Artifact 171c9a2831ebd23c53b9e1d5d6de449318fcb912c3aedf7deb27c399bad65a90:
- Executable file
r37/packages/specfn/sfbes.red
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 31403) [annotate] [blame] [check-ins using] [more...]
- Executable file
r38/packages/specfn/sfbes.red
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 31403) [annotate] [blame] [check-ins using]
module sfbes; % Procedures and Rules for the Bessel functions. % Author: Chris Cannam, October 1992. % % Firstly, procedures to compute values of the Bessel functions by % direct bigfloat manipulation; also procedures for large arguments, % using an asymptotic formula. % These are specific to the Schoepf/Beckingham binary bigfloats, though % easily adapted, and they should only be used with n and z both % numeric, real and non-negative. % Then follow procedures written in algebraic mode and used for certain % special cases such as complex arguments. Anybody who wishes to create % symbolic mode complex-rounded versions is welcome to do so, with my % blessing. % No functions are provided to compute bessel K, though for special % cases the ruleset handles it. imports complex!*on!*switch, complex!*off!*switch, complex!*restore!*switch, sq2bf!*, sf!*eval; % This module exports no functions. I want to keep it available only % through the algebraic operators, largely because the functions are % quite a complicated lot. If you want to use it from symbolic mode, % use a wrapper and use the algebraic operators- it's slower, but at % least that way you'll get the answers. global '(logten); algebraic operator besselJ, besselY, besselI, besselK, hankel1, hankel2; symbolic operator do!*j, do!*y, do!*i; algebraic (bessel!*rules := { besselJ(~n,0) => 1 when n=0, % We need this form to be sure rules % are in right order. besselJ(~n,0) => 0 when numberp n and n neq 0, besselY(~n,0) => infinity, besselJ(1/2,~z) => sqrt(2/(pi*z)) * sin(z), besselJ(-1/2,~z) => sqrt(2/(pi*z)) * cos(z), besselY(-1/2,~z) => sqrt(2/(pi*z)) * sin(z), besselY(1/2,~z) => - sqrt(2/(pi*z)) * cos(z), besselK(~n,~z) => sqrt(Pi/(2*z))*e^(-z) when (n = 1/2 or n=-1/2), besselI(1/2,~z) => 1/sqrt(Pi*2*z)*(e^z - e^(-z)), besselI(-1/2,~z) => 1/sqrt(pi*2*z)*(e^z + e^(-z)), % J and Y for negative values and indices. besselJ(~n,~z) => ((-1)**n) * besselJ(-n,z) when numberp n and impart n=0 and n=floor n and n < 0, besselJ(~n,~z) => ((-1)**n) * besselJ(n,-z) when numberp n and impart n=0 and n=floor n and numberp z and repart z < 0, besselY(~n,~z) => ((-1)**n) * besselY(-n,z) when numberp n and impart n=0 and n=floor n and n < 0, besselY(~n,~z) => ((besselJ(n,z)*cos(n*pi))-(besselJ(-n,z)))/sin(n*pi) when not symbolic !*rounded and numberp n and (impart n neq 0 or not (repart n = floor repart n)), % Hankel functions. hankel1(~n,~z) => sqrt(2/(pi*z)) * (exp(i*z)/i) when symbolic !*complex and n = 1/2, hankel2(~n,~z) => sqrt(2/(pi*z)) * (exp(-i*z)/(-i)) when symbolic !*complex and n = 1/2, hankel1(~n,~z) => besselJ(n,z) + i * besselY(n,z) when symbolic !*complex and not symbolic !*rounded, hankel2(~n,~z) => besselJ(n,z) - i * besselY(n,z) when symbolic !*complex and not symbolic !