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  2. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    This method is an efficient variant of the 2 k-ary method. For example, to calculate the exponent 398, which has binary expansion (110 001 110) 2, we take a window of length 3 using the 2 k-ary method algorithm and calculate 1, x 3, x 6, x 12, x 24, x 48, x 49, x 98, x 99, x 198, x 199, x 398.

  3. Exponentiation - Wikipedia

    en.wikipedia.org/wiki/Exponentiation

    So, denotes the power set of S, that is the set of the functions from S to {,}, which can be identified with the set of the subsets of S, by mapping each function to the inverse image of 1. This fits in with the exponentiation of cardinal numbers , in the sense that | S T | = | S | | T | , where | X | is the cardinality of X .

  4. List of open-source software for mathematics - Wikipedia

    en.wikipedia.org/wiki/List_of_open-source...

    PARI/GP is a computer algebra system that facilitates number-theory computation. Besides support of factoring, algebraic number theory, and analysis of elliptic curves, it works with mathematical objects like matrices, polynomials, power series, algebraic numbers, and transcendental functions. [3]

  5. List of numerical-analysis software - Wikipedia

    en.wikipedia.org/wiki/List_of_numerical-analysis...

    PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of ...

  6. Modular exponentiation - Wikipedia

    en.wikipedia.org/wiki/Modular_exponentiation

    The smallest counterexample is for a power of 15, when the binary method needs six multiplications. Instead, form x 3 in two multiplications, then x 6 by squaring x 3 , then x 12 by squaring x 6 , and finally x 15 by multiplying x 12 and x 3 , thereby achieving the desired result with only five multiplications.

  7. Power iteration - Wikipedia

    en.wikipedia.org/wiki/Power_iteration

    In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.

  8. Minnesota Parents Who Locked Their Kids in Cages for ‘Their ...

    www.aol.com/lifestyle/minnesota-parents-locked...

    A Minnesota couple has reportedly been sentenced to four years after they locked their children in cages for "their safety." Benjamin and Christina Cotton from Red Wing, were sentenced by a ...

  9. GOP report: Liz Cheney should be investigated by FBI ... - AOL

    www.aol.com/gop-report-liz-cheney-investigated...

    (The Center Square) – A new Republican oversight report accuses former Congresswoman Liz Cheney of colluding with witnesses in the Jan. 6 Select Committee investigation that she oversaw.