Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 3.1, Problem 7E
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Using Karnaugh maps and Gray coding, reduce the following circuit represented as a table and write the final circuit in simplest form (first in terms of number of gates then in terms of fan-in of those gates). HINT: Pay closeattention to both the 1’s and the 0’s of the function.
Recall the RSA encryption/decryption system. The following questions are based on RSA. Suppose n (=15) is the product of the two prime numbers 3 and 5.1. Find an encryption key e for for the pair (e, n)2. Find a decryption key d for for the pair (d, n)3. Given the plaintext message x = 3, find the ciphertext y = x^(e) (where x^e is the message x encoded with encryption key e)4. Given the ciphertext message y (which you found in previous part), Show that the original message x = 3 can be recovered using (d, n)
Theorem 1: A number n ∈ N is divisible by 3 if and only if when n is writtenin base 10 the sum of its digits is divisible by 3. As an example, 132 is divisible by 3 and 1 + 3 + 2 is divisible by 3.1. Prove Theorem 1
2. Using Theorem 1 construct an NFA over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}which recognizes the language {w ∈ Σ^(∗)| w = 3k, k ∈ N}.
Chapter 3 Solutions
Numerical Analysis
Ch. 3.1 - Use Lagrange interpolation to find a polynomial...Ch. 3.1 - Use Newtons divided differences to find the...Ch. 3.1 - How many degree d polynomials pass through the...Ch. 3.1 - (a) Find a polynomial P(x) of degree 3 or less...Ch. 3.1 - (a) Find a polynomial P(x) of degree 3 or less...Ch. 3.1 - Write down a polynomial of degree exactly 5 that...Ch. 3.1 - Find P(0), where P(x) is the degree 10 polynomial...Ch. 3.1 - Let P(x) be the degree 9 polynomial that takes the...Ch. 3.1 - Give an example of the following, or explain why...Ch. 3.1 - Let P(x) be the degree 5 polynomial that takes the...
Ch. 3.1 - Let P1, P2, P3, and P4 be four different points...Ch. 3.1 - Can a degree 3 polynomial intersect a degree 4...Ch. 3.1 - Let P(x) be the degree 10 polynomial through the...Ch. 3.1 - Write down 4 noncollinear points (1,y1), (2,y2),...Ch. 3.1 - Write down the degree 25 polynomial that passes...Ch. 3.1 - List all degree 42 polynomials that pass through...Ch. 3.1 - The estimated mean atmospheric concentration of...Ch. 3.1 - Prob. 18ECh. 3.1 - Apply the following world population figures to...Ch. 3.1 - Write a version of Program 3.2 that is a MATLAB...Ch. 3.1 - Write a MATLAB function polyinterp.m that takes as...Ch. 3.1 - Remodel the sin1 calculator key in Program 3.3 to...Ch. 3.1 - (a) Use the addition formulas for sin and cos to...Ch. 3.2 - Find the degree 2 interpolating polynomial P2(x)...Ch. 3.2 - (a) Given the data points (1,0), (2,In2), (4,In4),...Ch. 3.2 - Assume that the polynomial P9(x) interpolates the...Ch. 3.2 - Consider the interpolating polynomial for...Ch. 3.2 - Assume that a function f(x) has been approximated...Ch. 3.2 - Assume that the polynomial P5(x) interpolates a...Ch. 3.2 - (a) Use the method of divided differences to find...Ch. 3.2 - Plot the interpolation error of the sin1 key from...Ch. 3.2 - The total world oil production in millions of...Ch. 3.2 - Use the degree 3 polynomial through the first four...Ch. 3.3 - List the Chebyshev interpolation nodes x1,...,xn...Ch. 3.3 - Find the upper bound for | (xx1)...