doc/info/ja/diag.texi

   1 @menu
   2 * Functions and Variables for diag::
   3 @end menu
   4
   5 @node Functions and Variables for diag,  , diag, diag
   6 @section Functions and Variables for diag
   7
   8
   9 @deffn {関数} diag (@var{lm})
  10 @var{lm}の行列を対角に持つ平方行列を構成します。
  11 @var{lm}は行列かスカラーのリストです。
  12
  13 例:
  14 @example
  15 (%i1) load("diag")$
  16
  17 (%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$
  18
  19 (%i3) a2:matrix([1,1],[1,0])$
  20
  21 (%i4) diag([a1,x,a2]);
  22                    [ 1  2  3  0  0  0 ]
  23                    [                  ]
  24                    [ 0  4  5  0  0  0 ]
  25                    [                  ]
  26                    [ 0  0  6  0  0  0 ]
  27 (%o4)              [                  ]
  28                    [ 0  0  0  x  0  0 ]
  29                    [                  ]
  30                    [ 0  0  0  0  1  1 ]
  31                    [                  ]
  32                    [ 0  0  0  0  1  0 ]
  33 @end example
  34
  35 この関数を使うためには、最初に@code{load("diag")}を書いてください。
  36
  37 @opencatbox
  38 @category{Matrices} @category{Share packages} @category{Package diag}
  39 @closecatbox
  40
  41 @end deffn
  42
  43
  44 @deffn {関数} JF (@var{lambda},@var{n})
  45 固有値@var{lambda}を持つ次数@var{n}のJordan細胞を返します。
  46
  47 例:
  48 @example
  49 (%i1) load("diag")$
  50
  51 (%i2) JF(2,5);
  52                     [ 2  1  0  0  0 ]
  53                     [               ]
  54                     [ 0  2  1  0  0 ]
  55                     [               ]
  56 (%o2)               [ 0  0  2  1  0 ]
  57                     [               ]
  58                     [ 0  0  0  2  1 ]
  59                     [               ]
  60                     [ 0  0  0  0  2 ]
  61 (%i3) JF(3,2);
  62                          [ 3  1 ]
  63 (%o3)                    [      ]
  64                          [ 0  3 ]
  65 @end example
  66
  67 この関数を使うためには、最初に@code{load("diag")}を書いてください。
  68
  69 @opencatbox
  70 @category{Package diag}
  71 @closecatbox
  72
  73 @end deffn
  74
  75
  76 @deffn {関数} jordan (@var{mat})
  77 行列@var{mat}のJordan形を返しますが、それはMaximaリストでコード化されます。
  78 対応する行列を得るには、
  79 @code{jordan}の出力を引数として使って関数@code{dispJordan}をコールしてください。
  80
  81 例:
  82 @example
  83 (%i1) load("diag")$
  84
  85 (%i3) a:matrix([2,0,0,0,0,0,0,0],
  86                [1,2,0,0,0,0,0,0],
  87                [-4,1,2,0,0,0,0,0],
  88                [2,0,0,2,0,0,0,0],
  89                [-7,2,0,0,2,0,0,0],
  90                [9,0,-2,0,1,2,0,0],
  91                [-34,7,1,-2,-1,1,2,0],
  92                [145,-17,-16,3,9,-2,0,3])$
  93
  94 (%i34) jordan(a);
  95 (%o4)             [[2, 3, 3, 1], [3, 1]]
  96 (%i5) dispJordan(%);
  97                 [ 2  1  0  0  0  0  0  0 ]
  98                 [                        ]
  99                 [ 0  2  1  0  0  0  0  0 ]
 100                 [                        ]
 101                 [ 0  0  2  0  0  0  0  0 ]
 102                 [                        ]
 103                 [ 0  0  0  2  1  0  0  0 ]
 104 (%o5)           [                        ]
 105                 [ 0  0  0  0  2  1  0  0 ]
 106                 [                        ]
 107                 [ 0  0  0  0  0  2  0  0 ]
 108                 [                        ]
 109                 [ 0  0  0  0  0  0  2  0 ]
 110                 [                        ]
 111                 [ 0  0  0  0  0  0  0  3 ]
 112 @end example
 113
 114 この関数を使うためには、最初に@code{load("diag")}を書いてください。
 115 @code{dispJordan}と@code{minimalPoly}も参照してください。
 116
 117 @opencatbox
 118 @category{Package diag}
 119 @closecatbox
 120
 121 @end deffn
 122
 123
 124 @deffn {関数} dispJordan (@var{l})
 125 関数@code{jordan}によって与えられる出力である
 126 Maximaリスト@var{l}で与えられたコードに関連付けられた
 127 Jordan行列を返します。
 128
 129 例:
 130 @example
 131 (%i1) load("diag")$
 132
 133 (%i2) b1:matrix([0,0,1,1,1],
 134                 [0,0,0,1,1],
 135                 [0,0,0,0,1],
 136                 [0,0,0,0,0],
 137                 [0,0,0,0,0])$
 138
 139 (%i3) jordan(b1);
 140 (%o3)                  [[0, 3, 2]]
