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maint: merge stable to default.
author John W. Eaton <jwe@octave.org>
date Wed, 10 Feb 2021 10:10:40 -0500
parents a948253f9976 0a5b15007766
children 796f54d4ddbf
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2017-2021 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {@var{q} =} integral (@var{f}, @var{a}, @var{b})
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28 ## @deftypefnx {} {@var{q} =} integral (@var{f}, @var{a}, @var{b}, @var{prop}, @var{val}, @dots{})
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29 ##
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30 ## Numerically evaluate the integral of @var{f} from @var{a} to @var{b} using
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31 ## adaptive quadrature.
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32 ##
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33 ## @code{integral} is a wrapper for @code{quadcc} (general real-valued, scalar
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34 ## integrands and limits), @code{quadgk} (integrals with specified integration
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35 ## paths), and @code{quadv} (array-valued integrands) that is intended to
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36 ## provide @sc{matlab} compatibility. More control of the numerical
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37 ## integration may be achievable by calling the various quadrature functions
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38 ## directly.
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39 ##
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40 ## @var{f} is a function handle, inline function, or string containing the name
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41 ## of the function to evaluate. The function @var{f} must be vectorized and
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42 ## return a vector of output values when given a vector of input values.
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43 ##
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44 ## @var{a} and @var{b} are the lower and upper limits of integration. Either
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45 ## or both limits may be infinite or contain weak end singularities. If either
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46 ## or both limits are complex, @code{integral} will perform a straight line
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47 ## path integral. Alternatively, a complex domain path can be specified using
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48 ## the @qcode{"Waypoints"} option (see below).
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49 ##
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50 ## Additional optional parameters can be specified using
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51 ## @qcode{"@var{property}", @var{value}} pairs. Valid properties are:
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52 ##
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53 ## @table @code
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54 ## @item Waypoints
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55 ## Specifies points to be used in defining subintervals of the quadrature
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56 ## algorithm, or if @var{a}, @var{b}, or @var{waypoints} are complex then
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57 ## the quadrature is calculated as a contour integral along a piecewise
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58 ## continuous path. For more detail, @pxref{XREFquadgk,,@code{quadgk}}.
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59 ##
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60 ## @item ArrayValued
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61 ## @code{integral} expects @var{f} to return a scalar value unless
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62 ## @var{arrayvalued} is specified as true. This option will cause
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63 ## @code{integral} to perform the integration over the entire array and return
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64 ## @var{q} with the same dimensions as returned by @var{f}. For more detail
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65 ## @pxref{XREFquadv,,@code{quadv}}.
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66 ##
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67 ## @item AbsTol
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68 ## Define the absolute error tolerance for the quadrature. The default
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69 ## absolute tolerance is 1e-10 (1e-5 for single).
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70 ##
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71 ## @item RelTol
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72 ## Define the relative error tolerance for the quadrature. The default
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73 ## relative tolerance is 1e-6 (1e-4 for single).
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74 ## @end table
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75 ##
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76 ## Adaptive quadrature is used to minimize the estimate of error until the
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77 ## following is satisfied:
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78 ## @tex
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79 ## $$error \leq \max \left( AbsTol, RelTol\cdot\vert q\vert \right)$$
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80 ## @end tex
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81 ## @ifnottex
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82 ##
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83 ## @example
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84 ## @group
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85 ## @var{error} <= max (@var{AbsTol}, @var{RelTol}*|@var{q}|).
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86 ## @end group
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87 ## @end example
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88 ##
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89 ## @end ifnottex
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90 ##
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91 ## Known @sc{matlab} incompatibilities:
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92 ##
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93 ## @enumerate
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94 ## @item
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95 ## If tolerances are left unspecified, and any integration limits or waypoints
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96 ## are of type @code{single}, then Octave's integral functions automatically
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97 ## reduce the default absolute and relative error tolerances as specified
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98 ## above. If tighter tolerances are desired they must be specified.
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99 ## @sc{matlab} leaves the tighter tolerances appropriate for @code{double}
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100 ## inputs in place regardless of the class of the integration limits.
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101 ##
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102 ## @item
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103 ## As a consequence of using @code{quadcc}, @code{quadgk}, and @code{quadv},
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104 ## certain option combinations are not supported. Currently,
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105 ## @qcode{"ArrayValued"} cannot be combined with @qcode{"RelTol"} or
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106 ## @qcode{"Waypoints"}.
