Mercurial > hg > octave-lyh
annotate doc/interpreter/numbers.txi @ 17289:bc924baa2c4e
doc: Add new @qcode macro for code samples which are quoted.
Macro handles options ("on") or properties ("position") more elegantly
than @code{"text"}.
* doc/interpreter/macros.texi: Add new @qcode macro.
* doc/interpreter/tips.txi: Add documentation about @qcode macro.
* doc/interpreter/basics.txi, doc/interpreter/container.txi,
doc/interpreter/emacs.txi, doc/interpreter/errors.txi,
doc/interpreter/eval.txi, doc/interpreter/expr.txi,
doc/interpreter/external.txi, doc/interpreter/func.txi,
doc/interpreter/grammar.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/io.txi, doc/interpreter/matrix.txi,
doc/interpreter/numbers.txi, doc/interpreter/oop.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/quad.txi, doc/interpreter/sparse.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/vectorize.txi, libinterp/corefcn/balance.cc,
libinterp/corefcn/bitfcns.cc, libinterp/corefcn/cellfun.cc,
libinterp/corefcn/conv2.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/debug.cc, libinterp/corefcn/defaults.cc,
libinterp/corefcn/dirfns.cc, libinterp/corefcn/dlmread.cc,
libinterp/corefcn/error.cc, libinterp/corefcn/file-io.cc,
libinterp/corefcn/find.cc, libinterp/corefcn/gammainc.cc,
libinterp/corefcn/graphics.cc, libinterp/corefcn/help.cc,
libinterp/corefcn/hex2num.cc, libinterp/corefcn/input.cc,
libinterp/corefcn/load-path.cc, libinterp/corefcn/load-save.cc,
libinterp/corefcn/ls-oct-ascii.cc, libinterp/corefcn/lu.cc,
libinterp/corefcn/luinc.cc, libinterp/corefcn/matrix_type.cc,
libinterp/corefcn/oct-hist.cc, libinterp/corefcn/pager.cc,
libinterp/corefcn/pr-output.cc, libinterp/corefcn/pt-jit.cc,
libinterp/corefcn/qz.cc, libinterp/corefcn/rand.cc,
libinterp/corefcn/regexp.cc, libinterp/corefcn/schur.cc,
libinterp/corefcn/sighandlers.cc, libinterp/corefcn/sparse.cc,
libinterp/corefcn/spparms.cc, libinterp/corefcn/str2double.cc,
libinterp/corefcn/svd.cc, libinterp/corefcn/symtab.cc,
libinterp/corefcn/syscalls.cc, libinterp/corefcn/toplev.cc,
libinterp/corefcn/tril.cc, libinterp/corefcn/typecast.cc,
libinterp/corefcn/utils.cc, libinterp/corefcn/variables.cc,
libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/chol.cc,
libinterp/dldfcn/colamd.cc, libinterp/dldfcn/fftw.cc, libinterp/dldfcn/qr.cc,
libinterp/dldfcn/symbfact.cc, libinterp/octave-value/ov-base.cc,
libinterp/octave-value/ov-fcn-handle.cc,
libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-java.cc,
libinterp/octave-value/ov-range.cc, libinterp/octave-value/ov-struct.cc,
libinterp/octave-value/ov-usr-fcn.cc, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/pt-binop.cc, libinterp/parse-tree/pt-eval.cc,
libinterp/parse-tree/pt-mat.cc, scripts/@ftp/ftp.m,
scripts/deprecated/java_convert_matrix.m, scripts/deprecated/java_debug.m,
scripts/deprecated/java_unsigned_conversion.m, scripts/deprecated/shell_cmd.m,
scripts/general/dblquad.m, scripts/general/display.m,
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scripts/general/profexplore.m, scripts/general/profile.m,
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scripts/general/subsindex.m, scripts/general/triplequad.m,
scripts/geometry/griddata.m, scripts/geometry/griddata3.m,
scripts/geometry/griddatan.m, scripts/geometry/voronoi.m, scripts/help/help.m,
scripts/help/lookfor.m, scripts/image/cmpermute.m, scripts/image/colormap.m,
scripts/image/image.m, scripts/image/imagesc.m, scripts/image/imfinfo.m,
scripts/image/imformats.m, scripts/image/imread.m, scripts/image/imshow.m,
scripts/image/imwrite.m, scripts/image/ind2gray.m, scripts/image/lines.m,
scripts/image/rgb2ind.m, scripts/image/spinmap.m, scripts/io/dlmwrite.m,
scripts/io/strread.m, scripts/io/textread.m, scripts/io/textscan.m,
scripts/java/javaclasspath.m, scripts/java/usejava.m,
scripts/miscellaneous/bzip2.m, scripts/miscellaneous/computer.m,