*rounded, % Modified Bessel functions I and K. besselI(~n,0) => (if n = 0 then 1 else 0) when numberp n, besselI(~n,~z) => besselI(-n,z) when numberp n and impart n=0 and n=floor n and n < 0, besselK(~n,~z) => besselK(-n,z) when numberp n and impart n=0 and n=floor n and n < 0, besselK(~n,0) => infinity, besselK(~n,~z) => (pi/2)*((besselI(-n,z) - besselI(n,z))/(sin(n*pi))) when numberp n and impart n = 0 and not (n = floor n), % Derivatives. % df(besselJ(~n,~z),z) => -besselJ(1,z) when numberp n and n = 0, % df(besselY(~n,~z),z) => -besselY(1,z) when numberp n and n = 0, % df(besselI(~n,~z),z) => besselI(1,z) when numberp n and n = 0, % df(besselK(~n,~z),z) => -besselK(1,z) when numberp n and n = 0, % AS (9.1.26 and 27) df(besselJ(~n,~z),z) => besselJ(n-1,z) - (n/z) * besselJ(n,z), df(besselY(~n,~z),z) => besselY(n-1,z) - (n/z) * besselY(n,z), df(BesselK(~n,~z),z) => - BesselK(n-1,z) - (n/z) * BesselK(n,z), df(hankel1(~n,~z),z) => hankel1(n-1,z) - (n/z) * hankel1(n,z), df(hankel2(~n,~z),z) => hankel2(n-1,z) - (n/z) * hankel2(n,z), df(besselI(~n,~z),z) => (besselI(n-1,z) + besselI(n+1,z)) / 2, % Sending to be computed besselJ(~n,~z) => do!*j(n,z) when numberp n and numberp z and symbolic !*rounded, besselY(~n,~z) => do!*y(n,z) when numberp n and numberp z and symbolic !*rounded, besselI(~n,~z) => do!*i(n,z) when numberp n and numberp z and symbolic !*rounded })$ algebraic (let bessel!*rules); algebraic procedure do!*j(n,z); (if impart n = 0 and impart z = 0 and repart z > 0 then algebraic sf!*eval('j!*calc!*s,{n,z}) else algebraic sf!*eval('j!*calc, {n,z})); algebraic procedure do!*y(n,z); (if impart n = 0 and impart z = 0 and n = floor n then if repart z < 0 then algebraic sf!*eval('y!*calc!*sc, {n,z }) else algebraic sf!*eval('y!*calc!*s, {n,z,{}}) else if impart n neq 0 or n neq floor n then y!*reexpress(n,z) else algebraic sf!*eval('y!*calc, {n,z })); % What should be the value of BesselY(0,3i)? algebraic procedure do!*i(n,z); (if impart n = 0 and impart z = 0 and repart z > 0 then algebraic sf!*eval('i!*calc!*s, {n,z}) else algebraic sf!*eval('i!*calc, {n,z})); algebraic procedure j!*calc!*s(n,z); begin scalar n0, z0, fkgamnk, result, alglist!*; integer prepre, precom; precom := complex!*off!*switch(); prepre := precision 0; if z > (2*prepre) and z > 2*n and (result := algebraic sf!*eval('asymp!*j!*calc,{n,z})) neq {} then << precision prepre; complex!*restore!*switch(precom); return result >>; if prepre < !!nfpd then precision (!!nfpd+3+floor(abs n/10)) else precision (prepre+6+floor(abs n/10)); n0 := n; z0 := z; fkgamnk := gamma(n+1); result := algebraic sf!*eval('j!*calc!*s!*sub,{n0,z0,fkgamnk,prepre}); precision prepre; complex!*restore!*switch(precom); return result; end; symbolic procedure j!*calc!*s!*sub(n,z,fkgamnk,prepre); begin scalar result, admissable, this, modify, fkgamnk, zfsq, zfsqp, knk, azfsq, k; n := sq2bf!* n; z := sq2bf!* z; fkgamnk := sq2bf!* fkgamnk; modify := exp!:(timbf(log!:(divbf(z,bftwo!*), c!:prec!:()+2),n), c!:prec!:()); % modify := ((z/2)**n); zfsq := minus!:(divbf(timbf(z,z),i2bf!: 4)); % zfsq := (-(z**2)/4); azfsq := abs!: zfsq; result := divbf(bfone!*, fkgamnk); k := bfone!*; zfsqp := zfsq; fkgamnk := timbf(fkgamnk, plubf(n,bfone!*)); if lessp!:(abs!: result, bfone!