Mercurial > forge
changeset 11671:c5116717efd1 octave-forge
specfun: removing ellipj, ellipke, and expint (merged into octave core)
| author | carandraug |
|---|---|
| date | Mon, 29 Apr 2013 13:18:39 +0000 |
| parents | 0f16ee5611b8 |
| children | 2fd78b3a815d |
| files | main/specfun/NEWS main/specfun/inst/ellipke.m main/specfun/inst/expint.m main/specfun/src/.svnignore main/specfun/src/Makefile main/specfun/src/ellipj.cc |
| diffstat | 6 files changed, 3 insertions(+), 1068 deletions(-) [+] |
line wrap: on
line diff
--- a/main/specfun/NEWS Mon Apr 29 12:53:23 2013 +0000 +++ b/main/specfun/NEWS Mon Apr 29 13:18:39 2013 +0000 @@ -18,7 +18,10 @@ ** The following functions have removed since they now are part of Octave core: + ellipj + ellipke erfcinv + expint ** The following functions have been deprecated (see their help text for the recommended alternatives):
--- a/main/specfun/inst/ellipke.m Mon Apr 29 12:53:23 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,116 +0,0 @@ -## Copyright (C) 2001 David Billinghurst <David.Billinghurst@riotinto.com> -## Copyright (C) 2001 Paul Kienzle <pkienzle@users.sf.net> -## Copyright (C) 2003 Jaakko Ruohio -## -## This program is free software; you can redistribute it and/or modify it under -## the terms of the GNU General Public License as published by the Free Software -## Foundation; either version 3 of the License, or (at your option) any later -## version. -## -## This program is distributed in the hope that it will be useful, but WITHOUT -## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or -## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more -## details. -## -## You should have received a copy of the GNU General Public License along with -## this program; if not, see <http://www.gnu.org/licenses/>. - -## -*- texinfo -*- -## @deftypefn {Function File} {[@var{k}, @var{e}] =} ellipke (@var{m}[,@var{tol}]) -## Compute complete elliptic integral of first K(@var{m}) and second E(@var{m}). -## -## @var{m} is either real array or scalar with 0 <= m <= 1 -## -## @var{tol} will be ignored (@sc{Matlab} uses this to allow faster, less -## accurate approximation) -## -## Ref: Abramowitz, Milton and Stegun, Irene A. Handbook of Mathematical -## Functions, Dover, 1965, Chapter 17. -## @seealso{ellipj} -## @end deftypefn - -function [k,e] = ellipke( m ) - - if (nargin < 1 || nargin > 2) - print_usage; - endif - - k = e = zeros(size(m)); - m = m(:); - if any(~isreal(m)) - error("ellipke must have real m"); - endif - if any(m>1) - error("ellipke must have m <= 1"); - endif - - Nmax = 16; - idx = find(m == 1); - if (!isempty(idx)) - k(idx) = Inf; - e(idx) = 1.0; - endif - - idx = find(m == -Inf); - if (!isempty(idx)) - k(idx) = 0.0; - e(idx) = Inf; - endif - - ## Arithmetic-Geometric Mean (AGM) algorithm - ## ( Abramowitz and Stegun, Section 17.6 ) - idx = find(m != 1 & m != -Inf); - if (!isempty(idx)) - idx_neg = find(m < 0 & m != -Inf); - mult_k = 1./sqrt(1-m(idx_neg)); - mult_e = sqrt(1-m(idx_neg)); - m(idx_neg) = -m(idx_neg)./(1-m(idx_neg)); - a = ones(length(idx),1); - b = sqrt(1.0-m(idx)); - c = sqrt(m(idx)); - f = 0.5; - sum = f*c.*c; - for n = 2:Nmax - t = (a+b)/2; - c = (a-b)/2; - b = sqrt(a.*b); - a = t; - f = f * 2; - sum = sum + f*c.*c; - if all(c./a < eps), break; endif - endfor - if n >= Nmax, error("ellipke: not enough workspace"); endif - k(idx) = 0.5*pi./a; - e(idx) = 0.5*pi.*(1.0-sum)./a; - k(idx_neg) = mult_k.*k(idx_neg); - e(idx_neg) = mult_e.*e(idx_neg); - endif - -endfunction - -## Test complete elliptic functions of first and second kind -## against "exact" solution from Mathematica 3.0 -%!test -%! m = [0.0; 0.01; 0.1; 0.5; 0.9; 0.99; 1.0 ]; -%! [k,e] = ellipke(m); -%! -%! # K(1.0) is really infinity - see below -%! K = [ -%! 1.5707963267948966192; -%! 1.5747455615173559527; -%! 1.6124413487202193982; -%! 1.8540746773013719184; -%! 2.5780921133481731882; -%! 3.6956373629898746778; -%! 0.0 ]; -%! E = [ -%! 1.5707963267948966192; -%! 1.5668619420216682912; -%! 1.5307576368977632025; -%! 1.3506438810476755025; -%! 1.1047747327040733261; -%! 1.0159935450252239356; -%! 1.0 ]; -%! if k(7)==Inf, k(7)=0.0; endif; -%! assert(K,k,8*eps); -%! assert(E,e,8*eps);
