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author | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
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committer | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
commit | ab8cc85adde879fb963c94d15675783f2cf4b183 (patch) | |
tree | c202482327f474583b750b2c45dedfc4e4312b1d /src/algebra/special.spad.pamphlet | |
download | open-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz |
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diff --git a/src/algebra/special.spad.pamphlet b/src/algebra/special.spad.pamphlet new file mode 100644 index 00000000..1765eebd --- /dev/null +++ b/src/algebra/special.spad.pamphlet @@ -0,0 +1,456 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra special.spad} +\author{Bruce W. Char, Stephen M. Watt} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{package DFSFUN DoubleFloatSpecialFunctions} +<<package DFSFUN DoubleFloatSpecialFunctions>>= +)abbrev package DFSFUN DoubleFloatSpecialFunctions +++ Author: Bruce W. Char, Stephen M. Watt +++ Date Created: 1990 +++ Date Last Updated: June 25, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: +++ Examples: +++ References: +++ Description: +++ This package provides special functions for double precision +++ real and complex floating point. + +DoubleFloatSpecialFunctions(): Exports == Impl where + NNI ==> NonNegativeInteger + R ==> DoubleFloat + C ==> Complex DoubleFloat + + Exports ==> with + Gamma: R -> R + ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by + ++ \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}. + Gamma: C -> C + ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by + ++ \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}. + + Beta: (R, R) -> R + ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by + ++ \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}. + ++ This is related to \spad{Gamma(x)} by + ++ \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}. + Beta: (C, C) -> C + ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by + ++ \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}. + ++ This is related to \spad{Gamma(x)} by + ++ \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}. + + logGamma: R -> R + ++ logGamma(x) is the natural log of \spad{Gamma(x)}. + ++ This can often be computed even if \spad{Gamma(x)} cannot. + logGamma: C -> C + ++ logGamma(x) is the natural log of \spad{Gamma(x)}. + ++ This can often be computed even if \spad{Gamma(x)} cannot. + + digamma: R -> R + ++ digamma(x) is the function, \spad{psi(x)}, defined by + ++ \spad{psi(x) = Gamma'(x)/Gamma(x)}. + digamma: C -> C + ++ digamma(x) is the function, \spad{psi(x)}, defined by + ++ \spad{psi(x) = Gamma'(x)/Gamma(x)}. + + polygamma: (NNI, R) -> R + ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}. + polygamma: (NNI, C) -> C + ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}. + + + besselJ: (R,R) -> R + ++ besselJ(v,x) is the Bessel function of the first kind, + ++ \spad{J(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}. + besselJ: (C,C) -> C + ++ besselJ(v,x) is the Bessel function of the first kind, + ++ \spad{J(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}. + + besselY: (R, R) -> R + ++ besselY(v,x) is the Bessel function of the second kind, + ++ \spad{Y(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}. + ++ Note: The default implmentation uses the relation + ++ \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)} + ++ so is not valid for integer values of v. + besselY: (C, C) -> C + ++ besselY(v,x) is the Bessel function of the second kind, + ++ \spad{Y(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}. + ++ Note: The default implmentation uses the relation + ++ \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)} + ++ so is not valid for integer values of v. + + besselI: (R,R) -> R + ++ besselI(v,x) is the modified Bessel function of the first kind, + ++ \spad{I(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}. + besselI: (C,C) -> C + ++ besselI(v,x) is the modified Bessel function of the first kind, + ++ \spad{I(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}. + + besselK: (R, R) -> R + ++ besselK(v,x) is the modified Bessel function of the first kind, + ++ \spad{K(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}. + ++ Note: The default implmentation uses the relation + ++ \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)}. + ++ so is not valid for integer values of v. + besselK: (C, C) -> C + ++ besselK(v,x) is the modified Bessel function of the first kind, + ++ \spad{K(v,x)}. + ++ This function satisfies the differential equation: + ++ \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}. + ++ Note: The default implmentation uses the relation + ++ \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)} + ++ so is not valid for integer values of v. + + airyAi: C -> C + ++ airyAi(x) is the Airy function \spad{Ai(x)}. + ++ This function satisfies the differential equation: + ++ \spad{Ai''(x) - x * Ai(x) = 0}. + airyAi: R -> R + ++ airyAi(x) is the Airy function \spad{Ai(x)}. + ++ This function satisfies the differential equation: + ++ \spad{Ai''(x) - x * Ai(x) = 0}. + + airyBi: R -> R + ++ airyBi(x) is the Airy function \spad{Bi(x)}. + ++ This function satisfies the differential equation: + ++ \spad{Bi''(x) - x * Bi(x) = 0}. + airyBi: C -> C + ++ airyBi(x) is the Airy function \spad{Bi(x)}. + ++ This function satisfies the differential equation: + ++ \spad{Bi''(x) - x * Bi(x) = 0}. + + hypergeometric0F1: (R, R) -> R + ++ hypergeometric0F1(c,z) is the hypergeometric function + ++ \spad{0F1(; c; z)}. + hypergeometric0F1: (C, C) -> C + ++ hypergeometric0F1(c,z) is the hypergeometric function + ++ \spad{0F1(; c; z)}. + + + Impl ==> add + a, v, w, z: C + n, x, y: R + + -- These are hooks to Bruce's boot code. + Gamma z == CGAMMA(z)$Lisp + Gamma x == RGAMMA(x)$Lisp + + polygamma(k,z) == CPSI(k, z)$Lisp + polygamma(k,x) == RPSI(k, x)$Lisp + + logGamma z == CLNGAMMA(z)$Lisp + logGamma x == RLNGAMMA(x)$Lisp + + besselJ(v,z) == CBESSELJ(v,z)$Lisp + besselJ(n,x) == RBESSELJ(n,x)$Lisp + + besselI(v,z) == CBESSELI(v,z)$Lisp + besselI(n,x) == RBESSELI(n,x)$Lisp + + hypergeometric0F1(a,z) == CHYPER0F1(a, z)$Lisp + hypergeometric0F1(n,x) == retract hypergeometric0F1(n::C, x::C) + + + -- All others are defined in terms of these. + digamma x == polygamma(0, x) + digamma z == polygamma(0, z) + + Beta(x,y) == Gamma(x)*Gamma(y)/Gamma(x+y) + Beta(w,z) == Gamma(w)*Gamma(z)/Gamma(w+z) + + fuzz := (10::R)**(-7) + + import IntegerRetractions(R) + import IntegerRetractions(C) + + besselY(n,x) == + if integer? n then n := n + fuzz + vp := n * pi()$R + (cos(vp) * besselJ(n,x) - besselJ(-n,x) )/sin(vp) + besselY(v,z) == + if integer? v then v := v + fuzz::C + vp := v * pi()$C + (cos(vp) * besselJ(v,z) - besselJ(-v,z) )/sin(vp) + + besselK(n,x) == + if integer? n then n := n + fuzz + p := pi()$R + vp := n*p + ahalf:= 1/(2::R) + p * ahalf * ( besselI(-n,x) - besselI(n,x) )/sin(vp) + besselK(v,z) == + if integer? v then v := v + fuzz::C + p := pi()$C + vp := v*p + ahalf:= 1/(2::C) + p * ahalf * ( besselI(-v,z) - besselI(v,z) )/sin(vp) + + airyAi x == + ahalf := recip(2::R)::R + athird := recip(3::R)::R + eta := 2 * athird * (-x) ** (3*ahalf) + (-x)**ahalf * athird * (besselJ(-athird,eta) + besselJ(athird,eta)) + airyAi z == + ahalf := recip(2::C)::C + athird := recip(3::C)::C + eta := 2 * athird * (-z) ** (3*ahalf) + (-z)**ahalf * athird * (besselJ(-athird,eta) + besselJ(athird,eta)) + + airyBi x == + ahalf := recip(2::R)::R + athird := recip(3::R)::R + eta := 2 * athird * (-x) ** (3*ahalf) + (-x*athird)**ahalf * ( besselJ(-athird,eta) - besselJ(athird,eta) ) + + airyBi z == + ahalf := recip(2::C)::C + athird := recip(3::C)::C + eta := 2 * athird * (-z) ** (3*ahalf) + (-z*athird)**ahalf * ( besselJ(-athird,eta) - besselJ(athird,eta) ) + +@ +\section{package ORTHPOL OrthogonalPolynomialFunctions} +<<package ORTHPOL OrthogonalPolynomialFunctions>>= +)abbrev package ORTHPOL OrthogonalPolynomialFunctions +++ Author: Stephen M. Watt +++ Date Created: 1990 +++ Date Last Updated: June 25, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: +++ Examples: +++ References: +++ Description: +++ This package provides orthogonal polynomials as functions on a ring. + +OrthogonalPolynomialFunctions(R: CommutativeRing): Exports == Impl where + NNI ==> NonNegativeInteger + RN ==> Fraction Integer + + Exports ==> with + + chebyshevT: (NNI, R) -> R + ++ chebyshevT(n,x) is the n-th Chebyshev polynomial of the first + ++ kind, \spad{T[n](x)}. These are defined by + ++ \spad{(1-t*x)/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}. + + chebyshevU: (NNI, R) -> R + ++ chebyshevU(n,x) is the n-th Chebyshev polynomial of the second + ++ kind, \spad{U[n](x)}. These are defined by + ++ \spad{1/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}. + + hermiteH: (NNI, R) -> R + ++ hermiteH(n,x) is the n-th Hermite polynomial, \spad{H[n](x)}. + ++ These are defined by + ++ \spad{exp(2*t*x-t**2) = sum(H[n](x)*t**n/n!, n = 0..)