*) then admissable := abs!: divbf (bfone!*, timbf (exp!:(timbf(fl2bf logten, i2bf!:(prepre + length explode fkgamnk)), 8), modify)) else admissable := abs!: divbf (bfone!*, timbf (exp!:(timbf(fl2bf logten, i2bf!:(prepre + length explode (1 + conv!:bf2i abs!: result))), 8), modify)); this := plubf(admissable, bfone!*); while greaterp!:(abs!: this, admissable) do << this := divbf(zfsqp, fkgamnk); result := plubf (result, this); k := plubf(k,bfone!*); knk := timbf (k, plubf(n, k)); if greaterp!: (azfsq, knk) then precision (precision(0) + length explode(1 + conv!:bf2i divbf (azfsq, knk))); zfsqp := timbf(zfsqp,zfsq); fkgamnk := timbf(fkgamnk,knk) >>; result := timbf(result,modify); return mk!*sq !*f2q mkround result; end; flag('(j!*calc!*s!*sub), 'opfn); algebraic procedure asymp!*j!*calc(n,z); begin scalar result, admissable, alglist!*, modify, chi, mu, p, q, n0, z0; integer prepre; prepre := precision 0; if prepre < !!nfpd then precision (!!nfpd + 5) else precision (prepre+8); modify := sqrt(2/(pi*z)); admissable := 1 / (10 ** (prepre + 5)); chi := z - (n/2 + 1/4)*pi; mu := 4*(n**2); n0 := n; z0 := z; p := algebraic symbolic asymp!*p(n0,z0,mu,admissable); if p neq {} then << q := algebraic symbolic asymp!*q(n0,z0,mu,admissable); if q neq {} then result := modify*(first p * cos chi - first q * sin chi) else result := {} >> else result := {}; precision prepre; return result; end; algebraic procedure asymp!*y!*calc(n,z); begin scalar result, admissable, alglist!*, modify, chi, mu, p, q, n0, z0; integer prepre; prepre := precision 0; if prepre < !!nfpd then precision (!!nfpd + 5) else precision (prepre+8); modify := sqrt(2/(pi*z)); admissable := 1 / (10 ** (prepre + 5)); chi := z - (n/2 + 1/4)*pi; mu := 4*(n**2); n0 := n; z0 := z; p := algebraic symbolic asymp!*p(n0,z0,mu,admissable); if p neq {} then << q := algebraic symbolic asymp!*q(n0,z0,mu,admissable); if q neq {} then result := modify*(first p * sin chi + first q * cos chi) else result := {} >> else result := {}; precision prepre; return result; end; symbolic procedure asymp!*p(n,z,mu,admissable); begin scalar result, this, prev, zsq, zsqp, aj2t; integer k, f; n := sq2bf!* n; z := sq2bf!* z; mu := sq2bf!* mu; admissable := sq2bf!* admissable; k := 2; f := 1 + conv!:bf2i difbf(divbf(n,bftwo!*),divbf(bfone!*,i2bf!: 4)); this := plubf(admissable, bfone!*); result := bfone!*; aj2t := asymp!*j!*2term(2, mu); zsq := timbf(i2bf!: 4, timbf(z, z)); zsqp := zsq; while greaterp!:(abs!: this, admissable) do << prev := abs!: this; this := timbf(i2bf!: ((-1)**(k/2)), divbf(aj2t, zsqp)); if greaterp!: (abs!: this, prev) and (k > f) then result := this := bfz!* else << result := plubf(result, this); zsqp := timbf(zsqp, zsq); k := k + 2; aj2t := timbf(aj2t, asymp!*j!*2term!*modifier(k, mu)) >> >>; if result = bfz!* then return '(list) else return list('list, mk!*sq !*f2q mkround result); end; symbolic procedure asymp!*q(n,z,mu,admissable); begin scalar result, this, prev, zsq, zsqp, aj2t; integer k, f; n := sq2bf!* n; z := sq2bf!* z; mu := sq2bf!* mu; admissable := sq2bf!* admissable; k := 1; f := 1 + conv!:bf2i difbf(divbf(n,bftwo!*),divbf(i2bf!: 3, i2bf!: 4)); this := plubf(admissable, bfone!*); result := bfz!*; aj2t := asymp!