Real and Complex Fields
- Introduction
- Creation Functions
- Creation of Structures
- Creation of Elements
a . becpd: RngIntElt, RngIntElt, RngIntElt → FldReElta . bEcPd: RngIntElt, RngIntElt, RngIntElt → FldReElta.bec P d: RngIntElt, RngIntElt, RngIntElt → FldReElta.bec p d: RngIntElt, RngIntElt, RngIntElt → FldReElta.b E cpd: RngIntElt, RngIntElt, RngIntElt → FldReElta.b e cpd: RngIntElt, RngIntElt, RngIntElt → FldReEltelt<R | m, n>: FldRe, FldReElt, RngIntElt → FldReEltelt<C | x, y>: FldCom, FldReElt, FldReElt → FldComEltC ! [x, y]: FldCom, [FldReElt, FldReElt] → FldComEltR ! a: FldRe, RngElt → FldReEltC ! a: FldCom, RngElt → FldComElt- Example: Creating Real Numbers
One(R): FldRe → FldReEltOne(R): FldCom → FldComEltIdentity(R): FldRe → FldReEltIdentity(R): FldCom → FldComEltZero(R): FldRe → FldReEltZero(R): FldCom → FldComEltRepresentative(R): FldRe → FldReEltRepresentative(R): FldCom → FldComElt
- Structure Operations
- Element Operations
- Generic Element Functions and Predicates
- Comparison of and Membership
a eq b: FldReElt, FldReElt → BoolElta ne b: FldReElt, FldReElt → BoolElta eq b: FldComElt, FldComElt → BoolElta ne b: FldComElt, FldComElt → BoolElta in R: FldReElt, FldRe → BoolElta notin R: FldReElt, FldRe → BoolElta in R: FldComElt, FldCom → BoolElta notin R: FldComElt, FldCom → BoolElta gt b: FldReElt, FldReElt → BoolElta ge b: FldReElt, FldReElt → BoolElta lt b: FldReElt, FldReElt → BoolElta le b: FldReElt, FldReElt → BoolEltMaximum(a, b): FldReElt, FldReElt → FldReEltMinimum(a, b): FldReElt, FldReElt → FldReEltMaximum(Q): [FldReElt] → FldReEltMinimum(Q): [FldReElt] → FldReElt
- Other Predicates
- Arithmetic
+ r: FldReElt → FldReElt+ r: FldComElt → FldComElt- r: FldReElt → FldReElt- r: FldComElt → FldComEltr + s: FldReElt, FldReElt → FldReEltr + s: FldComElt, FldComElt → FldComEltr - s: FldReElt, FldReElt → FldReEltr - s: FldComElt, FldComElt → FldComEltr * s: FldReElt, FldReElt → FldReEltr * s: FldComElt, FldComElt → FldComEltr / s: FldReElt, FldReElt → FldReEltr / s: FldComElt, FldComElt → FldComEltr ^ k: FldReElt, FldReElt → FldReEltr ^ k: FldReElt, RngIntElt → FldReEltr ^ k: FldComElt, RngIntElt → FldComEltr ^ k: FldComElt, FldComElt → FldComEltr +:= s: FldReElt, FldReElt → FldReEltr +:= s: FldComElt, FldComElt → FldComEltr -:= s: FldReElt, FldReElt → FldReEltr -:= s: FldComElt, FldComElt → FldComEltr *:= s: FldReElt, FldReElt → FldReEltr *:= s: FldComElt, FldComElt → FldComEltr /:= s: FldReElt, FldReElt → FldReEltr /:= s: FldComElt, FldComElt → FldComEltr ^:= s: FldReElt, FldReElt → FldReEltr ^:= s: FldComElt, FldComElt → FldComElt
- Conversions
- Rounding
- Precision
- Constants
- Simple Element Functions
