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changeset 3276:edf6e4852287

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[project @ 1999-10-12 05:49:45 by jwe]
author jwe
date Tue, 12 Oct 1999 05:52:31 +0000
parents b9a024ee0312
children 9a1ac83591c0
files scripts/control/sysmin.m scripts/finance/Makefile.in scripts/general/randperm.m scripts/miscellaneous/pack.m scripts/quaternion/quaternion.ps scripts/specfun/bessel.m
diffstat 6 files changed, 9049 insertions(+), 0 deletions(-) [+]
line wrap: on
line diff
new file mode 100644
--- /dev/null
+++ b/scripts/control/sysmin.m
@@ -0,0 +1,80 @@
+function [retsys,nc,no,cflg,oflg] = sysmin(sys,flg);
+  # [retsys,nc,no] = sysmin(sys{,flg});
+  # return a minimal (or reduced order) system
+  # inputs:
+  #   sys: system data structure
+  #   flg: 0 [default] return minimal system; state names lost
+  #      : 1           return system with physical states removed that
+  #                    are either uncontrollable or unobservable
+  #                    (cannot reduce further without discarding physical
+  #                    meaning of states)
+  # outputs:
+  #   retsys: returned system
+  #   nc: number of controllable states in the returned system
+  #   no: number of observable states in the returned system
+  #   cflg: is_controllable(retsys)
+  #   oflg: is_observable(retsys)
+  
+  switch(nargin)
+  case(1), flg = 0;
+  case(2), jnk = flg;    # dummy operation
+  otherwise,
+    usage("[retsys,nc,no] = sysmin(sys{,flg})");
+  endswitch
+  dflg = is_digital(sys,flg);
+  Ts = sysgettsam(sys);
+  switch(flg)
+  case(0),
+    # reduce to a minimal system
+    [aa,bb,cc,dd] = sys2ss(sys);
+    [cflg,Uc] = is_controllable(aa,bb); 
+    if(!cflg)
+      # reduce to controllable states
+      if(!isempty(Uc))
+        aa = Uc'*aa*Uc;
+        bb = Uc'*bb;
+        cc = cc*Uc;
+      else
+        aa = bb = cc = [];
+      endif
+    endif
+    if(!isempty(aa))
+      [oflg,Uo] = is_observable(aa,cc);
+      if(!oflg)
+        if(!isempty(Uo))
+          aa = Uo'*aa*Uo;
+          bb = Uo'*bb;
+          cc = cc*Uo;
+        else
+          aa = bb = cc = [];
+        endif
+      endif
+    endif
+    switch(dflg)
+    case(0),
+      nc = no = nn = columns(aa);
+      nz = 0;
+    case(1),
+      nc = no = nz = columns(aa);
+      nn = 0;
+    endswitch
+    inname = sysgetsignals(sys,"in");
+    outname= sysgetsignals(sys,"out");
+    retsys = ss2sys(aa,bb,cc,dd,Ts,nn,nz,[],inname,outname);
+  case(1),
+    # reduced model with physical states
+    [cflg,Uc] = is_controllable(sys); xc = find(max(abs(Uc')) != 0);
+    [oflg,Uo] = is_observable(sys);   xo = find(max(abs(Uo')) != 0);
+    xx = intersection(xc,xo);
+    if(isempty(xx)) xx = 0;  endif    # signal no states in reduced model
+    retsys = sysprune(sys,[],[],xx);
+  otherwise,
+    error("illegal value of flg=%d",flg);
+  endswitch
+  if(sysdimensions(retsys,"st") > 0)
+    [cflg,Uc] = is_controllable(retsys); nc = columns(Uc);
+    [oflg,Uo] = is_observable(retsys);   no = columns(Uo);
+  else
+    nc = no = 0;
+  endif
+endfunction
new file mode 100644
--- /dev/null
+++ b/scripts/finance/Makefile.in
@@ -0,0 +1,75 @@
+
+# Makefile for octave's scripts/finance directory
+#
+# John W. Eaton
+# jwe@bevo.che.wisc.edu
+# University of Wisconsin-Madison
+# Department of Chemical Engineering
+
+TOPDIR = ../..
