Homework 7 solution sketches
Math 371, Fall 2011
Assigned: Saturday, October 28, 2011
Due: Thursday, November 3, 2011
Clearly label all plots using title, xlabel, ylabel, legend
Use the subplot command to compare multiple plots
Include printouts of all Ma
Homework 6 Solution Sketches
Math 371, Fall 2011
Assigned: Thursday, October 20, 2011
Due: Thursday, October 26, 2011
Clearly label all plots using title, xlabel, ylabel, legend
Use the subplot command to compare multiple plots
Include printouts of all
Homework 1
Math 371, Fall 2010 Assigned: Thursday, September 9, 2010 Due: Thursday, September 16, 2010 (1) (Finite precision numbers) The oating point representation of a real number takes the form x = (0.a1 a2 . . . an ) e , where a1 = 0, M e M . Suppose
Math 371 Winter 2013 Homework 5 due: Thursday March 14
announcement : on Feb 19/21 and March 12/14, both sections will meet in 133 Chrysler
This assignment consists of Matlab programming exercises. In class we considered the twopoint boundary value proble
Homework 9 Solutions
Math 371, Fall 2011
1. (Interpolating Polynomials) The function f (x) = 1/(x + 1) is given at the four points x0 = 1,
x1 = 2, x2 = 3, x3 = 4.
(a) Write the interpolating polynomial in Lagrange form.
(x 2)(x 3)(x 4) 1 (x 1)(x 3)(x 4) 1
1
chapter 4 : computing eigenvalues
4.1 introduction
problem : Given A, find and x 6= 0 such that Ax = x.
: e-value (e.g. frequency, growth rate, energy level)
x : e-vector (e.g. normal mode, principal component, bound state)
thm : Assume A is real and sy
1
chapter 6 : numerical integration
Z 1
0
f (x)dx
ex :
Z 1
0
e
x2
n
X
ci fi , ci : coefficients , fi = f (xi ) , xi : points
i=0
dx = 0.7468 . . . , consider xi = ih , h =
right-hand Riemann sum
1
n
, i = 0, . . . , n : uniform
trapezoid rule
R(h) = f1 h
1
20
Thurs
3/28
chapter 5 : polynomial approximation and interpolation
5.1 introduction
problem : Given a function f (x), find a polynomial approximation pn (x).
application :
Z b
a
f (x)dx !
Z b
a
pn (x)dx , . . .
one solution : The Taylor polynomial of
1
chapter 2 : root-finding
def : Given a function f (x), a root is a number r satisfying f (r) = 0.
ex : f (x) = x2 3 r = 3
question : How can we find the roots of a general function f (x)?
2.1 bisection method
idea : Find an interval [a, b] such that f (
Math 371
Winter 2013
Homework 1 due: Tuesday January 22
Please write neatly, explain your answers, and staple the sheets together.
0. (optional) Give a brief description of your academic background and interests. If you work
in a lab or research group, pl
Math 371
Winter 2013
Homework 9
due: Tuesday April 23
There are several Matlab exercises on this assignment; it is not necessary to turn in the code.
1. Consider f (x) = sin x for 4 x 4. Find the Taylor series for f (x) about x = 0
up to the x7 -term. Usi
Math 371
Winter 2013
Homework 4
due: Tuesday February 19
Solve the problems by hand, but you may use Matlab or a calculator to do arithmetic or
check your answers. All vector norms and matrix norms are the 1-norm.
1. Consider the equations,
2x1 + 3x2
x3 =
1
chapter 7 : time-dependent dierential equations
ordinary dierential equations
Let x(t) be the position of a particle moving on the x-axis at time t.
1st order ODE
dx
= f (x) : velocity is a function of position
dt
x(0) : initial position
The problem is
Math 371
Winter 2017
Homework 2
due: Tuesday January 24
Some problems have a yes/no answer, but to obtain full credit you need to explain your answer.
It is not necessary to submit the Matlab code (unless the assignment specifically requests it).