--- a/main/specfun/inst/expint.m Mon Apr 29 12:53:23 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,34 +0,0 @@ -## Copyright (C) 2006 Sylvain Pelissier <sylvain.pelissier@gmail.com> -## -## This program is free software; you can redistribute it and/or modify it under -## the terms of the GNU General Public License as published by the Free Software -## Foundation; either version 3 of the License, or (at your option) any later -## version. -## -## This program is distributed in the hope that it will be useful, but WITHOUT -## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or -## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more -## details. -## -## You should have received a copy of the GNU General Public License along with -## this program; if not, see <http://www.gnu.org/licenses/>. - -## -*- texinfo -*- -## @deftypefn {Function File} {@var{y} =} expint (@var{x}) -## Compute the exponential integral, -## @verbatim -## infinity -## / -## expint(x) = | exp(t)/t dt -## / -## x -## @end verbatim -## @seealso{expint_E1, expint_Ei} -## @end deftypefn - -function y = expint(x) - if (nargin != 1) - print_usage; - endif - y = expint_E1(x); -endfunction
--- a/main/specfun/src/.svnignore Mon Apr 29 12:53:23 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -PKG_ADD -*.octlink -*.oct
--- a/main/specfun/src/Makefile Mon Apr 29 12:53:23 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,6 +0,0 @@ -all: ellipj.oct - -%.oct: %.cc - mkoctfile -s $< - -clean: ; -rm *.o core octave-core *.oct *~
--- a/main/specfun/src/ellipj.cc Mon Apr 29 12:53:23 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,909 +0,0 @@ -/* - Copyright (C) 2001 Leopoldo Cerbaro <redbliss@libero.it> - - This program is free software; you can redistribute it and/or modify - it under the terms of the GNU General Public License as published by - the Free Software Foundation; either version 3 of the License, or - (at your option) any later version. - - This program is distributed in the hope that it will be useful, - but WITHOUT ANY WARRANTY; without even the implied warranty of - MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - GNU General Public License for more details. - - You should have received a copy of the GNU General Public License - along with this program; If not, see <http://www.gnu.org/licenses/>. - - Compute the Jacobi elliptic functions sn(u|m), cn(u|m) and dn(u|m) for - argument u (real or complex) and parameter m. - - usage: [sn,cn,dn] = ellipj(u,m[,tol]) - - u and can be complex. - m is restricted to 0 <= m <= 1. - They can be scalars, matrix and scalar, scalar and matrix, - column and row, conformant matrices. - - modified so u can be complex. Leopoldo Cerbaro redbliss@libero.it - - Ref: Abramowitz, Milton and Stegun, Irene A - Handbook of Mathematical Functions, Dover, 1965 - Chapter 16 (Sections 16.4, 16.13 and 16.15) - - Based upon ellipj.m made by David Billinghurst <David.Billinghurst@riotinto.com> - and besselj.cc - - Author: Leopoldo Cerbaro <redbliss@libero.it> - Created: 15 December 2001 -*/ - -#include "oct.h" -#include "lo-ieee.h" /* for octave_NaN */ - -static void -gripe_ellipj_arg ( const char *arg) -{ - error ("ellipj: expecting scalar or matrix as %s argument", arg); -} - -const double eps = 2.220446049e-16; -const int Nmax = 16; - -static void -sncndn ( double u, double m, double& sn, double& cn, double& dn, double& err) { -/* real */ -double sqrt_eps, m1, t=0., si_u, co_u, se_u, ta_u, b, c[Nmax], a[Nmax], phi; -int n, Nn, ii; - - if (m < 0. || m > 1.) { - warning ("ellipj: expecting 0. <= m <= 1."); /* -lc- */ - sn = cn = dn = lo_ieee_nan_value (); - return; - } - sqrt_eps = sqrt(eps); - if (m < sqrt_eps) { - /* # For small m, ( Abramowitz and Stegun, Section 16.13 ) */ - /*{{{*/ - si_u = sin(u); - co_u = cos(u); - t = 0.25*m*(u-si_u*co_u); - sn = si_u - t * co_u; - cn = co_u + t * si_u; - dn = 1.0 - 0.5*m*si_u*si_u; -/*}}}*/ - } else if ( (1.0 - m) < sqrt_eps ) { - /* For m1 = (1-m) small ( Abramowitz and Stegun, Section 16.15 ) */ - /*{{{*/ - m1 = 1.0-m; - si_u = sinh(u); - co_u = cosh(u); - ta_u = tanh(u); - se_u = 1.0/co_u; - sn = ta_u + 0.25*m1*(si_u*co_u-u)*se_u*se_u; - cn = se_u - 0.25*m1*(si_u*co_u-u)*ta_u*se_u; - dn = se_u + 0.25*m1*(si_u*co_u+u)*ta_u*se_u; -/*}}}*/ - } else { - /*{{{*/ - /* - // Arithmetic-Geometric Mean (AGM) algorithm - // ( Abramowitz and Stegun, Section 16.4 ) - */ - - a[0] = 1.0; - b = sqrt(1.0-m); - c[0] = sqrt(m); - for (n = 1; n<Nmax; ++n) { - a[n] = (a[n-1]+b)/2; - c[n] = (a[n-1]-b)/2; - b = sqrt(a[n-1]*b); - if ( c[n]/a[n] < eps) break; - } - if ( n >= Nmax-1) { - // fprintf(stderr, "Not enough workspace\n"); - err = 1.; - return; - } - Nn = n; - for ( ii = 1; n>0; ii = ii*2, --n) ; // pow(2, Nn) - phi = ii*a[Nn]*u; - for ( n = Nn; n > 0; --n) { - t = phi; - phi = (asin((c[n]/a[n])* sin(phi))+phi)/2.; - } - sn = sin(phi); - cn = cos(phi); - dn = cn/cos(t-phi); -/*}}}*/ - } - return; -} - -static void -sncndn ( Complex& u, double m, - Complex& sn, Complex& cn, Complex& dn, double& err) { -double m1 = 1.