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Raising and lowering is then done in coordinates. Given a vector with components , we can contract with the metric to obtain a covector: = and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:
That is, the array starts at 1 (the initial value), increments with each step from the previous value by 2 (the increment value), and stops once it reaches (or is about to exceed) 9 (the terminator value). The increment value can actually be left out of this syntax (along with one of the colons), to use a default value of 1. >>
The matrix–vector multiplication can be done in () arithmetical operations where is the average number of nonzero elements in a row. The total complexity is thus O ( d m n ) {\displaystyle O(dmn)} , or O ( d n 2 ) {\displaystyle O(dn^{2})} if m = n {\displaystyle m=n} ; the Lanczos algorithm can be very fast for sparse matrices.
The Nial example of the inner product of two arrays can be implemented using the native matrix multiplication operator. If a is a row vector of size [1 n] and b is a corresponding column vector of size [n 1]. a * b; By contrast, the entrywise product is implemented as: a .* b;
By analogy with the mathematical concepts vector and matrix, array types with one and two indices are often called vector type and matrix type, respectively. More generally, a multidimensional array type can be called a tensor type , by analogy with the physical concept, tensor .
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
The number of required evaluations is at least (/), where D is the length of the longest edge of the characteristic polyhedron. [8]: 11, Lemma.4.7 Note that Vrahatis and Iordanidis [8] prove a lower bound on the number of evaluations, and not an upper bound.
They may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods. [ citation needed ] A large class of useful Monte Carlo methods are the so-called Markov chain Monte Carlo algorithms, which include the Metropolis–Hastings algorithm and Gibbs sampling .