*j!*2term(1, mu); zsq := timbf(i2bf!: 4, timbf(z, z)); zsqp := timbf(bftwo!*, z); while greaterp!:(abs!: this, admissable) do << prev := abs!: this; this := timbf(i2bf!: ((-1)**((k-1)/2)), divbf(aj2t, zsqp)); if greaterp!: (abs!: this, prev) and (k > f) then result := this := bfz!* else << result := plubf(result, this); zsqp := timbf(zsqp, zsq); k := k + 2; aj2t := timbf(aj2t, asymp!*j!*2term!*modifier(k, mu)) >> >>; if result = bfz!* then return '(list) else return list('list, mk!*sq !*f2q mkround result); end; symbolic procedure asymp!*j!*2term(k, mu); begin scalar result; result := bfone!*; for j := 1 step 2 until (2*k - 1) do result := timbf(result, difbf(mu, i2bf!: (j**2))); result := divbf (result, i2bf!: (factorial k * (2**(2*k)))); return result; end; symbolic procedure asymp!*j!*2term!*modifier(k, mu); (timbf (difbf(mu, i2bf!: ((2*k-3)**2)), divbf (difbf(mu, i2bf!: ((2*k-1)**2)), i2bf!: ((k-1) * k * 16)))); algebraic procedure y!*calc!*s(n,z,st); begin scalar n0, z0, st0, ps, fkgamnk, result, alglist!*; integer prepre, precom; precom := complex!*off!*switch(); prepre := precision 0; if z > (2*prepre) and z > 2*n and (result := asymp!*y!*calc(n,z)) neq {} then << precision prepre; complex!*restore!*switch(precom); return result >>; if prepre < !!nfpd then precision (!!nfpd+5) else precision (prepre + 8); n0 := n; z0 := z; st0 := st; ps := psi 1 + psi(1+n); fkgamnk := gamma(n+1); result := algebraic symbolic y!*calc!*s!*sub(n0,z0,ps,fkgamnk,prepre,st0); precision prepre; complex!*restore!*switch(precom); return result; end; % The last arg to the next procedure is an algebraic list of the % modifier, start value and (factorial n) for the series. If this is % (LIST) (i.e. the nil algebraic list {}), the values will be computed % in this procedure; otherwise the values in st0 will be used. This % feature is used for decomposition of the computation of y at negative % real z. It is of course designed to make the code as hard to follow % as possible. Why else? % n must be a non-negative integer for this next procedure to work. symbolic procedure y!*calc!*s!*sub(n,z,ps,fkgamnk,prepre, st0); begin scalar start, result, this, ps, fc, modify, zfsq, zfsqp, nps, azfsq, bj, z0, n0, tpi, admissable; integer k, fk, fnk, difd, fcp; z0 := z; z := sq2bf!* z; ps := sq2bf!* ps; n := sq2bf!* n; n0 := conv!:bf2i n; tpi := pi!*(); if st0 = '(LIST) then << modify := divbf(exp!: (timbf(n, log!:(divbf(z, bftwo!*), c!:prec!:()+2)), c!:prec!:()), tpi); bj := retag cdr !*a2f sf!*eval('j!*calc!*s!*sub, list('list,n0,z0,fkgamnk,prepre)); if n0 < 1 then << start := timbf(timbf(divbf(bftwo!*,tpi), log!:(divbf(z,bftwo!*),c!:prec!:()+1)), bj); fc := factorial n0 >> else if (n0 < 100) then << start := bfz!*; zfsq := divbf(timbf(z,z), i2bf!: 4); for k := 0:(n0-1) do start := plubf(start, divbf (exptbf(zfsq, k, i2bf!: factorial (n0-k-1)), i2bf!: factorial k)); start := minus!: timbf(start, divbf(exp!: (timbf(minus!: n, log!:(divbf(z, bftwo!*), c!:prec!:()+2)), c!:prec!:()), tpi)); start := plubf (start, timbf(timbf(divbf(bftwo!*,tpi),bj), log!:(divbf(z,bftwo!*), c!:prec!:()+2))); fc := factorial n0 >> else << zfsq := divbf(timbf(z,z), i2bf!: 4); zfsqp := bfone!