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 pe
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
,
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
Math 343 - Fall, 2013
Homework 1
Solutions
1. Approximate using formulas (1)-(4) within 10 5 (without using MatLab programs). Find N first for
each formula.
(1) 4 tan 1 1 :
4 1 n1
n1
1 .
2n 1
1
2N 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 bi
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
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 Algorith
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
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
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 percen
Math 343 - Fall, 2013
Homework 9
Solutions
Exercises:
1. Consider the following data relating the amount of varnish additive and the resulting varnish drying time:
Additive (grams)
0
1. 0
2. 0
3. 0 4.
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 I
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
p
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 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
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
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
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 t
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 t
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
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 serie
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
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? Wh
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
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.