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107 ## @end enumerate
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108 ##
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109 ## @seealso{integral2, integral3, quad, quadgk, quadv, quadl, quadcc, trapz,
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110 ## dblquad, triplequad}
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111 ## @end deftypefn
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112
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113 function q = integral (f, a, b, varargin)
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114
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115 if (nargin < 3 || (mod (nargin, 2) == 0))
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116 print_usage ();
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117 endif
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
118
28731
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diff changeset
119 ## quadcc can't handle complex limits or integrands, but quadgk & quadv can.
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
120 ## Check for simple cases of complex limits and integrand.
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
121 f_is_complex = false;
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
122 if (iscomplex (a) || iscomplex (b))
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diff changeset
123 f_is_complex = true;
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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124 elseif (iscomplex (feval (f, a)) || iscomplex (feval (f, b)))
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
125 f_is_complex = true;
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diff changeset
126 endif
24055
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diff changeset
127
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
128 if (nargin == 3)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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129 ## Pass the simplest case directly to general integrator.
24155
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130 ## Let quadcc function handle input checks on function and limits.
28731
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131 if (! f_is_complex)
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diff changeset
132 try
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
133 q = quadcc (f, a, b);
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134 catch quaderror
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
135 if (strcmp (quaderror.message,
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diff changeset
136 "quadcc: integrand F must return a single, real-valued vector"))
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diff changeset
137 q = quadgk (f, a, b);
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138 else
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diff changeset
139 error (quaderror.message);
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
140 endif
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diff changeset
141 end_try_catch
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diff changeset
142
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
143 else
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diff changeset
144 ## Complex-valued integral
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
145 q = quadgk (f, a, b);
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diff changeset
146 endif
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
147
24055
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148 else
24062
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149 ## Parse options to determine how to call integrator.
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150 abstol = [];
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
151 reltol = [];
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diff changeset
152 waypoints = [];
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diff changeset
153 arrayvalued = false;
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diff changeset
154
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
155 idx = 1;
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
156 while (idx < nargin - 3)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
157 prop = varargin{idx++};
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
158 if (! ischar (prop))
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159 error ("integral: property PROP must be a string");
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160 endif
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161
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diff changeset
162 switch (tolower (prop))
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163 case "reltol"
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164 reltol = varargin{idx++};
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165 case "abstol"
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166 abstol = varargin{idx++};
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167 case "waypoints"
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168 waypoints = varargin{idx++}(:);
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169 case "arrayvalued"
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170 arrayvalued = varargin{idx++};
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171 otherwise
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172 error ("integral: unknown property '%s'", prop);
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173 endswitch
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174 endwhile
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175
24155
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176 issingle = (isa (a, "single") || isa (b, "single")
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177 || isa (waypoints, "single"));
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178
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diff changeset
179 if (arrayvalued)
24159
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180 ## Pass vector-valued function to quadv, checking for conflicting params
24155
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181
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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182 ## FIXME: Replace warning when have array compatible call with waypoints
24055
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183 if (! isempty (waypoints))
28912
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Rik <rik@octave.org>
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diff changeset
184 warning (["integral: array-valued quadrature routine currently ", ...
24055
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185 "unable to handle WayPoints. WayPoints are ignored."]);
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diff changeset
186 endif
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187
24155
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diff changeset
188 ## FIXME: Remove warning once we have reltol compatible arrayval'd quadfn
24055
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diff changeset
189 if (! isempty (reltol))
28912
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Rik <rik@octave.org>
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diff changeset
190 warning (["integral: array-valued quadrature only accepts AbsTol.", ...
24055
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diff changeset
191 " RelTol ignored."]);
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diff changeset
192 endif
24062
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diff changeset
193 if (isempty (abstol))
6570fdb7d3a0 integral.m: Pass tolerances to underlying integrators, correct precision in BIST tests (bug #42037).
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diff changeset
194 abstol = ifelse (issingle, 1e-5, 1e-10);
6570fdb7d3a0 integral.m: Pass tolerances to underlying integrators, correct precision in BIST tests (bug #42037).
