Mercurial > octave
view libinterp/corefcn/lsode.cc @ 31605:e88a07dec498 stable
maint: Use macros to begin/end C++ namespaces.
* oct-conf-post-public.in.h: Define two macros (OCTAVE_BEGIN_NAMESPACE,
OCTAVE_END_NAMESPACE) that can be used to start/end a namespace.
* mk-opts.pl, build-env.h, build-env.in.cc, __betainc__.cc, __contourc__.cc,
__dsearchn__.cc, __eigs__.cc, __expint__.cc, __ftp__.cc, __gammainc__.cc,
__ichol__.cc, __ilu__.cc, __isprimelarge__.cc, __lin_interpn__.cc,
__magick_read__.cc, __pchip_deriv__.cc, __qp__.cc, amd.cc, auto-shlib.cc,
auto-shlib.h, balance.cc, base-text-renderer.cc, base-text-renderer.h,
besselj.cc, bitfcns.cc, bsxfun.cc, c-file-ptr-stream.cc, c-file-ptr-stream.h,
call-stack.cc, call-stack.h, ccolamd.cc, cellfun.cc, chol.cc, colamd.cc,
colloc.cc, conv2.cc, daspk.cc, dasrt.cc, dassl.cc, data.cc, data.h, debug.cc,
defaults.cc, defaults.h, defun-int.h, defun.cc, det.cc, dirfns.cc, display.cc,
display.h, dlmread.cc, dmperm.cc, dot.cc, dynamic-ld.cc, dynamic-ld.h, eig.cc,
ellipj.cc, environment.cc, environment.h, error.cc, error.h, errwarn.h,
event-manager.cc, event-manager.h, event-queue.cc, event-queue.h, fcn-info.cc,
fcn-info.h, fft.cc, fft2.cc, fftn.cc, file-io.cc, filter.cc, find.cc,
ft-text-renderer.cc, ft-text-renderer.h, gcd.cc, getgrent.cc, getpwent.cc,
getrusage.cc, givens.cc, gl-render.cc, gl-render.h, gl2ps-print.cc,
gl2ps-print.h, graphics-toolkit.cc, graphics-toolkit.h, graphics.cc,
graphics.in.h, gsvd.cc, gtk-manager.cc, gtk-manager.h, hash.cc, help.cc,
help.h, hess.cc, hex2num.cc, hook-fcn.cc, hook-fcn.h, input.cc, input.h,
interpreter-private.cc, interpreter-private.h, interpreter.cc, interpreter.h,
inv.cc, jsondecode.cc, jsonencode.cc, kron.cc, latex-text-renderer.cc,
latex-text-renderer.h, load-path.cc, load-path.h, load-save.cc, load-save.h,
lookup.cc, ls-ascii-helper.cc, ls-ascii-helper.h, ls-oct-text.cc, ls-utils.cc,
ls-utils.h, lsode.cc, lu.cc, mappers.cc, matrix_type.cc, max.cc, mex-private.h,
mex.cc, mgorth.cc, nproc.cc, oct-fstrm.cc, oct-fstrm.h, oct-hdf5-types.cc,
oct-hdf5-types.h, oct-hist.cc, oct-hist.h, oct-iostrm.cc, oct-iostrm.h,
oct-opengl.h, oct-prcstrm.cc, oct-prcstrm.h, oct-procbuf.cc, oct-procbuf.h,
oct-process.cc, oct-process.h, oct-stdstrm.h, oct-stream.cc, oct-stream.h,
oct-strstrm.cc, oct-strstrm.h, oct-tex-lexer.in.ll, oct-tex-parser.yy,
ordqz.cc, ordschur.cc, pager.cc, pager.h, pinv.cc, pow2.cc, pr-flt-fmt.cc,
pr-output.cc, procstream.cc, procstream.h, psi.cc, qr.cc, quad.cc, quadcc.cc,
qz.cc, rand.cc, rcond.cc, regexp.cc, schur.cc, settings.cc, settings.h,
sighandlers.cc, sighandlers.h, sparse-xdiv.cc, sparse-xdiv.h, sparse-xpow.cc,
