## Transcribed Text

D Question 1
Convert the given binary number into decimal: (110000)2
0 64
0 34
0 46
0 48
3 ptsD Question 2 4 pts
For the function
when it is hard to compute, which equivalent function would you rather use to
compute f ( x) accurately?
0 f(x) == - ✓4x2 + 1 - 2lx l
0 None of the other answers.
0 f(x) == ✓4x2 + 1- 2lxl
0 f(x) - 1
~
2lx l + ✓4x2 + 1D Question 4
Write a Matlab script to plot the function
on the interval O -< x -< 2.
0 x=0:0.01:2; y=x/(x+1); plot(x,y)
0 x=0:0.01:2; y=x./(x+1); plot(x,y)
0 x=0:0.01:2;y=x./(x+1); plot(y,x)
0 x=0:0.01:2;y=x/(x+1);plot(y,x)
X
y= x+ l
3 ptsD Question 5 4pts
Let f ( x) be a function which has a Taylor series about x == -1, and let f 4 ( x)
be the partial sum of the first 5 terms. What is an estimated bound for the
error E5 == If ( x) - f 4 ( x) I ?
0 E5 < ;, f (5) (~)(x + 1)5 for some ~ between x and -1
0 E5 < ;, f (4) (~)(x + 1)4 for some ~ between x and -1
0 E5 < ;, f (5) (~)(x + 1)5 for some ~ between x and -1
) Question 6
Consider the data set:
X i 1 2 3
Yi 4 0 - 1
4 pts
Let p( x) == a 2 x 2 + a 1 x + a0 be a polynomial that interpolates the data set.
Which of the following linear system would give the solution for ( a0 , a1 , a2 )?
0 0 0 1 a2 0
4 2 1 a1 4
9 3 1 ao -1
0 1 1 1 a2 -1
4 2 1 a1 0
9 3 1 ao 4
0 9 4 1 a2 0
4 2 1 a1 0
2 3 1 ao 0
0 1 1 1 a2 4
4 2 1 a1 0
9 3 1 ao -1
D Question 7 4 pts
Given the interpolating points at ( - 1, 1, 2, 4), which of the following is the
cardinal function associated with the point x == - l ?
O
1
1
2
(x + l)(x - 2)(x - 4)
Q -
1
1
2
( X + 1) ( X - 1) ( X - 2)
Q -
1
1
2
( X + 1) ( X - 1) ( X - 2)
0 _ _!__(x - l)(x - 2)(x - 4)
30D Question 8
The polynomial p( x) == x4 + x2 - 8 has the values shown.
X 1-21-1 I O 11 12 p(x) 12 - 6 - 8 - 6 12
Find a polynomial q( x) of degree 5 that takes these values
X 1-2 1-l I O 11 I 2 I 4 q(x) 12 -6 -8 -6 12 -96
0 q(x) == p(x) - ~ (x + 2)(x + l)x(x - l)(x - 2)
0 q(x) == p(x) + ! (x + 2)(x + l)x(x - l)(x - 2)
0 q(x) == p(x) + ! (x + 2)(x + l)(x - l)(x - 2)(x - 3)
0 q(x) == p(x)
4 ptsD Question 9 3 pts
Let li ( x) be the cardinal function associated with the point X1 for a given set
interpolating points ( x0 , x 1 , x 2 , x 3 ) . Then we must have li ( x 3 ) == 1.
0 True
0 FalseD Question 10 4 pts
Which code will assign value 2 to t only if x is bigger than O?
0 if t>01 x=2; end
"'- - ' if x=O ' t=2·' end
0 if x>O, t=2; end
0 if x>=O, t=2D Question 11
Consider the piecewise polynomial,
S(x) <
to < X < t1,
t1 < X < t2,
Which of the following statement will make S( x) NOT a quadratic spline?
0 S' ( x) is a continuous function.
0 For some i , we have Si ( x) == constant.
Q For some i , we have S? ( ti+l) == 0, s;~l ( ti+l) == 1.
