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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 */ |
| 268 | 80 | storres | /* L'appel de note est r?alis? par un ?l?ment "a" de classe callNote */ |
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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 | 80 | storres | </style>
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| 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 | 81 | storres | degree and range, it also returns the approximation error (refered to, in the sequel, |
| 310 | 81 | storres | as the |
| 311 | 81 | storres | <span class="langElem">computedError</span> and the center of |
| 312 | 81 | storres | the interval as well). |
| 313 | 81 | storres | </p>
|
| 314 | 67 | storres | |
| 315 | 81 | storres | <p>Very often, we want to solve a slightly different problem: for a given
|
| 316 | 81 | storres | function, degree, lower bound and approximation error (referred to, in the |
| 317 | 81 | storres | sequel, as the |
| 318 | 81 | storres | <span class="langElem">targetError</span> ), compute the |
| 319 | 67 | storres | approximation polynomial and the upper bound of the interval comprised between |
| 320 | 81 | storres | it and the lower bound for which the approximation error stands. |
| 321 | 81 | storres | </p>
|
| 322 | 67 | storres | |
| 323 | 81 | storres | <p>The idea is to break a given initially “large” interval into a |
| 324 | 67 | storres | collection of smaller intervals because for the given function-degree-error the |
| 325 | 67 | storres | initial interval is too large.</p>
|
| 326 | 67 | storres | |
| 327 | 81 | storres | <p>The most naive approach is to:</p> |
| 328 | 67 | storres | <ol>
|
| 329 | 81 | storres | <li>give it a try with the initial data (a very probably too high upper bound
|
| 330 | 81 | storres | for the given <span class="langElem">targetError</span>);</li> |
| 331 | 69 | storres | <li>when you get a too large error, reduce (e. g. divide by 2) the interval by
|
| 332 | 69 | storres | lowering the upper bound until the computed error matches the target |
| 333 | 67 | storres | error;</li>
|
| 334 | 69 | storres | <li>this upper bound becomes the lower bound of the next interval, always
|
| 335 | 67 | storres | using the orginal upper bound and repeat the process until you can cover |
| 336 | 67 | storres | the all initial interval with the computed sub-interval.</li>
|
| 337 | 67 | storres | </ol>
|
| 338 | 67 | storres | |
| 339 | 67 | storres | <p>In step 2 you may have to compute several polynomials until the computed
|
| 340 | 81 | storres | error is below or equal to the expected error. But you must not decrease the |
| 341 | 81 | storres | interval width too agressively: there is no point in getting a too small |
| 342 | 81 | storres | approximation error since it multiplies the number of interval (and |
| 343 | 69 | storres | lengthens the subsequent manipulations that are made on each sub-interval).</p>
|
| 344 | 67 | storres | |
| 345 | 69 | storres | <p>In step 3 you probably inutily compute polynomials for too large intervals
|
| 346 | 67 | storres | and you could know it from the previous computations.</p>
|
| 347 | 67 | storres | |
| 348 | 69 | storres | <p>The whole idea is to reduce the number of polynomial computations and get the
|
| 349 | 81 | storres | maximum width (compatible with the target error).</p>
|
| 350 | 67 | storres | |
| 351 | 67 | storres | <p>This process has different aspects:</p> |
| 352 | 67 | storres | <ol>
|
| 353 | 67 | storres | <li>for a given interval (be it too small or too large) and the given
|
| 354 | 67 | storres | (computed) error, compute the new best upper bound;</li>
|
| 355 | 67 | storres | <li>reuse the computation already done in previous steps.</li> |
| 356 | 67 | storres | </ol>
|
| 357 | 67 | storres | |
| 358 | 81 | storres | <p>We use two strategies.</p> |
| 359 | 81 | storres | <p>The first is in the inner function that computes the first approximation
|
| 360 | 81 | storres | polynomial. If the upper bound is too high, we shrink the interval by a non |
| 361 | 81 | storres | linear (and complex) factor computed from the ratio beetwen the obtained |
| 362 | 81 | storres | error and the target error.<br>The whole idea is not to end up with a too
|
| 363 | 81 | storres | small interval.</p>
|
| 364 | 81 | storres | <p>This only takes into account aspect 1</p>. |
| 365 | 81 | storres | <p>The second strategy is in the outer function. In this function the
|
| 366 | 81 | storres | computed intervals returned by the inner function give all correct |
| 367 | 81 | storres | approximation errors. The computation of next upper bound will take into |
| 368 | 81 | storres | account the value of the previous interval and that of the previous |
| 369 | 81 | storres | computed error.</p>
|
| 370 | 81 | storres | <p>If the errors ratio is lower than 2, we try the same interval length
|
| 371 | 81 | storres | as given by the previous computation. If larger we increase the next |
| 372 | 81 | storres | interval but by a small margin (log2(errorsRatio).</p>
|
| 373 | 67 | storres | |
| 374 | 81 | storres | <p>In this case, aspects 1 and 2 are considered </p> |
| 375 | 81 | storres | |
| 376 | 81 | storres | <p>For the moment, we do not resort to more complex forcasting techniques
|
| 377 | 81 | storres | (using the derivative of the function?). This could be an option if the |
| 378 | 81 | storres | present strategies did not output enough performance. </p>
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| 379 | 67 | storres | </body>
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| 380 | 67 | storres | </html> |