*; fk := 1; fnk := factorial (n0-1); fc := fnk * n0; start := bfz!*; for k := 0:(n0-2) do << start := plubf(start, timbf(i2bf!: fnk, divbf(zfsqp, i2bf!: fk))); fk := fk * (k+1); fnk := fnk / (n0-k-1); zfsqp := timbf(zfsqp, zfsq) >>; start := plubf(start, timbf(i2bf!: fnk, divbf(zfsqp, i2bf!: fk))); start := minus!: plubf(timbf(start, divbf(bfone!*,timbf(modify,timbf(tpi,tpi)))), timbf(timbf(divbf(bftwo!*,tpi), bj), log!:(divbf(z,bftwo!*),c!:prec!:()+2))) >> >> else << start := sq2bf!* cadr st0; modify := sq2bf!* caddr st0; fc := cadddr st0 >>; zfsq := minus!: divbf(timbf(z,z),i2bf!: 4); azfsq := abs!: zfsq; result := divbf(ps, i2bf!: fc); k := 1; zfsqp := zfsq; fc := fc * (n0+1); ps := plubf(ps,plubf(bfone!*,divbf(bfone!*,plubf(n,bfone!*)))); % Note: we are assuming numberp start. Be sure to catch other cases % elsewhere. (Notably for z < 0). This goes for bessel J as well. if lessp!: (abs!: plubf(result, start), bfone!*) then admissable := abs!: divbf(divbf(bfone!*, exp!:(timbf(fl2bf logten, plubf(i2bf!:(prepre+2), divbf(log!:(divbf(bfone!*, plubf(abs!: result, abs!: start)), 5), fl2bf logten))), 5)), modify) else admissable := abs!: divbf(divbf(bfone!*, exp!:(timbf(fl2bf logten, plubf(i2bf!:(prepre+2), divbf(log!:(plubf(abs!: result, abs!: start), 5), fl2bf logten))), 5)), modify); this := plubf(admissable, bfone!*); while greaterp!: (abs!: this, admissable) do << this := timbf(ps, divbf(zfsqp, i2bf!: fc)); result := plubf(result, this); k := k + 1; zfsqp := timbf(zfsqp, zfsq); nps := plubf(ps, plubf(divbf(bfone!*,i2bf!: k), divbf(bfone!*,i2bf!:(k+n0)))); fcp := k * (n0+k); if greaterp!:(timbf(nps,azfsq),timbf(ps,i2bf!: fcp)) then << difd := 1 + conv!:bf2i divbf(timbf(nps,azfsq),timbf(ps,i2bf!: fcp)); precision (precision(0) + length explode difd) >>; fc := fc * fcp; ps := nps >>; result := difbf(start, timbf(result, modify)); return mk!*sq !*f2q mkround result; end; algebraic procedure i!*calc!*s(n,z); begin scalar n0, z0, ps, fkgamnk, result, alglist!*; integer prepre, precom; precom := complex!*off!*switch(); prepre := precision 0; if prepre < !!nfpd then precision (!!nfpd+3+floor(abs n/10)) else precision (prepre+8+floor(abs n/10)); n0 := n; z0 := z; fkgamnk := gamma(n+1); result := algebraic symbolic i!*calc!*s!*sub(n0,z0,fkgamnk,prepre); precision prepre; complex!*restore!*switch(precom); return result; end; symbolic procedure i!*calc!*s!*sub(n,z,fkgamnk,prepre); begin scalar result, admissable, this, modify, fkgamnk, zfsq, zfsqp, knk, azfsq, k; n := sq2bf!* n; z := sq2bf!* z; fkgamnk := sq2bf!* fkgamnk; modify := exp!:(timbf(log!:(divbf(z,bftwo!*), c!:prec!:()+2),n), c!:prec!:()); % modify := ((z/2)**n); zfsq := divbf(timbf(z,z),i2bf!:(4)); % zfsq := (-(z**2)/4); azfsq := abs!: zfsq; result := divbf(bfone!*, fkgamnk); k := bfone!*; zfsqp := zfsq; fkgamnk := timbf(fkgamnk, plubf(n,bfone!*)); if lessp!:(abs!: result, bfone!*) then admissable := abs!: divbf (bfone!*, timbf (exp!:(timbf(fl2bf logten, i2bf!:(prepre + length explode fkgamnk)), 8), modify)) else admissable := abs!: divbf (bfone!*, timbf (exp!:(timbf(fl2bf logten, i2bf!:(prepre + length explode (1 + conv!:bf2i abs!: result))), 8), modify)); this := plubf(admissable, bfone!