scripts/miscellaneous/copyfile.m, scripts/miscellaneous/debug.m,
scripts/miscellaneous/dos.m, scripts/miscellaneous/edit.m,
scripts/miscellaneous/gzip.m, scripts/miscellaneous/license.m,
scripts/miscellaneous/mkoctfile.m, scripts/miscellaneous/movefile.m,
scripts/miscellaneous/parseparams.m, scripts/miscellaneous/unix.m,
scripts/optimization/fminbnd.m, scripts/optimization/fminsearch.m,
scripts/optimization/fminunc.m, scripts/optimization/fsolve.m,
scripts/optimization/fzero.m, scripts/optimization/glpk.m,
scripts/optimization/lsqnonneg.m, scripts/optimization/optimset.m,
scripts/optimization/pqpnonneg.m, scripts/pkg/pkg.m, scripts/plot/allchild.m,
scripts/plot/ancestor.m, scripts/plot/area.m, scripts/plot/axis.m,
scripts/plot/bar.m, scripts/plot/barh.m, scripts/plot/box.m,
scripts/plot/caxis.m, scripts/plot/cla.m, scripts/plot/clabel.m,
scripts/plot/clf.m, scripts/plot/close.m, scripts/plot/colorbar.m,
scripts/plot/daspect.m, scripts/plot/ezmesh.m, scripts/plot/ezmeshc.m,
scripts/plot/ezsurf.m, scripts/plot/ezsurfc.m, scripts/plot/findall.m,
scripts/plot/findobj.m, scripts/plot/gcbo.m, scripts/plot/gcf.m,
scripts/plot/gco.m, scripts/plot/grid.m, scripts/plot/guihandles.m,
scripts/plot/hdl2struct.m, scripts/plot/hidden.m, scripts/plot/hold.m,
scripts/plot/isonormals.m, scripts/plot/isosurface.m, scripts/plot/legend.m,
scripts/plot/mesh.m, scripts/plot/meshc.m, scripts/plot/meshz.m,
scripts/plot/newplot.m, scripts/plot/orient.m, scripts/plot/pareto.m,
scripts/plot/patch.m, scripts/plot/pbaspect.m, scripts/plot/pcolor.m,
scripts/plot/plot.m, scripts/plot/print.m,
scripts/plot/private/__add_default_menu__.m, scripts/plot/quiver.m,
scripts/plot/quiver3.m, scripts/plot/refreshdata.m, scripts/plot/saveas.m,
scripts/plot/scatter.m, scripts/plot/scatter3.m, scripts/plot/shading.m,
scripts/plot/shrinkfaces.m, scripts/plot/slice.m, scripts/plot/stem.m,
scripts/plot/stem3.m, scripts/plot/struct2hdl.m, scripts/plot/subplot.m,
scripts/plot/surf.m, scripts/plot/surfc.m, scripts/plot/surfl.m,
scripts/plot/tetramesh.m, scripts/plot/uigetfile.m, scripts/plot/uimenu.m,
scripts/plot/uiputfile.m, scripts/plot/waterfall.m, scripts/plot/whitebg.m,
scripts/plot/xlim.m, scripts/plot/ylim.m, scripts/plot/zlim.m,
scripts/polynomial/conv.m, scripts/polynomial/polyout.m,
scripts/polynomial/splinefit.m, scripts/set/ismember.m, scripts/set/powerset.m,
scripts/set/setdiff.m, scripts/set/union.m, scripts/set/unique.m,
scripts/signal/detrend.m, scripts/signal/filter2.m, scripts/signal/freqz.m,
scripts/signal/periodogram.m, scripts/signal/spectral_adf.m,
scripts/signal/spectral_xdf.m, scripts/sparse/eigs.m, scripts/sparse/svds.m,
scripts/specfun/legendre.m, scripts/special-matrix/gallery.m,
scripts/statistics/base/mean.m, scripts/statistics/base/moment.m,
scripts/statistics/tests/cor_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test_2.m,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/sign_test.m,
scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m,
scripts/statistics/tests/t_test_regression.m,
scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m,
scripts/statistics/tests/welch_test.m,
scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m,
scripts/statistics/tests/z_test_2.m, scripts/strings/base2dec.m,
scripts/strings/index.m, scripts/strings/isstrprop.m,
scripts/strings/mat2str.m, scripts/strings/regexptranslate.m,
scripts/strings/rindex.m, scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjust.m, scripts/strings/strmatch.m,
scripts/strings/validatestring.m, scripts/testfun/demo.m,
scripts/testfun/example.m, scripts/testfun/test.m, scripts/time/addtodate.m,
scripts/time/asctime.m, scripts/time/datestr.m, scripts/time/datetick.m,
scripts/time/weekday.m, scripts/ui/errordlg.m, scripts/ui/helpdlg.m,
scripts/ui/inputdlg.m, scripts/ui/listdlg.m, scripts/ui/msgbox.m,
scripts/ui/questdlg.m, scripts/ui/warndlg.m: Use new @qcode macro.
