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maint: Periodic merge of gui-release to default.
author John W. Eaton <jwe@octave.org>
date Tue, 20 Jan 2015 10:24:46 -0500
parents 9e5b64b3c1fe 446c46af4b42
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4 @c 4 @c
5 @c Octave is free software; you can redistribute it and/or modify it 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 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 7 @c Free Software Foundation; either version 3 of the License, or (at
8 @c your option) any later version. 8 @c your option) any later version.
9 @c 9 @c
10 @c Octave is distributed in the hope that it will be useful, but WITHOUT 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 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 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
13 @c for more details. 13 @c for more details.
14 @c 14 @c
15 @c You should have received a copy of the GNU General Public License 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 16 @c along with Octave; see the file COPYING. If not, see
17 @c <http://www.gnu.org/licenses/>. 17 @c <http://www.gnu.org/licenses/>.
18 18
19 @node Diagonal and Permutation Matrices 19 @node Diagonal and Permutation Matrices
31 31
32 @node Basic Usage 32 @node Basic Usage
33 @section Creating and Manipulating Diagonal/Permutation Matrices 33 @section Creating and Manipulating Diagonal/Permutation Matrices
34 34
35 A diagonal matrix is defined as a matrix that has zero entries outside the main 35 A diagonal matrix is defined as a matrix that has zero entries outside the main
36 diagonal; that is, 36 diagonal; that is,
37 @tex 37 @tex
38 $D_{ij} = 0$ if $i \neq j$ 38 $D_{ij} = 0$ if $i \neq j$
39 @end tex 39 @end tex
40 @ifnottex 40 @ifnottex
41 @code{D(i,j) == 0} if @code{i != j}. 41 @code{D(i,j) == 0} if @code{i != j}.
44 equally be applied to non-square matrices, in which case we usually speak of a 44 equally be applied to non-square matrices, in which case we usually speak of a
45 rectangular diagonal matrix. 45 rectangular diagonal matrix.
46 46
47 A permutation matrix is defined as a square matrix that has a single element 47 A permutation matrix is defined as a square matrix that has a single element
48 equal to unity in each row and each column; all other elements are zero. That 48 equal to unity in each row and each column; all other elements are zero. That
49 is, there exists a permutation (vector) 49 is, there exists a permutation (vector)
50 @tex 50 @tex
51 $p$ such that $P_{ij}=1$ if $j = p_i$ and 51 $p$ such that $P_{ij}=1$ if $j = p_i$ and
52 $P_{ij}=0$ otherwise. 52 $P_{ij}=0$ otherwise.
53 @end tex 53 @end tex
54 @ifnottex 54 @ifnottex
55 @code{p} such that @code{P(i,j) == 1} if @code{j == p(i)} and 55 @code{p} such that @code{P(i,j) == 1} if @code{j == p(i)} and
56 @code{P(i,j) == 0} otherwise. 56 @code{P(i,j) == 0} otherwise.
57 @end ifnottex 57 @end ifnottex
58 58
59 Octave provides special treatment of real and complex rectangular diagonal 59 Octave provides special treatment of real and complex rectangular diagonal
60 matrices, as well as permutation matrices. They are stored as special objects, 60 matrices, as well as permutation matrices. They are stored as special objects,
61 using efficient storage and algorithms, facilitating writing both readable and 61 using efficient storage and algorithms, facilitating writing both readable and
122 1 0 0 122 1 0 0
123 0 2 0 123 0 2 0
124 0 0 3 124 0 0 3
125 0 0 0 125 0 0 0
126 0 0 0 126 0 0 0
127 @end example 127 @end example
128 128
129 @node Creating Permutation Matrices 129 @node Creating Permutation Matrices
130 @subsection Creating Permutation Matrices 130 @subsection Creating Permutation Matrices
131 131
132 For creating permutation matrices, Octave does not introduce a new function, but 132 For creating permutation matrices, Octave does not introduce a new function, but
140 140
141 @noindent 141 @noindent
142 will create a permutation matrix - a special matrix object. 142 will create a permutation matrix - a special matrix object.
143 143
144 @example 144 @example
145 eye (n) (q, :) 145 eye (n) (q, :)
146 @end example 146 @end example
147 147
148 @noindent 148 @noindent
149 will also work (and create a row permutation matrix), as well as 149 will also work (and create a row permutation matrix), as well as
150 150
151 @example 151 @example
152 eye (n) (q1, q2). 152 eye (n) (q1, q2).
153 @end example 153 @end example
154 154
248 @cindex diagonal matrix expressions 248 @cindex diagonal matrix expressions
249 249
250 Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix, 250 Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix,
251 then @code{D*M} will scale the rows of @var{M}. That means, 251 then @code{D*M} will scale the rows of @var{M}. That means,
252 if @code{S = D*M}, then for each pair of indices 252 if @code{S = D*M}, then for each pair of indices
253 i,j it holds 253 i,j it holds
254 @tex 254 @tex
255 $$S_{ij} = D_{ii} M_{ij}$$ 255 $$S_{ij} = D_{ii} M_{ij}$$
256 @end tex 256 @end tex
257 @ifnottex 257 @ifnottex
258 258
269 @example 269 @example
270 D(:,1:m) * M(1:m,:), 270 D(:,1:m) * M(1:m,:),
271 @end example 271 @end example
272 272
273 @noindent 273 @noindent
274 i.e., trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n}, 274 i.e., trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n},
275 then @code{D*M} is equivalent to 275 then @code{D*M} is equivalent to
276 276
277 @example 277 @example
278 [D(1:n,n) * M; zeros(m-n, columns (M))], 278 [D(1:n,n) * M; zeros(m-n, columns (M))],
279 @end example 279 @end example
280 280
288 in a least-squares minimum-norm sense. In exact arithmetic, this is 288 in a least-squares minimum-norm sense. In exact arithmetic, this is
289 equivalent to multiplying by a pseudoinverse. The pseudoinverse of 289 equivalent to multiplying by a pseudoinverse. The pseudoinverse of
290 a rectangular diagonal matrix is again a rectangular diagonal matrix 290 a rectangular diagonal matrix is again a rectangular diagonal matrix
291 with swapped dimensions, where each nonzero diagonal element is replaced 291 with swapped dimensions, where each nonzero diagonal element is replaced
292 by its reciprocal. 292 by its reciprocal.
293 The matrix division algorithms do, in fact, use division rather than 293 The matrix division algorithms do, in fact, use division rather than
294 multiplication by reciprocals for better numerical accuracy; otherwise, they 294 multiplication by reciprocals for better numerical accuracy; otherwise, they
295 honor the above definition. Note that a diagonal matrix is never truncated due 295 honor the above definition. Note that a diagonal matrix is never truncated due
296 to ill-conditioning; otherwise, it would not be of much use for scaling. This 296 to ill-conditioning; otherwise, it would not be of much use for scaling. This
297 is typically consistent with linear algebra needs. A full matrix that only 297 is typically consistent with linear algebra needs. A full matrix that only
298 happens to be diagonal (and is thus not a special object) is of course treated 298 happens to be diagonal (and is thus not a special object) is of course treated
307 If @var{D1} and @var{D2} are both diagonal matrices, then the expressions 307 If @var{D1} and @var{D2} are both diagonal matrices, then the expressions
308 308
309 @example 309 @example
310 @group 310 @group
311 D1 + D2 311 D1 + D2
312 D1 - D2 312 D1 - D2
313 D1 * D2 313 D1 * D2
314 D1 / D2 314 D1 / D2
315 D1 \ D2 315 D1 \ D2
316 @end group 316 @end group
317 @end example 317 @end example
318 318
319 @noindent 319 @noindent
321 dimension matching rules are obeyed. The relations used are same as described 321 dimension matching rules are obeyed. The relations used are same as described
322 above. 322 above.
