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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
[3] Dynamically valued constants originated as a language feature with ALGOL 68. [3] Studies of Ada and C++ code have shown that dynamically valued constants are used infrequently, typically for 1% or less of objects, when they could be used much more, as some 40–50% of local, non-class objects are actually invariant once created.
This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33. Decimal: Binary: 11 3 1011 11 5 6 101 110 2 12 10 1100 1 24 1 11000 —— —————— 33 100001 Describing the steps explicitly: 11 and 3 are written at the top
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 field. If the polynomials are encoded as binary numbers, carry-less multiplication can be used to perform the first step of this ...
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
A linear congruential generator with base b = 2 32 is implemented as + = (+) , where c is a constant. If a ≡ 1 (mod 4) and c is odd, the resulting base-2 32 congruential sequence will have period 2 32.
The Object class is the root of the class hierarchy, so that its methods and attributes are available for all instantiated objects of each class. The Class class (a.k.a. meta class) is used to maintain the properties of a class (like its method objects) and gets used for creating instances (a.k.a. objects, values).