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}
  39 @category{Share packages}
  40 @category{Package diag}
  41 @closecatbox
  42
  43 @end deffn
  44
  45
  46 @deffn {関数} JF (@var{lambda},@var{n})
  47 固有値@var{lambda}を持つ次数@var{n}のJordan細胞を返します。
  48
  49 例:
  50 @example
  51 (%i1) load("diag")$
  52
  53 (%i2) JF(2,5);
  54                     [ 2  1  0  0  0 ]
  55                     [               ]
  56                     [ 0  2  1  0  0 ]
  57                     [               ]
  58 (%o2)               [ 0  0  2  1  0 ]
  59                     [               ]
  60                     [ 0  0  0  2  1 ]
  61                     [               ]
  62                     [ 0  0  0  0  2 ]
  63 (%i3) JF(3,2);
  64                          [ 3  1 ]
  65 (%o3)                    [      ]
  66                          [ 0  3 ]
  67 @end example
  68
  69 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。
  70
  71 @opencatbox
  72 @category{Package diag}
  73 @closecatbox
  74
  75 @end deffn
  76
  77
  78 @deffn {関数} jordan (@var{mat})
  79 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。
  80 対応する行列を得るには、
  81 @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。
  82
  83 例:
  84 @example
  85 (%i1) load("diag")$
  86
  87 (%i3) a:matrix([2,0,0,0,0,0,0,0],
  88                [1,2,0,0,0,0,0,0],
  89                [-4,1,2,0,0,0,0,0],
  90                [2,0,0,2,0,0,0,0],
  91                [-7,2,0,0,2,0,0,0],
  92                [9,0,-2,0,1,2,0,0],
  93                [-34,7,1,-2,-1,1,2,0],
  94                [145,-17,-16,3,9,-2,0,3])$
  95
  96 (%i34) jordan(a);
  97 (%o4)             [[2, 3, 3, 1], [3, 1]]
  98 (%i5) dispJordan(%);
  99                 [ 2  1  0  0  0  0  0  0 ]
 100                 [                        ]
 101                 [ 0  2  1  0  0  0  0  0 ]
 102                 [                        ]
 103                 [ 0  0  2  0  0  0  0  0 ]
 104                 [                        ]
 105                 [ 0  0  0  2  1  0  0  0 ]
 106 (%o5)           [                        ]
 107                 [ 0  0  0  0  2  1  0  0 ]
 108                 [                        ]
 109                 [ 0  0  0  0  0  2  0  0 ]
 110                 [                        ]
 111                 [ 0  0  0  0  0  0  2  0 ]
 112                 [                        ]
 113                 [ 0  0  0  0  0  0  0  3 ]
 114 @end example
 115
 116 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。
 117 @code{dispJordan}と@code{minimalPoly}も参照してください。
 118
 119 @opencatbox
 120 @category{Package diag}
 121 @closecatbox
 122
 123 @end deffn
 124
 125
 126 @deffn {関数} dispJordan (@var{l})
 127 関数@code{jordan}によって与えられる出力である
 128 Maximaリスト@var{l}で与えられたコードに関連付けられた
 129 Jordan行列を返します。
 130
 131 例:
 132 @example
 133 (%i1) load("diag")$
 134
 135 (%i2) b1:matrix([0,0,1,1,1],
 136                 [0,0,0,1,1],
 137                 [0,0,0,0,1],
 138                 [0,0,0,0,0],
 139                 [0,0,0,0,0])$
 140
 141 (%i3) jordan(b1);
 142 (%o3)                  [[0, 3, 2]]
 143 (%i4) dispJordan(%);
 144                     [ 0  1  0  0  0 ]
 145                     [               ]
 146                     [ 0  0  1  0  0 ]
 147                     [               ]
 148 (%o4)               [ 0  0  0  0  0 ]
 149                     [               ]
 150                     [ 0  0  0  0  1 ]
 151                     [               ]
 152                     [ 0  0  0  0  0 ]
 153 @end example
 154
 155 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。
 156 @code{jordan}と@code{minimalPoly}も参照してください。
 157
 158 @opencatbox
 159 @category{Package diag}
 160 @closecatbox
 161
 162 @end deffn
 163
 164
 165 @deffn {関数} minimalPoly (@var{l})
 166 関数@code{jordan}によって与えられる出力である
 167 Maximaリスト@var{l}で与えられたコードに関連付けられた
 168 最小多項式を返します。
 169
 170 例:
 171 @example
 172 (%i1) load("diag")$
 173
 174 (%i2) a:matrix([2,1,2,0],
 175                [-2,2,1,2],
 176                [-2,-1,-1,1],
 177                [3,1,2,-1])$
 178
 179 (%i3) jordan(a);
 180 (%o3)               [[- 1, 1], [1, 3]]
 181 (%i4) minimalPoly(%);
 182                             3
 183 (%o4)                (x - 1)  (x + 1)
 184 @end example
 185
 186 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。
 187 @code{jordan}と@code{dispJordan}も参照してください。
 188
 189 @opencatbox
 190 @category{Package diag}
 191 @closecatbox
 192
 193 @end deffn
 194
 195 @deffn {関数} ModeMatrix (@var{A},@var{l})
 196 Returns the matrix
 197 @math{(M^^-1).A.M=J}―
 198 ただし@var{J}は@var{A}のJordan形とする―
 199 のような
 200 行列@var{M}を返します。
 201 Maximaリスト@var{l}は
 202 関数@code{jordan}が返すようなJordan形のコード化された形式です。
 203
 204 例:
 205 @example
 206 (%i1) load("diag")$
 207
 208 (%i2) a:matrix([2,1,2,0],
 209           [-2,2,1,2],
 210           [-2,-1,-1,1],
 211           [3,1,2,-1])$
 212
 213 (%i3) jordan(a);
 214 (%o3)               [[- 1, 1], [1, 3]]
 215 (%i4) M: ModeMatrix(a,%);
 216                   [  1    - 1   1   1 ]
 217                   [                   ]
 218                   [   1               ]
 219                   [ - -   - 1   0   0 ]
 220                   [   9               ]
 221                   [                   ]
 222 (%o4)             [   13              ]
 223                   [ - --   1   - 1  0 ]
 224                   [   9               ]
 225                   [                   ]
 226                   [  17               ]
 227                   [  --   - 1   1   1 ]
 228                   [  9                ]
 229 (%i5) is(  (M^^-1).a.M = dispJordan(%o3)  );
 230 (%o5)                      true
 231 @end example
 232 @code{dispJordan(%o3)}は
 233 行列@code{a}のJordan形であることに注意してください。
 234
 235 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。
 236 @code{jordan}と@code{dispJordan}も参照してください。
 237
 238 @opencatbox
 239 @category{Package diag}
 240 @closecatbox
 241
 242 @end deffn
 243
 244
 245 @deffn {関数} mat_function (@var{f},@var{mat})
 246 @math{f(mat)}を返します。
 247 ここで、@var{f}は解析関数で@var{mat}は行列です。
 248 この計算はCauchyの積分公式に基づきます。
 249 積分公式は、
 250 もし@code{f(x)}が解析的、かつ、
 251
 252 @example
 253 mat = diag([JF(m1,n1),...,JF(mk,nk)]),
 254 @end example
 255
 256 なら、
 257
 258 @example
 259 f(mat) = ModeMatrix*diag([f(JF(m1,n1)), ..., f(JF(mk,nk))])
 260                                               *ModeMatrix^^(-1)
 261 @end example
 262
 263 をはっきり述べます。
 264
 265 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。
 266
 267 いくつかの例が続きます。
 268
 269 例 1:
 270 @example
 271 (%i1) load("diag")$
 272
 273 (%i2) b2:matrix([0,1,0], [0,0,1], [-1,-3,-3])$
 274
 275 (%i3) mat_function(exp,t*b2);
 276                2   - t
 277               t  %e          - t     - t
 278 (%o3) matrix([-------- + t %e    + %e   ,
 279                  2
 280         - t     - t                           - t
 281  2    %e      %e        - t           - t   %e
 282 t  (- ----- - ----- + %e   ) + t (2 %e    - -----)
 283         t       2                             t
 284                t
 285                          - t          - t     - t
 286        - t       - t   %e        2  %e      %e
 287  + 2 %e   , t (%e    - -----) + t  (----- - -----)
 288                          t            2       t
 289                2   - t            - t     - t
 290      - t      t  %e        2    %e      %e        - t
 291  + %e   ], [- --------, - t  (- ----- - ----- + %e   ),
 292                  2                t       2
 293                                          t
 294         - t     - t      2   - t
 295    2  %e      %e        t  %e          - t
 296 - t  (----- - -----)], [-------- - t %e   ,
 297         2       t          2
 298         - t     - t                           - t
 299  2    %e      %e        - t           - t   %e
 300 t  (- ----- - ----- + %e   ) - t (2 %e    - -----),
 301         t       2                             t
 302                t
 303       - t     - t                 - t
 304  2  %e      %e            - t   %e
 305 t  (----- - -----) - t (%e    - -----)])
 306       2       t                   t
 307 (%i4) ratsimp(%);
 308                [   2              - t ]
 309                [ (t  + 2 t + 2) %e    ]
 310                [ -------------------- ]
 311                [          2           ]
 312                [                      ]
 313                [         2   - t      ]
 314 (%o4)  Col 1 = [        t  %e         ]
 315                [      - --------      ]
 316                [           2          ]
 317                [                      ]
 318                [     2          - t   ]
 319                [   (t  - 2 t) %e      ]
 320                [   ----------------   ]
 321                [          2           ]
 322          [      2        - t    ]
 323          [    (t  + t) %e       ]
 324          [                      ]
 325  Col 2 = [     2            - t ]
 326          [ - (t  - t - 1) %e    ]
 327          [                      ]
 328          [     2          - t   ]
 329          [   (t  - 3 t) %e      ]
 330          [        2   - t       ]
 331          [       t  %e          ]
 332          [       --------       ]
 333          [          2           ]
 334          [                      ]
 335          [      2          - t  ]
 336  Col 3 = [    (t  - 2 t) %e     ]
 337          [  - ----------------  ]
 338          [           2          ]
 339          [                      ]
 340          [   2              - t ]
 341          [ (t  - 4 t + 2) %e    ]
 342          [ -------------------- ]
 343          [          2           ]
 344
 345 @end example
 346
 347
 348 例 2:
 349 @example
 350 (%i5) b1:matrix([0,0,1,1,1],
 351                 [0,0,0,1,1],
 352                 [0,0,0,0,1],
 353                 [0,0,0,0,0],
 354                 [0,0,0,0,0])$
 355
 356 (%i6) mat_function(exp,t*b1);
 357                   [              2     ]
 358                   [             t      ]
 359                   [ 1  0  t  t  -- + t ]
 360                   [             2      ]
 361                   [                    ]
 362 (%o6)             [ 0  1  0  t    t    ]
 363                   [                    ]
 364                   [ 0  0  1  0    t    ]
 365                   [                    ]
 366                   [ 0  0  0  1    0    ]
 367                   [                    ]
 368                   [ 0  0  0  0    1    ]
 369 (%i7) minimalPoly(jordan(b1));
 370                              3
 371 (%o7)                       x
 372 (%i8) ident(5)+t*b1+1/2*(t^2)*b1^^2;
 373                   [              2     ]
 374                   [             t      ]
 375                   [ 1  0  t  t  -- + t ]
 376                   [             2      ]
 377                   [                    ]
 378 (%o8)             [ 0  1  0  t    t    ]
 379                   [                    ]
 380                   [ 0  0  1  0    t    ]
 381                   [                    ]
 382                   [ 0  0  0  1    0    ]
 383                   [                    ]
 384                   [ 0  0  0  0    1    ]
 385 (%i9) mat_function(exp,%i*t*b1);
 386              [                           2 ]
 387              [                          t  ]
 388              [ 1  0  %i t  %i t  %i t - -- ]
 389              [                          2  ]
 390              [                             ]
 391 (%o9)        [ 0  1   0    %i t    %i t    ]
 392              [                             ]
 393              [ 0  0   1     0      %i t    ]
 394              [                             ]
 395              [ 0  0   0     1        0     ]
 396              [                             ]
 397              [ 0  0   0     0        1     ]
 398 (%i10) mat_function(cos,t*b1)+%i*mat_function(sin,t*b1);
 399               [                           2 ]
 400               [                          t  ]
 401               [ 1  0  %i t  %i t  %i t - -- ]
 402               [                          2  ]
 403               [                             ]
 404 (%o10)        [ 0  1   0    %i t    %i t    ]
 405               [                             ]
 406               [ 0  0   1     0      %i t    ]
 407               [                             ]
 408               [ 0  0   0     1        0     ]
 409               [                             ]
 410               [ 0  0   0     0        1     ]
 411 @end example
 412
 413 例 3:
 414 @example
 415 (%i11) a1:matrix([2,1,0,0,0,0],
 416                  [-1,4,0,0,0,0],
 417                  [-1,1,2,1,0,0],
 418                  [-1,1,-1,4,0,0],
 419                  [-1,1,-1,1,3,0],
 420                  [-1,1,-1,1,1,2])$
 421
 422 (%i12) fpow(x):=block([k],declare(k,integer),x^k)$
 423
 424 (%i13) mat_function(fpow,a1);
 425                 [  k      k - 1 ]         [      k - 1    ]
 426                 [ 3  - k 3      ]         [   k 3         ]
 427                 [               ]         [               ]
 428                 [       k - 1   ]         [  k      k - 1 ]
 429                 [  - k 3        ]         [ 3  + k 3      ]
 430                 [               ]         [               ]
 431                 [       k - 1   ]         [      k - 1    ]
 432                 [  - k 3        ]         [   k 3         ]
 433 (%o13)  Col 1 = [               ] Col 2 = [               ]
 434                 [       k - 1   ]         [      k - 1    ]
 435                 [  - k 3        ]         [   k 3         ]
 436                 [               ]         [               ]
 437                 [       k - 1   ]         [      k - 1    ]
 438                 [  - k 3        ]         [   k 3         ]
 439                 [               ]         [               ]
 440                 [       k - 1   ]         [      k - 1    ]
 441                 [  - k 3        ]         [   k 3         ]
 442          [       0       ]         [       0       ]
 443          [               ]         [               ]
 444          [       0       ]         [       0       ]
 445          [               ]         [               ]
 446          [  k      k - 1 ]         [      k - 1    ]
 447          [ 3  - k 3      ]         [   k 3         ]
 448          [               ]         [               ]
 449  Col 3 = [       k - 1   ] Col 4 = [  k      k - 1 ]
 450          [  - k 3        ]         [ 3  + k 3      ]
 451          [               ]         [               ]
 452          [       k - 1   ]         [      k - 1    ]
 453          [  - k 3        ]         [   k 3         ]
 454          [               ]         [               ]
 455          [       k - 1   ]         [      k - 1    ]
 456          [  - k 3        ]         [   k 3         ]
 457          [    0    ]
 458          [         ]         [ 0  ]
 459          [    0    ]         [    ]
 460          [         ]         [ 0  ]
 461          [    0    ]         [    ]
 462          [         ]         [ 0  ]
 463  Col 5 = [    0    ] Col 6 = [    ]
 464          [         ]         [ 0  ]
 465          [    k    ]         [    ]
 466          [   3     ]         [ 0  ]
 467          [         ]         [    ]
 468          [  k    k ]         [  k ]
 469          [ 3  - 2  ]         [ 2  ]
 470 @end example
 471
 472 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。
 473
 474 @opencatbox
 475 @category{Package diag}
 476 @closecatbox
 477
 478 @end deffn
 479
 480