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 \Phi_{n+1}(z)=z\Phi_n(z)-\overline\alpha_n\Phi_n^*(z)}

${\displaystyle \Phi _{n+1}(z)=z\Phi _{n}(z)-{\overline {\alpha }}_{n}\Phi _{n}^{*}(z)}$

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

Semantic latex: \Phi_{n+1}(z) = z \Phi_n(z) - \conj{\alpha}

Confidence: 0.78713822532466

Mathematica

Translation: Subscript[\[CapitalPhi], n + 1][z] == z*Subscript[\[CapitalPhi], n][z]- Conjugate[\[Alpha]]

Information

Sub Equations

Subscript[\[CapitalPhi], n + 1][z] = z*Subscript[\[CapitalPhi], n][z]- Conjugate[\[Alpha]]

Free variables

Subscript[\[CapitalPhi], n + 1]

Subscript[\[CapitalPhi], n]

\[Alpha]

n

z

Symbol info

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

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

Conjugate complex number; Example: \conj{z}

Will be translated to: Conjugate[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.9#E11 Mathematica: https://reference.wolfram.com/language/ref/Conjugate.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: Phi[n + 1](z) = z*Phi[n](z)- conjugate(alpha)

Information

Sub Equations

Phi[n + 1](z) = z*Phi[n](z)- conjugate(alpha)

Free variables

Phi[n + 1]

Phi[n]

alpha

n

z

Symbol info

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

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

Conjugate complex number; Example: \conj{z}

Will be translated to: conjugate($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.9#E11 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=conjugate

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\Phi _{n}(z)$

$\alpha _{n}$

Description

polynomial

coefficient

complex number with absolute value

Szegő 's recurrence

Verblunsky coefficient

Complete translation information:

{
  "id" : "FORMULA_ae75280cde2032160f44943bd80b1862",
  "formula" : "\\Phi_{n+1}(z)=z\\Phi_n(z)-\\overline\\alpha_n\\Phi_n^*(z)",
  "semanticFormula" : "\\Phi_{n+1}(z) = z \\Phi_n(z) - \\conj{\\alpha}",
  "confidence" : 0.7871382253246626,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[\\[CapitalPhi], n + 1][z] == z*Subscript[\\[CapitalPhi], n][z]- Conjugate[\\[Alpha]]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[\\[CapitalPhi], n + 1][z] = z*Subscript[\\[CapitalPhi], n][z]- Conjugate[\\[Alpha]]" ],
        "freeVariables" : [ "Subscript[\\[CapitalPhi], n + 1]", "Subscript[\\[CapitalPhi], n]", "\\[Alpha]", "n", "z" ],
        "tokenTranslations" : {
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\Phi" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\conj" : "Conjugate complex number; Example: \\conj{z}\nWill be translated to: Conjugate[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/1.9#E11\nMathematica:  https://reference.wolfram.com/language/ref/Conjugate.html"
        }
      },
      "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: \\conj [\\conj]"
        }
      },
      "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" : "Phi[n + 1](z) = z*Phi[n](z)- conjugate(alpha)",
      "translationInformation" : {
        "subEquations" : [ "Phi[n + 1](z) = z*Phi[n](z)- conjugate(alpha)" ],
        "freeVariables" : [ "Phi[n + 1]", "Phi[n]", "alpha", "n", "z" ],
        "tokenTranslations" : {
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\Phi" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\conj" : "Conjugate complex number; Example: \\conj{z}\nWill be translated to: conjugate($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/1.9#E11\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=conjugate"
        }
      },
      "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" : 2,
    "sentence" : 0,
    "word" : 6
  } ],
  "includes" : [ "\\Phi_n(z)", "\\alpha_n" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "polynomial",
    "score" : 0.7125985104912714
  }, {
    "definition" : "coefficient",
    "score" : 0.6859086196238077
  }, {
    "definition" : "complex number with absolute value",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Szegő 's recurrence",
    "score" : 0.5988174995334326
  }, {
    "definition" : "Verblunsky coefficient",
    "score" : 0.5500952380952381
  } ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, '''orthogonal polynomials on the unit circle''' are families of [[Orthogonal polynomials\|polynomials that are orthogonal with respect to integration]] over the [[unit circle]] in the [[complex plane]], for some [[probability measure]] on the unit circle. They were introduced by {{harvs\|txt\|last=Szegő\|year1=1920\|year2=1921\|year3=1939}}. ==Definition== Suppose that <math>\mu</math> is a probability measure on the unit circle in the complex plane, whose [[support (mathematics)\|support]] is not finite. The orthogonal polynomials associated to <math>\mu</math> are the polynomials <math>\Phi_n(z)</math> with leading term <math>z^n</math> that are orthogonal with respect to the measure <math>\mu</math>. ==The Szegő recurrence == Szegő's recurrence states that :<math>\Phi_0(z) = 1</math> :<math>\Phi_{n+1}(z)=z\Phi_n(z)-\overline\alpha_n\Phi_n^(z)</math> where :<math>\Phi_n^(z)=z^n\overline{\Phi_n(1/\overline{z})}</math> is the polynomial with its coefficients reversed and complex conjugated, and where the '''Verblunsky coefficients''' <math>\alpha_n</math> are complex numbers with absolute values less than 1. ==Verblunsky's theorem== Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support. ==Geronimus's theorem== Geronimus's theorem states that the Verblunsky coefficients of the measure μ are the [[Schur parameters]] of the function <math>f</math> defined by the equations :<math> \frac{1+zf(z)}{1-zf(z)}=F(z)=\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu.</math> ==Baxter's theorem== Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of <math>\mu</math> form an absolutely convergent series and the weight function <math>w</math> is strictly positive everywhere. ==Szegő's theorem== Szegő's theorem states that :<math>\prod_{n = 1}^\infty(1-\|\alpha_n\|^2) = \exp\big(\int_0^{2\pi}\log(w(\theta))d\theta/2\pi\big)</math> where <math>wd\theta / 2\pi</math> is the absolutely continuous part of the measure <math>\mu</math>. ==Rakhmanov's theorem== Rakhmanov's theorem states that if the absolutely continuous part <math>w</math> of the measure <math>\mu</math> is positive almost everywhere then the Verblunsky coefficients <math>\alpha_n</math> tend to 0. ==Examples== The [[Rogers–Szegő polynomials]] are an example of orthogonal polynomials on the unit circle. ==References== {{dlmf\|id=18.33\|title=Orthogonal Polynomials on the unit circle\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Simon \| first1=Barry \| author1-link=Barry Simon \| title=Orthogonal polynomials on the unit circle. Part 1. Classical theory \| url=https://books.google.com/books?id=d94r7kOSnKcC \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=American Mathematical Society Colloquium Publications \| isbn=978-0-8218-3446-6 \| mr=2105088 \| year=2005 \| volume=54}} {{Citation \| last1=Simon \| first1=Barry \| author1-link=Barry Simon \| title=Orthogonal polynomials on the unit circle. Part 2. Spectral theory \| url=https://books.google.com/books?id=54juCPc3ulwC \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=American Mathematical Society Colloquium Publications \| isbn=978-0-8218-3675-0 \| mr=2105089 \| year=2005 \| volume=54}} {{Citation \| last1=Szegő \| first1=Gábor \| title=Beiträge zur Theorie der Toeplitzschen Formen \| doi=10.1007/BF01199955 \| year=1920 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=6 \| issue=3–4 \| pages=167–202\| s2cid=118147030 \| url=https://zenodo.org/record/1447413 }} {{Citation \| last1=Szegő \| first1=Gábor \| title=Beiträge zur Theorie der Toeplitzschen Formen \| doi=10.1007/BF01279027 \| year=1921 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=9 \| issue=3–4 \| pages=167–190\| s2cid=125157848 \| url=https://zenodo.org/record/1447419 }} {{Citation \| last1=Szegő \| first1=Gábor \| title=Orthogonal Polynomials \| url=https://books.google.com/books?id=3hcW8HBh7gsC \| publisher= American Mathematical Society \| series=Colloquium Publications \| isbn=978-0-8218-1023-1 \| mr=0372517 \| year=1939 \| volume=XXIII}} [[Category:Orthogonal polynomials]]
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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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