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In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have originally been developed to ...
If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these.
All loops must have fixed bounds. This prevents runaway code. Avoid heap memory allocation. Restrict functions to a single printed page. Use a minimum of two runtime assertions per function. Restrict the scope of data to the smallest possible. Check the return value of all non-void functions, or cast to void to indicate the return value is useless.
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.
Since [7, 4, 3] = [n, k, d] = [2 m − 1, 2 m − 1 − m, 3]. The parity-check matrix H of a Hamming code is constructed by listing all columns of length m that are pair-wise independent. Thus H is a matrix whose left side is all of the nonzero n-tuples where order of the n-tuples in the columns of matrix does not matter.
IronPython allows running Python 2.7 programs (and an alpha, released in 2021, is also available for "Python 3.4, although features and behaviors from later versions may be included" [170]) on the .NET Common Language Runtime. [171] Jython compiles Python 2.7 to Java bytecode, allowing the use of the Java libraries from a Python program. [172]
An example is the derivative operator of calculus, /, which is a linear operator acting on functions () to give a new function (/) = ′ (). The n th power of the differentiation operator is the n th derivative:
As one special case, it can be used to prove that if n is a positive integer then 4 divides () if and only if n is not a power of 2. It follows from Legendre's formula that the p-adic exponential function has radius of convergence / ().