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14-640: Users of the NAG Library call its routines from within their applications in order to incorporate its mathematical or statistical functionality and to solve numerical problems - for example, finding the minimum or maximum of a function , fitting a curve or surface to data , or solving a differential equation . The NAG Library can be accessed from a variety of languages and environments such as C/C++, Fortran, Python, AD, MATLAB, Java and .NET. The main supported systems are currently Windows , Linux and macOS running on x86-64 architectures; 32-bit Windows support

28-665: A continuous optimization problem is minimize x f ( x ) s u b j e c t t o g i ( x ) ≤ 0 , i = 1 , … , m h j ( x ) = 0 , j = 1 , … , p {\displaystyle {\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\\&&&h_{j}(x)=0,\quad j=1,\dots ,p\end{aligned}}} where If m = p = 0 ,

42-439: A simple 'yes' or 'no'. In the field of approximation algorithms , algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem. Additive inverse In mathematics,

56-476: Is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0 . For example, if there is a graph G which contains vertices u and v , an optimization problem might be "find a path from u to v that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from u to v that uses 10 or fewer edges?" This problem can be answered with

70-510: Is being phased out. Some NAG mathematical optimization solvers are accessible via the optimization modelling suite. The original version of the NAG Library was written in Algol 60 and Fortran . It contained 98 user-callable routines, and was released for the ICL 1906A and 1906S machines on October 1, 1971. Three further Marks of the library appeared in the following five years; during this time

84-1107: Is closely related to subtraction , which can be viewed as an addition using the inverse: Conversely, the additive inverse can be thought of as subtraction from zero: This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors. Given an algebraic structure defined under addition ( S , + ) {\displaystyle (S,+)} with an additive identity e ∈ S {\displaystyle e\in S} , an element x ∈ S {\displaystyle x\in S} has an additive inverse y {\displaystyle y} if and only if y ∈ S {\displaystyle y\in S} , x + y = e {\displaystyle x+y=e} , and y + x = e {\displaystyle y+x=e} . Addition

98-445: Is important in solving algebraic equations . Not all sets where addition is defined have an additive inverse, such as the natural numbers . When working with integers , rational numbers , real numbers , and complex numbers , the additive inverse of any number can be found by multiplying it by −1 . The concept can also be extended to algebraic expressions, which is often used when balancing equations . The additive inverse

112-445: Is typically only used to refer to a commutative operation, but it is not necessarily associative . When it is associative, so ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} , the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases,

126-403: The additive inverse of an element x , denoted -x , is the element that when added to x , yields the additive identity , 0 . In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element . In elementary mathematics , the additive inverse is often referred to as the opposite number. The concept is closely related to subtraction and

140-462: The shared memory parallelism of Symmetric Multi-Processors (SMP) and multicore processors , appeared in 1997 for multiprocessor machines built using the Dec Alpha and SPARC architectures. The NAG Library for .NET, which is a CLI DLL assembly containing methods and objects that give Common Language Infrastructure (CLI) users access to NAG algorithms, was first released in 2010. Mark 29 of

154-581: The Algol version was ported to Algol 68, with the following platforms being supported: CDC 7600 / CYBER ( CDC ALGOL 68 ), IBM 360 /370/AMDAHL ( FLACC ALGOL 68 ), ICL 1900 ( ALGOL 68R ), ICL 1906A/S ( ALGOL 68R ), ICL 2900 ( ALGOL 68RS ) and Telefunken TR440 ( ALGOL 68C ). The first partially vectorized implementation of the NAG Fortran Library for the Cray-1 was released in 1983, while

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168-453: The NAG Library includes mathematical and statistical algorithms organised into chapters. Optimization problem In mathematics , engineering , computer science and economics , an optimization problem is the problem of finding the best solution from all feasible solutions . Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete : The standard form of

182-660: The first release of the NAG Parallel Library (which was specially designed for distributed memory parallel computer architectures) was in the early 1990s. Mark 1 of the NAG C Library was released in 1990. In 1992, the Library incorporated LAPACK routines for the first time; NAG had been a collaborator in the LAPACK project since 1987. The first release of the NAG Library for SMP & Multicore, which takes advantage of

196-711: The problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem . A maximization problem can be treated by negating the objective function. Formally, a combinatorial optimization problem A is a quadruple ( I , f , m , g ) , where The goal is then to find for some instance x an optimal solution , that is, a feasible solution y with m ( x , y ) = g { m ( x , y ′ ) : y ′ ∈ f ( x ) } . {\displaystyle m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.} For each combinatorial optimization problem, there

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