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A visual memory tool can replace the FOIL mnemonic for a pair of polynomials with any number of terms. Make a table with the terms of the first polynomial on the left edge and the terms of the second on the top edge, then fill in the table with products of multiplication. The table equivalent to the FOIL rule looks like this:
First multiply the quarters by 47, the result 94 is written into the first workspace. Next, multiply cwt 12*47 = (2 + 10)*47 but don't add up the partial results (94, 470) yet. Likewise multiply 23 by 47 yielding (141, 940). The quarters column is totaled and the result placed in the second workspace (a trivial move in this case).
Since that work, Pan has returned to symbolic and numeric computation and to an earlier theme of his research, computations with polynomials. He developed fast algorithms for the numerical computation of polynomial roots, and, with Bernard Mourrain, algorithms for multivariate polynomials based on their relations to structured matrices. [5]
Here we consider operations over polynomials and n denotes their degree; for the coefficients we use a unit-cost model, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers.
Note that a polynomial of degree d is uniquely determined by d + 1 points (for example, a line - polynomial of degree one is specified by two points). The idea is to evaluate p(·) and q(·) at various points. Then multiply their values at these points to get points on the product polynomial. Finally interpolate to find its coefficients.
Applications of the Schönhage–Strassen algorithm include large computations done for their own sake such as the Great Internet Mersenne Prime Search and approximations of π, as well as practical applications such as Lenstra elliptic curve factorization via Kronecker substitution, which reduces polynomial multiplication to integer ...
The elements of GF(2 n), i.e. a finite field whose order is a power of two, are usually represented as polynomials in GF(2)[X]. Multiplication of two such field elements consists of multiplication of the corresponding polynomials, followed by a reduction with respect to some irreducible polynomial which is taken from the construction of the ...
In a doubly linked list, one can insert or delete a node in a constant number of operations given only that node's address. To do the same in a singly linked list, one must have the address of the pointer to that node, which is either the handle for the whole list (in case of the first node) or the link field in the previous node. Some ...