1. Let f
Math 371
Winter 2013
Homework 3
due: Thursday February 7
1. In class we discussed the equation of state of chlorine gas as an example of root-finding.
The example uses Newtons method to compute the gas volume, given the pressure and
temperature, determine
Math 371 Winter 2017 Homework 3
Garrett McPeek
f ( x)
f ' ( x)
2
n a
( P+ 2 )(V nb)nRT
V
In this case, V 3=V 2
2
n a
2n 2 a
(P+ 2 )+(
)(V nb)
V
V3
1. Newton's method:
g ( x)= x
Now, using V2 = 12.651099337119016 and the other parameters as given, we can
Math 371
Winter 2013
Homework 7
due: Tuesday April 2
This exercise concerns the two-dimensional BVP discussed in class. Consider a metal plate
on the unit square D = cfw_(x, y) : 0 x, y 1. The plate temperature (x, y) satisfies
the Laplace equation xx + y
Chapter 3
Initial-value problems of
ODE
Differential equations (DEs) are used to model real life problems that involve the change of some variables with respect to others. Many DE require
certain initial conditions to be satisfied, hence it is called init
Chapter 2
Nonlinear systems of
equations
The general form for nonlinear system of equations is as follows:
f1 (x1 , x2 , , xn ) = 0
f2 (x1 , x2 , , xn ) = 0
.
.
fn (x1 , x2 , , xn ) = 0
or in the matrix form
F~ (~x) = ~0,
where
F~ =
f1 ( ) = 0
f2 ( ) =
Chapter 1
Linear Systems of Equations
Consider the linear systems
Ax = b
where A = (aij )nn , x and b are n-vectors. If A is invertible, a unique
solution x exists and given by x = A1 b. We are concerned with alogirhtms
for obtaining numerical solutions o
Math 371 Review Sheet Solutions for Midterm Exam Winter 2017
1. True or False? Give a reason to justify your answer.
a) TRUE (10101.01)2 = 24 + 22 + 1 + 41 = 16 + 4 + 1 + 0.25 = (21.25)10
b) TRUE
D+ D f (x) = D+ (D f (x) = D+
=
1
h
f (x+h)f (x)
h
f (x)f (
Math 371
Review Sheet for Midterm Exam
Winter 2017
1. True or False? Give a reason to justify your answer.
a) (10101.01)2 = (21.25)10 b) D+ D f (x) = D D+ f (x) c) D+ D+ f (x) = f 00 (x) + O(h2 )
d) When the derivative f 0 (x) is approximated by the forwa
Math 371
Winter 2013
Homework 6
due: Thursday March 21
1. Consider the linear system, 2x1 + x2 = 1, x1 + 2x2 =
1, with solution x1 = 1, x2 =
1.
a) Write Jacobis method in component form and take three steps starting from initial guess
x0 = (0, 0)T . Prese
Math 371
Winter 2013
Homework 8
due: Tuesday April 9
1. Recall the matrix Ah in the finite-dierence solution of the boundary value problem discussed
in class (steady state heat conduction). It was stated in class that (BJ ) = cos h, (BGS ) =
sin h
cos2 h,
Math 371 Review Sheet Solutions for Midterm Exam Winter 2013
1. True or False? Give a reason to justify your answer.
a) TRUE (10101.01)2 = 24 + 22 + 1 + 14 = 16 + 4 + 1 + 0.25 = (21.25)10
b) TRUE
D+ D f (x) = D+ (D f (x) = D+
=
1
h
f (x+h) f (x)
h
f (x) f
Math 371 Review Sheet for Midterm Exam Winter 2013
The midterm exam is on Thursday February 28 in class. It will cover all the class material up to
and including Thursday February 21. You may use a calculator to do arithmetic and one sheet
of handwritten
Math 371
Winter 2013
Homework 2
due: Tuesday January 29
Some problems have a yes/no answer, but to obtain full credit you need to explain your answer.
1. Let f (x) = 1 + x2 1.
a) Evaluate f (x) for x = 0.1 using 4-digit arithmetic. Show all intermediate s