| [z,fnu,kode,n,cy,nz,ierr]=cbesj(z,fnu,kode,n,cy,nz,ierr); |
function [z,fnu,kode,n,cy,nz,ierr]=cbesj(z,fnu,kode,n,cy,nz,ierr);
%***BEGIN PROLOGUE CBESJ
%***PURPOSE Compute a sequence of the Bessel functions J(a,z) for
% complex argument z and real nonnegative orders a=b,b+1,
% b+2,... where b>0. A scaling option is available to
% help avoid overflow.
%***LIBRARY SLATEC
%***CATEGORY C10A4
%***TYPE COMPLEX (CBESJ-C, ZBESJ-C)
%***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
% BESSEL FUNCTIONS OF THE FIRST KIND, J BESSEL FUNCTIONS
%***AUTHOR Amos, D. E., (SNL)
%***DESCRIPTION
%
% On KODE=1, CBESJ computes an N member sequence of complex
% Bessel functions CY(L)=J(FNU+L-1,Z) for real nonnegative
% orders FNU+L-1, L=1,...,N and complex Z in the cut plane
% -pi<arg(Z)<=pi. On KODE=2, CBESJ returns the scaled functions
%
% CY(L) = exp(-abs(Y))*J(FNU+L-1,Z), L=1,...,N and Y=Im(Z)
%
% which remove the exponential growth in both the upper and
% lower half planes as Z goes to infinity. Definitions and
% notation are found in the NBS Handbook of Mathematical
% Functions (Ref. 1).
%
% Input
% Z - Argument of type COMPLEX
% FNU - Initial order of type REAL, FNU>=0
% KODE - A parameter to indicate the scaling option
% KODE=1 returns
% CY(L)=J(FNU+L-1,Z), L=1,...,N
% =2 returns
% CY(L)=J(FNU+L-1,Z)*exp(-abs(Y)), L=1,...,N
% where Y=Im(Z)
% N - Number of terms in the sequence, N>=1
%
% Output
% CY - Result vector of type COMPLEX
% NZ - Number of underflows set to zero
% NZ=0 Normal return
% NZ>0 CY(L)=0, L=N-NZ+1,...,N
% IERR - Error flag
% IERR=0 Normal return - COMPUTATION COMPLETED
% IERR=1 Input error - NO COMPUTATION
% IERR=2 Overflow - NO COMPUTATION
% (Im(Z) too large on KODE=1)
% IERR=3 Precision warning - COMPUTATION COMPLETED
% (Result has half precision or less
% because abs(Z) or FNU+N-1 is large)
% IERR=4 Precision error - NO COMPUTATION
% (Result has no precision because
% abs(Z) or FNU+N-1 is too large)
% IERR=5 Algorithmic error - NO COMPUTATION
% (Termination condition not met)
%
% *Long Description:
%
% The computation is carried out by the formulae
%
% J(a,z) = exp( a*pi*i/2)*I(a,-i*z), Im(z)>=0
%
% J(a,z) = exp(-a*pi*i/2)*I(a, i*z), Im(z)<0
%
% where the I Bessel function is computed as described in the
% prologue to CBESI.
%
% For negative orders, the formula
%
% J(-a,z) = J(a,z)*cos(a*pi) - Y(a,z)*sin(a*pi)
%
% can be used. However, for large orders close to integers, the
% the function changes radically. When a is a large positive
% integer, the magnitude of J(-a,z)=J(a,z)*cos(a*pi) is a
% large negative power of ten. But when a is not an integer,
% Y(a,z) dominates in magnitude with a large positive power of
% ten and the most that the second term can be reduced is by
% unit roundoff from the coefficient. Thus, wide changes can
% occur within unit roundoff of a large integer for a. Here,
% large means a>abs(z).
%
% In most complex variable computation, one must evaluate ele-
% mentary functions. When the magnitude of Z or FNU+N-1 is
% large, losses of significance by argument reduction occur.