*); while greaterp!:(abs!: this, admissable) do << this := divbf(zfsqp, fkgamnk); result := plubf (result, this); k := plubf(k,bfone!*); knk := timbf (k, plubf(n, k)); if greaterp!: (azfsq, knk) then precision (precision(0) + length explode (1 + conv!:bf2i divbf (azfsq, knk))); zfsqp := timbf(zfsqp, zfsq); fkgamnk := timbf(fkgamnk, knk) >>; result := timbf(result, modify); return mk!*sq !*f2q mkround result; end; % % algebraic procedure j!*calc(n,z); % % Given integer n and arbitrary (I hope) z, compute and return % the value of the Bessel J-function, order n, at z. Current % version mostly coded for speed rather than clarity. % % Does work for non-integral n. % algebraic procedure j!*calc(n,z); begin scalar result, admissable, this, alglist!*, modify, fkgamnk, zfsq, zfsqp, azfsq, knk; % bind alglist!* to integer prepre, k, difd; % stop global alglist being cleared prepre := precision 0; % Don't need to check if asymptotic expansion is valid; % if we're using this routine, it's not appropriate anyway. % if z > (2*prepre) and z > 2*n and % (result := asymp!*j!*calc(n,z)) neq {} % then return result; precision (prepre + 4); modify := ((z/2) ** n); zfsq := (-(z**2)/4); azfsq := abs zfsq; fkgamnk := gamma(n+1); result := (1 / (fkgamnk)); k := 1; zfsqp := zfsq; fkgamnk := fkgamnk * (n+1); if numberp modify and impart modify = 0 then if (abs result) < 1 then << difd := ceiling (1/abs result); admissable := abs ((1 / (10 ** (prepre + (symbolic length explode difd)))) / modify) >> else << difd := ceiling abs result; admissable := abs ((1 / (10 ** (prepre - (symbolic length explode difd)))) / modify) >> else if (abs result) < 1 then << difd := ceiling (1/abs result); admissable := abs (1 / (10 ** (prepre + 10 + (symbolic length explode difd)))) >> else << difd := ceiling abs result; admissable := abs (1 / (10 ** (prepre + 10 - (symbolic length explode difd)))) >>; this := admissable + 1; while (abs this > admissable) do << this := (zfsqp / (fkgamnk)); result := result + this; k := k + 1; % Maintain k as term counter, knk := k * (n+k); if azfsq > knk then <<difd := ceiling (azfsq / knk); precision(precision(0)+(lisp length explode difd))>>; zfsqp := zfsqp * zfsq; % zfsqp as ((-(z**2)/4)**k), and fkgamnk := fkgamnk * knk >>; % fkgamnk as k! * gamma(n+k+1). result := result * modify; precision prepre; return result; end; % % Procedure to compute (modified) start value for % Bessel Y computations. Also used to get imaginary % part for certain values % algebraic procedure y!*modifier!*calc(n,z); begin scalar modify, start, zfsq, zfsqp, fc; integer fk, fnk, prepre; prepre := precision 0; % if prepre < !!nfpd then precision (!!nfpd + 2) % else precision (prepre + 2); modify := ((z/2)**n) / pi; % Simple expression for start value when n<1. if (n < 1) then << start := ((2/pi) * log(z/2) * besselJ(n,z)); fc := factorial n >> % If n smallish, just sum using factorials. (REDUCE % seems to do smallish factorials quite quickly. In % fact it does largish factorials quite quickly as well, % but not quite as quickly as we can build them by % per-term multiplication.) else if (n < 100) then << start := - (((z/2) ** (-n)) / pi) * (for k := 0:(n-1) sum ((factorial (n-k-1) * (((z**2)/4) ** k)) / (factorial