Recursion
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Recursion in JAVA with programming examples. How we can write a shortcode with the use of "Recursion."
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Recursion
Recursion is a technique that involves defining a function in terms of itself via self-referential calls. It can be used to loop without an explicit loop statement. A recursive function must have a base case that does not involve further recursion in order to terminate after a finite number of calls. When a recursive function is called, information like function values and local variables are pushed onto a call stack to keep track of each nested call.
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This document discusses recursion in programming. Recursion is when a function calls itself, either directly or indirectly. For a recursive function to work properly it must have a base case, which provides the terminating condition, and make progress toward the base case with each recursive call. Examples of recursively defined functions include factorials, powers, Fibonacci numbers, and towers of Hanoi. Recursion allows for simple programming of some complex problems but iterative solutions may have better performance due to less function call overhead.
Iteration, induction, and recursion
The document discusses various topics related to analyzing algorithms, including:
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ii. Iteration, induction, and recursion as fundamental concepts in data structures and algorithms. Recursive programs can sometimes be simpler than iterative programs.
iii. Proving properties of programs formally or informally, such as proving statements are true for each iteration of a loop or recursive call of a function. This is often done using induction.
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The document discusses the Z-transform, which is used in digital signal processing to characterize discrete-time signals and systems. It presents the basic theory of the Z-transform, including its formulation and properties like convergence. Examples are given to illustrate region of convergence concepts like stable, causal, and two-sided sequences. The inverse Z-transform and MATLAB commands for analysis are also covered.
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This document provides an introduction to z-transforms, which are a major mathematical tool for analyzing discrete systems including digital control systems. It discusses sequences and difference equations, which are foundational topics for understanding z-transforms. Specifically, it defines what a sequence is, distinguishes between first and second order difference equations, and explains how sequences can be shifted to the left or right. It also introduces key concepts like causal sequences and the unit step sequence that are important for z-transform analysis.
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This document discusses applying z-transforms to discrete systems to obtain the system's transfer function and output response. It begins by introducing z-transforms and their use in obtaining transfer functions for discrete systems. Transfer functions allow analyzing combinations of systems. The document provides examples of obtaining transfer functions for first and second order systems. It also covers obtaining the unit impulse and step responses from the transfer function. Finally, it discusses analyzing series combinations of systems by multiplying their individual transfer functions.
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Recursion
Recursion is a technique that involves defining a function in terms of itself via self-referential calls. It can be used to loop without an explicit loop statement. A recursive function must have a base case that does not involve further recursion in order to terminate after a finite number of calls. When a recursive function is called, information like function values and local variables are pushed onto a call stack to keep track of each nested call.
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This document discusses recursion in programming. Recursion is when a function calls itself, either directly or indirectly. For a recursive function to work properly it must have a base case, which provides the terminating condition, and make progress toward the base case with each recursive call. Examples of recursively defined functions include factorials, powers, Fibonacci numbers, and towers of Hanoi. Recursion allows for simple programming of some complex problems but iterative solutions may have better performance due to less function call overhead.
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The document discusses various topics related to analyzing algorithms, including:
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This document provides an introduction to z-transforms, which are a major mathematical tool for analyzing discrete systems including digital control systems. It discusses sequences and difference equations, which are foundational topics for understanding z-transforms. Specifically, it defines what a sequence is, distinguishes between first and second order difference equations, and explains how sequences can be shifted to the left or right. It also introduces key concepts like causal sequences and the unit step sequence that are important for z-transform analysis.
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Sec4
Here is a program to calculate the summation from X to Y of i using tail recursive logic in Prolog:
predicates
sum(integer, integer, integer)
sum_aux(integer, integer, integer, integer)
clauses
sum(X, Y, Sum):- sum_aux(X, Y, Sum, 0).
sum_aux(X, X, Sum, Sum):- !.
sum_aux(X, Y, Sum, Acc):-
NewX is X + 1,
NewAcc is Acc + NewX,
NewY is Y,
sum_aux(NewX, NewY, Sum, NewAcc).
This defines sum/3
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This ppt tells u about what dynamic programming method is with example and comparison with other methods to help u understand it better.
Recursion
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Here is a program to calculate the summation from X to Y of i using tail recursive logic in Prolog:
predicates
sum(integer, integer, integer)
sum_aux(integer, integer, integer, integer)
clauses
sum(X, Y, Sum):- sum_aux(X, Y, Sum, 0).
sum_aux(X, X, Sum, Sum):- !.
sum_aux(X, Y, Sum, Acc):-
NewX is X + 1,
NewAcc is Acc + NewX,
NewY is Y,
sum_aux(NewX, NewY, Sum, NewAcc).
This defines sum/3
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Recursion
- 1. Recursion
- 2. Recursion • In mathematics, we often say that a recursive solution is a solution such that f(xk) depends on some of f(xk-1), f(xk-2),...f(x1), and f(x0) • A usable recursive solution usually has the following characteristics • –A base case. This is one which is easily calculated. It also prevents infinite loops! • –A recursive case. This is the case in which the function “calls itself” in order to reduce the calculations to the base case.
- 3. Simple problems solved recursively • In math class we learned that 1+2+3+...+n = n*(n+1)/2 • For example, 1+2+...+10 = 10*11/2 = 55 • It is also easy to find the sum iteratively by an O(n) algorithm int sum=0; for i = 1 to n sum=sum + i • We can also solve this problem by recursion.
- 4. • Recursion to find 1 + 2 + + … + n public sum(n) if n == 1, sum =1 if n > 1, sum = n +sum(n-1) Identify the base case Identify the recursive case • Example: sum(5) = 5 + sum(4) = 5+4+sum(3) =9 +sum(3) =9 +3+sum(2) = 12 = 12 = 14 = 14 +sum(2)= +2+sum(1) • On the next slide, we implement the recursive function in Java
- 5. • Recursive sum used to find the sum to n for n = 1, 2, ..., 10 • Note there is NO reason to compute the sums this way. This example is purely for illustration of recursion.
- 6. n! • From math class, we know that n! = n*(n-1)*(n-2)*...*3*2*1 3! = 3*2*1 = 6 5!= 5*4*3 *2 *1 • It is easy to compute n! iteratively int prod = 1; for i=2 to n prod = prod * i
- 7. Recursive n! • We easily observe that there is a recursive definition of n! n! = n*(n-1) ! • For example 4 ! = 4*3*2*1 = 4*(3*2*1) = 4 * 3! • This prompts the following function int nfact (int n) if n == 0 or n ==1 return 1 else return n * nfact (n-1)
- 8. Fibonacci Numbers • The Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, .... • The first two elements of the sequence are both 1 • Subsequent elements are formed by adding the two before it. • Recursive Fibonacci fib (n) if (n==1 or n==0) return 1; else return fib (n-1) + fib (n-2)
- 9. Recursive & Iterative Fibonacci Notice how short and compact the recursive version is.
- 10. Remarks about recursion • In general • A recursive solution is simpler to code than an iterative solution • Some problems are so difficult that recursion may be the best solution • Iterative solutions are often faster than recursive solutions
- 11. Recursive search • It is very simple to write an iterative search to determine if a given value (toFind) is in an array for i = 0 to the array length -1 if array[i] = to_find return true End i loop Return false • When we do it recursively • If the array has length 1, we return true if the single element and toFind are the same • If not • we check the first element and return true if it and toFind are equal • Otherwise, we repeat the search on the remainder of the array