Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for...

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Recursion

Transcript of Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for...

Page 1: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Recursion

Page 2: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Recursion A recursive function contains a call to itself

Example: the factorialn!=n*(n-1)! for n>1n!=1 for n=1

int factorial (int n) {

if (n == 0) { return 1; } else { return (n * factorial (n-1)); }}

Page 3: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Stacks and The Recursive Call Computers use a structure called a stack

to keep track of recursion A stack is a memory structure analogous to a

stack of paper: Last In First Out Whenever a function is called, the computer

uses a "clean sheet of paper“ called active frame The function definition is copied to the paper The arguments are plugged in for the parameters The computer starts to execute the function body

Page 4: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Recursion Key elements of a recursive function

The function calls itself There must be some terminal condition that

ends the recursion Otherwise you will call yourself infinitely and the

program will blow up The recursion should be getting simpler on each

call

Page 5: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Examples power function:

int power(int base, int exponent); power (2,4)=2*2*2*2

power(2,4)=2*power(2,3)power(x,n)=x*power(x,n-1) (for n>0)

=1 (for n=0)int power (int base, int exponent) {

fill in the space

};

Page 6: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

power function int power(int base, int exponent) {

If (exponet==0)return 1;else If (exponet>0)

return (base*power(base, exponent-1));else {cout<<“we only take care of positive

exponent”<<endl;exit(1) }}

Page 7: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Fibonacci sequence

Fibonacci sequence

12

1

0

.

.

,1

,1

nnn aaa

a

a

Page 8: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Fibonacci sequence functionint fibonacci(int n) {

if (n == 1) { return 1; } else if (n == 2) { return 1; } else { return fibonacci(n-1) + fibonacci(n-2); }

}

Page 9: Recursion. A recursive function contains a call to itself Example: the factorial n!=n*(n-1)! for n>1 n!=1 for n=1 int factorial (int n) { if (n == 0)

Recursion and iteration Any task that can be accomplished using

recursion can also be done without recursion A nonrecursive version of a function typically

contains a loop or loops A non-recursive version of a function is usually

called an iterative-version A recursive version of a function

Usually runs slower Uses more storage May use code that is easier to write and understand


FACTORIAL AND FRACTIONAL FACTORIAL … · Abstract Two-level factorial and fractional factorial designs have played a prominent role in the theory and practice of experimental design.

FACTORIAL AND FRACTIONAL FACTORIAL … · Abstract Two-level factorial and fractional factorial designs have played a prominent role in the theory and practice of experimental design.

Factorial Preparatory Exercise #include using namespace std; double fact(double); int fact(int); int main(void) { const int n=20; ofstream my_file("results.txt");

Factorial Preparatory Exercise #include using namespace std; double fact(double); int fact(int); int main(void) { const int n=20; ofstream my_file("results.txt");

Designing experiments with several factors Outlines: Factorial Experiments Two-factor factorial experiment General factorial experiments 2 k factorial.

Designing experiments with several factors Outlines: Factorial Experiments Two-factor factorial experiment General factorial experiments 2 k factorial.

Recursion! - unibo.itMath Induction = CS Recursion 0! = 1 n! = n [(n-1)!] Math inductive definition Python (Functional) recursive function # recursive factorial def factorial(n): if

Recursion! - unibo.itMath Induction = CS Recursion 0! = 1 n! = n [(n-1)!] Math inductive definition Python (Functional) recursive function # recursive factorial def factorial(n): if

 · FACTORIAL DE UN NÚMERO NATURAL Se define y denota el factorial de un número natural "n" como el producto que resulta de multiplicar los n …

 · FACTORIAL DE UN NÚMERO NATURAL Se define y denota el factorial de un número natural "n" como el producto que resulta de multiplicar los n …

Interfaces & Sub-types - Cornell University · public static int factorial ... copy signatures, fill in bodies, ... – Reversible r = new Reversible(…); // no such constructor