-m, ss1, cc1, dd1; - - sncndn( imag(u), m1, ss1, cc1, dd1, err); - if ( real(u) == 0.) { - /* u is pure imag: Jacoby imag. transf. */ - /*{{{*/ - sn = Complex (0. , ss1/cc1); - cn = 1/cc1; // cn.imag = 0.; - dn = dd1/cc1; // dn.imag = 0.; - /*}}}*/ - } else { - /* u is generic complex */ - /*{{{*/ - double ss, cc, dd, ddd; - - sncndn( real(u), m, ss, cc, dd, err); - ddd = cc1*cc1 + m*ss*ss*ss1*ss1; - sn = Complex (ss*dd1/ddd, cc*dd*ss1*cc1/ddd); - cn = Complex (cc*cc1/ddd, -ss*dd*ss1*dd1/ddd); - dn = Complex (dd*cc1*dd1/ddd, -m*ss*cc*ss1/ddd); - /*}}}*/ - } - return; -} - -DEFUN_DLD (ellipj, args, nargout, - "-*- texinfo -*-\n\ -@deftypefn {Loadable Function} {[@var{sn}, @var{cn}, @var{dn}] =} \ -ellipj (@var{u}, @var{m}, @var{err})\n\ -Compute the Jacobi elliptic functions sn, cn, dn of complex argument and real parameter.\n\ -\n\ -If @var{m} is a scalar, the results are the same size as @var{u}.\n\ -If @var{u} is a scalar, the results are the same size as @var{m}.\n\ -If @var{u} is a column vector and @var{m} is a row vector, the\n\ -results are matrices with @code{length (@var{u})} rows and\n\ -@code{length (@var{m})} columns. Otherwise, @var{u} and\n\ -@var{m} must conform and the results will be the same size.\n\ -\n\ -The value of @var{u} may be complex.\n\ -The value of @var{m} must be 0 <= m <= 1. .\n\ -\n\ -If requested, @var{err} contains the following status information\n\ -and is the same size as the result.\n\ -\n\ -@enumerate 0\n\ -@item\n\ -Normal return.\n\ -@item\n\ -Error---no computation, algorithm termination condition not met,\n\ -return @code{NaN}.\n\ -@end enumerate\n\ -@end deftypefn") -{ - octave_value_list retval; - - int nargin = args.length (); - - if (nargin == 2 ) { - octave_value u_arg = args(0); - octave_value m_arg = args(1); - - if (m_arg.is_scalar_type ()) { // m is scalar - double m = args(1).double_value (); - - if (! error_state) { - - if (u_arg.is_scalar_type ()) { /* u scalar */ - /*{{{*/ - if (u_arg.is_real_type ()) { // u real - double u = args(0).double_value (); - - if (! error_state) { - double sn, cn, dn; - double err=0; - octave_value result; - - sncndn(u, m, sn, cn, dn, err); - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) - retval(3) = err; - } else - gripe_ellipj_arg ( "first"); - - } else { // u complex - Complex u = u_arg.complex_value (); - - if (! error_state) { - Complex sn, cn, dn; - double err; - octave_value result; - - sncndn( u, m, sn, cn, dn, err); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - gripe_ellipj_arg ( "second"); - } - /*}}}*/ - } else { /* u is matrix ( m is scalar ) */ - /*{{{*/ - ComplexMatrix u = u_arg.complex_matrix_value (); - - if (! error_state) { - octave_value result; - int nr = u.rows (); - int nc = u.cols (); - - ComplexMatrix sn (nr, nc), cn (nr, nc), dn (nr, nc); - Matrix err (nr, nc); - - for (int j = 0; j < nc; j++) - for (int i = 0; i < nr; i++) - sncndn (u(i,j), m, sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - gripe_ellipj_arg ( "first"); - /*}}}*/ - } - } else - gripe_ellipj_arg ( "second"); - } else { // m is matrix - Matrix m = args(1).matrix_value (); - - if (! error_state) { - int mr = m.rows (); - int mc = m.cols (); - - if (u_arg.is_scalar_type ()) { /* u is scalar */ - /*{{{*/ - octave_value result; - int nr = m.rows (); - int nc = m.cols (); - Matrix err (nr, nc); - - if (u_arg.is_real_type ()) { - double u = u_arg.double_value (); - Matrix sn (nr, nc), cn (nr, nc), dn (nr, nc); - if (! error_state) { - for (int j = 0; j < nc; j++) - for (int i = 0; i < nr; i++) - sncndn (u, m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - gripe_ellipj_arg ( "first"); - } else { - Complex u = u_arg.complex_value (); - ComplexMatrix sn (nr, nc), cn (nr, nc), dn (nr, nc); - if (! error_state) { - for (int j = 0; j < nc; j++) - for (int i = 0; i < nr; i++) - sncndn (u, m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - gripe_ellipj_arg ( "first"); - } - /*}}}*/ - } else { // u is matrix (m is matrix) - /*{{{*/ - if (u_arg.is_real_type ()) { // u real matrix - - Matrix u = u_arg.matrix_value (); - if (! error_state) { - int ur = u.rows (); - int uc = u.cols (); - - if (mr == 1 && uc == 1) { // u column, m row - RowVector rm = m.row ((octave_idx_type)0); - ColumnVector cu = u.column ((octave_idx_type)0); - - Matrix sn (ur, mc), cn (ur, mc), dn (ur, mc); - Matrix err(ur,mc); -// octave_value result; - - for (int j = 0; j < mc; j++) - for (int i = 0; i < ur; i++) - sncndn (cu(i), rm(j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else if (ur == mr && uc == mc) { - Matrix sn (ur, mc), cn (ur, mc), dn (ur, mc); - Matrix err(ur,mc); -// octave_value result; - - for (int j = 0; j < uc; j++) - for (int i = 0; i < ur; i++) - sncndn (u(i,j), m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - error("u m invalid"); - } else - gripe_ellipj_arg ( "first "); - } else { // u complex matrix - ComplexMatrix u = u_arg.complex_matrix_value (); - if (! error_state) { - int ur = u.rows (); - int uc = u.cols (); - - if (mr == 1 && uc == 1) { - RowVector rm = m.row ((octave_idx_type)0); - ComplexColumnVector cu = u.column ((octave_idx_type)0); - - ComplexMatrix sn (ur, mc), cn (ur, mc), dn (ur, mc); - Matrix err(ur,mc); -// octave_value result; - - for (int j = 0; j < mc; j++) - for (int i = 0; i < ur; i++) - sncndn (cu(i), rm(j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else if (ur == mr && uc == mc) { - - ComplexMatrix sn (ur, mc), cn (ur, mc), dn (ur, mc); - Matrix err(ur,mc); -// octave_value result; - - for (int j = 0; j < uc; j++) - for (int i = 0; i < ur; i++) - sncndn (u(i,j), m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j)); - - retval (0) = sn; - retval (1) = cn; - retval (2) = dn; - if (nargout > 3) retval(3) = err; - } else - error("u m invalid"); - } else - gripe_ellipj_arg ( "second"); - } - /*}}}*/ - } - } else - gripe_ellipj_arg ( "second"); - } // m matrix - } else // wrong n. of argin - print_usage (); - return retval; -} - -/* -## demos taken from inst/ellipj.m - -%!demo -%! N = 150; -%! % m = [1-logspace(0,log(eps),N-1), 1]; ## m near 1 -%! % m = [0, logspace(log(eps),0,N-1)]; ## m near 0 -%! m = linspace(0,1,N); ## m equally spaced -%! u = linspace(-20,20,N); -%! M = ones(length(u),1) * m; -%! U = u' * ones(1, length(m)); -%! [sn, cn, dn] = ellipj(U,M); -%! -%! %% Plotting -%! figure(2) -%! c = colormap(hot(64)); -%! data = {sn,cn,dn}; -%! dname = {"sn","cn","dn"}; -%! for i=1:3 -%! subplot(1,3,i); -%! image(m,u,32*clip(data{i},[-1,1])+32); # clip function belongs to audio package -%! title(dname{i}); -%! end -%! colormap(c); - -%!demo -%! N = 200; -%! % m = [1-logspace(0,log(eps),N-1), 1]; ## m near 1 -%! % m = [0, logspace(log(eps),0,N-1)]; ## m near 0 -%! m = linspace(0,1,N); ## m equally spaced -%! u = linspace(0,20,5); -%! M = ones(length(u),1) * m; -%! U = u' * ones(1, length(m)); -%! [sn, cn, dn] = ellipj(U,M); -%! -%! %% Plotting -%! data = {sn,cn,dn}; -%! dname = {"sn","cn","dn"}; -%! for i=1:3 -%! subplot(1,3,i); -%! plot(m, data{i}); -%! title(dname{i}); -%! grid on; -%! end -*/ - -/* -## tests taken from inst/test_sncndn.m - -%!test -%! k = (tan(pi/8.))^2; m = k*k; -%! SN = [ -%! -1. + I * 0. , -0.8392965923 + 0. * I -%! -1. + I * 0.2 , -0.8559363407 + 0.108250955 * I -%! -1. + I * 0.4 , -0.906529758 + 0.2204040232 * I -%! -1. + I * 0.6 , -0.9931306727 + 0.3403783409 * I -%! -1. + I * 0.8 , -1.119268095 + 0.4720784944 * I -%! -1. + I * 1. , -1.29010951 + 0.6192468708 * I -%! -1. + I * 1.2 , -1.512691987 + 0.7850890595 * I -%! -1. + I * 1.4 , -1.796200374 + 0.9714821804 * I -%! -1. + I * 1.6 , -2.152201882 + 1.177446413 * I -%! -1. + I * 1.8 , -2.594547417 + 1.396378892 * I -%! -1. + I * 2. , -3.138145339 + 1.611394819 * I -%! -0.8 + I * 0. , -0.7158157937 + 0. * I -%! -0.8 + I * 0.2 , -0.7301746722 + 0.1394690862 * I -%! -0.8 + I * 0.4 , -0.7738940898 + 0.2841710966 * I -%! -0.8 + I * 0.6 , -0.8489542135 + 0.4394411376 * I -%! -0.8 + I * 0.8 , -0.9588386397 + 0.6107824358 * I -%! -0.8 + I * 1. , -1.108848724 + 0.8038415767 * I -%! -0.8 + I * 1.2 , -1.306629972 + 1.024193359 * I -%! -0.8 + I * 1.4 , -1.563010199 + 1.276740951 * I -%! -0.8 + I * 1.6 , -1.893274688 + 1.564345558 * I -%! -0.8 + I * 1.8 , -2.318944084 + 1.88491973 * I -%! -0.8 + I * 2. , -2.869716809 + 2.225506523 * I -%! -0.6 + I * 0. , -0.5638287208 + 0. * I -%! -0.6 + I * 0.2 , -0.5752723012 + 0.1654722474 * I -%! -0.6 + I * 0.4 , -0.610164314 + 0.3374004736 * I -%! -0.6 + I * 0.6 , -0.6702507087 + 0.5224614298 * I -%! -0.6 + I * 0.8 , -0.7586657365 + 0.7277663879 * I -%! -0.6 + I * 1. , -0.8803349115 + 0.9610513652 * I -%! -0.6 + I * 1.2 , -1.042696526 + 1.230800819 * I -%! -0.6 + I * 1.4 , -1.256964505 + 1.546195843 * I -%! -0.6 + I * 1.6 , -1.540333527 + 1.916612621 * I -%! -0.6 + I * 1.8 , -1.919816065 + 2.349972151 * I -%! -0.6 + I * 2. , -2.438761841 + 2.848129496 * I -%! -0.4 + I * 0. , -0.3891382858 + 0. * I -%! -0.4 + I * 0.2 , -0.3971152026 + 0.1850563793 * I -%! -0.4 + I * 0.4 , -0.4214662882 + 0.3775700801 * I -%! -0.4 + I * 0.6 , -0.4635087491 + 0.5853434119 * I -%! -0.4 + I * 0.8 , -0.5256432877 + 0.8168992398 * I -%! -0.4 + I * 1. , -0.611733177 + 1.081923504 * I -%! -0.4 + I * 1.2 , -0.7278102331 + 1.391822501 * I -%! -0.4 + I * 1.4 , -0.8833807998 + 1.760456461 * I -%! -0.4 + I * 1.6 , -1.093891878 + 2.205107766 * I -%! -0.4 + I * 1.8 , -1.385545188 + 2.747638761 * I -%! -0.4 + I * 2. , -1.805081271 + 3.41525351 * I -%! -0.2 + I * 0. , -0.1986311721 + 0. * I -%! -0.2 + I * 0.2 , -0.2027299916 + 0.1972398665 * I -%! -0.2 + I * 0.4 , -0.2152524522 + 0.402598347 * I -%! -0.2 + I * 0.6 , -0.2369100139 + 0.6246336356 * I -%! -0.2 + I * 0.8 , -0.2690115146 + 0.8728455227 * I -%! -0.2 + I * 1. , -0.3136938773 + 1.158323088 * I -%! -0.2 + I * 1.2 , -0.3743615191 + 1.494672508 * I -%! -0.2 + I * 1.4 , -0.4565255082 + 1.899466033 * I -%! -0.2 + I * 1.6 , -0.5694611346 + 2.39667232 * I -%! -0.2 + I * 1.8 , -0.7296612675 + 3.020990664 * I -%! -0.2 + I * 2. , -0.9685726188 + 3.826022536 * I -%! 