% Consequently, if either one exceeds U1=SQRT(0.5/UR), then
% losses exceeding half precision are likely and an error flag
% IERR=3 is triggered where UR=R1MACH(4)=UNIT ROUNDOFF. Also,
% if either is larger than U2=0.5/UR, then all significance is
% lost and IERR=4. In order to use the INT function, arguments
% must be further restricted not to exceed the largest machine
% integer, U3=I1MACH(9). Thus, the magnitude of Z and FNU+N-1
% is restricted by MIN(U2,U3). In IEEE arithmetic, U1,U2, and
% U3 approximate 2.0E+3, 4.2E+6, 2.1E+9 in single precision
% and 4.7E+7, 2.3E+15 and 2.1E+9 in doubleprecision. This
% makes U2 limiting in single precision and U3 limiting in
% doubleprecision. This means that one can expect to retain,
% in the worst cases on IEEE machines, no digits in single pre-
% cision and only 6 digits in doubleprecision. Similar con-
% siderations hold for other machines.
%
% The approximate relative error in the magnitude of a complex
% Bessel function can be expressed as P*10**S where P=MAX(UNIT
% ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
% sents the increase in error due to argument reduction in the
% elementary functions. Here, S=MAX(1,ABS(LOG10(ABS(Z))),
% ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
% ABS(Z),ABS(EXPONENT OF FNU)) ). However, the phase angle may
% have only absolute accuracy. This is most likely to occur
% when one component (in magnitude) is larger than the other by
% several orders of magnitude. If one component is 10**K larger
% than the other, then one can expect only MAX(ABS(LOG10(P))-K,
% 0) significant digits; or, stated another way, when K exceeds
% the exponent of P, no significant digits remain in the smaller
% component. However, the phase angle retains absolute accuracy
% because, in complex arithmetic with precision P, the smaller
% component will not (as a rule) decrease below P times the
% magnitude of the larger component. In these extreme cases,
% the principal phase angle is on the order of +P, -P, PI/2-P,
% or -PI/2+P.
%
%***REFERENCES 1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
% matical Functions, National Bureau of Standards
% Applied Mathematics Series 55, U. S. Department
% of Commerce, Tenth Printing (1972) or later.
% 2. D. E. Amos, Computation of Bessel Functions of
% Complex Argument, Report SAND83-0086, Sandia National
% Laboratories, Albuquerque, NM, May 1983.
% 3. D. E. Amos, Computation of Bessel Functions of
% Complex Argument and Large Order, Report SAND83-0643,
% Sandia National Laboratories, Albuquerque, NM, May
% 1983.
% 4. D. E. Amos, A Subroutine Package for Bessel Functions
% of a Complex Argument and Nonnegative Order, Report
% SAND85-1018, Sandia National Laboratory, Albuquerque,
% NM, May 1985.
% 5. D. E. Amos, A portable package for Bessel functions
% of a complex argument and nonnegative order, ACM
% Transactions on Mathematical Software, 12 (September
% 1986), pp. 265-273.
%
%***ROUTINES CALLED CBINU, I1MACH, R1MACH
%***REVISION HISTORY (YYMMDD)
% 830501 DATE WRITTEN
% 890801 REVISION DATE from Version 3.2
% 910415 Prologue converted to Version 4.0 format. (BAB)
% 920128 Category corrected. (WRB)
% 920811 Prologue revised. (DWL)
%***end PROLOGUE CBESJ
%
persistent aa alim arg ascle atol az bb ci csgn dig elim firstCall fn fnul hpi i inu inuh ir k k1 k2 nl r1 r1m5 r2 rl rtol tol yy zn ; if isempty(firstCall),firstCall=1;end;
if isempty(ci), ci=0; end;
if isempty(csgn), csgn=0; end;
if isempty(zn), zn=0; end;
if isempty(aa), aa=0; end;
if isempty(alim), alim=0; end;
if isempty(arg), arg=0; end;
if isempty(dig), dig=0; end;
if isempty(elim), elim=0; end;
if isempty(fnul), fnul=0; end;
if isempty(hpi), hpi=0; end;
if isempty(rl), rl=0; end;
if isempty(r1), r1=0; end;
if isempty(r1m5), r1m5=0; end;
if isempty(r2), r2=0; end;
if isempty(tol), tol=0; end;
if isempty(yy), yy=0; end;
if isempty(az), az=0; end;
if isempty(fn), fn=0; end;
if isempty(bb), bb=0; end;
if isempty(ascle), ascle=0; end;
if isempty(rtol), rtol=0; end;
if isempty(atol), atol=0; end;
if isempty(i), i=0; end;
if isempty(inu), inu=0; end;
if isempty(inuh), inuh=0; end;
if isempty(ir), ir=0; end;
if isempty(k1), k1=0; end;
if isempty(k2), k2=0; end;
if isempty(nl), nl=0; end;
if isempty(k), k=0; end;
if firstCall, hpi=[1.57079632679489662e0]; end;
firstCall=0;
%
%***FIRST EXECUTABLE STATEMENT CBESJ
ierr = 0;
nz = 0;
if( fnu<0.0e0 )
ierr = 1;
end;
if( kode<1 || kode>2 )
ierr = 1;
end;
if( n<1 )
ierr = 1;
end;
if( ierr~=0 )
return;
end;
%-----------------------------------------------------------------------
% SET PARAMETERS RELATED TO MACHINE CONSTANTS.
% TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
% ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
% EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND
% EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
% UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
% RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
% DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
% FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU.
%-----------------------------------------------------------------------
tol = max(r1mach(4),1.0e-18);
[k1 ]=i1mach(12);
[k2 ]=i1mach(13);
[r1m5 ]=r1mach(5);
k = fix(min(abs(k1),abs(k2)));
elim = 2.303e0.*(k.*r1m5-3.0e0);
k1 = fix(i1mach(11) - 1);
aa = r1m5.*k1;
dig = min(aa,18.0e0);
aa = aa.*2.303e0;
alim = elim + max(-aa,-41.45e0);
rl = 1.2e0.*dig + 3.0e0;
fnul = 10.0e0 + 6.0e0.*(dig-3.0e0);
ci = complex(0.0e0,1.0e0);
yy = imag(z);
az = abs(z);
%-----------------------------------------------------------------------
% TEST FOR RANGE
%-----------------------------------------------------------------------
aa = 0.5e0./tol;
bb = i1mach(9).*0.5e0;
aa = min(aa,bb);
fn = fnu +(n-1);
if( az<=aa )
if( fn<=aa )
aa = sqrt(aa);
if( az>aa )
ierr = 3;
end;
if( fn>aa )
ierr = 3;
end;
%-----------------------------------------------------------------------
% CALCULATE CSGN=EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
% WHEN FNU IS LARGE
%-----------------------------------------------------------------------
inu = fix(fnu);
inuh = fix(fix(inu./2));
ir = fix(inu - 2.*inuh);
arg =(fnu-(inu-ir)).*hpi;
r1 = cos(arg);
r2 = sin(arg);
csgn = complex(r1,r2);
if( rem(inuh,2)==1 )
csgn = -csgn;
end;
%-----------------------------------------------------------------------
% ZN IS IN THE RIGHT HALF PLANE
%-----------------------------------------------------------------------
zn = -z.*ci;
if( yy<0.0e0 )
zn = -zn;
csgn = conj(csgn);
ci = conj(ci);
end;
[zn,fnu,kode,n,cy,nz,rl,fnul,tol,elim,alim]=cbinu(zn,fnu,kode,n,cy,nz,rl,fnul,tol,elim,alim);
if( nz>=0 )
nl = fix(n - nz);
if( nl==0 )
return;
end;
rtol = 1.0e0./tol;
ascle = r1mach(1).*rtol.*1.0e+3;
for i = 1 : nl;
% CY(I)=CY(I)*CSGN
zn = cy(i);
aa = real(zn);
bb = imag(zn);
atol = 1.0e0;
if( max(abs(aa),abs(bb))<=ascle )
zn = zn.*complex(rtol,0.0e0);
atol = tol;
end;
zn = zn.*csgn;
cy(i) = zn.*complex(atol,0.0e0);
csgn = csgn.*ci;
end; i = fix(nl+1);
return;
elseif( nz==(-2) ) ;
nz = 0;
ierr = 5;
return;
else;
nz = 0;
ierr = 2;
return;
end;
end;
end;
nz = 0;
ierr = 4;
end
%DECK CBESK
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