Interfaces & Sub-types - Cornell University · public static int factorial ... copy signatures, fill in bodies, ... – Reversible r = new Reversible(…); // no such constructor

FACTORIAL AND FRACTIONAL FACTORIAL DESIGNS …pranjan/research/PHD_thesis.pdf · Abstract Two-level factorial and fractional factorial designs have played a prominent role in the

FACTORIAL AND FRACTIONAL FACTORIAL DESIGNS …pranjan/research/PHD_thesis.pdf · Abstract Two-level factorial and fractional factorial designs have played a prominent role in the

Factorial p n p kq n ìp ìq ìp q · Factorial Notations Traditional name Factorial Traditional notation n! Mathematica StandardForm notation Factorial@nD Specific values Specialized

Factorial p n p kq n ìp ìq ìp q · Factorial Notations Traditional name Factorial Traditional notation n! Mathematica StandardForm notation Factorial@nD Specific values Specialized

Centerpoint Designs Include n c center points (0,…,0) in a factorial design Include n c center points (0,…,0) in a factorial design –Obtains estimate of.

Centerpoint Designs Include n c center points (0,…,0) in a factorial design Include n c center points (0,…,0) in a factorial design –Obtains estimate of.

§3 Dynamic Programming Use a table instead of recursion 1. Fibonacci Numbers: F(N) = F(N – 1) + F(N – 2) int Fib( int N ) { if ( N

§3 Dynamic Programming Use a table instead of recursion 1. Fibonacci Numbers: F(N) = F(N – 1) + F(N – 2) int Fib( int N ) { if ( N

TOP T E N - CBMC Int

TOP T E N - CBMC Int

#include int main() { char *string1=”C++”; char *string2=”Program”; int m=strlen(string1); int n=strlen(string2); for(int i=1;i

#include int main() { char *string1=”C++”; char *string2=”Program”; int m=strlen(string1); int n=strlen(string2); for(int i=1;i

2k Factorial Designs - Wayne State Universityhzhang/courses/7290/Lectures/5 - 2^k Factorial... · 2k Factorial Designs Hongwei Zhang ... Chapter 18: 2 kr Factorial Design with replications

2k Factorial Designs - Wayne State Universityhzhang/courses/7290/Lectures/5 - 2^k Factorial... · 2k Factorial Designs Hongwei Zhang ... Chapter 18: 2 kr Factorial Design with replications

Independent Int e r n at i

Independent Int e r n at i

Factorial BG ANOVA Psy 420 Ainsworth. Topics in Factorial Designs Factorial? Crossing and Nesting Assumptions…

Factorial BG ANOVA Psy 420 Ainsworth. Topics in Factorial Designs Factorial? Crossing and Nesting Assumptions…

2^n CONFOUNDED FACTORIAL - Sas Institute · 2^n CONFOUNDED FACTORIAL Confounded factorial is a design technique for arranging a complete factorial experiment in block, Where the block

2^n CONFOUNDED FACTORIAL - Sas Institute · 2^n CONFOUNDED FACTORIAL Confounded factorial is a design technique for arranging a complete factorial experiment in block, Where the block

Part II: Atomicity for Software Model Checking. Class Account { int balance; static int MIN = 0, MAX = 100; bool synchronized deposit(int n) { int t =

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Module Fourteen: More Complicated Designs – Block Designs, 2 n Factorial Designs, etc.

Module Fourteen: More Complicated Designs – Block Designs, 2 n Factorial Designs, etc.

Recursion - skeoop.github.io · Recursive factorial(n) We can write a function that computes factorials by calling itself to compute factorial of a smaller number: long factorial(

Recursion - skeoop.github.io · Recursive factorial(n) We can write a function that computes factorials by calling itself to compute factorial of a smaller number: long factorial(

Ob Ifeel 9 Int Conflt n Negtn

Ob Ifeel 9 Int Conflt n Negtn