sparse-xpow.h, sparse.cc, spparms.cc, sqrtm.cc, stack-frame.cc, stack-frame.h,
stream-euler.cc, strfind.cc, strfns.cc, sub2ind.cc, svd.cc, sylvester.cc,
symbfact.cc, syminfo.cc, syminfo.h, symrcm.cc, symrec.cc, symrec.h,
symscope.cc, symscope.h, symtab.cc, symtab.h, syscalls.cc, sysdep.cc, sysdep.h,
text-engine.cc, text-engine.h, text-renderer.cc, text-renderer.h, time.cc,
toplev.cc, tril.cc, tsearch.cc, typecast.cc, url-handle-manager.cc,
url-handle-manager.h, urlwrite.cc, utils.cc, utils.h, variables.cc,
variables.h, xdiv.cc, xdiv.h, xnorm.cc, xnorm.h, xpow.cc, xpow.h,
__delaunayn__.cc, __fltk_uigetfile__.cc, __glpk__.cc, __init_fltk__.cc,
__init_gnuplot__.cc, __ode15__.cc, __voronoi__.cc, audiodevinfo.cc,
audioread.cc, convhulln.cc, fftw.cc, gzip.cc, mk-build-env-features.sh,
mk-builtins.pl, cdef-class.cc, cdef-class.h, cdef-fwd.h, cdef-manager.cc,
cdef-manager.h, cdef-method.cc, cdef-method.h, cdef-object.cc, cdef-object.h,
cdef-package.cc, cdef-package.h, cdef-property.cc, cdef-property.h,
cdef-utils.cc, cdef-utils.h, ov-base.cc, ov-base.h, ov-bool-mat.cc,
ov-builtin.h, ov-cell.cc, ov-class.cc, ov-class.h, ov-classdef.cc,
ov-classdef.h, ov-complex.cc, ov-fcn-handle.cc, ov-fcn-handle.h, ov-fcn.h,
ov-java.cc, ov-java.h, ov-mex-fcn.h, ov-null-mat.cc, ov-oncleanup.cc,
ov-struct.cc, ov-typeinfo.cc, ov-typeinfo.h, ov-usr-fcn.cc, ov-usr-fcn.h,
ov.cc, ov.h, octave.cc, octave.h, mk-ops.sh, op-b-b.cc, op-b-bm.cc,
op-b-sbm.cc, op-bm-b.cc, op-bm-bm.cc, op-bm-sbm.cc, op-cdm-cdm.cc, op-cell.cc,
op-chm.cc, op-class.cc, op-cm-cm.cc, op-cm-cs.cc, op-cm-m.cc, op-cm-s.cc,
op-cm-scm.cc, op-cm-sm.cc, op-cs-cm.cc, op-cs-cs.cc, op-cs-m.cc, op-cs-s.cc,
op-cs-scm.cc, op-cs-sm.cc, op-dm-dm.cc, op-dm-scm.cc, op-dm-sm.cc,
op-dm-template.cc, op-dms-template.cc, op-fcdm-fcdm.cc, op-fcm-fcm.cc,
op-fcm-fcs.cc, op-fcm-fm.cc, op-fcm-fs.cc, op-fcn.cc, op-fcs-fcm.cc,
op-fcs-fcs.cc, op-fcs-fm.cc, op-fcs-fs.cc, op-fdm-fdm.cc, op-fm-fcm.cc,
op-fm-fcs.cc, op-fm-fm.cc, op-fm-fs.cc, op-fs-fcm.cc, op-fs-fcs.cc,
op-fs-fm.cc, op-fs-fs.cc, op-i16-i16.cc, op-i32-i32.cc, op-i64-i64.cc,
op-i8-i8.cc, op-int-concat.cc, op-m-cm.cc, op-m-cs.cc, op-m-m.cc, op-m-s.cc,
op-m-scm.cc, op-m-sm.cc, op-mi.cc, op-pm-pm.cc, op-pm-scm.cc, op-pm-sm.cc,
op-pm-template.cc, op-range.cc, op-s-cm.cc, op-s-cs.cc, op-s-m.cc, op-s-s.cc,
op-s-scm.cc, op-s-sm.cc, op-sbm-b.cc, op-sbm-bm.cc, op-sbm-sbm.cc,
op-scm-cm.cc, op-scm-cs.cc, op-scm-m.cc, op-scm-s.cc, op-scm-scm.cc,
op-scm-sm.cc, op-sm-cm.cc, op-sm-cs.cc, op-sm-m.cc, op-sm-s.cc, op-sm-scm.cc,
op-sm-sm.cc, op-str-m.cc, op-str-s.cc, op-str-str.cc, op-struct.cc,
op-ui16-ui16.cc, op-ui32-ui32.cc, op-ui64-ui64.cc, op-ui8-ui8.cc, ops.h,
