First, recursive algorithm

Recursion algorithm (English: recursion algorithm) refers to a method to solve a problem by repeatedly decomposing it into similar subproblems. Recursive method can be used to solve many computer science problems, so it is a very important concept in computer science. Most programming languages support self-calling of functions, in which functions can be called recursively by themselves. Computational theory can prove that recursion can completely replace loop, so many functional programming languages (such as Scheme) use recursion to realize loop.

Second, the recursive program

In programming languages that support self-tuning, recursion can be accomplished by simple function calls. For example, the program for calculating factorial can be mathematically defined as:

This program can be written in Scheme language:

1

(define (factorial n) (if (= n 0) 1 (* n (factorial (-n 1)))

Fixed-point combiner

Even if a programming language does not support self-calling, recursion can be generated by fixed-point combiners (English) if a function is the first class in the language (that is, it can be created at runtime and treated as a variable). The following Scheme program does not use self-calling, but uses a fixed-point combiner called Z operator (English: Z combinator), so it can also achieve the purpose of recursion.

1

(definition of z (lambda (f) (lambda (recurve) (f (lambda arg (apply (recurve) arg)))) (lambda (recurve) (f (lambda arg (apply (recurve) arg)))) (definition of fact (z) = n 0) 1 (* n (f (- n 1))))))))

The idea of this program is that since the function can't call itself here, we can use the function obtained by combining function application with z to apply the parameters to be calculated.

Tail recursion

Tail recursion means that the recursive function returns the value directly after calling itself, and there is no need to add operations to it. Tail recursion is equivalent to a loop, which can be optimized as a loop instruction in some languages (such as Scheme). Therefore, in these languages, tail recursion does not occupy the call stack space. The following Scheme program also calculates the factorial of a number, but uses tail recursion:

1

(define (factorial n) (define (iter product counter) (if(& gt；; Counter n) product (ITER (* Counterproduct) (+counter1)) (ITER11))

Third, the problem that can be solved.

Data is defined recursively. Such as Fibonacci function.

The problem is solved by recursive algorithm. For example, Hanoi.

The structure of data is defined recursively. Such as binary tree, generalized table, etc.

Fourth, recursive data.

Data types can be defined by recursion, for example, a simple recursion is defined as the definition of natural numbers: "One natural number equals 0 or another natural number plus 1". Haskell can define a linked list as:

1

data list of strings = EmptyList | Cons String list of strings

This definition is equivalent to declaring "a linked list or an empty string column, or a linked list is preceded by a string". As you can see, this recursive definition can reach all linked lists.