aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/special.spad.pamphlet
diff options
context:
space:
mode:
Diffstat (limited to 'src/algebra/special.spad.pamphlet')
-rw-r--r--src/algebra/special.spad.pamphlet456
1 files changed, 456 insertions, 0 deletions
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}
generated by cgit v1.2.3 (git 2.39.1) at 2024-12-23 17:01:46 +0000