+
+script_sub_dir = finance
+
+srcdir = @srcdir@
+top_srcdir = @top_srcdir@
+VPATH = @srcdir@
+
+include $(TOPDIR)/Makeconf
+
+INSTALL = @INSTALL@
+INSTALL_PROGRAM = @INSTALL_PROGRAM@
+INSTALL_DATA = @INSTALL_DATA@
+
+SOURCES = *.m
+
+DISTFILES = Makefile.in $(SOURCES)
+
+FCN_FILES = $(wildcard $(srcdir)/*.m)
+FCN_FILES_NO_DIR = $(notdir $(FCN_FILES))
+
+BINDISTFILES = $(FCN_FILES)
+
+all:
+.PHONY: all
+
+install install-strip:
+	$(top_srcdir)/mkinstalldirs $(fcnfiledir)/$(script_sub_dir)
+	for f in $(FCN_FILES_NO_DIR); do \
+	  rm -f $(fcnfiledir)/$(script_sub_dir)/$$f; \
+	  $(INSTALL_DATA) $(srcdir)/$$f $(fcnfiledir)/$(script_sub_dir)/$$f; \
+	done
+.PHONY: install install-strip
+
+uninstall:
+	for f in $(FCN_FILES_NO_DIR); \
+	  do rm -f $(fcnfiledir)/$(script_sub_dir)/$$f; \
+	done
+.PHONY: uninstall
+
+clean:
+.PHONY: clean
+
+tags: $(SOURCES)
+	ctags $(SOURCES)
+
+TAGS: $(SOURCES)
+	etags $(SOURCES)
+
+mostlyclean: clean
+.PHONY: mostlyclean
+
+distclean: clean
+	rm -f Makefile
+.PHONY: distclean
+
+maintainer-clean: distclean
+	rm -f tags TAGS
+.PHONY: maintainer-clean
+
+dist:
+	ln $(DISTFILES) ../../`cat ../../.fname`/scripts/finance
+.PHONY: dist
+
+bin-dist:
+	ln $(BINDISTFILES) ../../`cat ../../.fname`/scripts/finance
+.PHONY: bin-dist
new file mode 100644
--- /dev/null
+++ b/scripts/general/randperm.m
@@ -0,0 +1,40 @@
+## Copyright (C) 1998 John W. Eaton
+##
+## This file is part of Octave.
+##
+## Octave 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 2, or (at your option)
+## any later version.
+##
+## Octave 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 Octave; see the file COPYING.  If not, write to the Free
+## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
+## 02111-1307, USA.
+
+## usage: randperm (N)
+##
+## Return a row vector containing a random permutation of the
+## integers from 1 to N.
+
+## Author: "James R. Van Zandt" <jrv@vanzandt.mv.com>
+## Adapted-By: jwe
+
+function retval = randperm (n)
+
+  if (nargin == 1 && is_scalar (n) && floor (n) == n)
+    if (n > 0)
+      [junk, retval] = sort (rand (1, n));
+    else
+      error ("randperm: argument must be positive");
+    endif
+  else
+    usage ("randperm (n)");
+  endif
+
+endfunction
new file mode 100644
--- /dev/null
+++ b/scripts/miscellaneous/pack.m
@@ -0,0 +1,29 @@
+## Copyright (C) 1999 John W. Eaton
+##
+## This file is part of Octave.
+##
+## Octave 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 2, or (at your option)
+## any later version.
+##
+## Octave 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 Octave; see the file COPYING.  If not, write to the Free
+## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
+## 02111-1307, USA.
+
+## usage: pack
+##
+## This function is provided for compatibility with Matlab, but it
+## doesn't actually do anything.