k))) + ((2/pi)*log(z/2)*besselJ(n,z)); fc := factorial n >> % If n largish, avoid computing factorials, and try % to do the minimum possible real work. else << zfsq := (z**2)/4; zfsqp := 1; fk := 1; fnk := factorial (n-1); fc := fnk * n; start := 0; for k := 0:(n-2) do << start := start + (fnk * zfsqp / fk); fk := fk * (k+1); fnk := floor(fnk/(n-k-1)); zfsqp := zfsqp * zfsq >>; start := start + (fnk * zfsqp / fk); start := - ((1/(modify*(pi**2)))*start)+ ((2/pi)*log(z/2)*besselJ(n,z)) >>; precision prepre; return {start, modify, fc}; end; % % algebraic procedure y!*calc(n,z); % % Given integer n and arbitrary (I hope) z, compute and return % the value of the Bessel Y-function, order n, at z. Current % version mostly coded for speed rather than clarity. % % Owing to its dependence upon factorials, doesn't work for % non-integral n. (But in any case it'd be very slow, particularly % for large non-integral n.) % algebraic procedure y!*calc(n,z); begin scalar start, result, this, ps, fc, smf, modify, zfsq, zfsqp, alglist!*, nps, azfsq; integer prepre, k, fk, fnk, difd, fcp; prepre := precision(0); precision (prepre + 8); smf := y!*modifier!*calc (n,z); start := first smf; modify := second smf; fc := third smf; % Now we have the starting value: prepare the loop for % the remaining terms. k will be our loop counter. p1 % will hold psi(k+1), and p2 psi(k+n+1); zfsqp is % maintained at ((-(z**2)/4)**k); fc is k! * (n+k)!. % The sum is of (p1 + p2) * zfsqp / fc, and we % precompute the first term in order to get an idea % of the general magnitude (it's a decreasing series). ps := psi 1 + psi(1+n); zfsq := (-(z**2)/4); azfsq := abs zfsq; result := ps / fc; k := 1; zfsqp := zfsq; fc := fc * (n+1); ps := ps + 1 + (1/(n+1)); % Having the first term and start, we check whether % they're small or large and modify the maximum % acceptable error accordingly. if numberp start then if (abs (result + start)) < 1 then admissable := abs ((1 / (10 ** (prepre+2 + log10(1/(abs result + abs start)))))/modify) else admissable := abs ((1 / (10 ** (prepre + 2))) * (log10(abs result + abs start)) / modify) else admissable := abs (1 / (10 ** (prepre + 10))); this := admissable + 1; % Now sum the series. while ((abs this) > admissable) do << this := ps * (zfsqp / fc); result := result + this; k := k + 1; zfsqp := zfsqp * zfsq; nps := ps + (1/k) + (1/(k+n)); fcp := k * (n+k); if (nps*azfsq) > (ps*fcp) then <<difd := ceiling ((nps*azfsq)/(ps*fcp)); precision(precision(0) + (lisp length explode difd))>>; fc := fc * fcp; % fc ends up as k! * (n+k)! ps := nps >>; % Amalgamate the start value and modification, and % return the answer. result := start - (result * modify); precision prepre; return result; end; % % algebraic procedure i!*calc(n,z); % % Given integer n and arbitrary (I hope) z, compute and return % the value of the (modified) Bessel I-function, order n, at z. % Current version mostly coded for speed rather than clarity. % % Does work for non-integral n. % algebraic procedure i!*calc(n,z); begin scalar result, admissable, this, prev, nprev, alglist!*, modify, fkgamnk, zfsq, zfsqp, knk; % bind alglist!