MathCom_Algorithm & Orders of Growth of Mathematic for Computing

Algorithm is the sequences step of dfference processing to solve the problems clearly .
Algorithm Characteristics :
Some important general features of algorithms :

• Input : Information or  data that comes in. 
• Output : Information or data that goes out.
• Definiteness : Algorithm is precisely defined.
• Correctness : Outputs correctly relate to inputs.
• Finiteness : Won't take forever to describe or run.
• Effectiveness : Individual steps are all do-able.
• Generality : Works for many possible inputs.
• Efficiency : Takes little time & memory to run.

=======================================================

Pseudocode Language :

1.variable := expression   ==> a := b + 1
2.{comment}                   ==> {for calculate เป็นภาษาอะไรก็ตาม}   

3.begin statements end   ==> begin a:=2 end
4.if condition then          ==> if b>3 then

       statements [else                  b=b+3  else

       statements]                          b=b-2

[else ......] ไม่เขียนก็ได้

5.for variable := initial value to final value ==> for a := 1 to 5 

   statements                                                          b = b+a

6.while condition                                     ==> while a < 4

   statements                                                           write I love mathematic

7.return expression                                 ==> return a      {copy value of a to the function above}

=======================================================

Search Algorithm

    1.Linear Search

     {An}=A1,A2,A3,A4.........An;

      We want to find location of x which =Ai ==> i = ?

      How to :

     //-------------------------------------------------------

      procedure linear search

      i := 1 {the initial location of sequences}

      while (i <= n & x != Ai)

               i := i+1

      if i <= n then location := i

      else locationon := 0

      return location

      //-------------------------------------------------------

    2.Binary Search

    {An}=A1,A2,A3,A4.........An;

      We want to find location of x which =Ai ==> i = ?

      How to :

     //-------------------------------------------------------

      procedure binary search

      i := 1 {the initial location of sequences}

      j := n {the lastest location of sequences}

      while i<j 

            begin 

               m := lowest ((i+j)/2)     {Ex: lowest of 2.3=2 ; 2.9=2}

            if x > Am then i := m+1 else j := m

      else 

      if x=Ai then location := i else location := 0

      return location

=======================================================

Time Complexity

     1.time which use for work 

     2.memory space 

     3.f(n) is the function of time working , n the data of time

     4.Operator :

              _Assignment ( := )

              _Comparison ( = , != , < , > , <= , >= )

              _Arithmetic operation (+ , - , * , / )

              _Logical operations ( and , or , not )

=======================================================

Orders of Growth

      *Big-O (Concept of Orders of Growth)

          f(n)=O(g) {mean f(n) is smaller then g(n)}

          n^a+/-n^(a-1)..........=O(n^a)

       *Big-O (Growth of Function)

         f =O(g) <=> f € O(g)  {f belong to O(g)}

         method of Big-O

               |f(n)| <= c×|g(n)| , n >= k ; c,k € N ===> f(n) € O(g(n))

               k is the highest intersection point of f(n) & g(n)

       *Big-O (Useful Rules) 

            1.if   f1(n)=O(g1(n)) & f2(n)=O(g2(n))  ====> (f1+f2)(n) =O(max(g1(n),g2(n))       

            2.if   f1(n)=O(g(n)) & f2(n)=O(g(n))      ====> (f1+f2)(n) =O(g(n))

            3.if   f1(n)=O(g1(n)) & f2(n)=O(g2(n))  ====> (f1×f2)(n) =O(g1(n)×g2(n)) 

                **Function comparison :  n € N

                n^a >= n^b ; a > b

                n×log(n) >= log(n!) 

                n² >= n×log(n)