Homework Set # 1 Math 371 Fall 2009 Quiz date: 9/1/2009
1. If a is an approximate value for a quantity whose true value is t, and a has relative error r ,
prove from the denitions of these terms that a = t(1 + r ).
2. Let p(x) = 1.01x4 4.62x3 3.11x2 + 12.
MATH 371001 Numerical Algorithm, Spring 2016
Lab 5 Polynomial Interpolation
Assigned: Wednesday, Feb. 17, 2016
Due: Wednesday, 11:59pm, Feb. 24, 2016
1. (10 Points) Write a code to compute the divided-difference coefficients in the Newton
interpolating po
MATH 371001 Numerical Algorithm, Spring 2016
Lab 6 Numerical Integration
Assigned: Thursday, Mar. 10, 2016
Due: Thursday, 11:59pm, Mar. 18, 2016
1. (15 Points) Write a code for the composite trapezoid rule with uniform spacing to
Rb
approximate a f (x) dx
MATH 371001 Numerical Algorithm, Spring 2016
Lab 10
Assigned: Thursday, Apr. 14, 2016
Due: Thursday, 11:59pm, Apr. 21, 2016
1. (15 Points) Write a code (in double precision) to compute the coefficients cfw_aj nj=0 of
the least Squares polynomial pn (x) =
MATH 371001 Numerical Algorithm, Spring 2016
Lab 2
Assigned: Tuesday, Jan. 26, 2016
Due: Wednesday, 4:40 pm, Feb. 3, 2016
Given the following functions
f (x) = x3 + 2x2 + 10x 20,
g(x) = x tan x.
1. (15 Points) Write a code (in double precision) to apply t
MATH 371001 Numerical Algorithm, Spring 2016
Lab 7 Linear Spline
Assigned: Thursday, Mar. 24, 2016
Due: Thursday, 11:59pm, Mar. 31, 2016
1. (15 Points) Write a subroutine which performs function evaluation S(x) of the linear
spline S at input x, n, cfw_ti
MATH 371001 Numerical Algorithm, Spring 2016
Lab 8 Initial Values Problems
Assigned: Thursday, Mar. 31, 2016
Due: Thursday, 11:59pm, Apr. 7, 2016
1. (20 Points) Consider the following initial-value problem
(
x0 = x + x2
on [1, 2.77]
e
x(1) = 16e
(1). Writ
MATH 371001 Numerical Algorithm, Spring 2016
Homework 3
Assigned: Tuesday, Mar. 29, 2016,
Submit #1, #2, #4
Due: Thursday, Apr. 7, 2016
Total Points: 30.
1. If the composite trapezoid rule is to be used to compute
Z 1
2
ex dx
0
with an error of at most
1
MATH 371001 Numerical Algorithm, Spring 2016
Homework 1
Assigned: Thursday, Feb. 4, 2016 Due in class: Tuesday, Feb. 19, 2016
Submit #3, #5, #7. Total Points: 30.
1. A real number x is represented approximately by 0.6032, and we are told that the relative
MATH 371001 Numerical Algorithm, Spring 2016
Lab 0
Assigned: Thursday, Jan. 14, 2016
Due: Warm up assignment
1. (10 Points) The derivative of a function f at a point x is defined by the equation
f 0 (x) = lim
h0
f (x + h) f (x)
.
h
Create an m-file called
AIMS Exercise Set # 4
Solutions
Peter J. Olver
1. Find the explicit formula for the solution to the following linear iterative system:
u(k+1) = u(k) 2 v (k) ,
Solution: u(k) =
v (k+1) = 2 u(k) + v (k) ,
u(0) = 1,
v (0) = 0.
3k + (1)k
3k + (1)k
, v (k) =
MATH 371001 Numerical Algorithm, Spring 2016
Homework 4
Due: Tuesday, Apr. 26, 2016
(Submit #2, #4, #6, #7.)
