [\\HermitepolyH]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "w[i] = ((2)^(n(-)*1)* factorial(n)(sqrt(Pi)))/(((n(HermiteH(n(-)*1, x[i])))^(2))^(2))",
      "translationInformation" : {
        "subEquations" : [ "w[i] = ((2)^(n(-)*1)* factorial(n)(sqrt(Pi)))/(((n(HermiteH(n(-)*1, x[i])))^(2))^(2))" ],
        "freeVariables" : [ "i", "w[i]", "x[i]" ],
        "tokenTranslations" : {
          "i" : "You use a typical letter for a constant [the imaginary unit == the principal square root of -1].\nWe keep it like it is! But you should know that Maple uses I for this constant.\nIf you want to translate it as a constant, use the corresponding DLMF macro \\iunit\n",
          "\\cpi" : "Pi was translated to: Pi",
          "\\HermitepolyH" : "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/18.3#T1.t1.r13\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=HermiteH",
          "n" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ {
    "section" : 0,
    "sentence" : 2,
    "word" : 24
  } ],
  "includes" : [ "x_{i}", "w_{i}", "n" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "associated weight",
    "score" : 0.6859086196238077
  }, {
    "definition" : "root of the physicists ' version",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Hermite polynomial",
    "score" : 0.5988174995334326
  } ]
}

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

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