Artificial Intelligence Programming Assignment
Problem Statements
Eight-Queens Puzzle
Is it possible to place eight Queens on a chessboard, so that none of the Queens occupy the same row, column, or diagonal?
Binary Search
Depth-First & Breadth-First Search
Newton’s Method
Take a number whose square root is to be calculated, any positive number.
Take a guess at the number’s square root.
Calculate the square root by improving on the current guess as indicated:
Next guess = (number/ current guess + current guess)/ 2
Repeat this process until the difference between the next guess and the current is within the accepted level of accuracy. The better your guess, the fewer the number of iterations needed to get the square root. A good first guess is typically half the number whose square root is to be calculated. The process is ten repeated until the desired accuracy is achieved.
Newton-Raphson Method
Determine a root of the equation f(x) = x^3-x^2-9x+9 = 0 using the Newton-Raphson method if the initial guess is x1 = 1.5.
Gauss-Siedel Method
Solve the following set of linear simultaneous equations using the Gauss-Seidel method:
10x1 + 2x2 + 3x3 = 11
X1 + 5x2 + 2x3 = 20
3x1 + 2x2 + 6x3 = -12
Theoretical Solutions
Eight-Queens Puzzle
1. Pick a position for the Queen
2. If legal, go to next row.
3. If illegal, pick the next position.
4. If no legal position is found, back up to one row.
If legal positions are found for all eight rows, the problem is solved.
Binary Search
• Search the current node value to see if it equals the search value.
• If the search value is smaller than the current value, make the current node the
left child node.
• Make the current node the right child node.
Depth-First & Breadth-First Search
Depth-First Search:
• Searches as far down the left side of the binary tree.
• When it encounters, NULL, the search switches to the bottom-most right child and
resumes.
Breadth-First Search:
• Remove a node from the queue. This becomes the current node.
• Place all child nodes of the current node onto the queue.
Newton’s Method
• Get a positive number whose square root is to be calculated from the user.
• Get the desired precision.
• While more numbers remain, calculate firs guess, x0.
• Repeat
Xn = 0.5 * (X (n-1) + Number/ X (n-1))
Until abs (Xn - X (n – 1)) *= Desired precision
Get a positive number whose square root is to be calculated from the user.
Get the desired precision.
End while
Newton-Raphson Method
1. Set number of iterations num_iter to zero.
Step Two: The next step, will be to get rid of the square root. So, to get rid of the square root we must square it. Remember with equations, what you do to one side you do it to the other.
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"My name is Dorothy," said the girl, "and I am going to the Emerald City, to ask the Oz to send me back to Kansas."
As, shown in the Fig. 8 below, for the 2nd queen we only have one place to put it on the chessboard.
Generates only a single point solution for each iteration, a sequence of those converging to the optimal solution.
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That is, the method of iteration will converge if in each equation of the given system, the absolute value of the largest coefficient is greater than the sum of the absolute values of all the remaining coefficients.