| author | Rik <rik@octave.org> |
|---|---|
| date | Mon, 19 Aug 2013 20:46:38 -0700 |
| parents | f2a8592b8fbd |
| children | 4bcd301754ce |
| rev | line source |
|---|---|
|
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1 @c Copyright (C) 1996-2012 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Numeric Data Types |
| 3294 | 20 @chapter Numeric Data Types |
| 21 @cindex numeric constant | |
| 22 @cindex numeric value | |
| 23 | |
| 24 A @dfn{numeric constant} may be a scalar, a vector, or a matrix, and it | |
| 25 may contain complex values. | |
| 26 | |
| 27 The simplest form of a numeric constant, a scalar, is a single number | |
| 28 that can be an integer, a decimal fraction, a number in scientific | |
| 6620 | 29 (exponential) notation, or a complex number. Note that by default numeric |
| 3294 | 30 constants are represented within Octave in double-precision floating |
| 31 point format (complex constants are stored as pairs of double-precision | |
|
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32 floating point values). It is, however, possible to represent real |
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33 integers as described in @ref{Integer Data Types}. Here are some |
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34 examples of real-valued numeric constants, which all have the same |
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35 value: |
| 3294 | 36 |
| 37 @example | |
| 38 @group | |
| 39 105 | |
| 40 1.05e+2 | |
| 41 1050e-1 | |
| 42 @end group | |
| 43 @end example | |
| 44 | |
| 45 To specify complex constants, you can write an expression of the form | |
| 46 | |
| 47 @example | |
| 48 @group | |
| 49 3 + 4i | |
| 50 3.0 + 4.0i | |
| 51 0.3e1 + 40e-1i | |
| 52 @end group | |
| 53 @end example | |
| 54 | |
| 6620 | 55 @noindent |
| 3294 | 56 all of which are equivalent. The letter @samp{i} in the previous example |
| 57 stands for the pure imaginary constant, defined as | |
| 58 @tex | |
| 59 $\sqrt{-1}$. | |
| 60 @end tex | |
| 6620 | 61 @ifnottex |
| 3294 | 62 @code{sqrt (-1)}. |
| 6620 | 63 @end ifnottex |
| 3294 | 64 |
| 65 For Octave to recognize a value as the imaginary part of a complex | |
| 66 constant, a space must not appear between the number and the @samp{i}. | |
| 67 If it does, Octave will print an error message, like this: | |
| 68 | |
| 69 @example | |
| 70 @group | |
| 71 octave:13> 3 + 4 i | |
| 72 | |
| 73 parse error: | |
| 74 | |
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75 syntax error |
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76 |
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77 >>> 3 + 4 i |
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78 ^ |
| 3294 | 79 @end group |
| 80 @end example | |
| 81 | |
| 6620 | 82 @noindent |
| 3294 | 83 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
| 84 @samp{i} above. All four forms are equivalent. | |
| 85 | |
| 6549 | 86 @DOCSTRING(double) |
| 87 | |
| 88 @DOCSTRING(complex) | |
| 89 | |
| 3294 | 90 @menu |
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91 * Matrices:: |
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92 * Ranges:: |
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93 * Single Precision Data Types:: |
| 6549 | 94 * Integer Data Types:: |
| 6781 | 95 * Bit Manipulations:: |
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96 * Logical Values:: |
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97 * Promotion and Demotion of Data Types:: |
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98 * Predicates for Numeric Objects:: |
| 3294 | 99 @end menu |
| 100 | |
| 4167 | 101 @node Matrices |
| 3294 | 102 @section Matrices |
| 103 @cindex matrices | |
| 104 | |
| 105 @opindex [ | |
| 106 @opindex ] | |
| 107 @opindex ; | |
| 108 @opindex , | |
| 109 | |
| 110 It is easy to define a matrix of values in Octave. The size of the | |
| 111 matrix is determined automatically, so it is not necessary to explicitly | |
| 112 state the dimensions. The expression | |
| 113 | |
| 114 @example | |
| 115 a = [1, 2; 3, 4] | |
| 116 @end example | |
| 117 | |
| 118 @noindent | |
| 119 results in the matrix | |
| 120 @tex | |
| 121 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ | |
| 122 @end tex | |
| 6620 | 123 @ifnottex |
| 3294 | 124 |
| 125 @example | |
| 126 @group | |
| 127 | |
| 128 / \ | |
| 129 | 1 2 | | |
| 130 a = | | | |
| 131 | 3 4 | | |
| 132 \ / | |
| 133 | |
| 134 @end group | |
| 135 @end example | |
|
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136 |
| 6620 | 137 @end ifnottex |
| 3294 | 138 |
| 139 Elements of a matrix may be arbitrary expressions, provided that the | |
| 140 dimensions all make sense when combining the various pieces. For | |
| 141 example, given the above matrix, the expression | |
| 142 | |
| 143 @example | |
| 144 [ a, a ] | |
| 145 @end example | |
| 146 | |
| 147 @noindent | |
| 148 produces the matrix | |
| 149 | |
| 150 @example | |
| 151 @group | |
| 152 ans = | |
| 153 | |
| 154 1 2 1 2 | |
| 155 3 4 3 4 | |
| 156 @end group | |
| 157 @end example | |
| 158 | |
| 159 @noindent | |
| 160 but the expression | |
| 161 | |
| 162 @example | |
| 163 [ a, 1 ] | |
| 164 @end example | |
| 165 | |
| 166 @noindent | |
| 167 produces the error | |
| 168 | |
| 169 @example | |
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170 error: number of rows must match (1 != 2) near line 13, column 6 |
| 3294 | 171 @end example |
| 172 | |
| 173 @noindent | |
| 174 (assuming that this expression was entered as the first thing on line | |
| 175 13, of course). | |
| 176 | |
| 177 Inside the square brackets that delimit a matrix expression, Octave | |
| 178 looks at the surrounding context to determine whether spaces and newline | |
| 179 characters should be converted into element and row separators, or | |
| 4476 | 180 simply ignored, so an expression like |
| 3294 | 181 |
| 182 @example | |
| 183 @group | |
| 184 a = [ 1 2 | |
| 185 3 4 ] | |
| 186 @end group | |
| 187 @end example | |
| 188 | |
| 189 @noindent | |
| 190 will work. However, some possible sources of confusion remain. For | |
| 191 example, in the expression | |
| 192 | |
| 193 @example | |
| 194 [ 1 - 1 ] | |
| 195 @end example | |
| 196 | |
| 197 @noindent | |
| 198 the @samp{-} is treated as a binary operator and the result is the | |
| 199 scalar 0, but in the expression | |
| 200 | |
| 201 @example | |
| 202 [ 1 -1 ] | |
| 203 @end example | |
| 204 | |
| 205 @noindent | |
| 206 the @samp{-} is treated as a unary operator and the result is the | |
| 4476 | 207 vector @code{[ 1, -1 ]}. Similarly, the expression |
| 208 | |
| 209 @example | |
| 210 [ sin (pi) ] | |
| 211 @end example | |
| 212 | |
| 213 @noindent | |
| 214 will be parsed as | |
| 215 | |
| 216 @example | |
| 217 [ sin, (pi) ] | |
| 218 @end example | |
| 3294 | 219 |
| 4476 | 220 @noindent |
| 221 and will result in an error since the @code{sin} function will be | |
| 222 called with no arguments. To get around this, you must omit the space | |
| 223 between @code{sin} and the opening parenthesis, or enclose the | |