anon-fcn-validator.cc, anon-fcn-validator.h, bp-table.cc, bp-table.h,
comment-list.cc, comment-list.h, filepos.h, lex.h, lex.ll, oct-lvalue.cc,
oct-lvalue.h, oct-parse.yy, parse.h, profiler.cc, profiler.h,
pt-anon-scopes.cc, pt-anon-scopes.h, pt-arg-list.cc, pt-arg-list.h,
pt-args-block.cc, pt-args-block.h, pt-array-list.cc, pt-array-list.h,
pt-assign.cc, pt-assign.h, pt-binop.cc, pt-binop.h, pt-bp.cc, pt-bp.h,
pt-cbinop.cc, pt-cbinop.h, pt-cell.cc, pt-cell.h, pt-check.cc, pt-check.h,
pt-classdef.cc, pt-classdef.h, pt-cmd.h, pt-colon.cc, pt-colon.h, pt-const.cc,
pt-const.h, pt-decl.cc, pt-decl.h, pt-eval.cc, pt-eval.h, pt-except.cc,
pt-except.h, pt-exp.cc, pt-exp.h, pt-fcn-handle.cc, pt-fcn-handle.h, pt-id.cc,
pt-id.h, pt-idx.cc, pt-idx.h, pt-jump.h, pt-loop.cc, pt-loop.h, pt-mat.cc,
pt-mat.h, pt-misc.cc, pt-misc.h, pt-pr-code.cc, pt-pr-code.h, pt-select.cc,
pt-select.h, pt-spmd.cc, pt-spmd.h, pt-stmt.cc, pt-stmt.h, pt-tm-const.cc,
pt-tm-const.h, pt-unop.cc, pt-unop.h, pt-vm-eval.cc, pt-walk.cc, pt-walk.h,
pt.cc, pt.h, token.cc, token.h, Range.cc, Range.h, idx-vector.cc, idx-vector.h,
range-fwd.h, CollocWt.cc, CollocWt.h, aepbalance.cc, aepbalance.h, chol.cc,
chol.h, gepbalance.cc, gepbalance.h, gsvd.cc, gsvd.h, hess.cc, hess.h,
lo-mappers.cc, lo-mappers.h, lo-specfun.cc, lo-specfun.h, lu.cc, lu.h,
oct-convn.cc, oct-convn.h, oct-fftw.cc, oct-fftw.h, oct-norm.cc, oct-norm.h,
oct-rand.cc, oct-rand.h, oct-spparms.cc, oct-spparms.h, qr.cc, qr.h, qrp.cc,
qrp.h, randgamma.cc, randgamma.h, randmtzig.cc, randmtzig.h, randpoisson.cc,
randpoisson.h, schur.cc, schur.h, sparse-chol.cc, sparse-chol.h, sparse-lu.cc,
sparse-lu.h, sparse-qr.cc, sparse-qr.h, svd.cc, svd.h, child-list.cc,
child-list.h, dir-ops.cc, dir-ops.h, file-ops.cc, file-ops.h, file-stat.cc,
file-stat.h, lo-sysdep.cc, lo-sysdep.h, lo-sysinfo.cc, lo-sysinfo.h,
mach-info.cc, mach-info.h, oct-env.cc, oct-env.h, oct-group.cc, oct-group.h,
oct-password.cc, oct-password.h, oct-syscalls.cc, oct-syscalls.h, oct-time.cc,
oct-time.h, oct-uname.cc, oct-uname.h, action-container.cc, action-container.h,
base-list.h, cmd-edit.cc, cmd-edit.h, cmd-hist.cc, cmd-hist.h, f77-fcn.h,
file-info.cc, file-info.h, lo-array-errwarn.cc, lo-array-errwarn.h, lo-hash.cc,
lo-hash.h, lo-ieee.h, lo-regexp.cc, lo-regexp.h, lo-utils.cc, lo-utils.h,
oct-base64.cc, oct-base64.h, oct-glob.cc, oct-glob.h, oct-inttypes.h,
oct-mutex.cc, oct-mutex.h, oct-refcount.h, oct-shlib.cc, oct-shlib.h,
oct-sparse.cc, oct-sparse.h, oct-string.h, octave-preserve-stream-state.h,
pathsearch.cc, pathsearch.h, quit.cc, quit.h, unwind-prot.cc, unwind-prot.h,
url-transfer.cc, url-transfer.h : Use new macros to begin/end C++ namespaces.
