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<title>sageSLZ Notes</title> |
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</head>
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<body>
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<h1>sageSLZ Notes</h1> |
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<h2>The approximation interval problem</h2> |
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<p>When Sollya computes an approximation polynomial for a given function,
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degree and range, it also returns the approximation error (and the center of |
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the interval as well).</p>
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<p>Very often we want to solve a slightly different problem: for a given
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function, degree, lower bound and approximation error, computed the |
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approximation polynomial and the upper bound of the interval comprised between |
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it and the upper bound for which the approximation error stands.</p>
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<p>Very often the idea is to break a given “large” into a |
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collection of smaller intervals because for the given function-degree-error the |
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initial interval is too large.</p>
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<p>The naive approach is to:</p> |
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<ol>
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<li>give it a try with the initial data;</li> |
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<li>when you get a too large error, reduce (e. g. divide by 2) the interval by
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lowering the upper bound until the computed error matches the target |
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error;</li>
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<li>this upper bound becomes the lower bound of the next interval, always
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using the orginal upper bound and repeat the process until you can cover |
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the all initial interval with the computed sub-interval.</li>
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</ol>
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<p>In step 2 you may have to compute several polynomials until the computed
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error is below or equal to the expected error. You must not decrease the |
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interval width too agressively. There is no point in getting a too small approximation error since it multiplies the number of interval (and |
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lengthens the subsequent manipulations that are made on each sub-interval).</p>
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<p>In step 3 you probably inutily compute polynomials for too large intervals
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and you could know it from the previous computations.</p>
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<p>The whole idea is to reduce the number of polynomial computations and get the
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maximum width (compatible with the expected approwimaiton error).</p>
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<p>This process has different aspects:</p> |
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<ol>
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<li>for a given interval (be it too small or too large) and the given
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(computed) error, compute the new best upper bound;</li>
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<li>reuse the computation already done in previous steps.</li> |
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</ol>
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<p>For point 1, can something be done with the derivative at the upper bound or
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at the center point of the interval (also given by Sollya).</p>
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<p>For point 2, can some type of data structure keep the informations (which
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ones ?) and allow for an efficient reuse.</p>
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<p>Adjacent question: is it possible, given a the result of a previous
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computation, to forecast the effect on the approximation precision of a degree |
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in(de)crease?</p>
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