| 224 expression in a set of parentheses: | |
| 225 | |
| 226 @example | |
| 227 [ (sin (pi)) ] | |
| 228 @end example | |
| 229 | |
| 230 Whitespace surrounding the single quote character (@samp{'}, used as a | |
| 231 transpose operator and for delimiting character strings) can also cause | |
| 232 confusion. Given @code{a = 1}, the expression | |
| 3294 | 233 |
| 234 @example | |
| 235 [ 1 a' ] | |
| 236 @end example | |
| 237 | |
| 238 @noindent | |
| 4476 | 239 results in the single quote character being treated as a |
| 3294 | 240 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
| 241 expression | |
| 242 | |
| 243 @example | |
| 244 [ 1 a ' ] | |
| 245 @end example | |
| 246 | |
| 247 @noindent | |
| 248 produces the error message | |
| 249 | |
| 250 @example | |
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251 @group |
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252 parse error: |
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253 |
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254 syntax error |
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255 |
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256 >>> [ 1 a ' ] |
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257 ^ |
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258 @end group |
| 3294 | 259 @end example |
| 260 | |
| 261 @noindent | |
| 6620 | 262 because not doing so would cause trouble when parsing the valid expression |
| 3294 | 263 |
| 264 @example | |
| 265 [ a 'foo' ] | |
| 266 @end example | |
| 267 | |
| 268 For clarity, it is probably best to always use commas and semicolons to | |
| 4476 | 269 separate matrix elements and rows. |
| 3294 | 270 |
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271 The maximum number of elements in a matrix is fixed when Octave is compiled. |
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272 The allowable number can be queried with the function @code{sizemax}. Note |
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273 that other factors, such as the amount of memory available on your machine, |
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274 may limit the maximum size of matrices to something smaller. |
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275 |
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276 @DOCSTRING(sizemax) |
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277 |
| 3294 | 278 When you type a matrix or the name of a variable whose value is a |
| 279 matrix, Octave responds by printing the matrix in with neatly aligned | |
| 280 rows and columns. If the rows of the matrix are too large to fit on the | |
| 281 screen, Octave splits the matrix and displays a header before each | |
| 282 section to indicate which columns are being displayed. You can use the | |
| 283 following variables to control the format of the output. | |
| 284 | |
| 3321 | 285 @DOCSTRING(output_max_field_width) |
| 3294 | 286 |
| 3321 | 287 @DOCSTRING(output_precision) |
| 3294 | 288 |
| 289 It is possible to achieve a wide range of output styles by using | |
| 290 different values of @code{output_precision} and | |
| 291 @code{output_max_field_width}. Reasonable combinations can be set using | |
| 292 the @code{format} function. @xref{Basic Input and Output}. | |
| 293 | |
| 3321 | 294 @DOCSTRING(split_long_rows) |
| 3294 | 295 |
| 296 Octave automatically switches to scientific notation when values become | |
| 297 very large or very small. This guarantees that you will see several | |
| 298 significant figures for every value in a matrix. If you would prefer to | |
| 299 see all values in a matrix printed in a fixed point format, you can set | |
| 300 the built-in variable @code{fixed_point_format} to a nonzero value. But | |
| 301 doing so is not recommended, because it can produce output that can | |
| 302 easily be misinterpreted. | |
| 303 | |
| 3321 | 304 @DOCSTRING(fixed_point_format) |
| 3294 | 305 |
| 306 @menu | |
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307 * Empty Matrices:: |
| 3294 | 308 @end menu |
| 309 | |
| 4167 | 310 @node Empty Matrices |
| 3294 | 311 @subsection Empty Matrices |
| 312 | |
| 313 A matrix may have one or both dimensions zero, and operations on empty | |
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314 matrices are handled as described by Carl @nospell{de} Boor in @cite{An Empty |
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315 Exercise}, SIGNUM, Volume 25, pages 2-6, 1990 and C. N. Nett and W. M. |
| 3294 | 316 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
| 317 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, | |
| 318 Number 5, May 1993. | |
| 319 @tex | |
| 320 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, | |
| 321 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or | |
| 322 both dimensions equal to zero), the following are true: | |
| 323 $$ | |
| 324 \eqalign{% | |
| 325 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr | |
| 326 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr | |
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327 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr |
| 3294 | 328 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr |
| 329 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} | |
| 330 $$ | |
| 331 @end tex | |
| 6620 | 332 @ifnottex |
| 3294 | 333 Briefly, given a scalar @var{s}, an @var{m} by |
| 334 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix | |
| 335 @code{[](mxn)} (with either one or both dimensions equal to zero), the | |
| 336 following are true: | |
| 337 | |
| 338 @example | |
| 339 @group | |
| 340 s * [](mxn) = [](mxn) * s = [](mxn) | |
| 341 | |
| 342 [](mxn) + [](mxn) = [](mxn) | |
| 343 | |
| 344 [](0xm) * M(mxn) = [](0xn) | |
| 345 | |
| 346 M(mxn) * [](nx0) = [](mx0) | |
| 347 | |
| 348 [](mx0) * [](0xn) = 0(mxn) | |
| 349 @end group | |
| 350 @end example | |
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351 |
| 6620 | 352 @end ifnottex |
| 3294 | 353 |
| 354 By default, dimensions of the empty matrix are printed along with the | |
| 355 empty matrix symbol, @samp{[]}. The built-in variable | |
| 356 @code{print_empty_dimensions} controls this behavior. | |
| 357 | |
| 3321 | 358 @DOCSTRING(print_empty_dimensions) |
| 3294 | 359 |
| 360 Empty matrices may also be used in assignment statements as a convenient | |
| 361 way to delete rows or columns of matrices. | |
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362 @xref{Assignment Ops,,Assignment Expressions}. |
| 3294 | 363 |
| 364 When Octave parses a matrix expression, it examines the elements of the | |
| 365 list to determine whether they are all constants. If they are, it | |
| 366 replaces the list with a single matrix constant. | |
| 367 | |
| 4167 | 368 @node Ranges |
| 3294 | 369 @section Ranges |
| 370 @cindex range expressions | |
| 371 @cindex expression, range | |
| 372 | |
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373 @opindex : |
| 3294 | 374 |
| 375 A @dfn{range} is a convenient way to write a row vector with evenly | |