0. + I * 0. , 0. + 0. * I -%! 0. + I * 0.2 , 0. + 0.201376364 * I -%! 0. + I * 0.4 , 0. + 0.4111029248 * I -%! 0. + I * 0.6 , 0. + 0.6380048435 * I -%! 0. + I * 0.8 , 0. + 0.8919321473 * I -%! 0. + I * 1. , 0. + 1.184486615 * I -%! 0. + I * 1.2 , 0. + 1.530096023 * I -%! 0. + I * 1.4 , 0. + 1.947754612 * I -%! 0. + I * 1.6 , 0. + 2.464074356 * I -%! 0. + I * 1.8 , 0. + 3.119049475 * I -%! 0. + I * 2. , 0. + 3.97786237 * I -%! 0.2 + I * 0. , 0.1986311721 + 0. * I -%! 0.2 + I * 0.2 , 0.2027299916 + 0.1972398665 * I -%! 0.2 + I * 0.4 , 0.2152524522 + 0.402598347 * I -%! 0.2 + I * 0.6 , 0.2369100139 + 0.6246336356 * I -%! 0.2 + I * 0.8 , 0.2690115146 + 0.8728455227 * I -%! 0.2 + I * 1. , 0.3136938773 + 1.158323088 * I -%! 0.2 + I * 1.2 , 0.3743615191 + 1.494672508 * I -%! 0.2 + I * 1.4 , 0.4565255082 + 1.899466033 * I -%! 0.2 + I * 1.6 , 0.5694611346 + 2.39667232 * I -%! 0.2 + I * 1.8 , 0.7296612675 + 3.020990664 * I -%! 0.2 + I * 2. , 0.9685726188 + 3.826022536 * I -%! 0.4 + I * 0. , 0.3891382858 + 0. * I -%! 0.4 + I * 0.2 , 0.3971152026 + 0.1850563793 * I -%! 0.4 + I * 0.4 , 0.4214662882 + 0.3775700801 * I -%! 0.4 + I * 0.6 , 0.4635087491 + 0.5853434119 * I -%! 0.4 + I * 0.8 , 0.5256432877 + 0.8168992398 * I -%! 0.4 + I * 1. , 0.611733177 + 1.081923504 * I -%! 0.4 + I * 1.2 , 0.7278102331 + 1.391822501 * I -%! 0.4 + I * 1.4 , 0.8833807998 + 1.760456461 * I -%! 0.4 + I * 1.6 , 1.093891878 + 2.205107766 * I -%! 0.4 + I * 1.8 , 1.385545188 + 2.747638761 * I -%! 0.4 + I * 2. , 1.805081271 + 3.41525351 * I -%! 0.6 + I * 0. , 0.5638287208 + 0. * I -%! 0.6 + I * 0.2 , 0.5752723012 + 0.1654722474 * I -%! 0.6 + I * 0.4 , 0.610164314 + 0.3374004736 * I -%! 0.6 + I * 0.6 , 0.6702507087 + 0.5224614298 * I -%! 0.6 + I * 0.8 , 0.7586657365 + 0.7277663879 * I -%! 0.6 + I * 1. , 0.8803349115 + 0.9610513652 * I -%! 0.6 + I * 1.2 , 1.042696526 + 1.230800819 * I -%! 0.6 + I * 1.4 , 1.256964505 + 1.546195843 * I -%! 0.6 + I * 1.6 , 1.540333527 + 1.916612621 * I -%! 0.6 + I * 1.8 , 1.919816065 + 2.349972151 * I -%! 0.6 + I * 2. , 2.438761841 + 2.848129496 * I -%! 0.8 + I * 0. , 0.7158157937 + 0. * I -%! 0.8 + I * 0.2 , 0.7301746722 + 0.1394690862 * I -%! 0.8 + I * 0.4 , 0.7738940898 + 0.2841710966 * I -%! 0.8 + I * 0.6 , 0.8489542135 + 0.4394411376 * I -%! 0.8 + I * 0.8 , 0.9588386397 + 0.6107824358 * I -%! 0.8 + I * 1. , 1.108848724 + 0.8038415767 * I -%! 0.8 + I * 1.2 , 1.306629972 + 1.024193359 * I -%! 0.8 + I * 1.4 , 1.563010199 + 1.276740951 * I -%! 0.8 + I * 1.6 , 1.893274688 + 1.564345558 * I -%! 0.8 + I * 1.8 , 2.318944084 + 1.88491973 * I -%! 0.8 + I * 2. , 2.869716809 + 2.225506523 * I -%! 1. + I * 0. , 0.8392965923 + 0. * I -%! 1. + I * 0.2 , 0.8559363407 + 0.108250955 * I -%! 1. + I * 0.4 , 0.906529758 + 0.2204040232 * I -%! 1. + I * 0.6 , 0.9931306727 + 0.3403783409 * I -%! 1. + I * 0.8 , 1.119268095 + 0.4720784944 * I -%! 1. + I * 1. , 1.29010951 + 0.6192468708 * I -%! 1. + I * 1.2 , 1.512691987 + 0.7850890595 * I -%! 1. + I * 1.4 , 1.796200374 + 0.9714821804 * I -%! 1. + I * 1.6 , 2.152201882 + 1.177446413 * I -%! 1. + I * 1.8 , 2.594547417 + 1.396378892 * I -%! 1. + I * 2. , 3.138145339 + 1.611394819 * I -%! ]; -%! CN = [ -%! -1. + I * 0. , 0.5436738271 + 0. * I -%! -1. + I * 0.2 , 0.5541219664 + 0.1672121517 * I -%! -1. + I * 0.4 , 0.5857703552 + 0.3410940893 * I -%! -1. + I * 0.6 , 0.6395034233 + 0.5285979063 * I -%! -1. + I * 0.8 , 0.716688504 + 0.7372552987 * I -%! -1. + I * 1. , 0.8189576795 + 0.9755037374 * I -%! -1. + I * 1.2 , 0.9477661951 + 1.253049471 * I -%! -1. + I * 1.4 , 1.103540657 + 1.581252712 * I -%! -1. + I * 1.6 , 1.284098214 + 1.973449038 * I -%! -1. + I * 1.8 , 1.481835651 + 2.4449211 * I -%! -1. + I * 2. , 1.679032464 + 3.011729224 * I -%! -0.8 + I * 0. , 0.6982891589 + 0. * I -%! -0.8 + I * 0.2 , 0.71187169 + 0.1430549855 * I -%! -0.8 + I * 0.4 , 0.7530744458 + 0.2920273465 * I -%! -0.8 + I * 0.6 , 0.8232501212 + 0.4531616768 * I -%! -0.8 + I * 0.8 , 0.9245978896 + 0.6334016187 * I -%! -0.8 + I * 1. , 1.060030206 + 0.8408616109 * I -%! -0.8 + I * 1.2 , 1.232861756 + 1.085475913 * I -%! -0.8 + I * 1.4 , 1.446126965 + 1.379933558 * I -%! -0.8 + I * 1.6 , 1.701139468 + 1.741030588 * I -%! -0.8 + I * 1.8 , 1.994526268 + 2.191509596 * I -%! -0.8 + I * 2. , 2.312257188 + 2.762051518 * I -%! -0.6 + I * 0. , 0.8258917445 + 0. * I -%! -0.6 + I * 0.2 , 0.842151698 + 0.1130337928 * I -%! -0.6 + I * 0.4 , 0.8915487431 + 0.2309124769 * I -%! -0.6 + I * 0.6 , 0.975948103 + 0.3588102098 * I -%! -0.6 + I * 0.8 , 1.098499209 + 0.5026234141 * I -%! -0.6 + I * 1. , 1.263676101 + 0.6695125973 * I -%! -0.6 + I * 1.2 , 1.477275851 + 0.8687285705 * I -%! -0.6 + I * 1.4 , 1.746262523 + 1.112955966 * I -%! -0.6 + I * 1.6 , 2.078179075 + 1.420581466 * I -%! -0.6 + I * 1.8 , 2.479425208 + 1.819580713 * I -%! -0.6 + I * 2. , 2.950586798 + 2.354077344 * I -%! -0.4 + I * 0. , 0.9211793498 + 0. * I -%! -0.4 + I * 0.2 , 0.9395019377 + 0.07822091534 * I -%! -0.4 + I * 0.4 , 0.9952345231 + 0.1598950363 * I -%! -0.4 + I * 0.6 , 1.090715991 + 0.2487465067 * I -%! -0.4 + I * 0.8 , 1.229998843 + 0.34910407 * I -%! -0.4 + I * 1. , 1.419103868 + 0.4663848201 * I -%! -0.4 + I * 1.2 , 1.666426377 + 0.607877235 * I -%! -0.4 + I * 1.4 , 1.983347336 + 0.7841054404 * I -%! -0.4 + I * 1.6 , 2.385101684 + 1.01134031 * I -%! -0.4 + I * 1.8 , 2.89185416 + 1.316448705 * I -%! -0.4 + I * 2. , 3.529393374 + 1.74670531 * I -%! -0.2 + I * 0. , 0.9800743122 + 0. * I -%! -0.2 + I * 0.2 , 0.9997019476 + 0.03999835809 * I -%! -0.2 + I * 0.4 , 1.059453907 + 0.08179712295 * I -%! -0.2 + I * 0.6 , 1.16200643 + 0.1273503824 * I -%! -0.2 + I * 0.8 , 1.312066413 + 0.1789585449 * I -%! -0.2 + I * 1. , 1.516804331 + 0.2395555269 * I -%! -0.2 + I * 1.2 , 1.786613221 + 0.313189147 * I -%! -0.2 + I * 1.4 , 2.136422971 + 0.405890925 * I -%! -0.2 + I * 1.6 , 2.588021972 + 0.527357091 * I -%! -0.2 + I * 1.8 , 3.174302819 + 0.6944201617 * I -%! -0.2 + I * 2. , 3.947361147 + 0.9387994989 * I -%! 0. + I * 0. , 1. + 0. * I -%! 0. + I * 0.2 , 1.020074723 + 0. * I -%! 0. + I * 0.4 , 1.08120563 + 0. * I -%! 0. + I * 0.6 , 1.18619146 + 0. * I -%! 0. + I * 0.8 , 1.339978715 + 0. * I -%! 0. + I * 1. , 1.550164037 + 0. * I -%! 0. + I * 1.2 , 1.827893279 + 0. * I -%! 0. + I * 1.4 , 2.189462954 + 0. * I -%! 0. + I * 1.6 , 2.659259752 + 0. * I -%! 0. + I * 1.8 , 3.275434266 + 0. * I -%! 0. + I * 2. , 4.101632484 + 0. * I -%! 0.2 + I * 0. , 0.9800743122 + 0. * I -%! 0.2 + I * 0.2 , 0.9997019476 - 0.03999835809 * I -%! 0.2 + I * 0.4 , 1.059453907 - 0.08179712295 * I -%! 0.2 + I * 0.6 , 1.16200643 - 0.1273503824 * I -%! 0.2 + I * 0.8 , 1.312066413 - 0.1789585449 * I -%! 0.2 + I * 1. , 1.516804331 - 0.2395555269 * I -%! 0.2 + I * 1.2 , 1.786613221 - 0.313189147 * I -%! 0.2 + I * 1.4 , 2.136422971 - 0.405890925 * I -%! 0.2 + I * 1.6 , 2.588021972 - 0.527357091 * I -%! 0.2 + I * 1.8 , 3.174302819 - 0.6944201617 * I -%! 0.2 + I * 2. , 3.947361147 - 0.9387994989 * I -%! 0.4 + I * 0. , 0.9211793498 + 0. * I -%! 0.4 + I * 0.2 , 0.9395019377 - 0.07822091534 * I -%! 0.4 + I * 0.4 , 0.9952345231 - 0.1598950363 * I -%! 0.4 + I * 0.6 , 1.090715991 - 0.2487465067 * I -%! 0.4 + I * 0.8 , 1.229998843 - 0.34910407 * I -%! 0.4 + I * 1. , 1.419103868 - 0.4663848201 * I -%! 0.4 + I * 1.2 , 1.666426377 - 0.607877235 * I -%! 0.4 + I * 1.4 , 1.983347336 - 0.7841054404 * I -%! 0.4 + I * 1.6 , 2.385101684 - 1.01134031 * I -%! 0.4 + I * 1.8 , 2.89185416 - 1.316448705 * I -%! 0.4 + I * 2. , 3.529393374 - 1.74670531 * I -%! 0.6 + I * 0. , 0.8258917445 + 0. * I -%! 0.6 + I * 0.2 , 0.842151698 - 0.1130337928 * I -%! 0.6 + I * 0.4 , 0.8915487431 - 0.2309124769 * I -%! 0.6 + I * 0.6 , 0.975948103 - 0.3588102098 * I -%! 0.6 + I * 0.8 , 1.098499209 - 0.5026234141 * I -%! 0.6 + I * 1. , 1.263676101 - 0.6695125973 * I -%! 0.6 + I * 1.2 , 1.477275851 - 0.8687285705 * I -%! 0.6 + I * 1.4 , 1.746262523 - 1.112955966 * I -%! 0.6 + I * 1.6 , 2.078179075 - 1.420581466 * I -%! 0.6 + I * 1.8 , 2.479425208 - 1.819580713 * I -%! 0.6 + I * 2. , 2.950586798 - 2.354077344 * I -%! 0.8 + I * 0. , 0.6982891589 + 0. * I -%! 0.8 + I * 0.2 , 0.71187169 - 0.1430549855 * I -%! 0.8 + I * 0.4 , 0.7530744458 - 0.2920273465 * I -%! 0.8 + I * 0.6 , 0.8232501212 - 0.4531616768 * I -%! 0.8 + I * 0.8 , 0.9245978896 - 0.6334016187 * I -%! 0.8 + I * 1. , 1.060030206 - 0.8408616109 * I -%! 0.8 + I * 1.2 , 1.232861756 - 1.085475913 * I -%! 0.8 + I * 1.4 , 1.446126965 - 1.379933558 * I -%! 0.8 + I * 1.6 , 1.701139468 - 1.741030588 * I -%! 0.8 + I * 1.8 , 1.994526268 - 2.191509596 * I -%! 0.8 + I * 2. , 2.312257188 - 2.762051518 * I -%! 1. + I * 0. , 0.5436738271 + 0. * I -%! 1. + I * 0.2 , 0.5541219664 - 0.1672121517 * I -%! 