Prof. Ming Gu, 861 Evans, tel: 2-3145 Email: mgu@math.berkeley.edu http:/www.math.berkeley.edu/mgu/MA128A
Math128A: Numerical Analysis Midterm I
This is a closed book, closed notes exam. You need to justify every one of your answers. Completely correct an
Math 128a, fall 2011, Chorin, computer homework 3
b
1. Write a computer program that evaluates a f (x)dx by the trapezoidal
rule with extrapolation. Apply it to f = ex on [0, 1] and make a graph
showing that the error in the trapezoidal rule is O(h2 ) and
UCB Math 128A, Fall 2012: Programming Assignment 1
Solutions
1. function [a,b]=findbracket(f,x0)
dx=1e-3;
a=x0;
b=x0;
while 1
fprintf(Bracket = [%12.8f , %12.8f ]\n,a,b);
a=a-dx;
if f(a)*f(b)<0, break; end
b=b+dx;
if f(a)*f(b)<0, break; end
dx=2*dx;
end
f
Math 128a, fall 2011, Chorin, theory homework 1, due in the week of Sept. 5.
1. Use the formula for the error in interpolation to show that if two polynomials Pn , Qn of degree n coincide at n + 1 distinct points, then they are
identical and coincide at a
Math 128a, fall 2011, Chorin, theory homework 3, due in the week of Sept.
19.
b
1. Verify that the inner product (f, g )W = a f (x)g (x)W (x)dx (where W (x)
is a continuous function that is positive on [a, b]) satises all the axioms
for an inner product,
Math 128a, fall 2011, Chorin, theory homework 5, due the week of Oct. 17.
1. Solve (analytically) the following initial initial value problems for i > 1:
(i) ui+1 ui + 0.25ui1 = 0, u0 = 1, u1 = 1. (ii)ui+1 + ui1 = 0, u0 =
1, u1 = 1.
2. Which of the follow
MATH 128A, FALL 2014, WILKENING
HW 6 SOLUTIONS
Homework 6: Due Wed, Oct 22
4.4: 1d, 3d, 14
14c: use our notation for the composite midpoint rule:
b
a = x0 , b = xn , a f (x) dx = 2h n/2 f (x2j1 ) + ba h2 f ()
j=1
6
4.5: 2a, 10, 13
4.6: 1d, 9
4.7: 1f, 4f,
MATH 128A, FALL 2014, WILKENING
HW 9 SOLUTION
Homework 9: Due Wed, Nov 12
1c, 4c: only use the 3-step AB and AM schemes
5.6: 1c, 4c, 12, 14
and only report the results at the nal time, t = 2
5.10: 1, 2, 5, 7, 8
section 5.6
1. Use the 3-step Adams-Bashfort
Prof. Ming Gu, 861 Evans, tel: 2-3145
Email: mgu@math.berkeley.edu
http:/www.math.berkeley.edu/mgu/MA128A2012S
Math128A: Numerical Analysis Sample Midterm
This is a closed book, closed notes exam. You need to justify every one of your
answers. Completely
MATH 128A, FALL 2014, WILKENING
HW 7 SOLUTIONS
Homework 7: Due Wed, Oct 29
4.8: 1bd, 2b, 5d, 13
on problem 8, use the result for problem 7 given in the back of the
book. Also, there is a typo in problem 7. The integral should be set to
4.9: 1a, 3b, 6, 8
MATH 128A, FALL 2014, WILKENING
HW 5 SOLUTIONS
Homework 5: Due Wed, Oct 15
4.1: 2b, 4b, 5a, 22, 29
4.2: 1d, 5, 9, 10
4.3: 2a, 4a, 6a, 8a, 14, 16
section 4.1
2. Use the forward-dierence and backward dierence formulas to determine each missing entry
in the
MATH 128A, SUMMER 2009, FINAL EXAM
BENJAMIN JOHNSON
Name:
SID:
Please draw a box around all of your nal answers.
You may not use a calculator on this exam.
The exam consists of 11 questions. The point values for each question are indicated below,
and a
Math128A: Numerical Analysis Final Exam
3
3. (16 Points) Consider a multi-step method of the form
wk+1 = wk + h (Bf (tk , wk ) + Cf (tk1 , wk1 ) ,
where tk = kh for all k.
(a) Dene what is meant by the local truncation error for the multi-step method.
