GAMS. Loops and Conditional Statements. x 2, x < 0. write a GAMS model that calculates the value of f(x) for the following set of x values:

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1 GAMS Loops and Conditional Statements 1. Examples: Example 1: The function f(x) is defined by: x 2, x < 0 f(x) = sin(x), 0 x < π 2 1, x π 2 write a GAMS model that calculates the value of f(x) for the following set of x values: { 3, 1, 1, 2, 4, 7}. Solution: The following GAMS code evaluates the function f(x) for the given values of x. set i index for values of x /1*6/; parameter x(i) values of x / /; parameter fx(i) values of f(x loop((i), if(x(i)<0, fx(i)=power(x(i),2) if(x(i)>=0 and x(i)<pi/2, fx(i)=sin(x(i)) if(x(i)>=(pi/2), fx(i)=1 display x,fx; The results are: PARAMETER x values of x , , , , , Prepared by Dr. Hussam Alshraideh

2 PARAMETER fx values of f(x) , , , , , Example 2: Write a GAMS code that computes the sum S = k. Solution: The following GAMS code finds the value of the summation required. scalar itrlim upper limit of the summation/100/ k summation variable S value of summation /0/; k=1 for(k=1 to itrlim by 1, S=S+k display S; The value of the sum is: PARAMETER S = value of summation Example 3: Write a program that computes the Hilbert matrix. That is an N N matrix H, with elements defined by the formula H(i, j) = 1/(i + j 1). Assign a value to N in the beginning of your program. Solution: The Hilbert matrix is generated using the following code. set N indices of matrix /1*5/; alias(n,n2) parameter H(N,N2) Hilbert matrix; loop(n, loop(n2, H(N,N2)=1/(ord(N)+ord(N2)-1 display H; 2 Prepared by Dr. Hussam Alshraideh

3 The Hilbert matrix is: PARAMETER H Hilbert matrix sin(k) Example 4: Write a program that computes an approximation of the sum S =. k 2 k=1 Stop adding more terms to the sum when next term has a smaller absolute value than a given tolerance of ɛ = Solution: The following code evaluates the summation. scalar val value of the last term /100/ S value of summation /0/ k summation index /1/ tol tolerance specified /1e-8/; while(val>=tol, val=sin(k)/power(k,2 S=S+val; k=k+1; display S; The value of the sum is: PARAMETER S = value of summation Example 5: Write a function that would take as input the following matrix of any length and converts each of its elements to either -1 or 1 according to the following rule: If the element is 3 Prepared by Dr. Hussam Alshraideh

4 a negative number, convert to -1, if it is zero or positive, convert to 1. Use the for and the if... else statement. A = [ ] Solution: The following GAMS codes solves the problem: set i index of elements in the matrix /1*11/ parameter A(i) the row matrix / / loop(i, if(a(i)lt 0, A(i)=-1; else A(i)=1; ) display A; The results of the code is PARAMETER A the row matrix , , , , , , , , , Prepared by Dr. Hussam Alshraideh

5 Example 6: Write a GAMS code that uses the Bisection Method to find the root for the following function: f(x) = x 3 x 2 within the interval [1 2]. Stop at iteration number 100. Solution: The following GAMS codes solves the problem: scalars a lower bound, f_a function value at a b upper bound, f_b function value at b c mid point, f_c function value at c itr iteration number; a=1; b=2; itr=0; while(itr le 100, c=(a+b)/2; f_a=power(a,3)-a-2; f_b=power(b,3)-b-2; f_c=power(c,3)-c-2; if(f_a le 0 and f_c le 0, a=c else b=c itr=itr+1; display c; The root is c = Example 7: Write a GAMS code that finds the product of two matrices. Use the following matrices to examine the code: A = B = Solution: The following GAMS codes solves the problem: set Ai index set of matrix A /A1*A3/ Prepared by Dr. Hussam Alshraideh

6 set Bi index set of matrix B /B1*B3/; alias(ai,aj) alias(bi,bj) scalar temp table A(Ai,Aj) matrix A A1 A2 A3 A A A ; table B(Bi,Bj) matrix B B1 B2 B3 B B B ; parameter AB(Ai,Bj loop(ai, loop(bj, temp=0; loop(aj, loop(bi, temp$(ord(aj)=ord(bi))=temp+a(ai,aj)*b(bi,bj AB(Ai,Bj)=temp; display A,B,AB; The result is: PARAMETER A matrix A A1 A2 A3 6 Prepared by Dr. Hussam Alshraideh

7 A A A PARAMETER B matrix B B1 B2 B3 B B B PARAMETER AB B1 B2 B3 A A A Exercises: Problem 1: Write a program that computes an approximation of the sum S = 100 k=1 sin(k) k 2. Problem 2: Starting with positive integer a sequence is generated using the following rules: 1. If a number in the sequence is odd then the next number is computed by multiplying by 3 and adding 1 to the product. 2. If the number is even the next number in the sequence is computed by dividing by 2. Starting with the number 6 the following sequence is generated: {6, 3, 10, 5, 16, 8, 4, 2, 1, }. It is assumed that every such sequence always ends with 1. Write a program that verifies this assumption for sequences starting with a number between 1 and Prepared by Dr. Hussam Alshraideh

8 Problem 3: A real root to the polynomial p(x) = x 3 3ax + 1 = 0, where a 1 Can be computed as the limit value of the sequence {x k } k=0 defined by x 0 = 0 x k+1 = x3 k +1, k = 0, 1, 2, 3a Assume a = 3, write a GAMS program that computes the limit value by computing x 1, x 2,, and so on. Stop the iteration when x k+1 x k < Prepared by Dr. Hussam Alshraideh