+
+## Author: jwe
+
+function pack ()
+
+endfunction
new file mode 100644
--- /dev/null
+++ b/scripts/quaternion/quaternion.ps
@@ -0,0 +1,8786 @@
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+%DVIPSParameters: dpi=600, compressed, comments removed
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+4559 y(\026)2368 4582 y(\012)h(=)g Fo(!)2612 4596 y Fk(1)2644
+4561 y Fq(^)2651 4582 y Fo(i)12 b Fq(+)f Fo(!)2833 4596
+y Fk(2)2871 4561 y Fq(^)2872 4582 y Fo(j)17 b Fq(+)11
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+Fo(k)29 b Fq(in)c(the)h(\014xed)g(frame.)0 4695 y(\()p
+Fj(qtrans)j Fq(ma)m(y)i(b)s(e)f(used)f(to)i(p)s(erform)e(the)i
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+b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
+Fo(:)p Fq(1)421 b Fn(\(c\))31 b(1998)871 b Fq(13)1829
+824 y Fo(!)p 3 setlinewidth np 1738 974 a 2342 974 li
+st 3 setlinewidth np 2281 990 a 2342 974 li 2281 959
+li st 3 setlinewidth np 1738 974 a 1285 1277 li st 3
+setlinewidth np 1327 1231 a 1285 1277 li 1343 1256 li
+st 3 setlinewidth np 1738 974 a 1738 371 li st 3 setlinewidth
+np 1753 431 a 1738 371 li 1723 431 li st 3 setlinewidth
+np 1738 975 a 1759 961 li st 3 setlinewidth np 1781 946
+a 1802 932 li st 3 setlinewidth np 1824 917 a 1845 903
+li st 3 setlinewidth np 1867 888 a 1888 874 li st 3 setlinewidth
+np 1911 860 a 1932 846 li st 3 setlinewidth np 1954 831
+a 1975 817 li st 3 setlinewidth np 1997 802 a 2018 788
+li st 3 setlinewidth np 2040 773 a 2061 759 li st 3 setlinewidth
+np 2083 745 a 2104 731 li st 3 setlinewidth np 2126 716
+a 2147 702 li st 3 setlinewidth np 2170 687 a 2191 673
+li st 3 setlinewidth np 2149 719 a 2191 673 li 2132 694
+li st 4 setlinewidth np [ 4 18.09 ] 0 setdash 2191 673
+a 2191 1163 li st [] 0 setdash [] 0 setdash 4 setlinewidth
+np [ 4 18.13 ] 0 setdash 2191 1163 a 1738 974 li st [] 0 setdash
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+np 2009 817 82 94.40 237.53 arc st 3 setlinewidth np
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+Fo(t)f Fq(seconds)g(is)g Fo(q)2462 1562 y Fh(r)2525 1548
+y Fq(=)2631 1525 y(\026)2621 1548 y(\012)14 b(sin)o(\()p
+Fo(!)s Fq(\001)p Fo(t=)p Fq(2\))19 b(+)f(cos)q(\()p Fo(!)s
+Fq(\001)p Fo(t=)p Fq(2\).)41 b(F)-8 b(or)0 1661 y(\001)p
+Fo(t)30 b Fq(su\016cien)m(tly)f(small,)g(cos)q(\()p Fo(!)s
+Fq(\001)p Fo(t=)p Fq(2\))d Fl(\031)f Fq(1)31 b(and)f(sin)n(\()p
+Fo(!)s Fq(\001)p Fo(t=)p Fq(2\))c Fl(\031)f Fo(!)s Fq(\001)p