(b)
Solutions to Midterm 2, Math 128A
Prof. John A. Strain
1. We know
N (h) = M + 3h + 5h2 + O(h3 ).
(1)
Replacing h by h/2 in (1)
h
3
5
N ( ) = M + h + h2 + O(h3 )
2
2
4
(2)
Replacing h by h/3 in (1)
h
5
N ( ) = M + h + h2 + O(h3 )
3
9
(3)
(1) 2(2) (cancelin
Math 128A Midterm 2, Fall 2014, Wilkening
your GSI: Danny, Jue, Weihua, Elan
your name: 5 olugmom
1. (7 points) Suppose f E CGUR). Find constants a, b, c so that
f(0) = % [¢f(2h) + beL) + af(0) + bf(h) + mm] + 0(h4).
Hint: N(h) 2 5%[f(h) 2f(0) + f(h)] s
MATH 128A, FALL 2014, WILKENING
Homework 1: Due Wed, Sep 10
9b: express the error bound as a function of x.
21: nd a bound independent of x that works for
all x in the given range.
15d: write the result in the form d.dddddddd + 2dd , where d is
a decimal
MATH 128A, SUMMER 2011, WILKENING, QUIZ 5
Date: August 4, 2011.
Name:
(1) (1 point) Suppose AB = I and BC = I. Show that A = C.
(2) (2 points) Write the following second order equation as a rst order system:
y + Ay + sin(y) = t2 .
(3) (2 points) Consider
Solutions to Midterm 2, Math 128A
Prof. John A. Strain
1. We know
N (h) = M + 3h + 5h2 + O(h3 ).
(1)
Replacing h by h/2 in (1)
h
3
5
N ( ) = M + h + h2 + O(h3 )
2
2
4
(2)
Replacing h by h/3 in (1)
h
5
N ( ) = M + h + h2 + O(h3 )
3
9
(3)
(1) 2(2) (cancelin
MATH 128A, SUMMER 2011, WILKENING, QUIZ 4
Date: July 28, 2011.
Name:
(1) (2 points) Consider the equation y = y. Use the 2 step Adams-Bashforth method wn+1 = wn +
h
[3f (tn , wn ) f (tn1 , wn1 )] to compute w2 when w0 = 1 and w1 = 1 + h.
2
(2) (2 points)
MATH 128A, SUMMER 2011, WILKENING, QUIZ 3
Name:
0 t1
1 t>1
D = cfw_(t, y) | 0 t 2, y R. (No justication needed).
(1) (1 point) True of False: the function f (t, y) =
satises a Lipschitz condition on
(2) (3 points) Find an integer n such that the composite
MATH 128A, SUMMER 2011, WILKENING, QUIZ 2
Name:
(1) (2 points) Do two steps of Newtons method on f (x) = x2 x 5, starting with p0 = 2.
1
(2) (2 points) Let pn = n . Compute pn and 2 pn .
(3) (3 points) Suppose P (x) = (xx0 )Q(x)+b0 , Q(x) = (xx0 )R(x)+c0
MATH 128A, SUMMER 2011, WILKENING, QUIZ 1
Name:
x3 x5 x7
+ + , which has radius of convergence 1.
3
5
7
One may also show that |f (5) (x)| 24 for x R.
(1) Let f (x) = tan1 x and recall that f (x) = x
(a) (1 point) True or False: Taylors theorem with remai
MATH 128A, mock midterm test.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be given but only for significant
progress towards a solutio
MATH 104-04: INTRODUCTION TO ANALYSIS
SUGGESTED PROBLEM: SECTION 21, 22
(1)
(2)
(3)
(4)
(5)
(6)
(7)
P176,
P176,
P177,
P177,
P184,
P184,
P184,
21.3.
21.4.
21.6.
21.8.
22.2.
22.3.
22.5.
(1) P176, 21.3.
Proof. For any > 0, we will show that for = and any two
MATH 104-04 MIDTERM 2 SOLUTION
1. (5 points) Determine whether the following statements are true of false, no
justification is required.
(1) For a sequence (sn ) of real numbers, if (sn ) has a subsequence converging to
2, then (sn ) also converges to 2.
MATH 104-04 PRACTICE OF FINAL
The problems here may or may not reflect what we will have in the actual exam,
these are just for practice.
1. Determine whether the following statements are true of false, no justification
is required.
(1) For a function f :