1. Determine x00 when x0 = xt2 + x3 + ex t.
2. Calculate an approximate value for x(0.2) (correct to two decimal places) using one step
of the Tay
58:111 Numerical Calculations
MATLAB EXAMPLES
Matrix Solution Methods
Department of Mechanical and Industrial Engineering
Some useful functions
det (A)
lu(A)
cond(A)
inv(A)
rank(A)
Determinant
LU decomposition
Matrix condition number
Matrix inverse
Matrix
MATH 371-001 — Numerical Algorithm, Spring 2016
Midterm Exam
Name: —_ ID:
There are 7 problems on 8 pages with 100 total points. Show all work for full credits.
1. (10 points) For some values of (1:, the function f = V332 + l—m cannot be accurately
comp
MATH 371001 Numerical Algorithm, Spring 2016
Lab 3
Assigned: Tue., Feb. 2, 2016 Due: Wednesday, 4:40pm, Feb. 10, 2016
1. (15 Points) Write a code to implement the (naive) Gauss elimination method. (First,
test it by using the example in class to check the
Homework Set # 2 Math 371 Fall 2009
Quiz date: 9/8/2009
1. Consider the system Ax = b, where A is an n n invertible matrix and x, b = 0 are in Rn .
consider the perturbed system
A(x + x) = b + b
Then show that
1 b
x
.
x
b
2. Let the matrix A be dened
Homework Set # 1 Math 371 Fall 2009 Quiz date: 9/1/2009
1. If a is an approximate value for a quantity whose true value is t, and a has relative error r ,
prove from the denitions of these terms that a = t(1 + r ).
2. Let p(x) = 1.01x4 4.62x3 3.11x2 + 12.
Homework Set # 3 Math 371 Fall 2009
Quiz date: 9/17/2009
1. Let
M =
10
01
0 2
01
2
03
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
What is the inverse of M ? (note: rather than diving into the row reduction, try thinking rst
in terms of elementary row operations. make a
Homework Set # 4 Math 371 Fall 2009
No Quiz! Will be included in exam on 9/22
1. Suppose you have the data set cfw_(1, 10), (2, 15), (4, 8), (5, 2), (7, 12). Use this data for the
following parts:
(a) Find the full interpolation polynomial of degree 4 det
Homework Set # 4 Math 371 Fall 2009
No Quiz! Will be included in exam on 9/29
1. Suppose you have the data set cfw_(1, 10), (2, 15), (4, 8), (5, 2), (7, 12). Use this data for the
following parts:
(a) Find the full interpolation polynomial of degree 4 det
Homework Set # 5 Math 371 Fall 2009
Quiz Date: 10/13/2009
1. Derive a recursive algorithm using Newtons method to calculate the pth root of a positive
number Q (i.e. - dene the appropriate f (x), and sub this particular function in to newtons
method - thi
Homework Set # 6 Math 371 Fall 2009
Quiz Date: 10/29/2009
1. The average scores reported by golfers of various handicaps on a dicult par-three hole are
Handicap 6
8 10 12 14 16 18 20 22 24
as follows:
Average 3.8 3.7 4.0 3.9 4.3 4.2 4.2 4.4 4.5 4.5
(a) Co
Homework Set # 6 Math 371 Fall 2009
Quiz Date: 10/29/2009
1. The average scores reported by golfers of various handicaps on a dicult par-three hole are
Handicap 6
8 10 12 14 16 18 20 22 24
as follows:
Average 3.8 3.7 4.0 3.9 4.3 4.2 4.2 4.4 4.5 4.5
(a) Co
Homework Set # 7 Math 371 Fall 2009
Quiz Date: none! practice for exam 2
1. Consider the integral
2
1 x cos(x) dx.
(a) Compute the integral exactly by hand.
(b) Approximate the integral using the Midpoint rule, Trapezoid rule, Simpsons Rule, composite Sim
Homework Set # 7 Math 371 Fall 2009
Quiz Date: none! practice for exam 2
1. Consider the integral
2
1 x cos(x) dx.
(a) Compute the integral exactly by hand.
(b) Approximate the integral using the Midpoint rule, Trapezoid rule, Simpsons Rule, composite Sim
Homework Set # 8 Math 371 Fall 2009
Quiz Date: 11/24
1. Convert the problem
y 0.1(1 y 2 )y + y = 0
with y (0) = 1 and y (0) = 1 to a rst order system.
2. Suppose that
dy
= f (t, y (t) .
dt
(a) Use a Taylor Series expansion with a remainder term to show th