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diff changeset
195 endif
24055
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diff changeset
196
24062
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diff changeset
197 q = quadv (f, a, b, abstol);
24155
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diff changeset
198
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
199 else
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
200 if (isempty (abstol))
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
201 abstol = ifelse (issingle, 1e-5, 1e-10);
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
202 endif
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
203 if (isempty (reltol))
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
204 reltol = ifelse (issingle, 1e-4, 1e-6);
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
205 endif
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
206
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
parents: 24102
diff changeset
207 if (! isempty (waypoints))
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
208 q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol,
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
209 "WayPoints", waypoints);
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
210 else
28731
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diff changeset
211 if (! f_is_complex)
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
212 try
29127
a948253f9976 maint: strip trailing spaces from code base.
Rik <rik@octave.org>
parents: 28958
diff changeset
213 q = quadcc (f, a, b, [abstol, reltol]);
28731
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
214 catch quaderror
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
215 if (strcmp (quaderror.message,
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
216 "quadcc: integrand F must return a single, real-valued vector"))
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
217 q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol);
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diff changeset
218 else
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
parents: 27961
diff changeset
219 error (quaderror.message);
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
220 endif
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
parents: 27961
diff changeset
221 end_try_catch
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
222 else
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diff changeset
223 ## Complex-valued integral
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diff changeset
224 q = quadgk (f, a, b, "AbsTol", abstol, "RelTol", reltol);
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
225 endif
24155
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
226 endif
24055
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
227 endif
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
228 endif
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
229
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
230 endfunction
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
231
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
232
2eae2ad53eb9 Add missing function integral.m (bug #42037).
Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
233 ## Matlab compatibility tests
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
234 %!test
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Nicholas R. Jankowski <jankowskin@asme.org>
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diff changeset
235 %! f = @(x) exp (-x.^2) .* log (x).^2;
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
236 %! emgamma = 0.57721566490153286;
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diff changeset
237 %! exact = (sqrt (pi)*(8*log (2)^2+8*emgamma*log (2)+pi^2+2*emgamma^2))/16;
24062
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238 %! assert (integral (f, 0, Inf), exact, -1e-6);
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parents: 24059
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239 %! assert (integral (f, 0, Inf, "RelTol", 1e-12), exact, -1e-12);
24055
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240
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241 %!test # with parameter
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242 %! f = @(x, c) 1 ./ (x.^3 - 2*x - c);
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243 %! assert (integral (@(x) f(x,5), 0, 2), -0.4605015338467329, 1e-10);
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244
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245 %!test # with tolerances
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246 %! f = @(x) log (x);
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247 %! assert (integral (@(x) f(x), 0, 1, "AbsTol", 1e-6), -1, 1e-6);
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248
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249 %!test # waypoints
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250 %! f = @(x) 1./(2.*x-1);
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251 %! assert (integral (f, 0, 0, "Waypoints", [1+1i, 1-1i]), -pi*1i, 1e-10);
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252
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253 %!test # an array-valued function
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254 %! f = @(x) sin ((1:5)*x);
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255 %! assert (integral (f, 0, 1, "ArrayValued", true), 1./[1:5]-cos(1:5)./[1:5],
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256 %! 1e-10);
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257
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258 %!test # test single input/output
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259 %! assert (integral (@sin, 0, 1), cos (0)-cos (1), 1e-10);
24155
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260 %! assert (class (integral (@sin, single (0), 1)), "single");
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261 %! assert (class (integral (@sin, 0, single (1))), "single");
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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262 %! assert (class (integral (@sin, single (0), single (1))), "single");
28912
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parents: 28753
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263 %! assert (integral (@sin, 0, 1, "Waypoints", 0.5), cos (0)-cos (1), 1e-10);
24155
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parents: 24102
diff changeset
264 %! assert (class (integral (@sin, 0, 1, "Waypoints", single (0.5))), "single");
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parents: 24102
diff changeset
265 %! assert (class (integral (@sin, single (0), 1, "Waypoints", 0.5)), "single");
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parents: 24102
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266 %! assert (class (integral (@sin, 0, single (1), "Waypoints", 0.5)), "single");
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parents: 24102
diff changeset
267
28731
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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parents: 27961
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268 %!test # test complex argument handling
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parents: 27961
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269 %! f = @(x) round (exp (i*x));
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parents: 27961
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270 %! assert (integral (f, 0, pi), quadgk (f, 0, pi), eps);
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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parents: 27961
diff changeset
271 %! assert (integral (f, -1, 1), 2, 5*eps);
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parents: 27961
diff changeset
272 %! assert (integral (@sin, -i, i), 0, eps);
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parents: 27961
diff changeset
273 %! assert (1.5 * integral (@sqrt, -1, 0), i, eps);
4f1e8a79bd44 integral.m: Fix input validation to handle complex integrands (bug #58636).
