Mercurial > octave

comparison libinterp/corefcn/qr.cc @ 31376:ad8a4102f910

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
doc: Spellcheck documentation for 8.1 release. * aspell-octave.en.pws: Update private dictionary with new Octave-specific words and remove words no longer required. * install.txi, preface.txi, vectorize.txi, interpreter.cc, qr.cc, uimenu.m, uiputfile.m: Correct spellings and/or use @nospell macro to remove words from spellchecking.
author Rik <rik@octave.org>
date Mon, 31 Oct 2022 10:01:50 -0700
parents e39634a768e7
children 23520a50d74d
comparison
equal deleted inserted replaced
31375:96f751f8392c 31376:ad8a4102f910
248 [@var{C}, @var{R}] = qr (@var{A}, @var{B}) 248 [@var{C}, @var{R}] = qr (@var{A}, @var{B})
249 @var{X} = @var{R} \ @var{C} 249 @var{X} = @var{R} \ @var{C}
250 @end group 250 @end group
251 @end example 251 @end example
252 252
253 If @var{A} is a sparse MxN matrix and an additional matrix @var{B} is 253 If @var{A} is a sparse @nospell{MxN} matrix and an additional matrix @var{B} is
254 supplied, one or two return values are possible. If one return value @var{X} 254 supplied, one or two return values are possible. If one return value @var{X}
255 is requested and M < N, then @var{X} is the minimum 2-norm solution of 255 is requested and M < N, then @var{X} is the minimum 2-norm solution of
256 @w{@code{@var{A} \ @var{B}}}. If M >= N, @var{X} is the least squares 256 @w{@code{@var{A} \ @var{B}}}. If M >= N, @var{X} is the least squares
257 approximation @w{of @code{@var{A} \ @var{B}}}. If two return values are 257 approximation @w{of @code{@var{A} \ @var{B}}}. If two return values are
258 requested, @var{C} and @var{R} have the same meaning as in the dense case 258 requested, @var{C} and @var{R} have the same meaning as in the dense case
259 (@var{C} is dense and @var{R} is sparse). 259 (@var{C} is dense and @var{R} is sparse). The version with one return
260 The version with one return parameter should be preferred because 260 parameter should be preferred because it uses less memory and can handle
261 it uses less memory and can handle rank-deficient matrices better. 261 rank-deficient matrices better.
262 262
263 If the final argument is the string @qcode{"vector"} then @var{P} is a 263 If the final argument is the string @qcode{"vector"} then @var{P} is a
264 permutation vector (of the columns of @var{A}) instead of a permutation 264 permutation vector (of the columns of @var{A}) instead of a permutation
265 matrix. In this case, the defining relationship is: 265 matrix. In this case, the defining relationship is:
266 266
269 @end example 269 @end example
270 270
271 The default, however, is to return a permutation matrix and this may be 271 The default, however, is to return a permutation matrix and this may be
272 explicitly specified by using a final argument of @qcode{"matrix"}. 272 explicitly specified by using a final argument of @qcode{"matrix"}.
273 273
274 If the final argument is the scalar 0 or the string @qcode{"econ"}, an 274 If the final argument is the scalar 0 or the string @qcode{"econ"}, an economy
275 economy factorization is returned. If the original matrix @var{A} has size 275 factorization is returned. If the original matrix @var{A} has size
276 MxN and M > N, then the economy factorization will calculate just N rows in 276 @nospell{MxN} and M > N, then the economy factorization will calculate just N
277 @var{R} and N columns in @var{Q} and omit the zeros in @var{R}. If M @leq{} 277 rows in @var{R} and N columns in @var{Q} and omit the zeros in @var{R}. If M
278 N, there is no difference between the economy and standard factorizations. 278 @leq{} N, there is no difference between the economy and standard
279 When calculating an economy factorization and @var{A} is dense, the output 279 factorizations. When calculating an economy factorization and @var{A} is
280 @var{P} is always a vector rather than a matrix. If @var{A} is sparse, 280 dense, the output @var{P} is always a vector rather than a matrix. If @var{A}
281 output @var{P} is a sparse permutation matrix. 281 is sparse, output @var{P} is a sparse permutation matrix.
282 282
283 Background: The QR factorization has applications in the solution of least 283 Background: The QR factorization has applications in the solution of least
284 squares problems 284 squares problems
285 @tex 285 @tex
286 $$ 286 $$