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prime), and set all the other elements true. Then for each i from 4 to SIZE-1, set isPrime[i] false if i is divisible by 2 (i.e., i%2 = 0). Then for each i from 6 to SIZE-1, set isPrime[i] false if i is divisible by 3. Repeat this process for each possible divisor from 2 to SIZE/2. When finished, all the is for which isPrime[i] is still true are the prime numbers. They are the numbers that have fallen through the sieve. 6.23 Write and test the following function:
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void reverse(int a[], int n);
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The function reverses the first n elements of the array. For example, the call reverse(a,5) would transform the array {22,33,44,55,66,77,88,99} into {66,55,44,33,22,77,88,99}. 6.24 Write and test the following function:
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bool isSymmetric(int a[], int n);
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The function returns true if and only if the array obtained by reversing the first n elements is the same as the original array. For example, if a is {22,33,44,55,44,33,22} then the call isSymmetric(a,7) would return true, but the call isSymmetric(a,4) would return false. Warning: The function should leave the array unchanged. 6.25 Write and test the following function:
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void add(float a[], int n, float b[]);
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The function adds the first n elements of b to the corresponding first n elements of a. For example, if a is {2.2,3.3,4.4,5.5,6.6,7.7,8.8,9.9} and b is {6.0,5.0,4.0,3.0,2.0,1.0}, then the call add(a,5,b) would transform a into {8.2,8.3,8.4,8.5,8.6,7.7,8.8,9.9}. 6.26 Write and test the following function:
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void multiply(float a[], int n, float b[]);
The function multiplies the first n elements of a by the corresponding first n elements of b. For example, if a is the array {2.2,3.3,4.4,5.5, 6.6,7.7, 8.8,9.9 } and b is the array {4.0, 3.0,2.0, 1.0,0.0, 0.0}, then the call multiply(a,5,b) would transform a into the array {8.8, 9.9,8.8, 5.5, 0.0,7.7, 8.8,9.9}. 6.27 Write and test the following function:
float innerProduct(float a[], int n, float b[]);
The function returns the inner product (also called the dot product or scalar product ) of the first n elements of a with the first n elements of b. This is defined as the sum of the products of corresponding terms. For example, if a is the array {2.2, 3.3,4.4, 5.5,6.6, 7.7,8.8, 9.9} and b is the array {4.0, 3.0, 2.0, 1.0,0.0,0.0}, then the call innerProduct(a,5,b) would return (2.2)(4.0) + (3.3)( 3.0) + (2.2)(4.0) + (5.5)( 1.0) + (6.6)(0.0) = 2.2. 6.28 Write and test the following function:
float outerProduct3(float p[][3], float a[], float b[]);
The function returns the outer product of the first 3 elements of a with the first 3 elements of b. For example, if a is the array {2.2,3.3,4.4} and b is the array {2.0, 1.0, 0.0 }, then the call outerProduct(p,a,b) would transform the two-dimensional array p into 4.4 2.2 0.0 6.6 3.3 0.0 8.8 4.4 0.0 Its element p[i][j] is the product of a[i] with b[j]. 6.29 Write and test a function that implements the Perfect Shuffle of a one-dimensional array with an even number of elements. For example, it would replace the array {11,22,33,44,55,66,77,88} with the array {11,55,22,66,33,77,44,88} .
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6.30 Write and test the function that rotates 90 clockwise a two-dimensional square array of ints. For example, it would transform the array 11 22 33 44 55 66 77 88 99 into the array 77 44 11 88 55 22 99 66 33 6.31 Write and run a program that reads an unspecified number of numbers and then prints them together with their deviations from their mean. 6.32 Write and test the following function:
double stdev(double x[], int n);
The function returns the standard deviation of a data set of n numbers x0, , xn 1 defined by the formula ( xi x ) i=0 s = --------------------------n 1 where x is the mean of the data. This formula says: square each deviation (x[i] - mean); sum those squares; divide that square root by n-1; take the square root of that sum. Extend the program from Problem 6.31 so that it also computes and prints the Z-scores of the input data. The Z-scores of the n numbers x0, , xn 1 are defined by zi = (xi x )/s. They normalize the given data so that they are centered about 0.0 and have standard deviation 1.0. Use the function defined in Problem 6.32. In the imaginary good old days when a grade of C was considered average, teachers of large classes would often curve their grades according to the following distribution: A: 1.5 z B: 0.5 z < 1.5 C: 0.5 z < 0.5 D: 1.5 z < 0.5 F: z < 1.5 If the grades were normally distributed (i.e., their density curve is bell-shaped), then this algorithm would produce about 7% A s, 24% B s, 38% C s, 24% D s, and 7% F s. Here the z values are the Z scores described in Problem 6.33. Extend the program from Problem 6.33 so that it prints the curved grade for each of the test scores read. Write and test a function that creates Pascal s Triangle in the square matrix that is passed to it. For example, if the two-dimensional array a and the integer 4 were passed to the function, then it would load the following into a: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 In the theory of games and economic behavior, founded by John von Neumann, certain two-person games can be represented by a single two-dimensional array, called the payoff matrix. Players can obtain optimal strategies when the payoff matrix has a saddle point. A saddle point is an entry in the matrix that is both the minimax and the maximin. The minimax
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