Math 344 - Homework Assignment 4 - Due Feb. 5
1 2 3
1. Let A 1
3 2 3
and A 2
2 5 4
. For each matrix, complete the following.
2 5 4
3 4 6
3 4 6
(1) Find the LDL T decomposition where L is a lower triangular matrix with 1s on its diagonal and D is a
diag
Math 344 - Homework Assignment 2 - Due Jan. 23 - Solutions
1. MatLab programs for solving Ax b and Ax
b1 br
and finding A 1 by Gaussian Elimination
and Backward Substitution:
(1) The MatLab program gaussbs.m is designed to solve Ax b where A is an n n ma
Math 344 - Homework Assignment 11 - Due April 9
1. Let yt be the solution of the initial-value problem: y ft, y, a t b, ya . The graph of yt is
given below. For each problem, estimate graphically y 1 and y 2 obtained by Eulers Method with h 0. 25
and h 0.
Math 344 - Spring, 2010
Homework 2
Section 4.1
fx 0 fx 0 h
1. (1) Use the backward difference formula: f x 0
to approximate the
h
derivative fx sinx at x 0 using h 0. 1, h 0. 01 and h 0. 001.
sin sin 0. 1
0. 998 334 166
0. 1
sin sin 0. 01
0. 999 983 33
Eulers Method - (7.2)
Let the initial-value problem for ordinary differential equation:
y t f t, y , a t b, ya ,
be well-posed. Let yt be the unique solution of the initial-value problem. Let h b a , where N is a
N
positive integer. Define t i t i1 h or t
Math 344 - Homework Assignment 9 - Due March 22 (Wednesday)
Noninear Shooting Method (11.2) and Finite Difference Methods (11.3)
1. Page 655: 1, 3(c). For each problem, complete the following:
a. Determine by Theorem 11.1 if the problem has a unique solut
Math 344 - Homework Assignment 5 - Due February 20 (Monday)
Multistep Methods (5.6)
1. Derive the 3-step Adams-Bashforth method.
t t i1 t t i
t t i2 t t i
t t i2 t t i1
f i1
fi
P 2 t f i2
t i2 t i1 t i2 t i
t i1 t i2 t i1 t i
t i t i2 t i t i1
t t
Math 344 - Homework Assignment 6 - Due February 27 (Monday)
Multistep Methods (5.6) and Higher Order Equations and Systems (5.9)
1. Write an Matlab program, called adamsbashforth2.m, for compute y i by the 2-step Adams-Bashforth
Method. Use the Midpoint M
Math 344 - Homework Assignment 9 - Due March 22 (Wednesday)
Nonlinear Shooting Method (11.2) and Finite Difference Methods (11.3) - Solutions.
1. Page 655: 1, 3(c). For each problem, complete the following:
1. y y 2 y ln x, 1 x 2, y1 0, y2 ln 2, solution:
Math 344 - Homework Assignment 10 - Due April 5 (Wednesday)
Finite Difference Methods (11.3)(11.4),
1. Page 661: 3(c), 4(c). For each problem, complete the following:
a. Determine by Theorem 11.3 if the system AY b has a unique solution.
b. Form the matri
Math 344 - Homework Assignment 7 - Due March 6 (Monday)
Higher Order Equations and Systems (5.9) and Stability (5.10)
1. Determine if the initial value problem 1(b) on Page 323 has a unique solution.
u 1 4u 1 2u 2 cos t 4 sin t
u 2 3u 1 u 2 3 sin t
, 0 t
Math 344 - Homework Assignment 4 - Due February 10 (Friday)
Runge-Kutta Methods (5.4)
1. Let yt be the solution of the initial-value problem: y ft, y, a t b, ya . The graph of
yt is given below. Estimate graphically y 1 obtained by the Midpoint Method and
Math 344 - Spring 2006
Hour Exam 1
Name:
1. (45pts) Consider the initial-value problem: y t y, 0 t 2, y0 1.
a. Determine if ft, y t y satisfies a Lipschitz condition in y on the set
D t, y; 0 t 2, y . Provide a reason to support your answer. Find the
Lips
Math 344 - Spring 2006
Hour Exam 2
Name: Solutions
1. (10pts) Derive Adams-Bashforth 2-step Method: y i1 y i h 3 ft i , y i 1 ft t1 , y i1 where t i t i1 h
2
2
for i 0, . . . , N 1, using the Lagrangian interpolating polynomials.
