# Heuristic Math

In mathematics, heuristic math a calculation method that quickly provides a solution!

The heuristic method is based on an almost continuous evaluation, mainly formative. The acquisition of concepts is assessed during the teacher’s observations. The assessment is based on explicit criteria shared with the students. The teacher specifies what he is evaluating (an achievement or the process used). This is the time to discuss, to verbalize successes and difficulties. The evaluations have a double objective: to allow the student to know where he is, and to allow the teacher to identify the difficulties in the students and to adjust the implementation of his teaching.

Heuristic is a computational method for solving a problem more quickly when classic methods are too slow

### In mathematics

Heuristic is a computational method that quickly provides a solution.

It distinguishes the operational, tactical and strategic levels. The first brings together basic skills, the last is the most intuitive and the most difficult. But experience makes the lower levels increasingly rich and effective.

Once the problem has been clearly identified (question, context: data, constraints, ins and outs), depending on the case:

• it is a known problem (or a special case);
• it is a problem that can be reduced to a combination of simpler problems;
• it is a problem resembling a problem that we know how to deal with.

In the world of computing , there are two fundamental goals, and they are to find algorithms with good execution times and good solutions, usually the optimal ones. Heuristic algorithms abandon one or both objectives, for example, they usually find good solutions, although there is no evidence that the solution cannot be arbitrarily wrong in some cases, or they run reasonably fast, although there is also no evidence that it will always be like this. Heuristics are generally used when there is no optimal solution under the given constraints (time, space, etc.), or when it does not exist at all.

Often, they can find specific instances of the problem where the heuristic will produce very bad results or run very slowly. However, these concrete instances can be ignored because they should never happen in practice to be of theoretical origin. Therefore, the use of heuristics is very common in the real world.

#### Heuristic algorithms to find the shortest path

For problems of finding the shortest path the term has a more specific meaning. In this case a heuristic is a mathematical function, h(n) defined on the nodes of a search tree, which serves as an estimate of the cost of the cheapest path from a given node to the target node. Heuristics are used in informed search algorithms such as selfish search. It will selfishly choose the node that has the lowest value in the heuristic function. A * will expand the nodes that have the lowest value for g(n)+h(n), where g(n) is the (exact) cost of the path from the initial state to the current node; when h(n) is admissible, that’s yes h(n) never overestimate the costs of finding the target; A * is probably optimal.

A classic problem that uses heuristics is the 15 game (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others). Counting the number of misplaced squares and finding the sum of the Manhattan distance between each block and its position on the target are often used heuristics for this problem.

#### Effect of heuristic algorithms on computational performance

In any search problem where there is{\displaystyle b}options at each node and a depth d at the target node, a naive search algorithm will potentially have to search between bd, nodes before finding the solution.

Heuristics improve the efficiency of search algorithms by reducing the branching factor of b to (ideally) a constant b*.

Although any admissible heuristic will return an optimal answer, a heuristic that returns a lower branching factor is computationally more efficient for the particular problem. It can be shown that a heuristic h2(n) is better than another h1(n), Yes h2(n) dominate h1(n), that means h1(n) < h2(n) for everything n.

#### Heuristics in Artificial Intelligence

Many algorithms in artificial intelligence are heuristic in nature, or use heuristic rules. A recent example is SpamAssassin which uses a wide variety of heuristic rules to determine when an email is spam. Any of the rules used independently can lead to classification errors, but when multiple heuristic rules are combined, the solution is more robust and credible. This is called high confidence in pattern recognition (extracted from the statisticson which it is based). When the word heuristic is used in rule-based language processing, pattern recognition, or image processing, it is used to refer to rules.

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