+Fo(t=)p Fq(2,)31 b(so)g Fo(q)2713 1675 y Fh(r)2781 1661
+y Fq(ma)m(y)g(b)s(e)f(appro)m(ximated)g(as)1576 1892
+y Fo(q)1617 1906 y Fh(r)1680 1892 y Fl(\031)25 b Fq(1)c(+)1932
+1773 y Fi(\022)2004 1830 y Fo(!)s Fq(\001)p Fo(t)p 2004
+1871 169 4 v 2065 1954 a Fq(2)2182 1773 y Fi(\023)2268
+1869 y Fq(\026)2258 1892 y(\012)0 2117 y(Th)m(us)1239
+2325 y Fo(q)1280 2340 y Fh(b)1314 2325 y Fq(\()p Fo(t)g
+Fq(+)e(\001)p Fo(t)p Fq(\))83 b(=)g Fo(q)1915 2340 y
+Fh(b)1949 2325 y Fq(\()p Fo(t)p Fq(\))2067 2206 y Fi(\022)2129
+2325 y Fq(1)21 b(+)2285 2206 y Fi(\022)2357 2264 y Fo(!)s
+Fq(\001)p Fo(t)p 2357 2304 V 2418 2388 a Fq(2)2535 2206
+y Fi(\023)2621 2302 y Fq(\026)2611 2325 y(\012)f(+)g
+Fl(\001)15 b(\001)g(\001)2894 2206 y Fi(\023)950 2559
+y Fo(q)991 2574 y Fh(b)1025 2559 y Fq(\()p Fo(t)20 b
+Fq(+)g(\001)p Fo(t)p Fq(\))g Fl(\000)g Fo(q)1500 2574
+y Fh(b)1534 2559 y Fq(\()p Fo(t)p Fq(\))83 b(=)g Fo(q)1915
+2574 y Fh(b)1949 2559 y Fq(\()p Fo(t)p Fq(\))2067 2440
+y Fi(\022)2139 2497 y Fo(!)s Fq(\001)p Fo(t)p 2139 2538
+V 2200 2621 a Fq(2)2317 2440 y Fi(\023)2404 2536 y Fq(\026)2393
+2559 y(\012)20 b(+)g Fl(\001)15 b(\001)g(\001)940 2737
+y Fo(q)981 2752 y Fh(b)1015 2737 y Fq(\()p Fo(t)20 b
+Fq(+)g(\001)p Fo(t)p Fq(\))g Fl(\000)g Fo(q)1490 2752
+y Fh(b)1524 2737 y Fq(\()p Fo(t)p Fq(\))p 940 2778 688
+4 v 1229 2861 a(\001)p Fo(t)1720 2799 y Fq(=)1874 2680
+y Fi(\022)1945 2737 y Fo(!)p 1945 2778 60 4 v 1952 2861
+a Fq(2)2015 2680 y Fi(\023)2091 2799 y Fo(q)2132 2814
+y Fh(b)2166 2799 y Fq(\()p Fo(t)p Fq(\))2279 2776 y(\026)2269
+2799 y(\012)h(+)f Fl(\001)15 b(\001)g(\001)0 3024 y Fq(W)-8
+b(e)32 b(tak)m(e)g(the)e(limit)e(at)j(\001)p Fo(t)25
+b Fl(!)g Fq(0)31 b(to)g(obtain)978 3171 y Fo(dq)1066
+3186 y Fh(b)p 978 3212 123 4 v 999 3295 a Fo(dt)1136
+3233 y Fq(=)1232 3114 y Fi(\022)1303 3171 y Fo(!)p 1303
+3212 60 4 v 1310 3295 a Fq(2)1373 3114 y Fi(\023)1449
+3233 y Fo(q)1490 3248 y Fh(b)1524 3233 y Fq(\()p Fo(t)p
+Fq(\))1637 3210 y(\026)1627 3233 y(\012)25 b(=)1824 3171
+y Fo(!)p 1824 3212 V 1831 3295 a Fq(2)1909 3164 y Fi(\000)1947
+3233 y Fo(q)1988 3248 y Fh(b;)p Fk(4)2087 3210 y Fq(\026)2077
+3233 y(\012)20 b Fl(\000)g Fq(\()7 b(\026)-52 b Fo(q)2330
+3248 y Fh(b)2384 3233 y Fl(\001)2440 3210 y Fq(\026)2430
+3233 y(\012)o(\))21 b Fl(\000)2652 3210 y Fq(\026)2642
+3233 y(\012)f Fl(\002)26 b Fq(\026)-51 b Fo(q)2860 3248
+y Fh(b)2894 3164 y Fi(\001)0 3592 y Fq(where)30 b Fo(x)21
+b Fl(\001)f Fo(y)29 b Fq(=)552 3528 y Fi(P)655 3592 y
+Fo(x)707 3607 y Fh(`)740 3592 y Fo(y)785 3607 y Fh(`)848