1. + I * 0.4 , 0.5857703552 - 0.3410940893 * I -%! 1. + I * 0.6 , 0.6395034233 - 0.5285979063 * I -%! 1. + I * 0.8 , 0.716688504 - 0.7372552987 * I -%! 1. + I * 1. , 0.8189576795 - 0.9755037374 * I -%! 1. + I * 1.2 , 0.9477661951 - 1.253049471 * I -%! 1. + I * 1.4 , 1.103540657 - 1.581252712 * I -%! 1. + I * 1.6 , 1.284098214 - 1.973449038 * I -%! 1. + I * 1.8 , 1.481835651 - 2.4449211 * I -%! 1. + I * 2. , 1.679032464 - 3.011729224 * I -%! ]; -%! DN = [ -%! -1. + I * 0. , 0.9895776106 + 0. * I -%! -1. + I * 0.2 , 0.9893361555 + 0.002756935338 * I -%! -1. + I * 0.4 , 0.9885716856 + 0.005949639805 * I -%! -1. + I * 0.6 , 0.9871564855 + 0.01008044183 * I -%! -1. + I * 0.8 , 0.9848512162 + 0.01579337596 * I -%! -1. + I * 1. , 0.9812582484 + 0.02396648455 * I -%! -1. + I * 1.2 , 0.9757399152 + 0.0358288294 * I -%! -1. + I * 1.4 , 0.9672786056 + 0.0531049859 * I -%! -1. + I * 1.6 , 0.954237868 + 0.0781744383 * I -%! -1. + I * 1.8 , 0.933957524 + 0.1141918269 * I -%! -1. + I * 2. , 0.9020917489 + 0.1650142936 * I -%! -0.8 + I * 0. , 0.992429635 + 0. * I -%! -0.8 + I * 0.2 , 0.9924147861 + 0.003020708044 * I -%! -0.8 + I * 0.4 , 0.99236555 + 0.00652359532 * I -%! -0.8 + I * 0.6 , 0.9922655715 + 0.0110676219 * I -%! -0.8 + I * 0.8 , 0.9920785856 + 0.01737733806 * I -%! -0.8 + I * 1. , 0.9917291795 + 0.02645738598 * I -%! -0.8 + I * 1.2 , 0.9910606387 + 0.03974949378 * I -%! -0.8 + I * 1.4 , 0.9897435004 + 0.05935252515 * I -%! -0.8 + I * 1.6 , 0.987077644 + 0.08832675281 * I -%! -0.8 + I * 1.8 , 0.9815667458 + 0.1310872821 * I -%! -0.8 + I * 2. , 0.970020127 + 0.1938136793 * I -%! -0.6 + I * 0. , 0.9953099088 + 0. * I -%! -0.6 + I * 0.2 , 0.995526009 + 0.002814772354 * I -%! -0.6 + I * 0.4 , 0.9962071136 + 0.006083312292 * I -%! -0.6 + I * 0.6 , 0.9974557125 + 0.01033463525 * I -%! -0.6 + I * 0.8 , 0.9994560563 + 0.01626207722 * I -%! -0.6 + I * 1. , 1.00249312 + 0.02484336286 * I -%! -0.6 + I * 1.2 , 1.006973922 + 0.0375167093 * I -%! -0.6 + I * 1.4 , 1.013436509 + 0.05645315628 * I -%! -0.6 + I * 1.6 , 1.022504295 + 0.08499262247 * I -%! -0.6 + I * 1.8 , 1.034670023 + 0.1283564595 * I -%! -0.6 + I * 2. , 1.049599899 + 0.194806122 * I -%! -0.4 + I * 0. , 0.9977686897 + 0. * I -%! -0.4 + I * 0.2 , 0.9981836165 + 0.002167241934 * I -%! -0.4 + I * 0.4 , 0.9994946045 + 0.004686808612 * I -%! -0.4 + I * 0.6 , 1.001910789 + 0.00797144174 * I -%! -0.4 + I * 0.8 , 1.005817375 + 0.01256717724 * I -%! -0.4 + I * 1. , 1.011836374 + 0.01925509038 * I -%! -0.4 + I * 1.2 , 1.020923572 + 0.02920828367 * I -%! -0.4 + I * 1.4 , 1.034513743 + 0.04425213602 * I -%! -0.4 + I * 1.6 , 1.054725746 + 0.06732276244 * I -%! -0.4 + I * 1.8 , 1.08462027 + 0.1033236812 * I -%! -0.4 + I * 2. , 1.128407402 + 0.1608240664 * I -%! -0.2 + I * 0. , 0.9994191176 + 0. * I -%! -0.2 + I * 0.2 , 0.9999683719 + 0.001177128019 * I -%! -0.2 + I * 0.4 , 1.001705496 + 0.00254669712 * I -%! -0.2 + I * 0.6 , 1.004913944 + 0.004334880912 * I -%! -0.2 + I * 0.8 , 1.010120575 + 0.006842775622 * I -%! -0.2 + I * 1. , 1.018189543 + 0.01050520136 * I -%! -0.2 + I * 1.2 , 1.030482479 + 0.01598431001 * I -%! -0.2 + I * 1.4 , 1.049126108 + 0.02433134655 * I -%! -0.2 + I * 1.6 , 1.077466003 + 0.0372877718 * I -%! -0.2 + I * 1.8 , 1.120863308 + 0.05789156398 * I -%! -0.2 + I * 2. , 1.188162088 + 0.09181238708 * I -%! 0. + I * 0. , 1. + 0. * I -%! 0. + I * 0.2 , 1.000596698 + 0. * I -%! 0. + I * 0.4 , 1.002484444 + 0. * I -%! 0. + I * 0.6 , 1.005973379 + 0. * I -%! 0. + I * 0.8 , 1.011641536 + 0. * I -%! 0. + I * 1. , 1.020441432 + 0. * I -%! 0. + I * 1.2 , 1.033885057 + 0. * I -%! 0. + I * 1.4 , 1.054361188 + 0. * I -%! 0. + I * 1.6 , 1.085694733 + 0. * I -%! 0. + I * 1.8 , 1.134186672 + 0. * I -%! 0. + I * 2. , 1.210701071 + 0. * I -%! 0.2 + I * 0. , 0.9994191176 + 0. * I -%! 0.2 + I * 0.2 , 0.9999683719 - 0.001177128019 * I -%! 0.2 + I * 0.4 , 1.001705496 - 0.00254669712 * I -%! 0.2 + I * 0.6 , 1.004913944 - 0.004334880912 * I -%! 0.2 + I * 0.8 , 1.010120575 - 0.006842775622 * I -%! 0.2 + I * 1. , 1.018189543 - 0.01050520136 * I -%! 0.2 + I * 1.2 , 1.030482479 - 0.01598431001 * I -%! 0.2 + I * 1.4 , 1.049126108 - 0.02433134655 * I -%! 0.2 + I * 1.6 , 1.077466003 - 0.0372877718 * I -%! 0.2 + I * 1.8 , 1.120863308 - 0.05789156398 * I -%! 0.2 + I * 2. , 1.188162088 - 0.09181238708 * I -%! 0.4 + I * 0. , 0.9977686897 + 0. * I -%! 0.4 + I * 0.2 , 0.9981836165 - 0.002167241934 * I -%! 0.4 + I * 0.4 , 0.9994946045 - 0.004686808612 * I -%! 0.4 + I * 0.6 , 1.001910789 - 0.00797144174 * I -%! 0.4 + I * 0.8 , 1.005817375 - 0.01256717724 * I -%! 0.4 + I * 1. , 1.011836374 - 0.01925509038 * I -%! 