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NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
To determine if a number is a power of two, conceptually we may repeatedly do integer divide by two until the number won't divide by 2 evenly; if the only factor left is 1, the original number was a power of 2. Using bit and logical operators, there is a simple expression which will return true (1) or false (0):
For example, John von Neumann constructs the number 0 as the empty set {}, and the successor of n, S(n), as the set n ∪ {n}. The axiom of infinity then guarantees the existence of a set that contains 0 and is closed with respect to S. The smallest such set is denoted by N, and its members are called natural numbers. [2]
For example, consider the following expression in which both variables are bound by logical quantifiers: ∀ y ∃ x ( x = y ) . {\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).} This expression evaluates to false if the domain of x {\displaystyle x} and y {\displaystyle y} is the real numbers, but true if the domain is the ...
The algorithm's given problem can be a “family of problems”. [10] There are two main types of these skeletons, ‘divide and conquer’ or ‘brand and bound’. ‘Divide and conquer’ uses a map skeleton as its basis, combining this with a while skeleton to solve the problem. In map algorithms, functions on data are applied simultaneously.
An example of a decision problem is deciding with the help of an algorithm whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?" A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
In 1960, Kolmogorov organized a seminar on mathematical problems in cybernetics at the Moscow State University, where he stated the () conjecture and other problems in the complexity of computation. Within a week, Karatsuba, then a 23-year-old student, found an algorithm that multiplies two n -digit numbers in O ( n log 2 3 ) {\displaystyle ...