* to prevent integer prepre, k, difd; % global alglist being cleared modify := ((z/2) ** n); prepre := precision 0; precision (prepre + 4); zfsq := (z**2)/4; azfsq := abs zfsq; fkgamnk := gamma(n+1); result := (1 / (fkgamnk)); k := 1; zfsqp := zfsq; fkgamnk := fkgamnk * (n+1); if numberp modify then if (abs result) < 1 then << difd := ceiling (1/abs result); admissable := abs ((1 / (10 ** (prepre + (symbolic length explode difd)))) / modify) >> else << difd := ceiling abs result; admissable := abs ((1 / (10 ** (prepre - (symbolic length explode difd)))) / modify) >> else if (abs result) < 1 then << difd := ceiling (1/abs result); admissable := abs (1 / (10 ** (prepre + 10 + (symbolic length explode difd)))) >> else << difd := ceiling abs result; admissable := abs (1 / (10 ** (prepre + 10 - (symbolic length explode difd)))) >>; this := admissable + 1; nprev := abs this; while (abs this > admissable) do << this := (zfsqp / (fkgamnk)); result := result + this; k := k + 1; % Maintain k as term counter, knk := k * (n+k); if azfsq > knk then <<difd := ceiling (azfsq / knk); precision(precision(0) + (lisp length explode difd))>>; zfsqp := zfsqp * zfsq; % zfsqp as ((-(z**2)/4)**k), and fkgamnk := fkgamnk * knk >>; % fkgamnk as k! * gamma(n+k+1). result := result * modify; precision prepre; return result; end; algebraic procedure k!*calc!*2(n,z); begin scalar result, precom; integer prepre; prepre := precision 0; precision (prepre + 8); precom := complex!*on!*switch(); result := (pi/2)*i*exp((pi/2)*n*i)*hankel1(n,z*exp((pi/2)*i)); complex!*restore!*switch(precom); precision prepre; return result; end; % % Function which simply rewrites bessely (with nonintegral % order) in terms of besselj. Turns off rounded mode to % do so, because if rounded is on, cos(n*pi) =/= 0 for % n*2 = floor (n*2), which can lead to some spectacular % inaccuracies. % algebraic procedure y!*reexpress(n,z); begin scalar result, premsg; premsg := lisp !*msg; off msg; off rounded; result := ((besselJ(n,z)*cos(n*pi))-(besselJ(-n,z)))/sin(n*pi); on rounded; if premsg then on msg; return result; end; % % Function to make an evil blend of the symbolic and % algebraic mode bessel-y functions where the order % is real and the arg is real and negative. Here the % result will be complex (probably), but most of the % computations involved will be with real numbers so % the symbolic mode version will do them better. % % Therefore this routine, which gets the modifier % and initial terms (the only complex bits) from the % algebraic procedure and then gets the rest from the % symbolic one. % algebraic procedure y!*calc!*sc(n,z); begin scalar st, ic, rc, md, fc, result, precom, prepre; prepre := precision 0; z := -z; if prepre < !!nfpd then precision (!!nfpd + 2) else precision (prepre + 4); st := y!*modifier!*calc(n,z); rc := - first st; precom := complex!*on!*switch(); ic := impart(log(-pi/2)); complex!*restore!*switch(precom); ic := ic*(2/pi)*besselj(n,-z); md := - second st; fc := third st; precision prepre; precom := complex!*off!*switch(); result := y!*calc!*s(n,z,{rc,md,fc}); complex!*restore!*switch(precom); if symbolic !*complex then result := result + i * ic else result := (if ic < 0 then 1 else -1) * sqrt(-(ic**2)) + result; return result; end; endmodule; end;