| 376 spaced elements. A range expression is defined by the value of the first | |
| 377 element in the range, an optional value for the increment between | |
| 378 elements, and a maximum value which the elements of the range will not | |
| 379 exceed. The base, increment, and limit are separated by colons (the | |
| 380 @samp{:} character) and may contain any arithmetic expressions and | |
| 381 function calls. If the increment is omitted, it is assumed to be 1. | |
| 382 For example, the range | |
| 383 | |
| 384 @example | |
| 385 1 : 5 | |
| 386 @end example | |
| 387 | |
| 388 @noindent | |
| 389 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range | |
| 390 | |
| 391 @example | |
| 392 1 : 3 : 5 | |
| 393 @end example | |
| 394 | |
| 395 @noindent | |
| 396 defines the set of values @samp{[ 1, 4 ]}. | |
| 397 | |
| 398 Although a range constant specifies a row vector, Octave does @emph{not} | |
| 399 convert range constants to vectors unless it is necessary to do so. | |
| 400 This allows you to write a constant like @samp{1 : 10000} without using | |
| 401 80,000 bytes of storage on a typical 32-bit workstation. | |
| 402 | |
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403 A common example of when it does become necessary to convert ranges into |
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404 vectors occurs when they appear within a vector (i.e., inside square |
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405 brackets). For instance, whereas |
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406 |
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407 @example |
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408 x = 0 : 0.1 : 1; |
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409 @end example |
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410 |
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411 @noindent |
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412 defines @var{x} to be a variable of type @code{range} and occupies 24 |
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413 bytes of memory, the expression |
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414 |
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415 @example |
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416 y = [ 0 : 0.1 : 1]; |
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417 @end example |
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418 |
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419 @noindent |
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420 defines @var{y} to be of type @code{matrix} and occupies 88 bytes of |
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421 memory. |
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422 |
| 3294 | 423 Note that the upper (or lower, if the increment is negative) bound on |
| 424 the range is not always included in the set of values, and that ranges | |
| 425 defined by floating point values can produce surprising results because | |
| 426 Octave uses floating point arithmetic to compute the values in the | |
| 427 range. If it is important to include the endpoints of a range and the | |
| 428 number of elements is known, you should use the @code{linspace} function | |
| 429 instead (@pxref{Special Utility Matrices}). | |
| 430 | |
| 9158 | 431 When adding a scalar to a range, subtracting a scalar from it (or subtracting a |
| 432 range from a scalar) and multiplying by scalar, Octave will attempt to avoid | |
| 433 unpacking the range and keep the result as a range, too, if it can determine | |
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434 that it is safe to do so. For instance, doing |
| 9158 | 435 |
| 436 @example | |
| 437 a = 2*(1:1e7) - 1; | |
| 438 @end example | |
| 439 | |
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440 @noindent |
| 9158 | 441 will produce the same result as @samp{1:2:2e7-1}, but without ever forming a |
| 442 vector with ten million elements. | |
| 443 | |
| 444 Using zero as an increment in the colon notation, as @samp{1:0:1} is not | |
| 445 allowed, because a division by zero would occur in determining the number of | |
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446 range elements. However, ranges with zero increment (i.e., all elements equal) |
| 9158 | 447 are useful, especially in indexing, and Octave allows them to be constructed |
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448 using the built-in function @dfn{ones}. Note that because a range must be a row |
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449 vector, @samp{ones (1, 10)} produces a range, while @samp{ones (10, 1)} does |
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450 not. |
| 9158 | 451 |
| 3294 | 452 When Octave parses a range expression, it examines the elements of the |
| 453 expression to determine whether they are all constants. If they are, it | |
| 454 replaces the range expression with a single range constant. | |
| 455 | |
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456 @node Single Precision Data Types |
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457 @section Single Precision Data Types |
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458 |
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459 Octave includes support for single precision data types, and most of the |
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460 functions in Octave accept single precision values and return single |
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461 precision answers. A single precision variable is created with the |
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462 @code{single} function. |
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463 |
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464 @DOCSTRING(single) |
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465 |
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466 for example: |
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467 |
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468 @example |
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469 @group |
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470 sngl = single (rand (2, 2)) |
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471 @result{} sngl = |
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472 0.37569 0.92982 |
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473 0.11962 0.50876 |
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474 class (sngl) |
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475 @result{} single |
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476 @end group |
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477 @end example |
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478 |
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479 Many functions can also return single precision values directly. For |
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480 example |
