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Homework assignment 8:
Suppose your background assumptions B are such that your degree of belief that an arbitrary
object a is not white is lower than your degree of belief that a is a swan say, because there
are more things that are not white than there
Homework assignment 7:
Consider two objects, today and tomorrow, and one property they can have, whether or not the
sun rises on them. There are four possibilities:
s1 = the sun rises today and the sun rises tomorrow
s2= the sun rises today, but the sun d
Homework assignment 4:
A partition P of an arbitrary non-empty set W is a set of subsets of W, P (W), such that any
two members B and C of P are mutually exclusive (have no members in common), B C = ,
and the members of P are jointly exhaustive (every mem
Homework assignment 3:
The strong party-hypothesis says that everybody who had time attended the party. The weak party
hypothesis says that somebody had time and attended the party. Parisa, Seya, and Franz are
notorious party-goers, but they do not always
Homework assignment 2:
The swan-hypothesis H is a universal if-then sentence and says that all swans are white. H is
logically equivalent to the universal if-then sentence H that everything that is not white is not a
swan. H is also logically equivalent t
Homework assignment 1:
Intuitively, a set is a collection of objects (things, entities). For instance, the set T of people
teaching this course is the collection of objects (or subjects, if you prefer) that is either a TA for
PHL 246 or an instructor for
Q UESTION 3 Construct a polynomial of degree at most 3 that interpolates (0, 1), (1, 3), (3, 13). Is it
CSC336 Tutorial 8 Interpolation
Q UESTION 1 Construct a polynomial of degree at most 2 that interpolates (0, 1), (1, 3), (3, 13). Is it
unique?
unique?
Q UESTION 2 Determine a, b, c and d so that the piecewise cubic polynomial
(x) = 1 + 2 x x3
S0
if 0 x < 1
S (x) =
1(x) = a + b(x 1)+ c(x 1)2 + d(x 1)3 if 1 x 2
S
CSC336 Tutorial 9 Splines
Q UESTION 1 Determine (if possible) constants a0, a1, a2, b0, b1,
Applying (2) to both triples of method (e), we get p = 2 (quadratic conv.). Applying (1) to all three
CSC336 Tutorial 7 Nonlinear equations
Q UESTION 1 Assume that ve iterative methods applied to a non-linear problem exhibit the conver-
pairs of method (e
CSC336 Tutorial 4 GE/LU, pivoting, scaling
Q UESTION 2 Do the same as in Question 1 with partial (row) pivoting. Furthermore, indicate the
pivotal vector ipiv at each step of GE, the elementary permutation matrix Pk associated with the k th
Q UESTION 1 Le
CSC336 Tutorial 5 Norms and condition numbers
Q UESTION 2 Let A =
Ax
x
Q UESTION 1 Prove that maxx=0
= max cfw_ Ax
x =1
(DA).
PROOF:
ANSWER: We have A
Also,
Ax
x
Ax
2. Note that max
= max A
= max
x=0
x=0
x=0
x
x
x
because x is just a positive scalar and
CSC336 Tutorial 2 Computer arithmetic
Q UESTION 1 Find the positive numbers in R2(4, 3) assuming normalized mantissa.
ANSWER: The numbers in R2(4, 3) are of the form f 2e, where f the mantissa and e the exponent.
1
The table below indicates all possible p
CSC336 Tutorial 3 Matrices, operation counts, GE/LU
Q UESTION 3 Show that the inverse of a l.t. matrix is a l.t. matrix.
Q UESTION 1 Show that the product of lower triangular (l.t.) matrices is a lower triangular matrix.
PROOF: Consider a l.t. matrix L an
CSC336S
Assignment 3
Due Wednesday, April 3, 2013, 6:10 PM
No late assignments (even with penalty) will be accepted. Please write your family and given names and underline your
family name on the front page of your paper.
1.
(a)
x2
sin x .
4
[2 points] U
CSC336S
Assignment 2
Due Monday, March 11, 2013
Please write your family and given names and underline your family name on the front page of your paper.
General note: Plotting quantity y versus quantity x , means that x is in the x -axis and y is on the y
CSC336S
Assignment 1
Due Monday, February 4, 2013
Please write your family and given names and underline your family name on the front page of your paper.
1.
(a)
[10 points] Find the condition number of f ( x ) =
conditioned.
(b)
(c)
(d)
(e)
1 cos( x )
,
Piecewise polynomials and splines
Piecewise polynomials and splines - dimension of pp space
Let = cfw_ a = x 0 , x 1 , . . . , x n = b be a set of distinct points, called knots or nodes or
breakpoints or gridpoints, partitioning the interval I = [ a, b] i
Polynomial interpolation with Lagrange basis
Polynomial interpolation with Lagrange basis
We (again) construct a polynomial p n ( x ) of degree at most n, that interpolates the data
( x i , f i ), i = 0, . . . , n. Instead of using the monomials 1, x , x