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CSc349~ AOI
Page 1
[18 marks]
1.
For each of the specified functions in the tablle below, place an X in all ofthe
appropriate boxes to indicate for which ofthe sets of values of x the evaluation of the
function may have a large relative error using
COMPUTER SCIENCE 349A SAMPLE FINAL EXAM WITH SOLUTIONS
(a) [4 marks] For what values of the real variable x , where x > 1, is the following expression subject to subtractive cancellation that will produce a very inaccurate result (in terms of relative err
COMPUTER SCIENCE 349A
SAMPLE EXAM QUESTIONS WITH SOLUTIONS
PARTS 3, 5, 6 , 7
PART 3.
3.1
Suppose that a computer program, using the Gaussian elimination algorithm, is to
be written to accurately solve a system of linear equations Ax = b , where A is an
ar
COMPUTER SCIENCE 349A
SAMPLE EXAM QUESTIONS WITH SOLUTIONS
PARTS 1, 2
PART 1.
1.1
(a) Define the term ill-conditioned problem.
(b) Give an example of a polynomial that has ill-conditioned zeros.
1.2
Consider evaluation of
f ( x) =
1
,
1 tanh( x)
where tan
COMPUTER SCIENCE 349A
Handout Number 15
NAVE GAUSSIAN ELIMINATION (Section 9.2)
Notation for a system of n linear equations in n unknowns: Ax = b , where
A is an n n nonsingular matrix
and
x and b are (column) vectors with n entries.
Equivalently,
a11 x1
COMPUTER SCIENCE 349A
Handout Number 14
ZEROS OF POLYNOMIALS USING NEWTON'S METHOD WITH
HORNERS ALGORITHM, AND POLYNOMIAL DEFLATION
Outline of a procedure to compute a zero of a polynomial f ( x) using Newton's method
and Horners algorithm:
Let x 0 be an
COMPUTER SCIENCE 349A
Handout Number 13
HORNERS ALGORITHM
(NESTED MULTIPLICATION, SYNTHETIC DIVISION)
n
Given a polynomial f ( x) = ai x i and a value x0 , this algorithm is used to efficiently
i =0
evaluate f ( x 0 ) and f ( x0 ) . To illustrate the basi
COMPUTER SCIENCE 349A
Handout Number 12
MULTIPLE ROOTS AND THE MULTIPLICITY OF A ZERO
(Section 6.4 in 5th ed. or Section 6.5 in 6th ed.)
If Newton's method converges to a zero xt of f ( x) , a necessary condition for quadratic
convergence is that f ( xt )
COMPUTER SCIENCE 349A
Handout Number 11
ORDER OF CONVERGENCE OF THE SECANT METHOD
AND BISECTION METHOD
The Secant method iterative formula is
xi +1 = xi f ( xi )
xi xi 1
.
f ( xi ) f ( xi 1 )
Therefore, if xt denotes an exact zero of f ( x) ,
xi +1 x t =
COMPUTER SCIENCE 349A
Handout Number 10
QUADRATIC CONVERGENCE OF NEWTON'S METHOD
The following result is proved on p. 141 of the 5th ed. (p. 150 of the 6th ed.)
Theorem If Newton's method is applied to f ( x) = 0 producing a sequence cfw_xi that
converge
COMPUTER SCIENCE 349A
Handout Number 9
An illustration of order of convergence when = 1 and = 2 . The limit of each
sequence is xt = 1.3652 3001 3414 097 .
Note: in the following computed approximations, the underlined digits are correct.
Example 1
Comput
CSc 349A A01/A02
Page 1
1. [18 marks]
For each of the specied functions in the table below, place an X in all of the
appropriate boxes to indicate for which of the sets of values of x the evaluation of the
function may have a large relative error using
CSC 349A PRACTICE MIDTERM EXAM
There are 5 questions of equal value. Time allowed 75 minutes.
Question 1.
(a) (12 marks) Compute the Taylor polynomial of degree 3 for the function f(x) = (1 + x)3/2
expanded about x0=0.
(b) (8 marks) Use the error term in
COMPUTER SCIENCE 349A
ASSIGNMENT #5 (Total: 20 points)
DUE MONDAY FEBRUARY 17, 2014
This is a really large class and the logistics of grading assignments are challenging. Me
and the markers require your help in making this process go smoothly. Please ensu
COMPUTER SCIENCE 349A
ASSIGNMENT #5 Solution Guide(Total: 20 points)
1
Solutions
1. (a) (3 points)
function p = PolyEval( n, a, y, x )
p = a(n+1);
for j = n:-1:1
p = a(j) + p*(x+y(j);
end
end
Common mistakes:
i. Using
p = a(j) + p*(x-y(j);
, which was use
CSC 349A Assignment#7
Suyang Hu
V00794127
6th March
Question 1
A) > x = [-1;-0.9;-0.8;-0.7;-0.6;-0.5;-0.4;-0.3;-0.2;-0.1;0];
> Y = sinh(cos(X);
> interp1(x, Y,-0.26,'linear')
CSC349A Assignment #5
Suyang Hu
V00794127
Question 1
function [ p ] = PolyEval( n, a, y, x )
for i = 1:1:n+1
ans1 = 1;
if i = 1
px = a(1);
else
for j = 1:1:i-1
ans1 = ans1 * (x + y(j);
end
px = px + a(i)*ans1;
end
end
fprintf ('p(x) = 0\n\n', px);
end
> P
CSC349A Assignment#3
Suyang Hu
V00794127
Question#1
a)
When P(X)=0 implies
Thus, we obtain four roots as following,
X1=1.51;
X3=1.50+0.01i;
b)
X2=1.49
X4=1.50-0.01i
By rounding the coefficients. We obtain
Then
X1=X2=X3=X4= 1.5
The relative change (By choo
P( x ) = a( x x2 ) + b( x x2 ) + c
f ( x0 )( x x0 ) 2
2!
f ( n ) ( x0 )
f ( n +1) ( ( x)
+
( x x0 ) n +
( x x0 ) ( n +1)
n!
(n + 1)!
where x0 ( x ) x
Pn ( x) = f ( x) Rn ( x) truncation error/remainder
2
f ( x ) = f ( x0 ) + f ( x0 )( x x0 ) +
*P(x) and f
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https:/www.coursehero.com/file/9377743/CSC-349-Sample-2009-Final-Exam-Solutions/
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