0.4 + I * 1.2 , 1.020923572 - 0.02920828367 * I -%! 0.4 + I * 1.4 , 1.034513743 - 0.04425213602 * I -%! 0.4 + I * 1.6 , 1.054725746 - 0.06732276244 * I -%! 0.4 + I * 1.8 , 1.08462027 - 0.1033236812 * I -%! 0.4 + I * 2. , 1.128407402 - 0.1608240664 * I -%! 0.6 + I * 0. , 0.9953099088 + 0. * I -%! 0.6 + I * 0.2 , 0.995526009 - 0.002814772354 * I -%! 0.6 + I * 0.4 , 0.9962071136 - 0.006083312292 * I -%! 0.6 + I * 0.6 , 0.9974557125 - 0.01033463525 * I -%! 0.6 + I * 0.8 , 0.9994560563 - 0.01626207722 * I -%! 0.6 + I * 1. , 1.00249312 - 0.02484336286 * I -%! 0.6 + I * 1.2 , 1.006973922 - 0.0375167093 * I -%! 0.6 + I * 1.4 , 1.013436509 - 0.05645315628 * I -%! 0.6 + I * 1.6 , 1.022504295 - 0.08499262247 * I -%! 0.6 + I * 1.8 , 1.034670023 - 0.1283564595 * I -%! 0.6 + I * 2. , 1.049599899 - 0.194806122 * I -%! 0.8 + I * 0. , 0.992429635 + 0. * I -%! 0.8 + I * 0.2 , 0.9924147861 - 0.003020708044 * I -%! 0.8 + I * 0.4 , 0.99236555 - 0.00652359532 * I -%! 0.8 + I * 0.6 , 0.9922655715 - 0.0110676219 * I -%! 0.8 + I * 0.8 , 0.9920785856 - 0.01737733806 * I -%! 0.8 + I * 1. , 0.9917291795 - 0.02645738598 * I -%! 0.8 + I * 1.2 , 0.9910606387 - 0.03974949378 * I -%! 0.8 + I * 1.4 , 0.9897435004 - 0.05935252515 * I -%! 0.8 + I * 1.6 , 0.987077644 - 0.08832675281 * I -%! 0.8 + I * 1.8 , 0.9815667458 - 0.1310872821 * I -%! 0.8 + I * 2. , 0.970020127 - 0.1938136793 * I -%! 1. + I * 0. , 0.9895776106 + 0. * I -%! 1. + I * 0.2 , 0.9893361555 - 0.002756935338 * I -%! 1. + I * 0.4 , 0.9885716856 - 0.005949639805 * I -%! 1. + I * 0.6 , 0.9871564855 - 0.01008044183 * I -%! 1. + I * 0.8 , 0.9848512162 - 0.01579337596 * I -%! 1. + I * 1. , 0.9812582484 - 0.02396648455 * I -%! 1. + I * 1.2 , 0.9757399152 - 0.0358288294 * I -%! 1. + I * 1.4 , 0.9672786056 - 0.0531049859 * I -%! 1. + I * 1.6 , 0.954237868 - 0.0781744383 * I -%! 1. + I * 1.8 , 0.933957524 - 0.1141918269 * I -%! 1. + I * 2. , 0.9020917489 - 0.1650142936 * I -%! ]; -%! tol = 1e-9; -%! for x = 0:10 -%! for y = 0:10 -%! ur = -1 + x * 0.2; -%! ui = y * 0.2; -%! ii = 1 + y + x*11; -%! [sn, cn, dn] = ellipj (ur + I * ui, m); -%! assert (SN (ii, 2), sn, tol); -%! assert (CN (ii, 2), cn, tol); -%! assert (DN (ii, 2), dn, tol); -%! endfor -%! endfor - -## tests taken from test_ellipj.m -%!test -%! u1 = pi/3; m1 = 0; -%! res1 = [sin(pi/3), cos(pi/3), 1]; -%! [sn,cn,dn]=ellipj(u1,m1); -%! assert([sn,cn,dn], res1, 10*eps); - -%!test -%! u2 = log(2); m2 = 1; -%! res2 = [ 3/5, 4/5, 4/5 ]; -%! [sn,cn,dn]=ellipj(u2,m2); -%! assert([sn,cn,dn], res2, 10*eps); - -%!test -%! u3 = log(2)*1i; m3 = 0; -%! res3 = [3i/4,5/4,1]; -%! [sn,cn,dn]=ellipj(u3,m3); -%! assert([sn,cn,dn], res3, 10*eps); - -%!test -%! u4 = -1; m4 = tan(pi/8)^4; -%! res4 = [-0.8392965923,0.5436738271,0.9895776106]; -%! [sn,cn,dn]=ellipj(u4, m4); -%! assert([sn,cn,dn], res4, 1e-10); - -%!test -%! u5 = -0.2 + 0.4i; m5 = tan(pi/8)^4; -%! res5 = [ -0.2152524522 + 0.402598347i, ... -%! 1.059453907 + 0.08179712295i, ... -%! 1.001705496 + 0.00254669712i ]; -%! [sn,cn,dn]=ellipj(u5,m5); -%! assert([sn,cn,dn], res5, 1e-9); - -%!test -%! u6 = 0.2 + 0.6i; m6 = tan(pi/8)^4; -%! res6 = [ 0.2369100139 + 0.624633635i, ... -%! 1.16200643 - 0.1273503824i, ... -%! 1.004913944 - 0.004334880912i ]; -%! [sn,cn,dn]=ellipj(u6,m6); -%! assert([sn,cn,dn], res6, 1e-8); - -%!test -%! u7 = 0.8 + 0.8i; m7 = tan(pi/8)^4; -%! res7 = [0.9588386397 + 0.6107824358i, ... -%! 0.9245978896 - 0.6334016187i, ... -%! 0.9920785856 - 0.01737733806i ]; -%! [sn,cn,dn]=ellipj(u7,m7); -%! assert([sn,cn,dn], res7, 1e-10); - -%!test -%! u=[0,pi/6,pi/4,pi/2]; m=0; -%! res = [0,1/2,1/sqrt(2),1;1,cos(pi/6),1/sqrt(2),0;1,1,1,1]; -%! [sn,cn,dn]=ellipj(u,m); -%! assert([sn;cn;dn],res, 100*eps); -%! [sn,cn,dn]=ellipj(u',0); -%! assert([sn,cn,dn],res', 100*eps); - -## XXX FIXME XXX -## need to check [real,complex]x[scalar,rowvec,colvec,matrix]x[u,m] - -%!test -%! ## Test Jacobi elliptic functions -%! ## against "exact" solution from Mathematica 3.0 -%! ## David Billinghurst <David.Billinghurst@riotinto.com> -%! ## 1 February 2001 -%! u = [ 0.25; 0.25; 0.20; 0.20; 0.672; 0.5]; -%! m = [ 0.0; 1.0; 0.19; 0.81; 0.36; 0.9999999999]; -%! S = [ sin(0.25); tanh(0.25); -%! 0.19842311013970879516; -%! 0.19762082367187648571; -%! 0.6095196917919021945; -%! 0.4621171572617320908 ]; -%! C = [ cos(0.25); sech(0.25); -%! 0.9801164570409401062; -%! 0.9802785369736752032; -%! 0.7927709286533560550; -%! 0.8868188839691764094 ]; -%! D = [ 1.0; sech(0.25); -%! 0.9962526643271134302; -%! 0.9840560289645665155; -%! 0.9307281387786906491; -%! 0.8868188839812167635 ]; -%! [sn,cn,dn] = ellipj(u,m); -%! assert(sn,S,8*eps); -%! assert(cn,C,8*eps); -%! assert(dn,D,8*eps); -*/