f
f
P 1 t f i1 t t i f i
Math 343 - Homework Assignment 1- Due January 20 (Friday)
(5.1) - Elementary Theory of Initial-Value Problems
1. Review the definition of a function f satisfying a Lipschitz condition.
(a) Suppose we know that the function ft, y satisfies the following in
Math 344 - Homework Assignment 3 - Due February 3 (Friday)
Taylor Method of Order n (5.3)
1. Let yt be the solution of the initial-value problem: y ft, y, a t b, ya . The
graph of yt is given below. For each problem, estimate graphically y 1 and y 2 obtai
Higher-Order Taylor Method - (7.3)
Consider the initial-value problem for ordinary differential equation:
y t f t, y , a t b, ya .
Suppose that the initial-value problem is well-posed and has a unique solution yt. Let h b a for a
N
given positive integer
Runge-Kutta Methods - (7.4)
Consider the initial-value problem for ordinary differential equation:
y t f t, y , a t b, ya .
Suppose that the initial-value problem is well-posed and has a unique solution yt. Let h b a for a
N
given positive integer N and l
Math 344 - Spring, 2014
Homework 4 - Solutions Section 5.1
Turn in: 1, 2a(ii)(iii), 2(b), 3(a), 4(a)(b), 5(a)(c)
Exercises:
1. Review the definition of a function f satisfying a Lipschitz condition.
(a) Suppose we know that the function ft, y satisfies th
Math 344, Fall 2014
Hour Exam 1
Name: Solutions
For each problem, please provide detailed derivations and/or explanations to support your
answer(s).
1. (10pts) Let fx e x/2 .
fx 0 h fx 0
to approximate the derivative of
(1) Use the forward difference for
Math 344 - Spring, 2014
Homework 3 (Sect. 4.3) - Solutions
Exercises:
1(d), 2(d), 3(d), 4, 6, 8, 9
1. Approximate the value of each of the following integrals using the Trapezoidal rule. Verify that the
theoretical error bound holds in each case.
1
d. tan
Math 344, Fall 2014
Hour Exam 2
Name:
Solutions
For each problem, please provide detailed derivations and/or explanations to support your
answer(s).
y
1. (25pts) Consider the initial value problem: y y 2 , 1 t 5, y1 1 . Suppose we
t
ln 2
know that |yt| 2
Math 344 - Spring, 2014
Homework 1 Solutions
fx 0 fx 0 h
to approximate the derivative
1. (1) Use the backward difference formula: f x 0
h
fx sinx at x 0 using h 0. 1, h 0. 01 and h 0. 001.
sin sin 0. 1
0. 998 334 166
0. 1
sin sin 0. 01
0. 999 983 333
5.2 - Eulers Method
Consider solving the initial-value problem for ordinary differential equation:
(*) y t f t, y , a t b, ya .
Let yt be the unique solution of the initial-value problem. In the previous section, two approximation
methods, Picards Method
5.1-The Initial-Value Problems For Ordinary Differential Equations
Consider solving initial-value problems for ordinary differential equations:
(*) y t f t, y , a t b, ya .
If we know the general solution yt of the ordinary differential equation
(*)
y t f
4.3 - Numerical Integration
Problem (1): Given a function fx continuous on a, b, approximate numerically the definite integral
a fx dx.
If we know an antiderivative Fx of fx, then by the Fundamental Theorem of Calculus we know
b
b
a fx dx Fb Fa.
In case
4.1 - Numerical Differentiation
1. Difference formulas derived using Taylor Theorem:
a. Difference formulas for f and their approximation errors:
Recall:
fx h fx
.
f x lim
h0
h
Consider h 0 small. Numerical Difference Formulas:
fx h fx
- forward differenc
4.2 - Richardson Extrapolation
1. Small-O Notation:
Recall that the big-O notation used to define the rate of convergence in Section 1.3:
Definition Let x n n1 converge to a number x . Suppose that n is a sequence known to
n1
converge to 0. The sequence x