author | Rik <rik@octave.org> |
---|---|
date | Thu, 01 Dec 2022 14:23:45 -0800 |
parents | 773a361bc476 |
children | aac27ad79be6 |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 1996-2022 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // 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 3 of the License, 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, see // <https://www.gnu.org/licenses/>. // //////////////////////////////////////////////////////////////////////// #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include <list> #include <string> #include "LSODE.h" #include "lo-mappers.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "interpreter-private.h" #include "ovl.h" #include "ov-fcn.h" #include "ov-cell.h" #include "pager.h" #include "parse.h" #include "pr-output.h" #include "unwind-prot.h" #include "utils.h" #include "variables.h" #include "LSODE-opts.cc" OCTAVE_BEGIN_NAMESPACE(octave) // Global pointer for user defined function required by lsode. static octave_value lsode_fcn; // Global pointer for optional user defined jacobian function used by lsode. static octave_value lsode_jac; // Have we warned about imaginary values returned from user function? static bool warned_fcn_imaginary = false; static bool warned_jac_imaginary = false; // Is this a recursive call? static int call_depth = 0; static ColumnVector lsode_user_function (const ColumnVector& x, double t) { ColumnVector retval; octave_value_list args; args(1) = t; args(0) = x; if (lsode_fcn.is_defined ()) { octave_value_list tmp; try { tmp = octave::feval (lsode_fcn, args, 1); } catch (octave::execution_exception& ee) { err_user_supplied_eval (ee, "lsode"); } if (tmp.empty () || ! tmp(0).is_defined ()) err_user_supplied_eval ("lsode"); if (! warned_fcn_imaginary && tmp(0).iscomplex ()) { warning ("lsode: ignoring imaginary part returned from user-supplied function"); warned_fcn_imaginary = true; } retval = tmp(0).xvector_value ("lsode: expecting user supplied function to return numeric vector"); if (retval.isempty ()) err_user_supplied_eval ("lsode"); } return retval; } static Matrix lsode_user_jacobian (const ColumnVector& x, double t) { Matrix retval; octave_value_list args; args(1) = t; args(0) = x; if (lsode_jac.is_defined ()) { octave_value_list tmp; try { tmp = octave::feval (lsode_jac, args, 1); } catch (octave::execution_exception& ee) { err_user_supplied_eval (ee, "lsode"); } if (tmp.empty () || ! tmp(0).is_defined ()) err_user_supplied_eval ("lsode"); if (! warned_jac_imaginary && tmp(0).iscomplex ()) { warning ("lsode: ignoring imaginary part returned from user-supplied jacobian function"); warned_jac_imaginary = true; } retval = tmp(0).xmatrix_value ("lsode: expecting user supplied jacobian function to return numeric array"); if (retval.isempty ()) err_user_supplied_eval ("lsode"); } return retval; } DEFMETHOD (lsode, interp, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {[@var{x}, @var{istate}, @var{msg}] =} lsode (@var{fcn}, @var{x_0}, @var{t}) @deftypefnx {} {[@var{x}, @var{istate}, @var{msg}] =} lsode (@var{fcn}, @var{x_0}, @var{t}, @var{t_crit}) Ordinary Differential Equation (ODE) solver. The set of differential equations to solve is @tex $$ {dx \over dt} = f (x, t) $$ with $$ x(t_0) = x_0 $$ @end tex @ifnottex @example @group dx -- = f (x, t) dt @end group @end example @noindent with @example x(t_0) = x_0 @end example @end ifnottex The