0 S' ( x) is a discontinuous function.
4 ptscD Question 13
If S( x) is a cubic spline function, then S" ( x) is a linear spline function.
0 True
0 False
3 ptsD Question 15
Find the value a that will make the following a linear spline:
S(x) = {x,
X - a,
O a = O
n Such value does not exist.
0 a= - 2
0 < X < 1,
1 < X < 2.
4ptsD Question 16 4pts
Consider the integral
fl 1
Jo 2x+l dx.
What is the value you would get if you compute the integral with Simpson's
rule using only the values at x = 0, 0.25, 0.5, 0. 75, 1, using 4 decimal places?
0 0.5500
0 0.6897
0 1.1023
0 1.5634D Question 17
Consider the integral
fl 1
Jo 2x+2 dx.
To compute the integral with trapezoid rule with error tolerance of 10- 6,
approximately how many points are needed?
0 812
0 290
0 443
0 40
4ptsD Question 18 4 pts
Find the degree of precision for the following rule.
J2 1 f(x) dx = 2f(O)
0 2
n 3
Q o
n 1D Question 19
We use Simpson's rule to computer an integral
I ( h) ~ S == f0
2 f ( x) dx
where h is the sub-interval length. Assume that the error is estimated by
E(h) == I(h) - S ~ C1h4 + C2h6 + · · ·
Given that
J(0.2) == 5.4678, J(O.l) == 5.4076
Find a better approximation to the integral.
0 5.4327
0 5.4098
0 5.4036
0 5.4670
4 ptsD Question 20
Given the Matlab code for the function
function v=fv(x)
v=(x+l).A(-1);
end
4 pts
---------------------------'
Will the following Matlab script compute the integral from Oto 2 using n + 1
points, using trapezoid rule?
I a=0; b=Z; h=(b-a)/n ; x=[a:h :b]; V=sum(fv(x)))*h;
0 True
0 FalseD Question 21
The function f(x) = x - ex+ 2 has a zero on the interval [-2, O].
0 True
0 False
3 ptsvD Question 22 4 pts
We consider bisection method for finding the root of the function
f ( x) == x 3 - 6 on the interval [-2, 2], so x 0 == 0. We perform 2 steps, and
our approximations X1 and x2 from these two steps are:
C X1 == 0D Question 23
Perform 1 step of secant iteration to find the root of
f ( X) === ex - 2
with xo === 2, x 1 === 1, we would get:
0 X2 === 1.7459
0 X1 === 0.8462
0 X2 === 0. 7358
Q X2 === 0.5820
4 ptsD Question 24
Fixed point iteration sometimes has linear convergence.
0 True
0 False
3 pts.3, X2 == 0.6
D Question 25 4pts
Assuming that we are performing a fixed point iteration which converges
linearly. Let ek be the absolute error of iteration step k. Which of the following
could be possible errors for the first 3 iterations?
n e0 = 0.04, e1 = 0.02, e2 = 0.04
0 eo = 0.005, e1 = 0.005, e2 = 0.005
0 eo = 1.2, e1 = 0.6, e2 = 2.3
C1 eo = 0.11, e1 = 0.055, e2 = 0.0275D Question 26 3 pts
What is the solution to the following 2 x 2 system of linear equations?
0 X = 3, y= - 2
0 X = 2, y=4
C'1 X = l, y= - 1
0 X = 2, y= - 5D Question 27 3 pts
The following matrix is tri-diagonal:
-5 1 0 0 0
1 -5 1 0 0
0 1 -5 1 0
0 0 1 - 5 1
0 0 0 1 - 5
0 True
0 FalseD Question 28
Consider the system of linear equations
{
4x + y
x + 4y
5
5
3 pts
Performing one step Jacobi iteration with initial guess x0 = 0, y0 = 0 , we will
get:
0 x1 = 1.25, y1 = 1.25
0 x1 = 2.2, y1 = 4.5
0 x1 = 1 '
yl = 1
0 x1 = 0.25, yl = 1D Question 29
The largest condition number for a matrix is 1.