AbsoluteValue(r): FldReElt → FldReEltAbsoluteValue(r): FldComElt → FldReEltAbs(r): FldReElt → FldReEltAbs(r): FldComElt → FldReEltSign(r): FldReElt → RngIntEltComplexConjugate(r): FldReElt → FldReEltComplexConjugate(c): FldComElt → FldComEltConjugate(c): FldComElt → FldComEltNorm(c): FldComElt → FldReEltNorm(r): FldReElt → FldReEltRoot(r, n): FldReElt, RngIntElt → FldReEltRoot(r, n): FldComElt, RngIntElt → FldComEltSquareRoot(c): FldComElt → FldComEltSqrt(c): FldComElt → FldComEltSquareRoot(r): FldReElt → FldReEltSqrt(r): FldReElt → FldReEltDistance(x, L): FldReElt, [FldReElt] → FldReElt, RngIntEltDistance(x, L): FldReElt, [FldComElt] → FldReElt, RngIntEltDistance(x, L): FldComElt, [FldReElt] → FldReElt, RngIntEltDistance(x, L): FldComElt, [FldComElt] → FldReElt, RngIntEltDiameter(L): [FldReElt] → FldReEltDiameter(L): [FldComElt] → FldReElt
- Roots
Roots(p): RngUPolElt → [ <FldComElt, RngIntElt> ]- Example: Roots of a Polynomial with a Multiple Root
RootsNonExact(p): RngUPolElt[FldRe] → [ FldComElt ], [ FldComElt ]RootsNonExact(p): RngUPolElt[FldCom] → [ FldComElt ], [ FldComElt ]- Example: Roots with Error Estimates
HenselLift(f, R, k): RngUPolElt, FldReElt, RngIntElt → FldReEltHenselLift(f, R, k): RngUPolElt, FldComElt, RngIntElt → FldComElt
- Continued Fractions
- Linear and Algebraic Dependencies
LinearRelation(q: parameters): [ FldComElt ] → [ RngIntElt ]LinearRelation(v: parameters): ModTupRngElt → ModTupRngEltAllLinearRelations(q,p): SeqEnum, RngIntElt → LatIntegerRelation(q): SeqEnum → SeqEnum, FldReEltIntegerRelation(q,N): SeqEnum, RngIntElt → SeqEnum, FldReEltPowerRelation(r, k: parameters): FldReElt, RngIntElt → RngUPolEltPowerRelation(r, k: parameters): FldComElt, RngIntElt → RngUPolEltMinimalPolynomial(r,d,N): FldReElt, RngIntElt, RngIntElt → RngUPolElt, FldReEltMinimalPolynomial(r,d,N): FldComElt, RngIntElt, RngIntElt → RngUPolElt, FldReEltMinimalPolynomial(r,d): FldReElt, RngIntElt → RngUPolElt -> FldReEltMinimalPolynomial(r,d): FldComElt, RngIntElt → RngUPolElt -> FldReElt- Example: Polynomial Factorization via LLL
- Transcendental Functions
- Exponential, Logarithmic, and Polylogarithmic Functions
Exp(f): RngSerElt → RngSerEltExp(c): FldComElt → FldComEltExp(r): FldReElt → FldReEltLog(f): RngSerElt → RngSerEltLog(r): FldReElt → FldReEltLog(c): FldComElt → FldComEltLog(b, r): FldReElt → FldReEltDilog(s): FldComElt → FldComEltDilog(s): FldReElt → FldReEltPolylog(m, f): RngIntElt, RngSerElt → RngSerEltPolylog(m, s): RngIntElt, FldComElt → FldComEltPolylogD(m, s): RngIntElt, FldComElt → FldComEltPolylogDold(m, s): RngIntElt, FldComElt → FldComEltPolylogP(m, s): RngIntElt, FldComElt → FldComElt
- Trigonometric Functions
- Inverse Trigonometric Functions
- Hyperbolic Functions
- Inverse Hyperbolic Functions
- Elliptic and Modular Functions
- Eisenstein Series
- Weierstrass Series
- The Jacobi \(\theta\) and Dedekind \(\eta\)-functions
