Math 343 - Numerical Methods I - Homework Assignments

Assignment 1 | Assignment 2 | Assignment 3 | Assignment 4 | Assignment 5 | Assignment 6 | Assignment 7 | Assignment 8 | Assignment 9 | Assignment 10 | Assignment 11 | Assignment 12 | Assignment 13 |
Sect. # Assignment
Assignment 1 - Turn in on Sept. 10
Sect. (1.1) Page 14: 1(b), 2(a), 3(a), 5, 6, 7, 10, 12, 13
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Assignment 2 - Turn in on Sept. 17
Sect. (1.2) Page 25: 2(c), 3(b), 4(b), 5(b), 6(c), 7(c), 14(a)
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Sect. (1.3) Page 39: 1(a), 5(a) (compute also ln(x+1)-ln(x) and its equivalent formula for x=100 and x=1000 in 4 decimal digits accuracy; and then compare these values with the true values), 9, 13(a), Show that sin(h)/h = 1 + O(h2)
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Assignment 3 - Turn in on September 24
Sect. (2.1) Page 52: 2, 3, 7, 8, 10, 14

Additional problems:
(1) Let g1(x)=(1-x3)/2 and g2=(1-2x)/x2.
(1.a) Verify that a solution of x3+2x-1=0 is also a fixed point of g1 and g2.
(1.b) For each of g1(x) and g2(x), find the interval(s) over which |g'i(x)|<1 where i=1,2.
(1.c) Show that the equation x3+2x-1=0 has a solution in [1/4, 1/2]. To find this solution using the fixed-point algorithm, which of g1(x) and g2(x) should be used? Can both be used? Should none of them be used?

(2) Consider the problem of solving the equation x5-5x-1=0.
(2.a) Find an interval [a,b] in which the equation x5-5x-1=0 has a solution x*
(2.b) Find a function g(x) such that x* is a fixed-point of g and the sequence {xn} generated by xn+1=g(xn) converges to x* with given x0 in [a,b] obtained in (2.a). Explain why the function g of your choice will do the job.

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Assignment 4 - Turn in on October 1
Sect. (2.2) Page 63: 4, 5 (Compute n only)

Consider solving the equation xsin2(x)-1=0.
(a) Show that there is a solution in [0, pi/2].
(b) Determine the least number of iterations needed if the Bisection Method is used to approximate the solution in [0, pi/2] with the error no larger than 10-6.
(c) Compute c1 and c2 by the Bisection Method.
(d) Compute c1 and c2 by the False Position Method.
(e) Choose an initial point x0 in [0, pi/2]. Compute x1 by the Newton Method.
(f) This equation has infinitely many solutions. (Extra points for proving it.)
Graph the equation for x in [-2pi, 10pi] using the following MatLab commands. 
   >> clg
   >> x=-2*pi:.1:10*pi;
   >> y=x.*(sin(x)).^2-1;
   >> plot(x,y)
   >> hold
   >> plot([-2*pi 10*pi],[0 0],':')
   >> hold off
Give an interval [a,b] other than [0, pi/2] containing a solution. Explain how your choice of the interval is made.
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Assignment 5 - Turn in on October 8
Sect. (2.4) & (2.5) Page 86: 2, 4(b), 34(c)(d) (Verify only for each given function that p is a root of order M.)
Page 98: 1, What does the result given in Problem 1 tell us? 2, 3