0 True
0 False
4 ptsD Question 30 4pts
Let x == ( x1 , x2 , • • • , Xn )t be a column vector and A == { aij} be an n x n
square matrix. Let llxll denote certain vector norm and IIAII be
the associated matrix norm. Then
IIAII == maxllx ll=l IIAx ll
0 True
0 FalseD Question 31
Consider the system of linear equations
{
3x + y
x + 3y
9
6
4pts
Performing one step Jacobi iteration with initial guess x0 = 0, y0 = 0, we will
get:
0 x1 = 2, yl = 3
0 x1 = 3, y 1 = 3.5
0 x1 = 2.2, y 1 = 4.5
0 x1 = 3, yl = 2D Question 32
Consider the system of linear equations
4x + y+ z = 8
X + 5y+ Z = 4
X + y + 6z = 2
Which of the following is the Gauss-Seidel iteration for step k ?
0 ~ 4xk+ l = 8 + yk + zk
◄ 5yk+ 1 = 4 + xk + zk
"6zk+ l = 2 + xk + yk
0 ~ 4xk+ l = 8 - yk - zk
◄ 5yk+l = 4 - xk - zk
"6zk+ l = 2 - xk - yk
0 ,.. 4xk+ l = 8 + yk+ l + zk+ l
◄ 5yk+ 1 = 4 + xk+ 1 + zk
\. 6zk+ l = 2 + xk+ l + yk+ l
0 ,.. 4xk+l = 8 - yk - zk
◄ 5yk+ 1 = 4 - xk+ 1 - zk
, 6zk+ l = 2 _ xk+l _ yk+ l
4 ptsD Question 33 4 pts
Given the splitting of a square matrix A = L + D + U where L is lower
triangular, D is diagonal, and U is upper triangular. The Jacobi iteration can be
written in the general form
with which choice of ( M, y)?
0 M = -(D + L)-1U, y = (D + L)-1b
n M = (D + L)-1L, y = (D+L)-1b
0 M = -D- 1(L + U), y = D-1b
0 M = D-1(L-U), y = D-1bD Question 34 3 pts
Consider the iteration of the form
where Xk, b are column vectors of length n , and M is an n x n matrix.
If IIM II > 1 in some matrix norm, then the iteration always converges with
any initial guess.
0 True
C FalseD Question 35 3 pts
If A == tri-diag( -1, -4, 1) , then solving Ax == b with SOR iteration with
w == l.2 will always converge for any initial guess.
(; True
0 FalseD Question 37 4 pts
Find a linear function y == ex that best fits the data with least square error
X i 1 4
Yi -2 2
8 0 y == 17x
0 y == 197 X
0 10
ITX
0 y == 167 XD Question 38 4 pts
Given the error function
the normal equations that will find the values of ( a, b) which minimize the
error will look like:
0 m m
a · L ( sin x k) + b · L sin x k - 0,
k=O k=O
0 m m
a· L(sin x k)2 + b · L sin Xk = 0,
k=O k=O
0 m m
a · L ( sin x k) + b · L sin x k = 0,
k=O k=O
0
m
a · L sin Xk + mb - 0.
k=O
m
a· L sinxk + (m + l)b = 0.
k=O
m
a·Lsinxk + (m + l)b = O
k=OD Question 39 3 pts
In the linear least squares method, there are as many normal equations as there
are basis functions.
0 True
Q FalseD Question 40
The functions
f(x) = ~ x and g(x) =cos(! x)
are orthogonal on [- 1r, 1r], such that
J:1r f(x)g(x)dx = 0
0 True
0 False
4ptsD Question 41
Which of the following is for sure a uniform grid for time { t k} f O ?
0 to < t1 < · · · < t N
C1 tk+l - tk = 0.lk
n tk+l - tk = 0.1
0 t3 = to + 0.03
3 ptsD Question 42 4 pts
Consider the ODE
X1
( t) == X - 2t
To solve it with Taylor Series method of m == 1 with uniform grid size h , which
of the following will be the iteration step?
