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Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
For example, if we want to round 1.2459 to 3 significant figures, then this step results in 1.25. If the n + 1 digit is 5 not followed by other digits or followed by only zeros, then rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures: Round half away from zero rounds up to 1.3.
The area of the blue region converges to Euler's constant. Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
A single-layer neural network computes a continuous output instead of a step function. The derivative of p i with respect to X = (x 1, ..., x k) is computed from the general form: = + where f(X) is an analytic function in X. With this choice, the single-layer neural network is identical to the logistic regression model.
The second step is possible due to the fact that, if AB = BA, then e At B = Be At. So, calculating e At leads to the solution to the system, by simply integrating the third step with respect to t . A solution to this can be obtained by integrating and multiplying by e A t {\displaystyle e^{{\textbf {A}}t}} to eliminate the exponent in the LHS.
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix.
Newton iteration starting anywhere left of the zero will converge, as will Fourier's modified Newton iteration starting anywhere right of the zero. The accuracy at any step of the iteration can be determined directly from the difference between the location of the iteration from the left and the location of the iteration from the right.
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.