Additional Problems:
Let f(x*)=0 and x* be the root of order 2. Define h(x)=f(x)/f'(x). Show that
(a) h(x*)=0;
(b) h'(x*) ><0.
(Hint: Show first that h(x)=x-gNT(x) where gNT(x)=x-f(x)/f'(x) is the generating function of the Fixed-point iteration for the Newton Method. You may use any results for gNT(x) derived in the class.)
Now we locate x* by applying the Newton Method to solve h(x)=0 for x0 in a proper chosen interval. What is the rate of convergence?
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Assignment 6 - Turn in on October 15
Sect. (3.1)-(3.4) Page 129: #1(iv), #2(a), #3(a)(c), #4
Page 142: #2, #3(b)(c), #5(a), #5(d) (use the definition of the determinant of a square matrix), #11, #12, #15 (check the sizes of XX' and X'X before computing).
Page 157: #1, (Carry necessary elementary row operations to show Ax=b and Ux=v are equivalent.) Compute also the determinant of the coefficient matrix, #13 (Use Gaussian Elimination to reduce Ax=b to Ux=v first and then use backward substitution to solve Ux=v.) Compute also the determinant of the coefficient matrix.
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Assignment 7 - Turn in on October 22
Sect. (3.4)-(3.6) Let A=[1.003 58.09;5.550 321.8] and b=[68.12;377.30]. Solve the system Ax=b in four decimal digits accuracy
(1) without pivoting;
(2) with the partial pivoting;
(3) with the scaled partial pivoting; and
(4) with the complete pivoting.
Which algorithm yields the most accurate results if the exact solution is [10;1]?
Page 177: #1, #4(a)(b).
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Assignment 8 - Turn in on October 29
Sect. (3.4)-(3.6)
  1. Let A be an nxn matrix and be nonsingular. Consider the following two methods to find the inverse of A. Let A-1=[c_1,..., c_n] and I be an nxn identity matrix.
    • (i) [A I] --> [U B] by the Gaussian Elimination.
      (ii) Let B=[b1,...,bn]. Solve Uci=bi, by the Backward subsititution for i=1,...,n.
    • [A I] --> [I A-1] by elementary row operations.

    Find the least number of multiplications needed for each method.
  2. Page 178 #9
  3. Let A=[1 -2 3 0;1 -2 3 1;1 -2 2 -2;2 1 3 -1]. Find matrice P, L, and U such that PA=LU where P is a permutation matrix, L is a lower triangular matrix with 1's on its diagonal and U is an upper triangular matrix.
  4. Let A=[1 0 -1;0 1 1;-1 1 k]. Find all values of k for which
    • A is singular.
    • A is strictly diagonal dominant.
    • A is symmetric.
    • A is positive definite.
  5. Let A=[4 1 -1 0;1 3 -1 0;-1 -1 5 2;0 0 2 4].
    • Show that A is symmetric positive definite.
    • Find matrices L and D such that A=LDLT where L is a lower triangular matrix with 1's on its diagonal and D is a diagonal matrix.
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Assignment 9 - Turn in on November 5
Sect. (4.1)-(4.3)
  1. Page 204: #4, #7, #9, #12
  2. Page 213: #2
  3. Page 225: #2, #4
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Assignment 10 - Turn in on November 12
Sect. (4.3)-(4.4)
  1. Page 225: #6(a)(b), #10
  2. Page 235: #5, #9
  3. Consider the problem of simulating the US population f(x) (in millions) where x is the number of years since 1997 for next 50 years by a nth degree interpolating polynomial Pn(x) with n+1 collected data on population including the current population. Suppose that |f(k)(x)| < (0.1)k for any k and x.
    • Suppose that the census will be done in every 10 years. Find the smallest possible error if the population in 2010 is approximated by Pn(2010).
    • Suppose that the census will be done in every 5 years. Find the smallest possible error if the population in 2010 is approximated by Pn(2010).
    • In order to have the approximation error less than 0.0000001 (in million) for any year between 1997 and 2047, what is the least number (n) of data that need to be collected?
Assignment 11 - Turn in on November 24
Sect. (4.3) & (5.3)
  1. Suppost that we have the following data and want to approximate f(1.06) by P2(1.06) where P2 is the interpolating polynomial of f(x) using Lagrange polynomials.
    (xi, yi=f(xi)): (1.00, 0.7656), (1.05, 0.8354), (1.07,0.8589)
    • Compute P2(1.06) by Neville's Method.
    • Compute P2(1.06) by Aitken's Method.
    • Suppose that we know f(x)=3xex-e2x. Estimate the approximation error for f(1.06) by P2(1.06). Compare it to the true error.
  2. Page 299: #2, #5(a)(b), #7(i)(ii), 9(a)(c)
  3. Page 299: #14
    • Find the clamped spline for the points.
    • Estimate the distance when t=5.
    • Estimate the velocity when t=2 and t=7.
Assignment 12 - Turn in on December 3
Sect. (6.1) & (6.2) Page 328: #2, #4, #7(a), #10
Page 342: #4, #5, #7
Assignment 13 - Turn in on December 10
Sect. (7.1) & (7.2) Page 355: #2(b), #3, #4, #5
Page 366: #1(i)(ii)(d), #2(b), #5(b), #10(a)(c)
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