 141 (%i4) dispJordan(%);
 142                     [ 0  1  0  0  0 ]
 143                     [               ]
 144                     [ 0  0  1  0  0 ]
 145                     [               ]
 146 (%o4)               [ 0  0  0  0  0 ]
 147                     [               ]
 148                     [ 0  0  0  0  1 ]
 149                     [               ]
 150                     [ 0  0  0  0  0 ]
 151 @end example
 152
 153 この関数を使うためには、最初に@code{load("diag")}を書いてください。
 154 @code{jordan}と@code{minimalPoly}も参照してください。
 155
 156 @opencatbox
 157 @category{Package diag}
 158 @closecatbox
 159
 160 @end deffn
 161
 162
 163 @deffn {関数} minimalPoly (@var{l})
 164 関数@code{jordan}によって与えられる出力である
 165 Maximaリスト@var{l}で与えられたコードに関連付けられた
 166 最小多項式を返します。
 167
 168 例:
 169 @example
 170 (%i1) load("diag")$
 171
 172 (%i2) a:matrix([2,1,2,0],
 173                [-2,2,1,2],
 174                [-2,-1,-1,1],
 175                [3,1,2,-1])$
 176
 177 (%i3) jordan(a);
 178 (%o3)               [[- 1, 1], [1, 3]]
 179 (%i4) minimalPoly(%);
 180                             3
 181 (%o4)                (x - 1)  (x + 1)
 182 @end example
 183
 184 この関数を使うためには、最初に@code{load("diag")}を書いてください。
 185 @code{jordan}と@code{dispJordan}も参照してください。
 186
 187 @opencatbox
 188 @category{Package diag}
 189 @closecatbox
 190
 191 @end deffn
 192
 193 @deffn {関数} ModeMatrix (@var{A},@var{l})
 194 Returns the matrix
 195 @math{(M^^-1).A.M=J}―
 196 ただし@var{J}は@var{A}のJordan形とする―
 197 のような
 198 行列@var{M}を返します。
 199 Maximaリスト@var{l}は
 200 関数@code{jordan}が返すようなJordan形のコード化された形式です。
 201
 202 例:
 203 @example
 204 (%i1) load("diag")$
 205
 206 (%i2) a:matrix([2,1,2,0],
 207           [-2,2,1,2],
 208           [-2,-1,-1,1],
 209           [3,1,2,-1])$
 210
 211 (%i3) jordan(a);
 212 (%o3)               [[- 1, 1], [1, 3]]
 213 (%i4) M: ModeMatrix(a,%);
 214                   [  1    - 1   1   1 ]
 215                   [                   ]
 216                   [   1               ]
 217                   [ - -   - 1   0   0 ]
 218                   [   9               ]
 219                   [                   ]
 220 (%o4)             [   13              ]
 221                   [ - --   1   - 1  0 ]
 222                   [   9               ]
 223                   [                   ]
 224                   [  17               ]
 225                   [  --   - 1   1   1 ]
 226                   [  9                ]
 227 (%i5) is(  (M^^-1).a.M = dispJordan(%o3)  );
 228 (%o5)                      true
 229 @end example
 230 @code{dispJordan(%o3)}は
 231 行列@code{a}のJordan形であることに注意してください。
 232
 233 この関数を使うためには、最初に@code{load("diag")}を書いてください。
 234 @code{jordan}と@code{dispJordan}も参照してください。
 235
 236 @opencatbox
 237 @category{Package diag}
 238 @closecatbox
 239
 240 @end deffn
 241
 242
 243 @deffn {関数} mat_function (@var{f},@var{mat})
 244 @math{f(mat)}を返します。
 245 ここで、@var{f}は解析関数で@var{mat}は行列です。
 246 この計算はCauchyの積分公式に基づきます。
 247 積分公式は、
 248 もし@code{f(x)}が解析的、かつ、
 249
 250 @example
 251 mat = diag([JF(m1,n1),...,JF(mk,nk)]),
 252 @end example
 253
 254 なら、
 255
 256 @example
 257 f(mat) = ModeMatrix*diag([f(JF(m1,n1)), ..., f(JF(mk,nk))])
 258                                               *ModeMatrix^^(-1)
 259 @end example
 260
 261 をはっきり述べます。
 262
 263 この計算に関して約6か8の別の方法があることに注意してください。
 264
 265 いくつかの例が続きます。
 266
 267 例 1:
 268 @example
 269 (%i1) load("diag")$
 270
 271 (%i2) b2:matrix([0,1,0], [0,0,1], [-1,-3,-3])$
 272
 273 (%i3) mat_function(exp,t*b2);
 274                2   - t
 275               t  %e          - t     - t
 276 (%o3) matrix([-------- + t %e    + %e   ,
 277                  2
 278         - t     - t                           - t
 279  2    %e      %e        - t           - t   %e
 280 t  (- ----- - ----- + %e   ) + t (2 %e    - -----)
 281         t       2                             t
 282                t
 283                          - t          - t     - t
 284        - t       - t   %e        2  %e      %e
 285  + 2 %e   , t (%e    - -----) + t  (----- - -----)
 286                          t            2       t