+3592 y Fq(is)h(the)h(v)m(ector)h(inner)d(\(dot\))j(pro)s(duct)d(and)h
+Fo(x)21 b Fl(\002)f Fo(y)29 b Fq(=)2689 3395 y Fi(\014)2689
+3445 y(\014)2689 3494 y(\014)2689 3544 y(\014)2689 3594
+y(\014)2689 3644 y(\014)2689 3694 y(\014)2716 3398 y(2)2716
+3544 y(6)2716 3597 y(4)2836 3462 y Fq(^)2843 3483 y Fo(i)3010
+3462 y Fq(^)3012 3483 y Fo(j)3185 3459 y Fq(^)3182 3483
+y Fo(k)2813 3596 y(x)2865 3610 y Fk(1)2987 3596 y Fo(x)3039
+3610 y Fk(2)3162 3596 y Fo(x)3214 3610 y Fk(3)2817 3709
+y Fo(y)2862 3723 y Fk(1)2991 3709 y Fo(y)3036 3723 y
+Fk(2)3166 3709 y Fo(y)3211 3723 y Fk(3)3295 3398 y Fi(3)3295
+3544 y(7)3295 3597 y(5)3350 3395 y(\014)3350 3445 y(\014)3350
+3494 y(\014)3350 3544 y(\014)3350 3594 y(\014)3350 3644
+y(\014)3350 3694 y(\014)3409 3592 y Fq(is)g(the)i(v)m(ector)0
+3815 y(cross)f(pro)s(duct.)40 b(In)30 b(matrix)f(form)h(this)g(is)1011
+3922 y Fi(2)1011 4068 y(6)1011 4118 y(6)1011 4168 y(6)1011
+4221 y(4)1125 3996 y Fq(_)-42 b Fo(q)1149 4011 y Fh(b)p
+Fk(1)1125 4109 y Fq(_)g Fo(q)1149 4124 y Fh(b)p Fk(2)1125
+4222 y Fq(_)g Fo(q)1149 4237 y Fh(b)p Fk(3)1125 4335
+y Fq(_)g Fo(q)1149 4350 y Fh(b)p Fk(4)1260 3922 y Fi(3)1260
+4068 y(7)1260 4118 y(7)1260 4168 y(7)1260 4221 y(5)1340
+4166 y Fq(=)1446 4104 y Fo(!)p 1446 4145 V 1453 4228
+a Fq(2)1531 3922 y Fi(2)1531 4068 y(6)1531 4118 y(6)1531
+4168 y(6)1531 4221 y(4)1688 3996 y Fq(0)180 b Fo(!)1970
+4010 y Fk(3)2127 3996 y Fl(\000)p Fo(!)2255 4010 y Fk(2)2377
+3996 y Fo(!)2434 4010 y Fk(1)1628 4109 y Fl(\000)p Fo(!)1756
+4123 y Fk(3)1938 4109 y Fq(0)g Fo(!)2220 4123 y Fk(1)2377
+4109 y Fo(!)2434 4123 y Fk(2)1663 4222 y Fo(!)1720 4236
+y Fk(2)1878 4222 y Fl(\000)p Fo(!)2006 4236 y Fk(1)2188
+4222 y Fq(0)144 b Fo(!)2434 4236 y Fk(3)1628 4335 y Fl(\000)p
+Fo(!)1756 4349 y Fk(1)1878 4335 y Fl(\000)p Fo(!)2006
+4349 y Fk(2)2127 4335 y Fl(\000)p Fo(!)2255 4349 y Fk(3)2402
+4335 y Fq(0)2515 3922 y Fi(3)2515 4068 y(7)2515 4118
+y(7)2515 4168 y(7)2515 4221 y(5)2585 3922 y(2)2585 4068
+y(6)2585 4118 y(6)2585 4168 y(6)2585 4221 y(4)2682 3996
+y Fo(q)2723 4011 y Fh(b)p Fk(1)2682 4109 y Fo(q)2723
+4124 y Fh(b)p Fk(2)2682 4222 y Fo(q)2723 4237 y Fh(b)p
+Fk(3)2682 4335 y Fo(q)2723 4350 y Fh(b)p Fk(4)2834 3922
+y Fi(3)2834 4068 y(7)2834 4118 y(7)2834 4168 y(7)2834
+4221 y(5)0 4522 y Fq(Notice)31 b(that)g(the)g(4)21 b
+Fl(\002)e Fq(4)31 b(matrix)f(is)f(sk)m(ew)i(symmetric)f(\(hence)h(its)e
+(matrix)h(exp)s(onen)m(tial)f(is)h(orthogonal\).)0 4806
+y Fp(References)0 5009 y Fq([Mul])127 b(Larry)30 b(Mullins.)37