JacobiTheta(q, z): FldReElt, RngSerElt[FldRe] → RngSerEltJacobiTheta(q, z): FldComElt, RngSerElt[FldCom] → RngSerEltJacobiTheta(q, z): FldReElt, FldReElt → FldReEltJacobiTheta(q, z): FldComElt, FldComElt → FldComEltJacobiThetaNullK(q, k): FldReElt, RngIntElt → FldReEltDedekindEta(z): RngSerElt → RngSerEltDedekindEta(s): FldComElt → FldComElt
- The \(j\)-Invariant and the Discriminant
- Weber’s Functions
- Theta Functions
- Gamma, Bessel, and Associated Functions
Gamma(f): RngSerElt → RngSerEltGamma(r): FldReElt → FldReEltGamma(r): FldComElt → FldComEltGamma(s, t): FldReElt, FldReElt → FldReEltGammaD(s): FldReElt → FldReEltLogGamma(f): RngSerElt → RngSerEltLogGamma(r): FldReElt → FldReEltLogGamma(r): FldComElt → FldComEltLogDerivative(s): FldReElt → FldReEltLogDerivative(s): FldComElt → FldComEltPsi(s): FldReElt → FldReEltPsi(s): FldComElt → FldComEltBesselFunction(n, r): RngIntElt, FldReElt → FldReEltBesselFunctionSecondKind(n, r): RngIntElt, FldReElt → FldReEltJBessel(n, s): RngIntElt, FldReElt → FldReEltJBessel(n, s): FldReElt, FldReElt → FldReEltKBessel(n, s): FldReElt, FldReElt → FldReEltKBessel(n, s): FldComElt, FldReElt → FldComEltKBessel2(n, s): FldReElt, FldReElt → FldReEltKBessel2(n, s): FldComElt, FldReElt → FldComElt
- The Hypergeometric Function
HypergeometricSeries(a, b, c, z): RngElt, RngElt, RngElt, RngElt → RngEltHypergeometricU(a, b, s): FldReElt, FldReElt, FldReElt → FldReElt
- Other Special Functions
ArithmeticGeometricMean(x, y): RngSerElt, RngSerElt → RngSerEltAGM(f, g): RngSerElt, RngSerElt → RngSerEltArithmeticGeometricMean(x, y): FldReElt, FldReElt → FldReEltAGM(x, y): FldReElt, FldReElt → FldReEltBernoulliNumber(n): RngIntElt → FldRatEltBernoulliApproximation(n): RngIntElt → FldReEltDawsonIntegral(r): FldReElt → FldReEltErrorFunction(r): FldReElt → FldReEltErf(r): FldReElt → FldReEltComplementaryErrorFunction(r): FldReElt → FldReEltErfc(r): FldReElt → FldReEltExponentialIntegral(r): FldReElt → FldReEltExponentialIntegralE1(r): FldReElt → FldReEltLogIntegral(r): FldReElt → FldReEltZetaFunction(s): FldReElt → FldReEltZetaFunction(R, n): FldRe, RngIntElt → FldReElt
- Summation of Infinite Series
- Numerical Integration
- Polynomial Interpolation
- Discrete Fourier Transform
- Integration of Complex Functions
- Gaussian Quadratures
GaussLegendreIntegrationPoints(N,D): RngIntElt, RngIntElt → SeqEnum[FldReElt], SeqEnum[FldReElt]GaussJacobiIntegrationPoints(N,D,a,b): RngIntElt, RngIntElt, RngReSubElt, RngReSubElt → SeqEnum, SeqEnum
- Clenshaw–Curtis Quadrature
- Tanh–Sinh Quadrature
- Romberg-Type Integration
RombergQuadrature(f, a, b: parameters): Program, FldReElt, FldReElt → FldReEltSimpsonQuadrature(f, a, b, n): Program, FldReElt, FldReElt, RngIntElt → FldReEltTrapezoidalQuadrature(f, a, b, n): Program, FldReElt, FldReElt, RngIntElt → FldReElt
- Numerical Derivatives
- Bibliography