 287                2   - t            - t     - t
 288      - t      t  %e        2    %e      %e        - t
 289  + %e   ], [- --------, - t  (- ----- - ----- + %e   ),
 290                  2                t       2
 291                                          t
 292         - t     - t      2   - t
 293    2  %e      %e        t  %e          - t
 294 - t  (----- - -----)], [-------- - t %e   ,
 295         2       t          2
 296         - t     - t                           - t
 297  2    %e      %e        - t           - t   %e
 298 t  (- ----- - ----- + %e   ) - t (2 %e    - -----),
 299         t       2                             t
 300                t
 301       - t     - t                 - t
 302  2  %e      %e            - t   %e
 303 t  (----- - -----) - t (%e    - -----)])
 304       2       t                   t
 305 (%i4) ratsimp(%);
 306                [   2              - t ]
 307                [ (t  + 2 t + 2) %e    ]
 308                [ -------------------- ]
 309                [          2           ]
 310                [                      ]
 311                [         2   - t      ]
 312 (%o4)  Col 1 = [        t  %e         ]
 313                [      - --------      ]
 314                [           2          ]
 315                [                      ]
 316                [     2          - t   ]
 317                [   (t  - 2 t) %e      ]
 318                [   ----------------   ]
 319                [          2           ]
 320          [      2        - t    ]
 321          [    (t  + t) %e       ]
 322          [                      ]
 323  Col 2 = [     2            - t ]
 324          [ - (t  - t - 1) %e    ]
 325          [                      ]
 326          [     2          - t   ]
 327          [   (t  - 3 t) %e      ]
 328          [        2   - t       ]
 329          [       t  %e          ]
 330          [       --------       ]
 331          [          2           ]
 332          [                      ]
 333          [      2          - t  ]
 334  Col 3 = [    (t  - 2 t) %e     ]
 335          [  - ----------------  ]
 336          [           2          ]
 337          [                      ]
 338          [   2              - t ]
 339          [ (t  - 4 t + 2) %e    ]
 340          [ -------------------- ]
 341          [          2           ]
 342
 343 @end example
 344
 345
 346 例 2:
 347 @example
 348 (%i5) b1:matrix([0,0,1,1,1],
 349                 [0,0,0,1,1],
 350                 [0,0,0,0,1],
 351                 [0,0,0,0,0],
 352                 [0,0,0,0,0])$
 353
 354 (%i6) mat_function(exp,t*b1);
 355                   [              2     ]
 356                   [             t      ]
 357                   [ 1  0  t  t  -- + t ]
 358                   [             2      ]
 359                   [                    ]
 360 (%o6)             [ 0  1  0  t    t    ]
 361                   [                    ]
 362                   [ 0  0  1  0    t    ]
 363                   [                    ]
 364                   [ 0  0  0  1    0    ]
 365                   [                    ]
 366                   [ 0  0  0  0    1    ]
 367 (%i7) minimalPoly(jordan(b1));
 368                              3
 369 (%o7)                       x
 370 (%i8) ident(5)+t*b1+1/2*(t^2)*b1^^2;
 371                   [              2     ]
 372                   [             t      ]
 373                   [ 1  0  t  t  -- + t ]
 374                   [             2      ]
 375                   [                    ]
 376 (%o8)             [ 0  1  0  t    t    ]
 377                   [                    ]
 378                   [ 0  0  1  0    t    ]
 379                   [                    ]
 380                   [ 0  0  0  1    0    ]
 381                   [                    ]
 382                   [ 0  0  0  0    1    ]
 383 (%i9) mat_function(exp,%i*t*b1);
 384              [                           2 ]
 385              [                          t  ]
 386              [ 1  0  %i t  %i t  %i t - -- ]
 387              [                          2  ]
 388              [                             ]
 389 (%o9)        [ 0  1   0    %i t    %i t    ]
 390              [                             ]
 391              [ 0  0   1     0      %i t    ]
 392              [                             ]
 393              [ 0  0   0     1        0     ]
 394              [                             ]
 395              [ 0  0   0     0        1     ]
 396 (%i10) mat_function(cos,t*b1)+%i*mat_function(sin,t*b1);
 397               [                           2 ]
 398               [                          t  ]
 399               [ 1  0  %i t  %i t  %i t - -- ]
 400               [                          2  ]
 401               [                             ]
 402 (%o10)        [ 0  1   0    %i t    %i t    ]
 403               [                             ]
 404               [ 0  0   1     0      %i t    ]
 405               [                             ]
 406               [ 0  0   0     1        0     ]
 407               [                             ]
 408               [ 0  0   0     0        1     ]
 409 @end example
 410
 411 例 3:
 412 @example
 413 (%i11) a1:matrix([2,1,0,0,0,0],
 414                  [-1,4,0,0,0,0],
 415                  [-1,1,2,1,0,0],
 416                  [-1,1,-1,4,0,0],
 417                  [-1,1,-1,1,3,0],
 418                  [-1,1,-1,1,1,2])$
 419
 420 (%i12) fpow(x):=block([k],declare(k,integer),x^k)$
 421
 422 (%i13) mat_function(fpow,a1);
 423                 [  k      k - 1 ]         [      k - 1    ]
 424                 [ 3  - k 3      ]         [   k 3         ]
 425                 [               ]         [               ]
 426                 [       k - 1   ]         [  k      k - 1 ]
 427                 [  - k 3        ]         [ 3  + k 3      ]
 428                 [               ]         [               ]
 429                 [       k - 1   ]         [      k - 1    ]
 430                 [  - k 3        ]         [   k 3         ]
 431 (%o13)  Col 1 = [               ] Col 2 = [               ]
 432                 [       k - 1   ]         [      k - 1    ]
 433                 [  - k 3        ]         [   k 3         ]
 434                 [               ]         [               ]
 435                 [       k - 1   ]         [      k - 1    ]
 436                 [  - k 3        ]         [   k 3         ]
 437                 [               ]         [               ]
 438                 [       k - 1   ]         [      k - 1    ]
 439                 [  - k 3        ]         [   k 3         ]
 440          [       0       ]         [       0       ]
 441          [               ]         [               ]
 442          [       0       ]         [       0       ]
 443          [               ]         [               ]
 444          [  k      k - 1 ]         [      k - 1    ]
 445          [ 3  - k 3      ]         [   k 3         ]
 446          [               ]         [               ]
 447  Col 3 = [       k - 1   ] Col 4 = [  k      k - 1 ]
 448          [  - k 3        ]         [ 3  + k 3      ]
 449          [               ]         [               ]
 450          [       k - 1   ]         [      k - 1    ]
 451          [  - k 3        ]         [   k 3         ]
 452          [               ]         [               ]
 453          [       k - 1   ]         [      k - 1    ]
 454          [  - k 3        ]         [   k 3         ]
 455          [    0    ]
 456          [         ]         [ 0  ]
 457          [    0    ]         [    ]
 458          [         ]         [ 0  ]
 459          [    0    ]         [    ]
 460          [         ]         [ 0  ]
 461  Col 5 = [    0    ] Col 6 = [    ]
 462          [         ]         [ 0  ]
 463          [    k    ]         [    ]
 464          [   3     ]         [ 0  ]
 465          [         ]         [    ]
 466          [  k    k ]         [  k ]
 467          [ 3  - 2  ]         [ 2  ]
 468 @end example
 469
 470 この関数を使うためには、最初に@code{load("diag")}を書いてください。
 471
 472 @opencatbox
 473 @category{Package diag}
 474 @closecatbox
 475
 476 @end deffn
 477
 478