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481 |
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482 @example |
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483 @group |
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484 ones (2, 2, "single") |
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485 zeros (2, 2, "single") |
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486 eye (2, 2, "single") |
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487 rand (2, 2, "single") |
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488 NaN (2, 2, "single") |
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489 NA (2, 2, "single") |
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490 Inf (2, 2, "single") |
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491 @end group |
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492 @end example |
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493 |
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494 @noindent |
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495 will all return single precision matrices. |
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496 |
| 6549 | 497 @node Integer Data Types |
| 498 @section Integer Data Types | |
| 499 | |
| 6620 | 500 Octave supports integer matrices as an alternative to using double |
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501 precision. It is possible to use both signed and unsigned integers |
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502 represented by 8, 16, 32, or 64 bits. It should be noted that most |
| 6620 | 503 computations require floating point data, meaning that integers will |
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504 often change type when involved in numeric computations. For this |
| 6620 | 505 reason integers are most often used to store data, and not for |
| 506 calculations. | |
| 507 | |
| 508 In general most integer matrices are created by casting | |
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509 existing matrices to integers. The following example shows how to cast |
| 6620 | 510 a matrix into 32 bit integers. |
| 511 | |
| 512 @example | |
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513 @group |
| 6620 | 514 float = rand (2, 2) |
| 515 @result{} float = 0.37569 0.92982 | |
| 516 0.11962 0.50876 | |
| 517 integer = int32 (float) | |
| 518 @result{} integer = 0 1 | |
| 519 0 1 | |
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520 @end group |
| 6620 | 521 @end example |
| 522 | |
| 523 @noindent | |
| 524 As can be seen, floating point values are rounded to the nearest integer | |
| 525 when converted. | |
| 526 | |
| 6549 | 527 @DOCSTRING(isinteger) |
| 528 | |
| 529 @DOCSTRING(int8) | |
| 530 | |
| 531 @DOCSTRING(uint8) | |
| 532 | |
| 533 @DOCSTRING(int16) | |
| 534 | |
| 535 @DOCSTRING(uint16) | |
| 536 | |
| 537 @DOCSTRING(int32) | |
| 538 | |
| 539 @DOCSTRING(uint32) | |
| 540 | |
| 541 @DOCSTRING(int64) | |
| 542 | |
| 543 @DOCSTRING(uint64) | |
| 544 | |
| 545 @DOCSTRING(intmax) | |
| 546 | |
| 547 @DOCSTRING(intmin) | |
| 548 | |
| 6624 | 549 @menu |
| 550 * Integer Arithmetic:: | |
| 551 @end menu | |
| 552 | |
| 6620 | 553 @node Integer Arithmetic |
| 554 @subsection Integer Arithmetic | |
| 555 | |
| 556 While many numerical computations can't be carried out in integers, | |
| 557 Octave does support basic operations like addition and multiplication | |
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558 on integers. The operators @code{+}, @code{-}, @code{.*}, and @code{./} |
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559 work on integers of the same type. So, it is possible to add two 32 bit |
| 6620 | 560 integers, but not to add a 32 bit integer and a 16 bit integer. |
| 561 | |
| 562 When doing integer arithmetic one should consider the possibility of | |
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563 underflow and overflow. This happens when the result of the computation |
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564 can't be represented using the chosen integer type. As an example it is |
| 6620 | 565 not possible to represent the result of @math{10 - 20} when using |
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566 unsigned integers. Octave makes sure that the result of integer |
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567 computations is the integer that is closest to the true result. So, the |
| 6620 | 568 result of @math{10 - 20} when using unsigned integers is zero. |
| 569 | |
| 570 When doing integer division Octave will round the result to the nearest | |
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571 integer. This is different from most programming languages, where the |
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572 result is often floored to the nearest integer. So, the result of |
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573 @code{int32 (5) ./ int32 (8)} is @code{1}. |
| 6620 | 574 |
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575 @DOCSTRING(idivide) |
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576 |
| 6781 | 577 @node Bit Manipulations |
| 578 @section Bit Manipulations | |
| 579 | |
| 580 Octave provides a number of functions for the manipulation of numeric | |
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581 values on a bit by bit basis. The basic functions to set and obtain the |
| 6781 | 582 values of individual bits are @code{bitset} and @code{bitget}. |
| 583 | |
| 584 @DOCSTRING(bitset) | |
| 585 | |
| 586 @DOCSTRING(bitget) | |
| 587 | |
| 588 The arguments to all of Octave's bitwise operations can be scalar or | |
| 589 arrays, except for @code{bitcmp}, whose @var{k} argument must a | |
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590 scalar. In the case where more than one argument is an array, then all |
| 6781 | 591 arguments must have the same shape, and the bitwise operator is applied |
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592 to each of the elements of the argument individually. If at least one |
| 6781 | 593 argument is a scalar and one an array, then the scalar argument is |
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594 duplicated. Therefore |
| 6781 | 595 |
| 596 @example | |
| 597 bitget (100, 8:-1:1) | |
| 598 @end example | |
| 599 | |
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600 @noindent |
| 6781 | 601 is the same as |
| 602 | |
| 603 @example | |
| 604 bitget (100 * ones (1, 8), 8:-1:1) | |
| 605 @end example | |
| 606 | |
| 607 It should be noted that all values passed to the bit manipulation | |
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608 functions of Octave are treated as integers. Therefore, even though the |
| 6781 | 609 example for @code{bitset} above passes the floating point value |
| 610 @code{10}, it is treated as the bits @code{[1, 0, 1, 0]} rather than the | |
| 611 bits of the native floating point format representation of @code{10}. | |
| 612 | |
| 8828 | 613 As the maximum value that can be represented by a number is important |
