68 DESIGN OF EFFICIENT ALGORITHMS 2.8 where the dimensions of each M, are shown in the brackets. Evaluating M in the order M₁ X (MX (M3 × M₁)) requires 125.000 operations, while evaluating M in the order (M, X (MM;)) × M₁ requires only 2200 operations. ☐ Trying all possible orderings in which to evaluate the product of n matrices so as to minimize the number of operations is an exponential process (see Ex- ercise 2.31), which is impractical when n is large. However, dynamic program- ming provides an O(n³) algorithm. Let m₁, be the minimum cost of computing M₁ × Mi+1 × ·× M; for 1 ≤ i ≤ j ≤n. Clearly, (0, if i=j (2.9) mij = (MIN (mik + M+1, +1-1), if j>i ish i, is the minimum, taken over all possible values of k between i and j- 1, of the sum of these three terms. The dynamic programming approach calculates the m₁'s in order of in- creasing difference in the subscripts. We begin by calculating m₁ for all i, then mi,i+1 for all i, next mi,i+2, and so on. In this way, the terms m₁ and mk+1,j in (2.9) will be available when we calculate mij. This follows since j - i must be strictly greater than either of k-i and j― (k+1) if k is in the range i ≤ k 1, the recursive algorithm is likely to be polynomial in time complexity. However, if the obvious division of a prob- lem of size n results in n problems of size n - 1, then a recursive algorithm is likely to have exponential growth. In this case a tabular technique called dynamic programming often results in a more efficient algorithm. In essence, dynamic programming calculates the solution to all subprob- lems. The computation proceeds from the small subproblems to the larger subproblems, storing the answers in a table. The advantage of the method lies in the fact that once a subproblem is solved, the answer is stored and never recalculated. The technique is easily understood from a simple example. Consider the evaluation of the product of n matrices M = M₁x M₂ × · · · × M„ where each M, is a matrix with 1-1 rows and r; columns. The order in which the matrices are multiplied together can have a significant effect on the total number of operations required to evaluate M, no matter what matrix multi- plication algorithm is used. Example 2.7. Assume that the multiplication of a p× q matrix by a q× r matrix requires pqr operations, as it does in the "usual" algorithm. and con- sider the product M = M₁ [10 × 20] M2 [20 x 50] M3 [501] MA (2.8) [1 × 100]

(Practice) State whether the following are valid function names and if so, whether they’re mnemonic names that convey some idea of the function’s purpose. If they are invalid names, state why. powerdensity m1234 newamp 1234 abcd total tangent absval computed b34a 34ab volts$ a2B3 while minVal sine $sine cosine speed netdistance sum return stack

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.

(Program) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters, such as zerocount, onecount, twocount, and so forth, to 0. Then generate a large number of pseudorandom integers between 0 and 9. Each time 0 occurs, increment zerocount; when 1 occurs, increment onecount; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of time they occurred.