+b(Course)30 b(4160:)43 b(Quaternions.)c(Course)30 b(notes.)0
+5192 y([R)-10 b(W91])47 b(M.)41 b(H.)g(Rheinfurth)d(and)h(H.)i(B.)g
+(Wilson.)70 b(Metho)s(ds)40 b(of)h(applied)d(dynamics.)69
+b(T)-8 b(ec)m(hnical)40 b(Rep)s(ort)337 5305 y(NASA)30
+b(RP-1262,)j(NASA,)e(George)h(C.)e(Marshall)f(Space)h(Fligh)m(t)g(Cen)m
+(ter,)h(1991.)p eop
+14 13 bop 0 527 a Fp(Index)0 709 y Fq(conjugate)166 821
+y(quaternion,)30 b(3)0 1017 y(Euler)f(angles)166 1130
+y(quaternions,)g(6)0 1326 y(Octa)m(v)m(e)166 1439 y Fj(qinv)p
+Fq(,)h(3)166 1552 y Fj(qmult)p Fq(,)f(3)166 1665 y Fj(qtrans)p
+Fq(,)g(4)166 1778 y(qtransv,)h(8)166 1891 y(qtransvmat,)h(8)166
+2004 y Fj(quaternion)p Fq(,)d(2)0 2200 y(quaternions)166
+2313 y(as)j(co)s(ordinate)f(rotations,)h(3)166 2425 y(co)s(ordinate)f
+(transformation,)g(4)166 2538 y(co)s(ordinate)g(transformation)g
+(matrices,)h(6)166 2651 y(cross)g(pro)s(ducts,)e(2)166
+2764 y(de\014nition,)f(2)166 2877 y(deriv)-5 b(ativ)m(es,)30
+b(12)166 2990 y(Euler)f(angles,)h(6)166 3103 y(m)m(ultiplication,)e(2)
+166 3216 y(unit)h(quaternions,)g(3)1905 5656 y(14)p eop
+end
+userdict /end-hook known{end-hook}if
new file mode 100644
--- /dev/null
+++ b/scripts/specfun/bessel.m
@@ -0,0 +1,39 @@
+## Copyright (C) 1996, 1997 John W. Eaton
+##
+## This file is part of Octave.
+##
+## Octave 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 2, or (at your option)
+## any later version.
+##
+## Octave 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 Octave; see the file COPYING.  If not, write to the Free
+## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
+## 02111-1307, USA.
+
+## The following functions are available for computing the values of
+## Bessel functions:
+##
+##   besselj (alpha, x)   Bessel functions of the first kind
+##   bessely (alpha, x)   Bessel functions of the second kind
+##   besseli (alpha, x)   modified Bessel functions of the first kind
+##   besselk (alpha, x)   modified Bessel functions of the second kind
+##
+## X must be a real matrix, vector or scalar.
+##
+## If ALPHA is a scalar, the result is the same size as X.  If ALPHA is
+## a range, X must be a vector or scalar, and the result is a matrix
+## with length(X) rows and length(ALPHA) columns.
+##
+## ALPHA must be greater than or equal to zero.  If ALPHA is a range, it
+## must have an increment equal to one.
+
+function bessel ()
+  error ("bessel: you must use besselj, bessely, besseli, or besselk");
+endfunction