| 6781 | 614 for bit manipulation, particularly when forming masks, Octave supplies |
| 615 the function @code{bitmax}. | |
| 616 | |
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617 @anchor{XREFflintmax} |
| 6781 | 618 @DOCSTRING(bitmax) |
| 619 | |
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620 This is the double precision version of the function @code{intmax}, |
| 6781 | 621 previously discussed. |
| 622 | |
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623 Octave also includes the basic bitwise 'and', 'or', and 'exclusive or' |
| 6781 | 624 operators. |
| 625 | |
| 626 @DOCSTRING(bitand) | |
| 627 | |
| 628 @DOCSTRING(bitor) | |
| 629 | |
| 630 @DOCSTRING(bitxor) | |
| 631 | |
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632 The bitwise 'not' operator is a unary operator that performs a logical |
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633 negation of each of the bits of the value. For this to make sense, the |
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634 mask against which the value is negated must be defined. Octave's |
| 6781 | 635 bitwise 'not' operator is @code{bitcmp}. |
| 636 | |
| 637 @DOCSTRING(bitcmp) | |
| 638 | |
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639 Octave also includes the ability to left-shift and right-shift values bitwise. |
| 6781 | 640 |
| 641 @DOCSTRING(bitshift) | |
| 642 | |
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643 Bits that are shifted out of either end of the value are lost. Octave |
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644 also uses arithmetic shifts, where the sign bit of the value is kept |
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645 during a right shift. For example: |
| 6781 | 646 |
| 647 @example | |
| 648 @group | |
| 649 bitshift (-10, -1) | |
| 650 @result{} -5 | |
| 651 bitshift (int8 (-1), -1) | |
| 652 @result{} -1 | |
| 653 @end group | |
| 654 @end example | |
| 655 | |
| 656 Note that @code{bitshift (int8 (-1), -1)} is @code{-1} since the bit | |
| 657 representation of @code{-1} in the @code{int8} data type is @code{[1, 1, | |
| 658 1, 1, 1, 1, 1, 1]}. | |
| 659 | |
| 4167 | 660 @node Logical Values |
| 3428 | 661 @section Logical Values |
| 662 | |
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663 Octave has built-in support for logical values, i.e., variables that |
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664 are either @code{true} or @code{false}. When comparing two variables, |
| 6620 | 665 the result will be a logical value whose value depends on whether or |
| 666 not the comparison is true. | |
| 667 | |
| 668 The basic logical operations are @code{&}, @code{|}, and @code{!}, | |
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669 which correspond to ``Logical And'', ``Logical Or'', and ``Logical |
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670 Negation''. These operations all follow the usual rules of logic. |
| 6620 | 671 |
| 672 It is also possible to use logical values as part of standard numerical | |
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673 calculations. In this case @code{true} is converted to @code{1}, and |
| 6620 | 674 @code{false} to 0, both represented using double precision floating |
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675 point numbers. So, the result of @code{true*22 - false/6} is @code{22}. |
| 6620 | 676 |
| 677 Logical values can also be used to index matrices and cell arrays. | |
| 678 When indexing with a logical array the result will be a vector containing | |
| 679 the values corresponding to @code{true} parts of the logical array. | |
| 680 The following example illustrates this. | |
| 681 | |
| 682 @example | |
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683 @group |
| 6620 | 684 data = [ 1, 2; 3, 4 ]; |
| 685 idx = (data <= 2); | |
| 686 data(idx) | |
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687 @result{} ans = [ 1; 2 ] |
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688 @end group |
| 6620 | 689 @end example |
| 690 | |
| 691 @noindent | |
| 692 Instead of creating the @code{idx} array it is possible to replace | |
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693 @code{data(idx)} with @w{@code{data( data <= 2 )}} in the above code. |
| 6620 | 694 |
| 8828 | 695 Logical values can also be constructed by |
| 6620 | 696 casting numeric objects to logical values, or by using the @code{true} |
| 697 or @code{false} functions. | |
| 698 | |
| 6549 | 699 @DOCSTRING(logical) |
| 700 | |
| 3428 | 701 @DOCSTRING(true) |
| 702 | |
| 703 @DOCSTRING(false) | |
| 704 | |
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705 @node Promotion and Demotion of Data Types |
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706 @section Promotion and Demotion of Data Types |
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707 |
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708 Many operators and functions can work with mixed data types. For example, |
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709 |
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710 @example |
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711 @group |
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712 uint8 (1) + 1 |
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713 @result{} 2 |
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714 @end group |
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715 @end example |
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716 |
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717 @noindent |
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718 where the above operator works with an 8-bit integer and a double precision |
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719 value and returns an 8-bit integer value. Note that the type is demoted |
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720 to an 8-bit integer, rather than promoted to a double precision value as |
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721 might be expected. The reason is that if Octave promoted values in |
|
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722 expressions like the above with all numerical constants would need to be |
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723 explicitly cast to the appropriate data type like |
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724 |
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725 @example |
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726 @group |
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727 uint8 (1) + uint8 (1) |
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728 @result{} 2 |
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729 @end group |
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730 @end example |
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731 |
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732 @noindent |