(xxn) | on the...Ch. 3.3 - Assume that Chebyshev interpolation is used to...Ch. 3.3 - Answer the same questions as in Exercise 3, but...Ch. 3.3 - Find an upper bound for the error on [ 0,2 ] when...Ch. 3.3 - Assume that you are to use Chebyshev interpolation...Ch. 3.3 - Suppose you are designing the In key for a...Ch. 3.3 - Let Tn(x) denote the degree n Chebyshev...Ch. 3.3 - Determine the following values: (a) T999(1) (b)...Ch. 3.3 - Prob. 1CPCh. 3.3 - Prob. 2CPCh. 3.3 - Carry out the steps of Computer Problem 2 forIn x,...Ch. 3.3 - Let f(x)=e| x |, Compare evenly spaced...Ch. 3.3 - Prob. 5CPCh. 3.4 - Decide whether the equations form a cubic spline....Ch. 3.4 - Check the spline conditions for {...Ch. 3.4 - Find c in the following cubic splines. Which of...Ch. 3.4 - Find k1,k2,k3 in the following cubic spline. Which...Ch. 3.4 - How many natural cubic splines on [ 0,2 ] are...Ch. 3.4 - Find the parabolically terminated cubic spline...Ch. 3.4 - Solve equations 3.26 to find the natural cubic...Ch. 3.4 - Solve equations 3.26 to find the natural cubic...Ch. 3.4 - Prob. 9ECh. 3.4 - True or false: Given n=3 data points, the...Ch. 3.4 - (a) How many parabolically terminated cubic...Ch. 3.4 - How many not-a-knot cubic splines are there for...Ch. 3.4 - Find b1 and c3 in the cubic spline S(x)={...Ch. 3.4 - Prob. 14ECh. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.4 - Prob. 17ECh. 3.4 - Prob. 18ECh. 3.4 - Prob. 19ECh. 3.4 - Discuss the existence and uniqueness of a...Ch. 3.4 - Prob. 21ECh. 3.4 - Prob. 1CPCh. 3.4 - Find and plot the not-a-knot cubic spline that...Ch. 3.4 - Find and plot the cubic spline S satisfying...Ch. 3.4 - Prob. 4CPCh. 3.4 - Prob. 5CPCh. 3.4 - Find and plot the cubic spline S satisfying...Ch. 3.4 - Prob. 7CPCh. 3.4 - Prob. 8CPCh. 3.4 - Find the clamped cubic spline that interpolates...Ch. 3.4 - Find the number of interpolation nodes in Computer...Ch. 3.4 - (a) Consider the natural cubic spline through the...Ch. 3.4 - Prob. 12CPCh. 3.4 - In a single plot, show the natural, not-a-knot,...Ch. 3.4 - Prob. 14CPCh. 3.4 - Prob. 15CPCh. 3.5 - Find the one-piece BĂ©zier curve (x(t),y(t))...Ch. 3.5 - Find the first endpoint two control points, and...Ch. 3.5 - Find the three-piece BĂ©zier curve forming the...Ch. 3.5 - Build a four-piece BĂ©zier spline that forms a...Ch. 3.5 - Describe the character drawn by the following...Ch. 3.5 - Describe the character drawn by the following...Ch. 3.5 - Find a one-piece BĂ©zier spline that has vertical...Ch. 3.5 - Find a one-piece Bezier spline that has a...Ch. 3.5 - Prob. 9ECh. 3.5 - Find the knots and control points for the...Ch. 3.5 - Prove the facts in (3.27), and explain how they...Ch. 3.5 - Given (x1,y1), (x2,y2), (x3,y3), and (x4,y4), show...Ch. 3.5 - Plot the cure in Exercise 7.Ch. 3.5 - Prob. 2CPCh. 3.5 - Plot the letter from BĂ©zier curves: (a) W (b) B...Ch. 3.5 - Use the bezierdraw.m program of Section 3.5 to...Ch. 3.5 - Revise the draw program to accept an n8 matrix of...Ch. 3.5 - Using the template above and your favorite text...Ch. 3.5 - Prob. 4SACh. 3.5 - Although font information was a closely guarded...Ch. 3.5 - Prob. 6SA
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