}. + + laguerreL: (NNI, R) -> R + ++ laguerreL(n,x) is the n-th Laguerre polynomial, \spad{L[n](x)}. + ++ These are defined by + ++ \spad{exp(-t*x/(1-t))/(1-t) = sum(L[n](x)*t**n/n!, n = 0..)}. + + laguerreL: (NNI, NNI, R) -> R + ++ laguerreL(m,n,x) is the associated Laguerre polynomial, + ++ \spad{L<m>[n](x)}. This is the m-th derivative of \spad{L[n](x)}. + + if R has Algebra RN then + legendreP: (NNI, R) -> R + ++ legendreP(n,x) is the n-th Legendre polynomial, + ++ \spad{P[n](x)}. These are defined by + ++ \spad{1/sqrt(1-2*x*t+t**2) = sum(P[n](x)*t**n, n = 0..)}. + + Impl ==> add + p0, p1: R + cx: Integer + + import IntegerCombinatoricFunctions() + + laguerreL(n, x) == + n = 0 => 1 + (p1, p0) := (-x + 1, 1) + for i in 1..n-1 repeat + (p1, p0) := ((2*i::R + 1 - x)*p1 - i**2*p0, p1) + p1 + laguerreL(m, n, x) == + ni := n::Integer + mi := m::Integer + cx := (-1)**m * binomial(ni,ni-mi) * factorial(ni) + p0 := 1 + p1 := cx::R + for j in 1..ni-mi repeat + cx := -cx*(ni-mi-j+1) + cx := (cx exquo ((mi+j)*j))::Integer + p0 := p0 * x + p1 := p1 + cx*p0 + p1 + chebyshevT(n, x) == + n = 0 => 1 + (p1, p0) := (x, 1) + for i in 1..n-1 repeat + (p1, p0) := (2*x*p1 - p0, p1) + p1 + chebyshevU(n, x) == + n = 0 => 1 + (p1, p0) := (2*x, 1) + for i in 1..n-1 repeat + (p1, p0) := (2*x*p1 - p0, p1) + p1 + hermiteH(n, x) == + n = 0 => 1 + (p1, p0) := (2*x, 1) + for i in 1..n-1 repeat + (p1, p0) := (2*x*p1 - 2*i*p0, p1) + p1 + if R has Algebra RN then + legendreP(n, x) == + n = 0 => 1 + p0 := 1 + p1 := x + for i in 1..n-1 repeat + c: RN := 1/(i+1) + (p1, p0) := (c*((2*i+1)*x*p1 - i*p0), p1) + p1 + +@ +\section{package NTPOLFN NumberTheoreticPolynomialFunctions} +<<package NTPOLFN NumberTheoreticPolynomialFunctions>>= +)abbrev package NTPOLFN NumberTheoreticPolynomialFunctions +++ Author: Stephen M. Watt +++ Date Created: 1990 +++ Date Last Updated: June 25, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: +++ Examples: +++ References: +++ Description: +++ This package provides polynomials as functions on a ring. + +NumberTheoreticPolynomialFunctions(R: CommutativeRing): Exports == Impl where + NNI ==> NonNegativeInteger + RN ==> Fraction Integer + + Exports ==> with + + cyclotomic: (NNI, R) -> R + ++ cyclotomic(n,r) \undocumented + + if R has Algebra RN then + bernoulliB: (NNI, R) -> R + ++ bernoulliB(n,r) \undocumented + eulerE: (NNI, R) -> R + ++ eulerE(n,r) \undocumented + + Impl ==> add + + import PolynomialNumberTheoryFunctions() + + I ==> Integer + SUP ==> SparseUnivariatePolynomial + + -- This is the wrong way to evaluate the polynomial. + cyclotomic(k, x) == + p: SUP(I) := cyclotomic(k) + r: R := 0 + while p ^= 0 repeat + d := degree p + c := leadingCoefficient p + p := reductum p + r := c*x**d + r + r + + if R has Algebra RN then + eulerE(k, x) == + p: SUP(RN) := euler(k) + r: R := 0 + while p ^= 0 repeat + d := degree p + c := leadingCoefficient p + p := reductum p + r := c*x**d + r + r + bernoulliB(k, x) == + p: SUP(RN) := bernoulli(k) + r: R := 0 + while p ^= 0 repeat + d := degree p + c := leadingCoefficient p + p := reductum p + r := c*x**d + r + r + +@ +\section{License} +<<license>>= +--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +--All rights reserved. +-- +--Redistribution and use in source and binary forms, with or without +--modification, are permitted provided that the following conditions are +--met: +-- +-- - Redistributions of source code must retain the above copyright +-- notice, this list of conditions and the following disclaimer. +-- +-- - Redistributions in binary form must reproduce the above copyright +-- notice, this list of conditions and the following disclaimer in +-- the documentation and/or other materials provided with the +-- distribution. +-- +-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the +-- names of its contributors may be used to endorse or promote products +-- derived from this software without specific prior written permission. +-- +--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +@ +<<*>>= +<<license>> + +<<package DFSFUN DoubleFloatSpecialFunctions>> +<<package ORTHPOL OrthogonalPolynomialFunctions>> +<<package NTPOLFN NumberTheoreticPolynomialFunctions>> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |