Solving Ordinary Differential Equations II: Stiff and Differential ...

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Solving Ordinary Differential Equations II: Stiff and Differential

  • Page 2: Springer Series in
  • Page 8: Ernst Hairer
  • Page 14: From the Preface to the First Editi
  • Page 18: Contents
  • Page 22: Contents XI
  • Page 26: Contents XIII
  • Page 30: Contents XV
  • Page 36: IV.1 Examples of Stiff Equations
  • Page 40: 4 IV. Stiff Problems - One-Step Met
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    10 IV. Stiff Problems - One-Step Me

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    Fig. 1.9. DOP853 on the beam

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    14 IV. Stiff Problems - One-Step Me

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    16 IV. Stiff Problems - One-Step Me

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    18 IV. Stiff Problems - One-Step Me

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    20 IV. Stiff Problems - One-Step Me

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    22 IV. Stiff Problems - One-Step Me

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    24 IV. Stiff Problems - One-Step Me

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    26 IV. Stiff Problems - One-Step Me

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    28 IV. Stiff Problems - One-Step Me

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    30 IV. Stiff Problems - One-Step Me

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    32 IV. Stiff Problems - One-Step Me

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    34 IV. Stiff Problems - One-Step Me

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    36 IV. Stiff Problems - One-Step Me

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    38 IV. Stiff Problems - One-Step Me

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    IV.3 Stability Function of Implicit

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    42 IV Stiff Problems - One-Step Met

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    44 IV Stiff Problems - One-Step Met

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    46 IV. Stiff Problems - One-Step Me

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    48 IV. Stiff Problems - One-Step Me

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    50 IV. Stiff Problems - One-Step Me

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    52 IV. Stiff Problems - One-Step Me

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    54 IV. Stiff Problems - One-Step Me

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    56 IV. Stiff Problems - One-Step Me

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    58 IV. Stiff Problems - One-Step Me

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    60 IV. Stiff Problems - One-Step Me

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    62 IV. Stiff Problems - One-Step Me

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    64 IV. Stiff Problems - One-Step Me

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    66 IV. Stiff Problems - One-Step Me

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    68 IV. Stiff Problems - One-Step Me

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    70 IV. Stiff Problems - One-Step Me

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    72 IV. Stiff Problems - One-Step Me

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    74 IV. Stiff Problems - One-Step Me

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    76 IV. Stiff Problems - One-Step Me

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    78 IV. Stiff Problems - One-Step Me

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    80 IV. Stiff Problems - One-Step Me

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    82 IV. Stiff Problems - One-Step Me

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    84 IV. Stiff Problems - One-Step Me

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    86 IV. Stiff Problems - One-Step Me

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    88 IV. Stiff Problems - One-Step Me

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    90 IV. Stiff Problems - One-Step Me

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    92 rv. Stiff Problems - One-Step Me

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    94 IV. Stiff Problems - One-Step Me

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    96 IV. Stiff Problems - One-Step Me

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    98 IV. Stiff Problems - One-Step Me

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    100 IV. Stiff Problems - One-Step M

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    IV.7 Rosenbrock-Type Methods

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    104 IV. Stiff Problems - One-Step M

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    106 IV. Stiff Problems - One-Step M

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    108 IV. Stiff Problems - One-Step M

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    110 IV. Stiff Problems - One-Step M

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    112 IV. Stiff Problems - One-Step M

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    114 IV. Stiff Problems - One-Step M

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    116 IV. Stiff Problems - One-Step M

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    IV.8 Implementation of Implicit Run

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    120 IV. Stiff Problems - One-Step M

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    122 IV. Stiff Problems - One-Step M

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    124 IV. Stiff Problems - One-Step M

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    126 IV. Stiff Problems - One-Step M

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    128 IV. Stiff Problems - One-Step M

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    130 Iv. Stiff Problems - One-Step M

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    132 Iv. Stiff Problems - One-Step M

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    134 IV. Stiff Problems - One-Step M

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    136 IV. Stiff Problems - One-Step M

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    138 IV. Stiff Problems - One-Step M

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    140 IV. Stiff Problems - One-Step M

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    142 IV. Stiff Problems - One-Step M

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    144 IV. Stiff Problems - One-Step M

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    146 IV. Stiff Problems - One-Step M

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    148 IV. Stiff Problems - One-Step M

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    150 IV. Stiff Problems - One-Step M

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    152 IV. Stiff Problems - One-Step M

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    154 IV. Stiff Problems - One-Step M

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    156 IV. Stiff Problems - One-Step M

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    158 IV. Stiff Problems - One-Step M

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    160 IV Stiff Problems - One-Step Me

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    162 IV. Stiff Problems - One-Step M

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    164 IV. Stiff Problems - One-Step M

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    166 IV. Stiff Problems - One-Step M

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    168 IV. Stiff Problems - One-Step M

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    170 IV. Stiff Problems - One-Step M

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    172 IV Stiff Problems - One-Step Me

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    174 IV. Stiff Problems - One-Step M

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    176 IV. Stiff Problems - One-Step M

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    178 IV. Stiff Problems - One-Step M

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    IV.12 B-Stability and Contractivity

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    182 IV. Stiff Problems - One-Step M

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    184 IV. Stiff Problems - One-Step M

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    186 IV. Stiff Problems - One-Step M

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    188 IV. Stiff Problems - One-Step M

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    190 IV. Stiff Problems - One-Step M

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    192 IV. Stiff Problems - One-Step M

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    194 IV. Stiff Problems - One-Step M

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    196 IV. Stiff Problems - One-Step M

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    198 IV. Stiff Problems - One-Step M

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    200 IV. Stiff Problems - One-Step M

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    202 IV. Stiff Problems - One-Step M

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    204 IV. Stiff Problems - One-Step M

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    206 IV. Stiff Problems - One-Step M

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    208 IV. Stiff Problems - One-Step M

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    210 IV. Stiff Problems - One-Step M

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    212 IV. Stiff Problems - One-Step M

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    214 IV. Stiff Problems - One-Step M

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    216 IV. Stiff Problems - One-Step M

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    218 IV Stiff Problems - One-Step Me

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    220 IV. Stiff Problems - One-Step M

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    222 IV. Stiff Problems - One-Step M

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    224 IV. Stiff Problems - One-Step M

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    226 IV. Stiff Problems - One-Step M

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    228 IV. Stiff Problems - One-Step M

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    230 IV. Stiff Problems - One-Step M

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    232 IV. Stiff Problems - One-Step M

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    234 IV. Stiff Problems - One-Step M

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    236 IV. Stiff Problems - One-Step M

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    238 IV. Stiff Problems - One-Step M

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    V.I Stability of Multistep Methods

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    242 V. Multistep Methods for Stiff

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    244 V. Multistep Methods for Stiff

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    246 V. Multistep Methods for Stiff

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    248 V. Multistep Methods for Stiff

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    V.2 "Nearly" A-Stable Multistep Met

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    252 V. Multistep Methods for Stiff

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    254 V. Multistep Methods for Stiff

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    256 V. Multistep Methods for Stiff

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    258 V. Multistep Methods for Stiff

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    260 V. Multistep Methods for Stiff

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    262 V. Multistep Methods for Stiff

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    264 V. Multistep Methods for Stiff

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    266 V. Multistep Methods for Stiff

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    268 V. Multistep Methods for Stiff

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    270 V. Multistep Methods for Stiff

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    272 V. Multistep Methods for Stiff

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    274 V. Multistep Methods for Stiff

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    276 V. Multistep Methods for Stiff

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    278 V. Multistep Methods for Stiff

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    280 V. Multistep Methods for Stiff

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    282 V. Multistep Methods for Stiff

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    284 V. Multistep Methods for Stiff

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    286 V. Multistep Methods for Stiff

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    288 V. Multistep Methods for Stiff

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    290 V. Multistep Methods for Stiff

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    292 V. Multistep Methods for Stiff

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    294 V. Multistep Methods for Stiff

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    296 V. Multistep Methods for Stiff

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    298 V. Multistep Methods for Stiff

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    V.S Experiments with Multistep Code

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    302 V. Multistep Methods for Stiff

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    304 V. Multistep Methods for Stiff

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    306 V. Multistep Methods for Stiff

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    308 V. Multistep Methods for Stiff

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    310 V. Multistep Methods for Stiff

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    312 V. Multistep Methods for Stiff

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    314 V. Multistep Methods for Stiff

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    316 V Multistep Methods for Stiff P

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    318 V. Multistep Methods for Stiff

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    320 V. Multistep Methods for Stiff

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    322 V. Multistep Methods for Stiff

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    324 V. Multistep Methods for Stiff

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    326 V. Multistep Methods for Stiff

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    328 V. Multistep Methods for Stiff

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    330 V. Multistep Methods for Stiff

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    332 V. Multistep Methods for Stiff

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    334 V Multistep Methods for Stiff P

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    336 V. Multistep Methods for Stiff

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    338 V. Multistep Methods for Stiff

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    340 V. Multistep Methods for Stiff

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    342 V. Multistep Methods for Stiff

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    344 V. Multistep Methods for Stiff

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    346 V. Multistep Methods for Stiff

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    348 V. Multistep Methods for Stiff

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    350 V. Multistep Methods for Stiff

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    352 V. Multistep Methods for Stiff

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    354 V. Multistep Methods for Stiff

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    V.9 Algebraic Stability of General

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    358 V. Multistep Methods for Stiff

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    360 V. Multistep Methods for Stiff

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    362 V. Multistep Methods for Stiff

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    364 V. Multistep Methods for Stiff

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    366 V. Multistep Methods for Stiff

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    368 V. Multistep Methods for Stiff

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    370 V. Multistep Methods for Stiff

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    VI.l Solving Index 1 Problems

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    374 VI. Singular Perturbation Probl

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    376 VI. Singular Perturbation Probl

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    378 VI. Singular Perturbation Probl

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    380 VI. Singular Perturbation Probl

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    VI.2 Multistep Methods

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    384 VI. Singular Perturbation Probl

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    386 VI. Singular Perturbation Probl

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    VI.3 Epsilon Expansions for Exact a

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    390 VI. Singular Perturbation Probl

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    392 VI. Singular Perturbation Probl

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    394 VI. Singular Perturbation Probl

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    396 VI. Singular Perturbation Probl

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    398 VI. Singular Perturbation Probl

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    400 VI. Singular Perturbation Probl

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    402 VI. Singular Perturbation Probl

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    404 VI. Singular Perturbation Probl

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    406 VI. Singular Perturbation Probl

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    408 VI. Singular Perturbation Probl

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    410 VI. Singular Perturbation Probl

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    412 VI. Singular Perturbation Probl

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    414 VI.

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    416 VI. Singular Perturbation Probl

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    418 VI. Singular Perturbation Probl

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    420 VI. Singular Perturbation Probl

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    422 VI. Singular Perturbation Probl

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    424 VI. Singular Perturbation Probl

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    VI.S Extrapolation Methods

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    428 VI. Singular Perturbation Probl

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    430 VI. Singular Perturbation Probl

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    432 VI. Singular Perturbation Probl

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    434 VI. Singular Perturbation Probl

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    436 VI. Singular Perturbation Probl

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    438 VI. Singular Perturbation Probl

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    440 VI. Singular Perturbation Probl

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    VI.6 Quasilinear Problems

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    444 VI. Singular Perturbation Probl

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    446 VI. Singular Perturbation Probl

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    448 VI. Singular Perturbation Probl

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    Chapter VII. Differential-Algebraic

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    VILI The Index and Various Examples

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    Fig.1.1a. The vector field (1.9a,d)

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    VII. I The Index and Various Exampl

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    The Perturbation Index

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    VII. 1 The Index and Various Exampl

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    VII. 1 The Index and Various Exampl

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    VII.l The Index and Various Example

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    VII. 1 The Index and Various Exampl

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    .0002 .0001

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    VII.2 Index Reduction Methods 471

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    VII.2 Index Reduction Methods 473

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    VII.2 Index Reduction Methods 475

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    VII.2 Index Reduction Methods 477

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    VII.2 Index Reduction Methods 479

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    VII.3 Multistep Methods for Index 2

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    We next show that

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    VII.3 Multistep Methods for Index 2

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    Vll.3 Multistep Methods for Index 2

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    VII.3 Multistep Methods for Index 2

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    VII.3 Multistep Methods for Index 2

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    VIl.4 Runge-Kutta Methods for Index

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    can be estimated as follows:

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    VII.4 Runge-Kutta Methods for Index

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    VII.4 Runge-Kutta Methods for Index

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    where VIl.4 Runge-Kutta Methods for

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    VIlA Runge-Kutta Methods for Index

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    Exercises

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    VII.5 Order Conditions for Index 2

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    VII.5 Order Conditions for Index 2

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    and VII.5 Order Conditions for Inde

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    VII.5 Order Conditions for Index 2

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    VII.S Order Conditions forIndex 2 D

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    VII.5 Order Conditions for Index 2

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    VII.6 Half-Explicit Methods for Ind

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    VII.6 Half .. Explicit Methods for

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    VII.6 Half-Explicit Methods for Ind

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    Extrapolation Methods

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    VII.6 Half-Explicit Methods for Ind

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    Vll.6 Half-Explicit Methods for Ind

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    VII.7 Computation of Multibody Mech

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    VII.7 Computation of Multibody Mech

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    VIT.7 Computation of Multibody Mech

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    VII.7 Computation of Multibody Mech

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    10-1 sec

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    A Stiff Mechanical System

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    VII.8 Symplectic Methods for

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    VII.8 Constrained Hamiltonian Syste

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    found a one-step method

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    VII.8 Constrained Hamiltonian Syste

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    VII.8 Constrained Hamiltonian Syste

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    VII.8 Constrained Hamiltonian Syste

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    VII.8 Constrained Hamiltonian Syste

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    2 4 6 VII.8 Constrained Hamiltonian

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    VII.8 Constrained Hamiltonian Syste

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    VII.8 Constrained Hamiltonian Syste

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    VII.8 Constrained Hamiltonian Syste

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    566 Appendix. Fortran Codes

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    568 Appendix. Fortran Codes

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    570 Appendix. Fortran Codes

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    572 Appendix. Fortran Codes

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    574 Appendix. Fortran Codes

  • Page 1184:

    576 Appendix. Fortran Codes

  • Page 1188:

    578 Bibliography

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    580 Bibliography

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    582 Bibliography

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    584 Bibliography

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    586 Bibliography

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    588 Bibliography

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    590 Bibliography

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    592 Bibliography

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    594 Bibliography

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    596 Bibliography

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    598 Bibliography

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    600 Bibliography

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    602 Bibliography

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    604 Bibliography

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    606 Symbol Index

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    608 Subject Index

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    610 Subject Index

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    612 Subject Index

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    614 Subject Index

  • Page 6: E. Hairer
  • Page 10: To Evi and Myriam
  • Page 16: VIII Preface
  • Page 20: X Contents
  • Page 24: XII Contents
  • Page 28: XIV Contents
  • Page 34: Chapter IV. Stiff Problems - One-St
  • Page 38: IV.l Examples of Stiff Equations 3
  • Page 42: -.5 -.6
  • Page 46: .12 .12
  • Page 50: IV.l Examples of Stiff Equations 9
  • Page 54:

    IV 1 Examples of Stiff Equations 11

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    IVI Examples of Stiff Equations 13

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    IV.2 Stability Analysis for Explici

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    IY.2 Stability Analysis for Explici

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    IY.2 Stability Analysis for Explici

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    Automatic Stiffness Detection

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    -I IV2 Stability Analysis for Expli

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    .0020 h

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    o 2

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    IV.2 Stability Analysis for Explici

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    IY.2 Stability Analysis for Explici

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    IY.2 Stability Analysis for Explici

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    y.2 Stability Analysis for Explicit

  • Page 106:

    Exercises

  • Page 110:

    IY.2 Stability Analysis for Explici

  • Page 114:

    IV3 Stability Function of Implicit

  • Page 118:

    if and

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    IV. 3 Stability Function of Implici

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    Stability Functions of Order ~ s

  • Page 130:

    with error

  • Page 134:

    IV.4 Order Stars

  • Page 138:

    Fig. 4.2. Order stars for Pade appr

  • Page 142:

    IVA Order Stars 55

  • Page 146:

    IV.4 Order Stars 57

  • Page 150:

    IVA Order Stars 59

  • Page 154:

    of the same degree. Theorem 4.15 no

  • Page 158:

    A consequence of Theorem 4.18 is

  • Page 162:

    IV.4 Order Stars 65

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    IVA Order Stars 67

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    Rnew left pictures: Cold < 0

  • Page 174:

    IV.S Construction of Implicit Runge

  • Page 178:

    IV,5 Construction ofImplicit Runge-

  • Page 182:

    Lobatto IlIA, IIIB and IIIC Methods

  • Page 186:

    and B(2s - 2) then yield

  • Page 190:

    IY.S Construction of Implicit Runge

  • Page 194:

    IY.S Construction of Implicit Runge

  • Page 198:

    IV.S Construction ofImplicit Runge-

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    IY.5 Construction of Implicit Runge

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    IV.5 Construction of Implicit Runge

  • Page 210:

    Exercises

  • Page 214:

    IV.6 Diagonally Implicit RK Methods

  • Page 218:

    'j IY.6 Diagonally Implicit RK Meth

  • Page 222:

    IY.6 Diagonally Implicit RK Methods

  • Page 226:

    IV,6 Diagonally Implicit RK Methods

  • Page 230:

    Choice of Method

  • Page 234:

    IY.6 Diagonally Implicit RK Methods

  • Page 238:

    IV. 7 Rosenbrock -Type Methods 103

  • Page 242:

    etc. Inserting this into (7.6) we o

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    IV.7 Rosenbrock-Type Methods 107

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    Here we have used the abbreviations

  • Page 254:

    Higher Order Methods

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    IY.7 Rosenbrock-Type Methods 113

  • Page 262:

    the second must be zero. Similarly,

  • Page 266:

    Exercises

  • Page 270:

    IVS Implementation ofImplicit Runge

  • Page 274:

    IV8 Implementation ofImplicit Runge

  • Page 278:

    Step Size Selection

  • Page 282:

    IY.S Implementation ofImplicit Rung

  • Page 286:

    IV.8 Implementation ofImplicit Rung

  • Page 290:

    IY.8 Implementation ofImplicit Rung

  • Page 294:

    IV.9 Extrapolation Methods

  • Page 298:

    IV.9 Extrapolation Methods 133

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    IY.9 Extrapolation Methods 135

  • Page 306:

    IY.9 Extrapolation Methods 137

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    IY.9 Extrapolation Methods 139

  • Page 314:

    IV.9 Extrapolation Methods 141

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    [V.10 Numerical Experiments

  • Page 322:

    IY.lO Numerical Experiments 145

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    Fig. 10.3. The cusp catastrophe wit

  • Page 330:

    IY.lO Numerical Experiments 149

  • Page 334:

    IV.lO Numerical Experiments 151

  • Page 338:

    IY.lO Numerical Experiments 153

  • Page 342:

    sec BECKDO

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    IY.lO Numerical Experiments 157

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    lOU ROBER

  • Page 354:

    IY.lO Numerical Experiments 161

  • Page 358:

    IY.lO Numerical Experiments 163

  • Page 362:

    tion T-ICT, we have

  • Page 366:

    IV.II Contractivity for Linear Prob

  • Page 370:

    IV.ll Contractivity for Linear Prob

  • Page 374:

    IY.11 Contractivity for Linear Prob

  • Page 378:

    0 as a free parameter. It">with "Y > 0 as a free parameter. It

  • Page 382:

    Contractivity in 11·1100 and 11·1

  • Page 386:

    IVll Contractivity for Linear Probl

  • Page 390:

    IV.ll Contractivity for Linear Prob

  • Page 394:

    IV.l2 B -Stability and Contractivit

  • Page 398:

    IV.12 B -Stability and Contractivit

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    IV.l2 B -Stability and Contractivit

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    Reducible Runge-Kutta Methods

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    IY.12 B -Stability and Contractivit

  • Page 414:

    IV.l2 B -Stability and Contractivit

  • Page 418:

    Error Growth Function

  • Page 422:

    IV.12 B -Stability and Contractivit

  • Page 426:

    IY.12 B -Stability and Contractivit

  • Page 430:

    Exercises

  • Page 434:

    IV.13 Positive Quadrature Formulas

  • Page 438:

    IY.13 Positive Quadrature Fonnulas

  • Page 442:

    IV13 Positive Quadrature Fonnulas a

  • Page 446:

    We further introduce

  • Page 450:

    IY.13 Positive Quadrature Formulas

  • Page 454:

    This means that (see (13.17))

  • Page 458:

    IY.13 Positive Quadrature Formulas

  • Page 462:

    IV.14 Existence and Uniqueness of I

  • Page 466:

    IY.14 Existence and Uniqueness of I

  • Page 470:

    IV14 Existence and Uniqueness oflRK

  • Page 474:

    IY.14 Existence and Uniqueness ofIR

  • Page 478:

    IY.14 Existence and Uniqueness ofIR

  • Page 482:

    IV.IS B-Convergence

  • Page 486:

    TY.15 B -Convergence 227

  • Page 490:

    IY.15 B -Convergence 229

  • Page 494:

    yields the desired estimate for v ~

  • Page 498:

    isfies (with h = liN)

  • Page 502:

    The main idea is now to introduce t

  • Page 506:

    IY.15 B -Convergence 237

  • Page 510:

    Chapter V. Multistep Methods for St

  • Page 514:

    Definition 1.1. The set

  • Page 518:

    Hence the root locus curve becomes

  • Page 522:

    V.1 Stability of Multistep Methods

  • Page 526:

    V.l Stability of Multistep Methods

  • Page 530:

    Exercises

  • Page 534:

    Y.2 "Nearly" A-Stable Multistep Met

  • Page 538:

    Y.2 "Nearly" A -Stable Multistep Me

  • Page 542:

    Y.2 "Nearly" A -Stable Multistep Me

  • Page 546:

    esulting function again satisfies

  • Page 550:

    Exercises

  • Page 554:

    V.3 Generalized Multistep Methods

  • Page 558:

    Y.3 Generalized Multistep Methods 2

  • Page 562:

    Second Derivative BDF Methods

  • Page 566:

    Y.3 Generalized Multistep Methods 2

  • Page 570:

    V.3 Generalized Multistep Methods 2

  • Page 574:

    V,3 Generalized Multistep Methods 2

  • Page 578:

    Methods of "Radau" Type

  • Page 582:

    or . ····:··········

  • Page 586:

    Y.3 Generalized Multistep Methods 2

  • Page 590:

    V.4 Order Stars on Riemann Surfaces

  • Page 594:

    VA Order Stars on Riemann Surfaces

  • Page 598:

    V4 Order Stars on Riemann Surfaces

  • Page 602:

    V.4 Order Stars on Riemann Surfaces

  • Page 606:

    VA Order Stars on Riemann Surfaces

  • Page 610:

    VA Order Stars on Riemann Surfaces

  • Page 614:

    VA Order Stars on Riemann Surfaees

  • Page 618:

    VA Order Stars on Riemann Surfaces

  • Page 622:

    V.4 Order Stars on Riemann Surfaces

  • Page 626:

    VA Order Stars on Riemann Surfaces

  • Page 630:

    V4 Order Stars on Riemann Surfaces

  • Page 634:

    0- 1 WllJJlllll.LJUIIIlLLJIIllll.LJ

  • Page 638:

    sec Y.S Experiments with Multistep

  • Page 642:

    V.6 One-Leg Methods and G-Stability

  • Page 646:

    V6 One-Leg Methods and G -stability

  • Page 650:

    Y.6 One-Leg Methods and G -stabilit

  • Page 654:

    which is of the form

  • Page 658:

    The function E( () (Formula (6.22»

  • Page 662:

    Taylor expansion of (6.37) and (6.3

  • Page 666:

    Y.6 One-Leg Methods and G -stabilit

  • Page 670:

    This proves the statement, because

  • Page 674:

    V.7 Convergence for Linear Problems

  • Page 678:

    V.7 Convergence for Linear Problems

  • Page 682:

    V.7 Convergence for Linear Problems

  • Page 686:

    Y.7 Convergence for Linear Problems

  • Page 690:

    Y.7 Convergence for Linear Problems

  • Page 694:

    Y.7 Convergence for Linear Problems

  • Page 698:

    Y.7 Convergence for Linear Problems

  • Page 702:

    V.7 Convergence for Linear Problems

  • Page 706:

    Y.7 Convergence for Linear Problems

  • Page 710:

    V.8 Convergence for Nonlinear Probl

  • Page 714:

    Y.S Convergence for Nonlinear Probl

  • Page 718:

    Y.S Convergence for Nonlinear Probl

  • Page 722:

    Y.S Convergence for Nonlinear Probl

  • Page 726:

    Y.8 Convergence for Nonlinear Probl

  • Page 730:

    depend on L) such that

  • Page 734:

    Y.8 Convergence for Nonlinear Probl

  • Page 738:

    (f) of the proof) so that

  • Page 742:

    Prove that for hL :::; 1 - C-1

  • Page 746:

    V9 Algebraic Stability of General L

  • Page 750:

    Y.9 Algebraic Stability of General

  • Page 754:

    Y.9 Algebraic Stability of General

  • Page 758:

    Further, the matrices A and B have

  • Page 762:

    Y.9 Algebraic Stability of General

  • Page 766:

    Y.9 Algebraic Stability of General

  • Page 770:

    Y.9 Algebraic Stability of General

  • Page 774:

    Chapter VI. Singular Perturbation P

  • Page 778:

    Vr.l Solving Index I Problems 373

  • Page 782:

    VI. 1 Solving Index 1 Problems 375

  • Page 786:

    node 1:

  • Page 790:

    gives yr.l Solving Index I Problems

  • Page 794:

    VI. 1 Solving Index 1 Problems 381

  • Page 798:

    VI.2 Multistep Methods 383

  • Page 802:

    also holds for negative n. Solving

  • Page 806:

    VI.2 Multistep Methods 387

  • Page 810:

    VI.3 Epsilon Expansions for Exact a

  • Page 814:

    VI.3 Epsilon Expansions for Exact a

  • Page 818:

    and the internal stages are given b

  • Page 822:

    VI.3 Epsilon Expansions for Exact a

  • Page 826:

    VI.3 Epsilon Expansions for Exact a

  • Page 830:

    VI.3 Epsilon Expansions for Exact a

  • Page 834:

    VI.3 Epsilon Expansions for Exact a

  • Page 838:

    VI.3 Epsilon Expansions for Exact a

  • Page 842:

    Perturbed Initial Values

  • Page 846:

    VI.4 Rosenbrock Methods

  • Page 850:

    VI.4 Rosenbrock Methods 409

  • Page 854:

    Taylor Expansion of the Exact Solut

  • Page 858:

    VIA Rosenbrock Methods 413

  • Page 862:

    VIA Rosenbrock Methods 415

  • Page 866:

    Table 4.1. Trees and elementary dif

  • Page 870:

    VI.4 Rosenbrock Methods 4 J 9

  • Page 874:

    VIA Rosenbrock Methods

  • Page 878:

    VIA Rosenbrock Methods 423

  • Page 882:

    3. State the order condition for th

  • Page 886:

    Example 5.1. Consider the test prob

  • Page 890:

    VI. 5 Extrapolation Methods 429

  • Page 894:

    V1.5 Extrapolation Methods 431

  • Page 898:

    VI.S Extrapolation Methods 433

  • Page 902:

    such that

  • Page 906:

    gives ( ~~::~ ) = (~~:)

  • Page 910:

    VI.5 Extrapolation Methods 439

  • Page 914:

    Exercises

  • Page 918:

    VI.6 Quasilinear Problems 443

  • Page 922:

    Problems of Index One

  • Page 926:

    VI.6 Quasilinear Problems 447

  • Page 930:

    VI.6 Quasilinear Problems 449

  • Page 936:

    VII.1 The Index and Various Example

  • Page 940:

    454 VII. Differential-Algebraic Equ

  • Page 944:

    456 VII. Differential-Algebraic Equ

  • Page 948:

    458 VII. Differential-Algebraic Equ

  • Page 952:

    460 VII. Differential-Algebraic Equ

  • Page 956:

    462 VII. Differential-Algebraic Equ

  • Page 960:

    464 VII. Differential-Algebraic Equ

  • Page 964:

    466 VII. Differential-Algebraic Equ

  • Page 968:

    VII.2 Index Reduction Methods

  • Page 972:

    470 VII. Differential-Algebraic Equ

  • Page 976:

    472 VII. Differential-Algebraic Equ

  • Page 980:

    474 VII. Differential-Algebraic Equ

  • Page 984:

    476 VII. Differential-Algebraic Equ

  • Page 988:

    478 VII. Differential-Algebraic Equ

  • Page 992:

    480 VII. Differential-Algebraic Equ

  • Page 996:

    482 VII. Differential-Algebraic Equ

  • Page 1000:

    484 VII. Differential-Algebraic Equ

  • Page 1004:

    486 VII. Differential-Algebraic Equ

  • Page 1008:

    488 VII. Differential-Algebraic Equ

  • Page 1012:

    490 VIT. Differential-Algebraic Equ

  • Page 1016:

    VII.4 Runge-Kutta Methods for Index

  • Page 1020:

    494 VII. Differential-Algebraic Equ

  • Page 1024:

    496 VII. Differential-Algebraic Equ

  • Page 1028:

    498 VII. Differential-Algebraic Equ

  • Page 1032:

    500 VII. Differential-Algebraic Equ

  • Page 1036:

    502 VII. Differential-Algebraic Equ

  • Page 1040:

    504 VII. Differential-Algebraic Equ

  • Page 1044:

    VII.S Order Conditions for Index 2

  • Page 1048:

    508 VII. Differential-Algebraic Equ

  • Page 1052:

    510 VII. Differential-Algebraic Equ

  • Page 1056:

    512 VII. Differential-Algebraic Equ

  • Page 1060:

    514 VII. Differential-Algebraic Equ

  • Page 1064:

    516 VII. Differential-Algebraic Equ

  • Page 1068:

    518 VII. Differential-Algebraic Equ

  • Page 1072:

    520 VII. Differential-Algebraic Equ

  • Page 1076:

    522 VII. Differential-Algebraic Equ

  • Page 1080:

    524 VII. Differential-Algebraic Equ

  • Page 1084:

    526 VII. Differential-Algebraic Equ

  • Page 1088:

    528 VII. Differential-Algebraic Equ

  • Page 1092:

    VII.7 Computation of Multibody Mech

  • Page 1096:

    532 VII. Differential-Algebraic Equ

  • Page 1100:

    534 VII. Differential-Algebraic Equ

  • Page 1104:

    536 VII. Differential-Algebraic Equ

  • Page 1108:

    538 VII. Differential-Algebraic Equ

  • Page 1112:

    540 VII. Differential-Algebraic Equ

  • Page 1116:

    542 VIT. Differential-Algebraic Equ

  • Page 1120:

    544 VII. Differential-Algebraic Equ

  • Page 1124:

    546 VII. Differential-Algebraic Equ

  • Page 1128:

    548 VII. Differential-Algebraic Equ

  • Page 1132:

    550 VII. Differential-Algebraic Equ

  • Page 1136:

    552 VII. Differential-Algebraic Equ

  • Page 1140:

    554 VII. Differential-Algebraic Equ

  • Page 1144:

    556 VII. Differential-Algebraic Equ

  • Page 1148:

    558 VII. Differential-Algebraic Equ

  • Page 1152:

    560 VII. Differential-Algebraic Equ

  • Page 1156:

    562 VII. Differential-Algebraic Equ

  • Page 1162:

    Appendix. Fortran Codes

  • Page 1166:

    Appendix. Fortran Codes 567

  • Page 1170:

    C C

  • Page 1174:

    C C

  • Page 1178:

    Appendix. Fortran Codes 573

  • Page 1182:

    Appendix. Fortran Codes 575

  • Page 1186:

    Bibliography

  • Page 1190:

    Bibliography 579

  • Page 1194:

    Bibliography 581

  • Page 1198:

    Bibliography 583

  • Page 1202:

    Bibliography 585

  • Page 1206:

    Bibliography 587

  • Page 1210:

    R.W. HansonSmith, see D.S. Watkins

  • Page 1214:

    Bibliography 591

  • Page 1218:

    Bibliography 593

  • Page 1222:

    Bibliography 595

  • Page 1226:

    Bibliography 597

  • Page 1230:

    Bibliography 599

  • Page 1234:

    Bibliography 601

  • Page 1238:

    Bibliography 603

  • Page 1242:

    Symbol Index

  • Page 1246:

    Subject Index

  • Page 1250:

    with invariants, 472f.

  • Page 1254:

    manifold, 457.

  • Page 1258:

    second derivative BDF methods, 265.

Contents

Contents

file1.pdf

file1.pdf

Developmental Genetics and Plant Evolution

Developmental Genetics and Plant Evolution

Jörg Grabner · Richard Nothhaft Konstruieren von Pkw-Karosserien

Jörg Grabner · Richard Nothhaft Konstruieren von Pkw-Karosserien

Why History

Why History

Немецкий для начинающих

Немецкий для начинающих

Surrealism and Architecture

Surrealism and Architecture

Untitled

Untitled

Untitled

Untitled

ИНЖЕНЕРНАЯ ГРАФИКА

ИНЖЕНЕРНАЯ ГРАФИКА

Untitled

Untitled

№1 (20) 2009

№1 (20) 2009

Front Matter

Front Matter

INSIGHTS INTO HITTITE HISTORY AND ARCHAEOLOGY

INSIGHTS INTO HITTITE HISTORY AND ARCHAEOLOGY

für Wort

für Wort

il E=!=68

il E=!=68

Fatigue as a Phenomenon in the Material

Fatigue as a Phenomenon in the Material

Power System Analysis & Design, SI Version, 5th ed.

Power System Analysis & Design, SI Version, 5th ed.

Organizations: Behavior, Structure, Processes

Organizations: Behavior, Structure, Processes

О рекламе

О рекламе

Computational Fluid Dynamics 2010

Computational Fluid Dynamics 2010

Андрей Дикий

Андрей Дикий

histopathology

histopathology

Multi-scale Quantum Models for Biocatalysis: Modern Techniques ...

Multi-scale Quantum Models for Biocatalysis: Modern Techniques ...

Thermodynamics of Systems Containing Flexible - Chain Polymers

Thermodynamics of Systems Containing Flexible - Chain Polymers

AFRICA

AFRICA

Probability, Random Processes, and Statistical Analysis

Probability, Random Processes, and Statistical Analysis

Introductory Calculations

Introductory Calculations

Untitled

Untitled

Ocular Trauma : Principles and Practice

Ocular Trauma : Principles and Practice

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