solution is returned in the matrix @var{x}, with each row corresponding to an element of the vector @var{t}. The first element of @var{t} should be @math{t_0} and should correspond to the initial state of the system @var{x_0}, so that the first row of the output is @var{x_0}. The first argument, @var{fcn}, is a string, inline, or function handle that names the function @math{f} to call to compute the vector of right hand sides for the set of equations. The function must have the form @example @var{xdot} = f (@var{x}, @var{t}) @end example @noindent in which @var{xdot} and @var{x} are vectors and @var{t} is a scalar. If @var{fcn} is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function @math{f} described above, and the second element names a function to compute the Jacobian of @math{f}. The Jacobian function must have the form @example @var{jac} = j (@var{x}, @var{t}) @end example @noindent in which @var{jac} is the matrix of partial derivatives @tex $$ J = {\partial f_i \over \partial x_j} = \left[\matrix{ {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} & \cdots & {\partial f_1 \over \partial x_N} \cr {\partial f_2 \over \partial x_1} & {\partial f_2 \over \partial x_2} & \cdots & {\partial f_2 \over \partial x_N} \cr \vdots & \vdots & \ddots & \vdots \cr {\partial f_M \over \partial x_1} & {\partial f_M \over \partial x_2} & \cdots & {\partial f_M \over \partial x_N} \cr}\right]$$ @end tex @ifnottex @example @group | df_1 df_1 df_1 | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | | | | df_2 df_2 df_2 | | ---- ---- ... ---- | df_i | dx_1 dx_2 dx_N | jac = ---- = | | dx_j | . . . . | | . . . . | | . . . . | | | | df_M df_M df_M | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | @end group @end example @end ifnottex The second argument specifies the initial state of the system @math{x_0}. The third argument is a vector, @var{t}, specifying the time values for which a solution is sought. The fourth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of @var{istate} will be 2 (consistent with the Fortran version of @sc{lsode}). If the computation is not successful, @var{istate} will be something other than 2 and @var{msg} will contain additional information. You can use the function @code{lsode_options} to set optional parameters for @code{lsode}. See @nospell{Alan C. Hindmarsh}, @cite{ODEPACK, A Systematized Collection of ODE Solvers}, in Scientific Computing, @nospell{R. S. Stepleman}, editor, (1983) or @url{https://computing.llnl.gov/projects/odepack} for more information about the inner workings of @code{lsode}. Example: Solve the @nospell{Van der Pol} equation @example @group fvdp = @@(@var{y},@var{t}) [@var{y}(2); (1 - @var{y}(1)^2) * @var{y}(2) - @var{y}(1)]; @var{t} = linspace (0, 20, 100); @var{y} = lsode (fvdp, [2; 0], @var{t}); @end group @end example @seealso{daspk, dassl, dasrt} @end deftypefn */) { int nargin = args.length (); if (nargin < 3 || nargin > 4) print_usage (); warned_fcn_imaginary = false; warned_jac_imaginary = false; unwind_protect_var<int> restore_var (call_depth); call_depth++; if (call_depth > 1) error ("lsode: invalid recursive call"); symbol_table& symtab = interp.get_symbol_table (); std::string fcn_name, fname, jac_name, jname; lsode_fcn = octave_value (); lsode_jac = octave_value (); octave_value f_arg = args(0); std::list<std::string> parameter_names ({"x", "t"}); if (f_arg.iscell ()) { Cell c = f_arg.cell_value (); if (c.numel () == 1) f_arg = c(0); else if (c.numel () == 2) { lsode_fcn = get_function_handle (interp, c(0), parameter_names); if (lsode_fcn.is_defined ()) { lsode_jac = get_function_handle (interp, c(1), parameter_names); if (lsode_jac.is_undefined ()) lsode_fcn = octave_value (); } } else error ("lsode: incorrect number of elements in cell array"); } if (lsode_fcn.is_undefined () && ! f_arg.iscell ()) { if (f_arg.is_function_handle () || f_arg.is_inline_function ()) lsode_fcn = f_arg; else { switch (f_arg.rows ()) { case 1: lsode_fcn = get_function_handle (interp, f_arg, parameter_names); break; case 2: { string_vector tmp = f_arg.string_vector_value (); lsode_fcn = get_function_handle (interp, tmp(0), parameter_names); if (lsode_fcn.is_defined ()) { lsode_jac = get_function_handle (interp, tmp(1), parameter_names); if (lsode_jac.is_undefined ()) lsode_fcn = octave_value (); } } break; default: error ("lsode: first arg should be a string or 2-element string array"); } } } if (lsode_fcn.is_undefined ()) error ("lsode: FCN argument is not a valid function name or handle"); ColumnVector state = args(1).xvector_value ("lsode: initial state X_0 must be a vector"); ColumnVector out_times = args(2).xvector_value ("lsode: output time variable T must be a vector"); ColumnVector crit_times; int crit_times_set = 0; if (nargin > 3) { crit_times = args(3).xvector_value ("lsode: list of critical times T_CRIT must be a vector"); crit_times_set = 1; } double tzero = out_times (0); ODEFunc fcn (lsode_user_function); if (lsode_jac.is_defined ()) fcn.set_jacobian_function (lsode_user_jacobian); LSODE ode (state, tzero, fcn); ode.set_options (lsode_opts); Matrix output; if (crit_times_set) output = ode.integrate (out_times, crit_times); else output = ode.integrate (out_times); if (fcn_name.length ()) symtab.clear_function (fcn_name); if (jac_name.length ()) symtab.clear_function (jac_name); std::string msg = ode.error_message (); octave_value_list retval (3); if (ode.integration_ok ()) retval(0) = output; else if (nargout < 2) error ("lsode: %s", msg.c_str ()); else retval(0) = Matrix (); retval(1) = static_cast<double> (ode.integration_state ()); retval(2) = msg; return retval; } /* ## dassl-1.m ## ## Test lsode() function ## ## Author: David Billinghurst (David.Billinghurst@riotinto.com.au) ## Comalco Research and Technology ## 20 May 1998 ## ## Problem ## ## y1' = -y2, y1(0) = 1 ## y2' = y1, y2(0) = 0 ## ## Solution ## ## y1(t) = cos(t) ## y2(t) = sin(t) ## %!function xdot = __f (x, t) %! xdot = [-x(2); x(1)]; %!endfunction %!test %! %! x0 = [1; 0]; %! xdot0 = [0; 1]; %! t = (0:1:10)'; %! %! tol = 500 * lsode_options ("relative tolerance"); %! %! x = lsode ("__f", x0, t); %! %! y = [cos(t), sin(t)]; %! %! assert (x, y, tol); %!function xdotdot = __f (x, t) %! xdotdot = [x(2); -x(1)]; %!endfunction %!test %! %! x0 = [1; 0]; %! t = [0; 2*pi]; %! tol = 100 * dassl_options ("relative tolerance"); %! %! x = lsode ("__f", x0, t); %! %! y = [1, 0; 1, 0]; %! %! assert (x, y, tol); %!function xdot = __f (x, t) %! xdot = x; %!endfunction %!test %! %! x0 = 1; %! t = [0; 1]; %! tol = 100 * dassl_options ("relative tolerance"); %! %! x = lsode ("__f", x0, t); %! %! y = [1; e]; %! %! assert (x, y, tol); %!test %! lsode_options ("absolute tolerance", eps); %! assert (lsode_options ("absolute tolerance") == eps); %!error lsode_options ("foo", 1, 2) */ OCTAVE_END_NAMESPACE(octave)