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733 which becomes difficult for the user to apply uniformly and might allow |
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734 hard to find bugs to be introduced. The same applies to single precision |
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735 values where a mixed operation such as |
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736 |
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737 @example |
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738 @group |
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739 single (1) + 1 |
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740 @result{} 2 |
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741 @end group |
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742 @end example |
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743 |
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744 @noindent |
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745 returns a single precision value. The mixed operations that are valid |
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746 and their returned data types are |
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747 |
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748 @multitable @columnfractions .2 .3 .3 .2 |
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749 @headitem @tab Mixed Operation @tab Result @tab |
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750 @item @tab double OP single @tab single @tab |
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751 @item @tab double OP integer @tab integer @tab |
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752 @item @tab double OP char @tab double @tab |
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753 @item @tab double OP logical @tab double @tab |
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754 @item @tab single OP integer @tab integer @tab |
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755 @item @tab single OP char @tab single @tab |
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756 @item @tab single OP logical @tab single @tab |
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757 @end multitable |
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758 |
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759 The same logic applies to functions with mixed arguments such as |
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760 |
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761 @example |
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762 @group |
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763 min (single (1), 0) |
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764 @result{} 0 |
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765 @end group |
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766 @end example |
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767 |
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768 @noindent |
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769 where the returned value is single precision. |
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770 |
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771 In the case of mixed type indexed assignments, the type is not |
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772 changed. For example, |
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773 |
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774 @example |
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775 @group |
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776 x = ones (2, 2); |
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777 x(1, 1) = single (2) |
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778 @result{} x = 2 1 |
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779 1 1 |
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780 @end group |
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781 @end example |
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782 |
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783 @noindent |
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784 where @code{x} remains of the double precision type. |
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785 |
| 4167 | 786 @node Predicates for Numeric Objects |
| 3294 | 787 @section Predicates for Numeric Objects |
| 788 | |
| 6620 | 789 Since the type of a variable may change during the execution of a |
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790 program, it can be necessary to do type checking at run-time. Doing this |
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791 also allows you to change the behavior of a function depending on the |
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792 type of the input. As an example, this naive implementation of @code{abs} |
| 7001 | 793 returns the absolute value of the input if it is a real number, and the |
| 6620 | 794 length of the input if it is a complex number. |
| 795 | |
| 796 @example | |
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797 @group |
| 6620 | 798 function a = abs (x) |
| 799 if (isreal (x)) | |
| 800 a = sign (x) .* x; | |
| 801 elseif (iscomplex (x)) | |
| 802 a = sqrt (real(x).^2 + imag(x).^2); | |
| 803 endif | |
| 804 endfunction | |
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805 @end group |
| 6620 | 806 @end example |
| 807 | |
| 808 The following functions are available for determining the type of a | |
| 809 variable. | |
| 810 | |
| 3428 | 811 @DOCSTRING(isnumeric) |
| 812 | |
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813 @DOCSTRING(isfloat) |
| 3428 | 814 |
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815 @DOCSTRING(isreal) |
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816 |
| 4029 | 817 @DOCSTRING(iscomplex) |
| 3428 | 818 |
| 4029 | 819 @DOCSTRING(ismatrix) |
| 3294 | 820 |
| 4029 | 821 @DOCSTRING(isvector) |
| 3294 | 822 |
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823 @DOCSTRING(isrow) |
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824 |
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825 @DOCSTRING(iscolumn) |
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826 |
| 4029 | 827 @DOCSTRING(isscalar) |
| 3294 | 828 |
| 4029 | 829 @DOCSTRING(issquare) |
| 3294 | 830 |
| 4029 | 831 @DOCSTRING(issymmetric) |
| 3428 | 832 |
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833 @DOCSTRING(ishermitian) |
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834 |
| 6550 | 835 @DOCSTRING(isdefinite) |
| 836 | |
| 6688 | 837 @DOCSTRING(islogical) |
| 6550 | 838 |
| 839 @DOCSTRING(isprime) | |
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840 |
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841 If instead of knowing properties of variables, you wish to know which |
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842 variables are defined and to gather other information about the |
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843 workspace itself, @pxref{Status of Variables}. |