323 323
324 Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or 324 Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or
325 raised to a scalar power if it is square, producing diagonal matrix result in 325 raised to a scalar power if it is square, producing diagonal matrix result in
326 all cases. 326 all cases.
327 327
328 A diagonal matrix can also be transposed or conjugate-transposed, giving the 328 A diagonal matrix can also be transposed or conjugate-transposed, giving the
329 expected result. Extracting a leading submatrix of a diagonal matrix, i.e., 329 expected result. Extracting a leading submatrix of a diagonal matrix, i.e.,
330 @code{D(1:m,1:n)}, will produce a diagonal matrix, other indexing expressions 330 @code{D(1:m,1:n)}, will produce a diagonal matrix, other indexing expressions
331 will implicitly convert to full matrix. 331 will implicitly convert to full matrix.
349 @node Expressions Involving Permutation Matrices 349 @node Expressions Involving Permutation Matrices
350 @subsection Expressions Involving Permutation Matrices 350 @subsection Expressions Involving Permutation Matrices
351 351
352 If @var{P} is a permutation matrix and @var{M} a matrix, the expression 352 If @var{P} is a permutation matrix and @var{M} a matrix, the expression
353 @code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will 353 @code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will
354 yield a column permutation. 354 yield a column permutation.
355 Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation. 355 Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation.
356 356
357 The previously described syntax for creating permutation matrices can actually 357 The previously described syntax for creating permutation matrices can actually
358 help an user to understand the connection between a permutation matrix and 358 help an user to understand the connection between a permutation matrix and
359 a permuting vector. Namely, the following holds, where @code{I = eye (n)} 359 a permuting vector. Namely, the following holds, where @code{I = eye (n)}
412 @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again 412 @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again
413 a diagonal matrix. @dfn{det} will use an efficient straightforward calculation 413 a diagonal matrix. @dfn{det} will use an efficient straightforward calculation
414 when given a diagonal matrix, as well as @dfn{cond}. 414 when given a diagonal matrix, as well as @dfn{cond}.
415 The following mapper functions can be applied to a diagonal matrix 415 The following mapper functions can be applied to a diagonal matrix
416 without converting it to a full one: 416 without converting it to a full one:
417 @dfn{abs}, @dfn{real}, @dfn{imag}, @dfn{conj}, @dfn{sqrt}. 417 @dfn{abs}, @dfn{real}, @dfn{imag}, @dfn{conj}, @dfn{sqrt}.
418 A diagonal matrix can also be returned from the @dfn{balance} 418 A diagonal matrix can also be returned from the @dfn{balance}
419 and @dfn{svd} functions. 419 and @dfn{svd} functions.
420 The @dfn{sparse} function will convert a diagonal matrix efficiently to a 420 The @dfn{sparse} function will convert a diagonal matrix efficiently to a
421 sparse matrix. 421 sparse matrix.
422 422
520 normally stored somewhere in memory as an explicit value. An "assumed zero", on 520 normally stored somewhere in memory as an explicit value. An "assumed zero", on
521 the contrary, is a zero matrix element implied by the matrix structure 521 the contrary, is a zero matrix element implied by the matrix structure
522 (diagonal, triangular) or a sparsity pattern; its value is usually not stored 522 (diagonal, triangular) or a sparsity pattern; its value is usually not stored
523 explicitly anywhere, but is implied by the underlying data structure. 523 explicitly anywhere, but is implied by the underlying data structure.
524 524
525 The primary distinction is that an assumed zero, when multiplied 525 The primary distinction is that an assumed zero, when multiplied
526 by any number, or divided by any nonzero number, 526 by any number, or divided by any nonzero number,
527 yields *always* a zero, even when, e.g., multiplied by @code{Inf} 527 yields *always* a zero, even when, e.g., multiplied by @code{Inf}
528 or divided by @code{NaN}. 528 or divided by @code{NaN}.
529 The reason for this behavior is that the numerical multiplication is not 529 The reason for this behavior is that the numerical multiplication is not
530 actually performed anywhere by the underlying algorithm; the result is 530 actually performed anywhere by the underlying algorithm; the result is