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In numerical mathematics, hierarchical matrices (H-matrices) [1] [2] [3] are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension can be represented efficiently in () units of storage by storing only its non-zero entries, a non-sparse matrix would require () units of storage, and using this type of matrices for large problems would therefore be ...
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
Numeric literals in Python are of the normal sort, e.g. 0, -1, 3.4, 3.5e-8. Python has arbitrary-length integers and automatically increases their storage size as necessary. Prior to Python 3, there were two kinds of integral numbers: traditional fixed size integers and "long" integers of arbitrary size.
Python's Guido van Rossum summarizes C3 superclass linearization thus: [11] Basically, the idea behind C3 is that if you write down all of the ordering rules imposed by inheritance relationships in a complex class hierarchy, the algorithm will determine a monotonic ordering of the classes that satisfies all of them.
In Python, namespaces are defined by the individual modules, and since modules can be contained in hierarchical packages, then namespaces are hierarchical too. [ 12 ] [ 13 ] In general when a module is imported then the names defined in the module are defined via that module's namespace, and are accessed in from the calling modules by using the ...
This unsorted tree has non-unique values (e.g., the value 2 existing in different nodes, not in a single node only) and is non-binary (only up to two children nodes per parent node in a binary tree). The root node at the top (with the value 2 here), has no parent as it is the highest in the tree hierarchy.
The latter corresponds to a hierarchical structure ("syntax tree") which is unique for the given expression. The compiler generates machine code from the tree in such a way that operations originating at the lowest hierarchy level are executed first. Examples: !A + !B is interpreted as (!A) + (!B) ++A + !B is interpreted as (++A) + (!B)
Vandermonde matrix: A row consists of 1, a, a 2, a 3, etc., and each row uses a different variable. Walsh matrix: A square matrix, with dimensions a power of 2, the entries of which are +1 or −1, and the property that the dot product of any two distinct rows (or columns) is zero. Z-matrix: A matrix with all off-diagonal entries less than zero.