Homework assignments for Math 304
Zhonggang Zeng
June 19, Thursday, Homework 01
(Due Tuesday, June 24)
Problems #1, 2, 3, 4 at the end of
A simple introduction to numerical programming with Maple
(This is a mws file, which can be read by Maple only. Save the file
as a:mplintro.mws, open Maple V.4, and open a:mplintro.mws inside
Maple)
Note: if you choose to take "C-only option", you need to have my
permission and skip this assignment.
June 24, Tuesday, Homework 02
(Due Thursday, June 26)
Problem #1: Write the correct program to solve the quadratic equation
a*x^2 + b*x + c = 0
and test your program for the following equations
(10^(-15))*x^2+x+1 = 0
(10^(-15))*x^2-x+1 = 0
Problem #2: The following two functions are equal mathematically:
f(x) = ( sqrt(1+x)-1) / x
g(x) = 1 /( sqrt(1+x)+1 )
Write a program to calculate the two equivalent functions
for i = 0.1, 0.01, 0.001, ..., 10^(-n), for n up to 12
and compare the behavior of their numerical calculation.
Problem #3: The following functions contain subtractions of close
numbers and result in inaccurate numerical calculation.
Manipulate the functions and propose equivalent ways of
calculation without subtraction:
(i) x^2 / ( 1-cos(x) ) for small x
(ii) x * ( sqrt(1+x^2)-x ) for large x
(iii) ln(1+x)-ln(x) for large x
June 26, Thursday, Homework 03
Problem #1: Write a bisection program to solve the
equation
x^2-3 = 0
Problem #2: Mary can afford $250/month contributed to her 401(k). She plans to work for 30 years and retire a millionair. What annual return should she look for? Use bisection method (program) to solve the problem.
Problem #3: Given
x(t) = (g/(2*w^2))*( (exp(w*t)-exp(-w*t))/2 - sin(w*t))
and data
t = 1
x(1) = 1.7
g = -32.17
use bisection program to solve for w, which is negative.
July 1, Tuesday Homework 04
Write a Newton's iteration program and a secant method
to find all solutions the following equations
(1) sin(x)-exp(-x) = 0 in [0,4] to 10^(-5)
(2) 4*x*cos(2*x) - (x-2)^2 = 0 in [0,8] to 10^(-5)
(3) 0.5+0.25*x^2-x*sin(x)-0.5*cos(2*x) = 0
in [0,Pi] to 10^(-5)
Test Newton's iteration on (1), (2), and (3)
Test Secant methd on (1) and (2)
Compare bisection methed, secant method and Newton's
iteration on (1) with epsilon = 10^(-8)
July 3, 1997 Thursday
Homework 05: Investigate
(1) existance
(2) uniqueness
(3) obtainability
(4) speed to accuracy 10^(-5)
of the following fixed point problems
(i) x = (2-exp(x)+x^2)/3
(ii) x = sqrt(exp(x)/3)
July 8, 1997, Tuesday
Homework 06:
#1: Investigate the convergence rate of the
following fixed point iterations solving
x^5 = 7
and rank them in the order from the fastest
to the slowest:
(1) x[n] = sqrt( 1 + (7-x[n-1]^3)/x[n-1]^2 )
(2) x[n]=x[n-1]-(x[n-1]^5-7)/x[n-1]^2
(3) x[n]=x[n-1]-(x[n-1]^5-7)/(5*x[n-1]^4)
(4) x[n]=x[n-1]-(x[n-1]^5-7)/12
#2: From the given data:
x 2.9 3.0 3.1 3.2
f(x) -4.83 -4.24 -3.50 -2.57
estimate f(Pi)
#3: From the given data:
x 1.0 1.1 1.2 1.3 1.4
f(x) 1.68 1.95 2.20 2.44 2.67
estimate f(1.15)
#4: Write down the Lagrange interpolation
polynomial in the unsimplified form using
the data from #3
July 11, Thursday
Homework 07
#1 Write a Maple program for Lagrange interpolating polynomial
Input
n --- the degree of the polynomial
u --- array(0..n) containing nodes on x-axis
v --- array(0..n) function values on nodes
x --- variable for the polynomial
output
the value of the polynomial at x
#2 Determine a polynomial to fit the following data
x 1.2 1.3 1.4 1.5
f(x) 2.57 3.60 5.79 14.1
f'(x) 7.62 13.98 34.62
and use the polynomial to check a few values compared
with the original function tan(x)
#3 For the given data
x 3.0 3.1 3.2
f(x) -4.24 -3.497 -2.597
(1) (pencil work) define the natural cubic spline and
write down the necessary equations used to determine
the coeficients in the piecewise polynomial
(2) Use Maple to determine the cubic spline and check accuracy
against the original function
f(x) = x*cos(x)-x^2*sin(x)
July 15, 1997
Homework 08
#1. For the functions x*ln(x) and exp(x) on the interval [8,8.7] with
nodes 8.0, 8.1, 8.3, 8.6 and 8.7,
(i) investigate the theoretical error bound of the the Lagrange
interpolation polynomial to the function;
(ii) investigate the actual error bound.