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diff changeset
274
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275 %!test
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276 %! f = @(x) x.^5 .* exp (-x) .* sin (x);
24062
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parents: 24059
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277 %! assert (integral (f, 0, inf, "RelTol", 1e-8, "AbsTol", 1e-12), -15, -1e-8);
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278
24155
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279 ## tests from quadcc
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280 %!assert (integral (@sin, -pi, pi), 0, 1e-10)
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281 %!assert (integral (inline ("sin"), -pi, pi), 0, 1e-10)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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282 %!assert (integral ("sin", -pi, pi), 0, 1e-10)
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283 %!assert (integral (@sin, -pi, 0), -2, 1e-10)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
284 %!assert (integral (@sin, 0, pi), 2, 1e-10)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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285 %!assert (integral (@(x) 1./(sqrt (x).*(x+1)), 0, Inf), pi, -1e-6)
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parents: 24102
diff changeset
286 %!assert (integral (@(x) 1./(sqrt (x).*(x+1)), 0, Inf,
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
287 %! "AbsTol", 0, "RelTol", 1e-8),
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parents: 24102
diff changeset
288 %! pi, -1e-8)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
Nicholas R. Jankowski <jankowskin@asme.org>
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289 %!assert (integral (@(x) exp (-x .^ 2), -Inf, Inf), sqrt (pi), 1e-10)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
290 %!assert (integral (@(x) exp (-x .^ 2), -Inf, 0), sqrt (pi)/2, 1e-10)
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
291
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292 ## tests from quadgk
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293 %!assert (integral (@sin,-pi,pi, "WayPoints",0, "AbsTol",1e-6, "RelTol",1e-3),
47dd094a6239 integral.m: Update to use recently changed quadcc and preserve type single (bug #42073).
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diff changeset
294 %! 0, 1e-6)
27961
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295 %!assert (integral (@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1), 2, 1e-10)
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diff changeset
296 %!assert (integral (@(z) log (z),1+1i,1+1i, "WayPoints", [1-1i, -1,-1i, -1+1i]),
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297 %! complex (0, pi), 1e-10)
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diff changeset
298
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299 ## tests from quadv
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300 ## Test vector-valued functions
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301 %!assert (integral (@(x) [(sin (x)), (sin (2*x))], 0, pi, "ArrayValued", 1),
27961
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302 %! [2, 0], 1e-10)
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diff changeset
303
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
304 ## Test matrix-valued functions
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305 %!test
24062
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306 %! assert (integral (@(x) [x,x,x; x,exp(x),x; x,x,x], 0, 1, "ArrayValued", 1),
6570fdb7d3a0 integral.m: Pass tolerances to underlying integrators, correct precision in BIST tests (bug #42037).
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diff changeset
307 %! [0.5,0.5,0.5; 0.5,(exp (1) - 1),0.5; 0.5,0.5,0.5], 1e-10);
24055
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diff changeset
308
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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309 ## Test input validation
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310 %!error integral (@sin)
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diff changeset
311 %!error integral (@sin, 0)
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diff changeset
312 %!error integral (@sin, 0, 1, 1e-6, true, 4)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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313 %!error integral (@sin, 0, 1, "DummyArg")
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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diff changeset
314 %!error <property PROP must be a string> integral (@sin, 0, 1, 2, 3)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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parents:
diff changeset
315 %!error <unknown property 'foo'> integral (@sin, 0, 1, "foo", 3)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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parents:
diff changeset
316 %!error integral (@sin, 0, 1, "AbsTol", ones (2,2))
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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parents:
diff changeset
317 %!error integral (@sin, 0, 1, "AbsTol", -1)
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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parents:
diff changeset
318 %!error integral (@sin, 0, 1, "RelTol", ones (2,2))
2eae2ad53eb9 Add missing function integral.m (bug #42037).
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parents:
diff changeset
319 %!error integral (@sin, 0, 1, "RelTol", -1)