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| 230 | 67 | storres | /* Introduction ? une (x*sous) section */ |
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| 258 | 67 | storres | /* Sommaire */ |
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| 267 | 67 | storres | /* Notes */ |
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| 272 | 67 | storres | /* Liste des notes proprement dite */ |
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| 282 | 67 | storres | /* ?l?ments divers (? trier)*/ |
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| 299 | 67 | storres | </a></abbr></style> |
| 300 | 67 | storres | <title>sageSLZ Notes</title> |
| 301 | 67 | storres | </head>
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| 302 | 67 | storres | |
| 303 | 67 | storres | <body>
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| 304 | 67 | storres | <h1>sageSLZ Notes</h1> |
| 305 | 67 | storres | |
| 306 | 67 | storres | <h2>The approximation interval problem</h2> |
| 307 | 67 | storres | |
| 308 | 67 | storres | <p>When Sollya computes an approximation polynomial for a given function,
|
| 309 | 67 | storres | degree and range, it also returns the approximation error (and the center of |
| 310 | 67 | storres | the interval as well).</p>
|
| 311 | 67 | storres | |
| 312 | 67 | storres | <p>Very often we want to solve a slightly different problem: for a given
|
| 313 | 67 | storres | function, degree, lower bound and approximation error, computed the |
| 314 | 67 | storres | approximation polynomial and the upper bound of the interval comprised between |
| 315 | 67 | storres | it and the upper bound for which the approximation error stands.</p>
|
| 316 | 67 | storres | |
| 317 | 67 | storres | <p>Very often the idea is to break a given “large” into a |
| 318 | 67 | storres | collection of smaller intervals because for the given function-degree-error the |
| 319 | 67 | storres | initial interval is too large.</p>
|
| 320 | 67 | storres | |
| 321 | 67 | storres | <p>The naive approach is to:</p> |
| 322 | 67 | storres | <ol>
|
| 323 | 67 | storres | <li>give it a try with the initial data;</li> |
| 324 | 69 | storres | <li>when you get a too large error, reduce (e. g. divide by 2) the interval by
|
| 325 | 69 | storres | lowering the upper bound until the computed error matches the target |
| 326 | 67 | storres | error;</li>
|
| 327 | 69 | storres | <li>this upper bound becomes the lower bound of the next interval, always
|
| 328 | 67 | storres | using the orginal upper bound and repeat the process until you can cover |
| 329 | 67 | storres | the all initial interval with the computed sub-interval.</li>
|
| 330 | 67 | storres | </ol>
|
| 331 | 67 | storres | |
| 332 | 67 | storres | <p>In step 2 you may have to compute several polynomials until the computed
|
| 333 | 67 | storres | error is below or equal to the expected error. You must not decrease the |
| 334 | 69 | storres | interval width too agressively. There is no point in getting a too small approximation error since it multiplies the number of interval (and |
| 335 | 69 | storres | lengthens the subsequent manipulations that are made on each sub-interval).</p>
|
| 336 | 67 | storres | |
| 337 | 69 | storres | <p>In step 3 you probably inutily compute polynomials for too large intervals
|
| 338 | 67 | storres | and you could know it from the previous computations.</p>
|
| 339 | 67 | storres | |
| 340 | 69 | storres | <p>The whole idea is to reduce the number of polynomial computations and get the
|
| 341 | 69 | storres | maximum width (compatible with the expected approwimaiton error).</p>
|
| 342 | 67 | storres | |
| 343 | 67 | storres | <p>This process has different aspects:</p> |
| 344 | 67 | storres | <ol>
|
| 345 | 67 | storres | <li>for a given interval (be it too small or too large) and the given
|
| 346 | 67 | storres | (computed) error, compute the new best upper bound;</li>
|
| 347 | 67 | storres | <li>reuse the computation already done in previous steps.</li> |
| 348 | 67 | storres | </ol>
|
| 349 | 67 | storres | |
| 350 | 67 | storres | <p>For point 1, can something be done with the derivative at the upper bound or
|
| 351 | 67 | storres | at the center point of the interval (also given by Sollya).</p>
|
| 352 | 67 | storres | |
| 353 | 67 | storres | <p>For point 2, can some type of data structure keep the informations (which
|
| 354 | 67 | storres | ones ?) and allow for an efficient reuse.</p>
|
| 355 | 69 | storres | |
| 356 | 69 | storres | <p>Adjacent question: is it possible, given a the result of a previous
|
| 357 | 69 | storres | computation, to forecast the effect on the approximation precision of a degree |
| 358 | 69 | storres | in(de)crease?</p>
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| 359 | 67 | storres | </body>
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| 360 | 67 | storres | </html> |