The Variability of the seasonal Light Curves of Magnetic Chemically Peculiar Stars
N
We are given u i , v i , d i , y i i1 which are discrete versions of periodic functions
ut, vt, dt and yt with a period 2L 1 over time t i N . We want to find a way to de
Math 343 - Fall, 2013
Homework 3
Solutions
Exercise:
1. (3pts) Consider real sequences: x n , y n , z n , u n and w n . Suppose that they all converge to a
real number L.
a. Suppose that
xn L O
1
n
, y n L O 1 , z n L O 1n
n
3
, un L O
1
en
and w n L O
Ra
Math 343 - Fall, 2013
Homework 4
Solutions
The graph of fx for x in 0, 2 is given at
the right. Let x be the solution of the
y
5
equation fx 0 for x in 0, 2 .
4
(a) Compute graphically x 1 and x 2 by the
3
Bisection Method.
1.
x1
1
2
2
0 2 1, x 2
1
2
1
Math 343 - Fall, 2013
Homework 2
Solutions
Exercise:
1. In the N-bit format, how many real numbers that can be represented exactly?
2N
2. Use the 64-bit format to find the decimal equivalent of the binary floating-point machine
number x. For each, find al
Math 343 - Fall, 2013
Homework 6 - Section 2.3 Solutions
Exercises:
3
1. Let gx x x 2 5 and p 0 1. Compute (without using the MatLab program fixpt.m) p 1 and p 2 .
3x
3
5
p 1 g1 1 1 2 7
3
31
p2 g
7
3
1
7
3
3
3
7
3
5
2
233
441
0. 528 344 671
2. The graph
Math 343 - Fall, 2013
Homework 7- Sections 2.3 & 2.4 Solutions
Exercises: - Section 2.3
5. Consider the function gx 1 x 1 x 3 .
8
c. What is the order of convergence if we use the Fixed-Point Algorithm to find this fixed-point? Show
your work in detail.
Math 343 - Fall, 2013
Homework 10
Solutions
Exercises:
1. Given x i , y i 0, 1, 1 , 1, 1, 2 where y i fx i for some fx, construct the polynomial
2
P 2 x that agrees with fx at x 0 , x 1 and x 2 in the following two ways.
(1) P 2 x a 0 a 1 x a 2 x 2 a stan
Math 343 - Fall, 2013
Homework 11
Solutions
Exercises:
1. Given x i , y i 0, 1, 1 , 1, 1, 2 where y i fx i for some fx, construct the interpolating
2
polynomial P 2 x that agrees with fx at x 0 , x 1 and x 2 in the Newtons forward divided-difference form.
Math 343 - Fall, 2013
Homework 12
Solutions
Exercises:
1. The following table gives the viscosity of sulfuric acid, in millipascal-seconds (centipoises), as a function
of concentration, in mass percent:
Concentration (C) 0
Viscosity (V)
20
40
60
80
100
0.
Math 343 - Fall, 2013
Homework 5
Solutions
1. Let fx e x x. Show by the Intermediate Value Theorem that the equation fx 0 has a solution in
0, 1.
f0 1 0 1 0 and f1 e 1 1 0. 632 120 559 0. So, by the IVT, fx 0 has a solution
in 0, 1.
a. Compute (without us
2.6 - Accelerating Convergence Aitkens 2 Method
1. Aitkens 2 Method:
Let p n be a sequence which converges to its limit p linearly. That is, there exists a positive number
such that
p n1 p
.
lim
n
pn p
Can the order of convergence of this sequence be im
2.1 - Bisection Method
The idea of the Bisection Method is based on the Intermediate Value Theorem that you studied
in Calculus I. Recall that:
Intermediate Value Theorem:
If f is continuous on a, b and K is a number between fa and fb, then there exists a
3.1 - Polynomial Regression:
Problem: Given n 1 pairs of data points x i , y i , i 0, 1, . . . , n, find a polynomial
P k x a 0 a 1 x . . . a k x k where k n
such that the error function
n
Ea 0 , a 1 , . . . , a k
y i P k x i 2
i0
is minimized. Note that
3.4 - Piecewise Linear-Quadratic Interpolation
Piecewise-polynomial Approximation:
Problem: Given n 1 pairs of data points x i , y i , i 0, 1, . . . , n, find a piecewise-polynomial Sx
S 0 x
Sx
if x 0 x x 1
S 1 x
if x 1 x x 2
:
:
S n1 x if x n1 x x n
whe
3.5 - Cubic Spline Interpolation
Cubic Spline Approximation:
Problem: Given n 1 pairs of data points x i , y i , i 0, 1, . . . , n, find a piecewise-cubic polynomial
Sx
S 0 x a 0 b 0 x x i c 0 x x 0 2 d 0 x x 0 3
if x 0 x x 1
S 1 x a 1 b 1 x x 1 c 1 x x 1
2.3 The Fixed-Point Algorithm
1. Mean Value Theorem:
Theorem Rolles Theorem : Suppose that f is continuous on a, b and is differentiable on a, b .
If fa fb, then there exists a number c in a, b such that f c 0.
Theorem Mean Value Theorem: Suppose that f i
2.4 - Convergence of the Newton Method
and Modified Newton Method
Newton Method:
Given fx, x 0 in a, b and , for n 1, 2, . . . ,
fx n1
(1) compute x n x n1
;
f x n1
(2) if x n x n1 or fx n , then the algorithm is
terminated and x x n ; otherwise goto S
1.2 - Computer Arithmetic and Round-off Errors
1. Binary Floating Point Arithmetic Standard:
How does a calculator or computer store a real number? Calculators and computers stores a machine
number x of a given real number x using four parameters: sign s,
Math 343, Fall, 2013 Hour Exam 1 - Part I Name: _Solutions_
For each problem, please provide detailed derivations and computations to support your
answer(s).
1. (10pts) Suppose we know the power series for
1 n
x 3 x 5 . . . 1 n x 2n1 . . .
x 2n1 .
sinx x
Math 343, Fall, 2013 Hour Exam 3 - Part I Name: _
For each problem, please provide detailed derivations and computations to support your answer(s).
1. (15pts) Let x 0 1, x 1 1. 3, x 2 1. 8, and y i fx i for i 0, 1, 2 where fx lnx.
a. Find the Lagrangian i
1.3 - Algorithms and their Convergence
1. Algorithms
What is an algorithm?
Definition An algorithm is a precisely defined sequence of steps for performing a specified task.
What are algorithms for? What are our tasks that need algorithms? In Lectures (1.2
Math 343, Fall, 2013 Hour Exam 2 - Part I Name: _Solutions_
For each problem, please provide detailed derivations and computations to support your answer(s).
y
1.0
(6pts) The graph of g for x in 0, 1 is given
1.
at the left. Let p 0 0. Label p 1 , p 2 , p
Math 343 - Fall, 2013
Homework 8
Solutions
Exercises: - Section 2.5
1. Let fx = x 3 cosx. Consider solving fx = 0.
(1) Solve a zero of fx within 10 10 using the Secant Method with x 0 = 1 and x 1 = 0.
>
>format long
> fun=@(x) -x.^3-cos(x);
> [xv,flag,ct]