#2. For data
x | 0.9 1.3 1.9 2.1 2.6 3.0 3.9 4.4 4.7 5.0 6.0 7.0 8.0 9.2 10.5 11.3 11.6 12.0 12.6 13.0 13.3
f(x)| 1.3 1.5 1.85 2.1 2.6 2.7 2.4 2.15 2.05 2.1 2.25 2.3 2.25 1.95 1.4 0.9 0.7 0.6 0.5 0.4 0.25
Plot the Lagrange interpolation polynomial. The data is the top profile
of a flying duck. Use the graph to demonstrate the shortcomings of the
Lagrange interpolation with large amount of data
July 17, 1997
Homework 09
#1. Propose a quadratic spline function that approximates a function via
fitting data
(x0,f(x0)), (x1,f(x1)), ..., (xn,f(xn))
#2. Use the program ntrlspln.txt and your cubic spline evaluation
to plot data in Problem #2, Homework 08.
#3. For a function f(x) = x - (ln(x))^x and nodes
x = 3.6, 3.8, 4.0, 4.2
visually investigate the actually error function on [3.6,4.2] of
(i) the Lagrange interpolation
(ii) the cubic spline
(iii) the quadratic spline you proposed in Problem #1. (Hint: use
Maple "solve" to compute parameters of the quadratic spline
and use "piecewise" to define it, then plot)
July 22, 1997
Homework 10
#1: For data
x 1.1 1.2 1.3 1.4
f(x) 1.95 2.20 2.44 2.67
(1) Derive a four point formula for approximating
(i) f'(1.1) and f'(1.4)
(ii) f''(1.1)
(2) carry out the approximation using your formulas and
check with f(x) = ln(exp(2*x)-2) for actual errors.
#2: For data
x 0.2 0.4 0.6 0.8 1.0
f(x) 0.98 0.918 0.808 0.639 0.384
(1) Derive a five point formula for approximating
f'(0.4) and f'(0.8)
(2) carry out the approximation using your formula
#3: Write a Maple program for input
n, f, a, b
calculate the numerical integration of f(x) over
interval [a,b], that is divided into n subintervals.
July 24, 1997
Homework 11
#1 Find a formula to approximate f'(x0) with
a*f(x0-h)+b*f(x0)+c*f(x0+s)+d*f'(x0-h)+e*f'(x+s)
and use your formula to approximate f'(3.05) with
x0-h=2.91, x0=3.05, x0+s=3.23
and f(x) = x*ln(x). Check actual accuracy
#2. Find a, b, c, d such that
a*f(x0)+b*f(x1)+c*f(x2)+d*f(x3)
approximate the definite integral of f(x) over [x0,x3], where
x1=x0+h, x2=x1+h, x3=x2+h
#3. Write a program for input n, f, a, b, output the approximate
definite integral of f(x) on [a,b] using the simpson's rule
on 2*n subintervals.
#4. Write a backward substitution program that solves an upper-
triangular linear system. Test your program for the system
| 2 3 -1 -6 | |x1| |-2|
| 0 5 3 -4 | |x2| = | 4|
| 0 0 -3 7 | |x3| | 4|
| 0 0 0 4 | |x4| | 4|
July 29, 1977
Homework #12
#1 Program the straightforward Gaussian elimination program
#2 Program the Gaussian elimination with pivoting
Test the above two programs on
(1) 0.000001 x1 + x2 = 1.000001
x1 + x2 = 2
with Digits:=5;
(2) | Pi -exp(1) sqrt(2) -sqrt(3) | | x1 | |sqrt(11)|
| Pi^2 exp(1) -exp(2) 3/7 | | x2 | | 0 |
|sqrt(5) -sqrt(6) 1 -sqrt(2) | | x3 | = | Pi |
| Pi^3 exp(2) -sqrt(7) 1/9 | | x4 | |sqrt(2) |
with Digits:=3; and Digits:=10
#3 How many multiplication is needed for straightforward
Gaussian elimination? (pencil work)
July 31, 1997
Homework #13
#1 Estimate the relative error of the linear system
x+2y+3z = 10
2x+3y+4z = -10
x+4y+6z = 20
if the right-hand side entries have a plus or minus 0.01
absolute error.
#2 Program the Jacobi iterative method and test
3x - y + z = 1
3x + 6y + 2z = 0
3x + 3y + 7z = 4
with tolerance 0.00001
July 5, 1997
Homework 14
#1 For iterative method x = Bx + c, prove that the relative error
of x^(k) is less than or equals to ||B||^k:
||x^(k) - x^* ||/||x^*|| =< ||B||^k
if the iteration starts from the zero vector
#2 For the linear system
10w + x - 2y = 6
w + 10x - y + 3z = 25
-2w - x + 8y - z = -11
3x - y + 5z = -11
construct matrix B and vector c such that the system is
equivalent to x = Bx + c. You need to construct two sets of
(B,c) using Jacobi and Gauss-Seidel methods.
#3 Estimate the number of iteration needed for solving the above
linear system to the RELATIVE error 10^(-5) using (i) Jacobi,
(ii) Gauss-Seidel.