0 Xk+l == Xk + h(xk - 2k)D Question 43 3 pts
Consider the system of OD Es x' == Ax, where A is an n x n symmetric
matrix and x is a vector of length n . Then, if the eigenvalues of the matrix A
satisfy Ai == 0.01, A2 == 1890, the system is rather stable with respect to
perturbation and thus easy to solve numerically.
C True
0 FalseD Question 44 4 pts
Consider the following ODE with initial condition
X
1 = X + t, x(l) = 1
We solve it with implicit backward Euler method, with uniform time step h and
initial values t0 = 1, x0 = 1. Which of the following is the formula for step n
?D Question 45
Given the ODE with initial condition
X
1
( t) == X + t, x(l) == - 1
We solve it with Heun's method, using time step h == 0.02. What is the
approximate solution for x(l.02)?
0 -0.9998
0 -0.9908
0 -1.0000
0 -1.0002
4 ptsD Question 46
Let y(x) and y(x) solve the equations
y"' + xy' = ex ,
Then, the function
solves which equation?
0 y"' + xy' = ex
0 y"' + 2xy' = ex
0 None of the other answers here.
0 y"' + xy' = 2ex
3 pts
f/" + 2xy' = ex .D Question 48
The following finite difference method
Xi == ih, Yi-l - 2yi + Yi+l == h2
Xi (1 - Xi ).
for the equation
is first order.
0 True
0 False
y" == x(l - x), y(O) == 0, y(l) == 0
3 ptsD Question 49
Consider the two-point boundary value problem
y" ( X) == y + x2
, y' (0) == 1, y(2) == 2.
To solve it with shooting method, we let y(.x) be the solution of
j} 11
( X) == j} + x2
,
and let fj ( x) be the solution of
f} 11
( X) == fj + x2
,
Let now
y' (0) == 1, y(O) == 0,
y' (0) == 1, fj(0) == 2.
y == ,,\y + (1 - --\)fj
4pts
For what value of A will this y be the solution of the two-point boundary value
problem?
0 None of these functions.
0 ,,\ == -2- y~2) .
y(2)-y(2)
-,
0 A == -1-y ~2) .
y(2)- y(2)
0 ,,\ == 2- y(2) .
-, -, y (0)-y (0)D Question 50 4 pts
Consider the two-point boundary value problem
y"(x) -y'(x) == 2xy, y(O) == 0, y(l) == 1.
We discretize the equation using central finite difference, with uniform grid
of mesh size h == l , such that y0 == 0, Yn == l, X i == ih. Which of the
n
following is the correct discretization, for i == 1, 2, • • • , n - l ?D Question 51
The following computational stencil
is for Finite difference method for Heat equation Ut = Uxx with CrankNicholson
time step.
0 True
0 False
3 ptsD Question 52
Consider the discrete equation
U·-1 · - 2u · · + U ·+1 · 'l, ,J i,J 'l, ,J
h2
2u·i, J· -1 - 4ui·, J· + 2u·i, J· + 1
+ h2
Which PDE could correspond to this discretization?
0 U xx + Uyy - 4ux + U == 0.
0 U xx + 2uyy - U x + U == 0 .
0 U xx + Uyy - 2ux + U == 0.
0 U xx + 2uyy - 2ux + u == 0.
4 pts
Ui+l,j - Ui-1,j ___2_ h _ + Ui,j = 0.D Question 53 3 pts
We solve the heat equation
with finite difference method using forward Euler time step. What is the CFL
stability condition?
0 1
~t < -~x. - 8
0 1
~t < 2(~x)2.
0 1
~t < -~x. - 2
0 1
~t < 8(~x)2.D Question 54 4 pts
Consider the equation
with the numerical scheme
u~+l - u":
1, 1,
u": 1
- 2u7: + u":+1 u~+l - 2u~+l + u~+l
i - i i + i -1 i i +l
2(~x)2 2(~x)2
What method is this?
Q Forward Euler, an explicit method
0 Backward Euler, an implicit method
Q None of these
0 Crank-Nicolson schemeD Question 55
The forward Euler step for solving heat equation is unconditionally stable.
0 True
0 False
3 pts

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