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APMA 4300: Introduction to Numerical Methods

创建日期：2024-05-22 17:27:50

所属分类：数学mathematics

标签： APMA 4300

Introduction to Numerical Methods

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APMA 4300: Introduction to Numerical Methods

Midterm (Thursday, March 23, 2023)
Name and UNI:
Problem 1. Given n ≥ 1 and m ≥ 1, let A1 ∈ Rn×n, A2 ∈ Rm×m, and
B ∈ Rn×m be given matrices. We are interested in solving the following
block linear system [
A1 B
BT A2
] [
x
y
]
=
[
z1
z2
]
.
Consider the following two methods
(i) A1x
(k+1) +By(k) = z1, B
Tx(k) +A2y
(k+1) = z2 ;
(ii) A1x
(k+1) +By(k) = z1, B
Tx(k+1) +A2y
(k+1) = z2 .
(a) Write a function to solve the system using method (i). Your function has
to take the matrices A1, A2, and B, the right hand side
[
z1
z2
]
, and an initial
guess
[
x(0)
y(0)
]
as input and output the solution
[
x
y
]
. Your algorithm should
stop if
∥∥∥ [x(k+1)
y(k+1)
]
−
[
x(k)
y(k)
] ∥∥∥
ℓ2
≤ ε with ε = 10−8.
(b) Write a function to solve the system using method (ii). The input and
output of the function should be the same as those of the function in part
(a).
(c) Take the system where
A1 =
62 24 123 50 7
4 6 58
 , A2 = [66 32 54
]
, B =
 8 1514 16
20 22
 ,
z1 =
11
1
 , z2 = [11
]
.
Run the algorithms in (a) and (b) on this system starting from initial guess[
x(0)
y(0)
]
= 0. Stop the algorithm at 100 iterations if they don’t stop earlier.
Output the final results.
(d) Can you find sufficient conditions in order for the two schemes to be
convergent for any choice of the initial data
[
x(0)
y(0)
]
? Provide some justification
for your answer.
Problem 2. Consider the function f : R→ R defined as
f(x) = −e−x
2
2 +
1
4
cos(2x) .
We are interested in finding the minimum of f(x).
(a) Design a fixed point iteration for this problem. Your fixed point iteration
should converge if the initial guess is sufficiently close to the true solution
x = 0.
(b) Write a function that solves the problem using the following fixed-point
iteration:
x(k+1) = g(x(k)), g(x) = sin(2x)
[ex22 (x sin(2x) + 2 cos(2x))− 2
2
(
x sin(2x) + 2 cos(2x)
) ]
Does this iteration converge if the initial guess is sufficiently close to the true
solution?
(c) Try to reproduce the convergent analysis we had in class to find the speed
of convergence for the fixed-point iteration you designed. Your calculations
should be done assuming you are close to the true solution x = 0. If the g
function you designed has non-zero derivative around the true solution, you
should have exactly what we had in class.
(d) Can you do the analysis in part (c) for the fixed-point algorithm in (b)?
Problem 3. In a chemical experiment, we are able to measure a quantity y
that depends on two other quantities x1 and x2. For simplicity, we use the
notation x = (x1, x2). We are interested in learning the relation between x
and y, that is, y = f(x). Our knowledge in chemistry says that y should
depend on x in the following way:
y = a0 + a1 cos(πI · x) + a2 cos(2πI · x) + · · ·+ a10 cos(10πI · x) ,
with I =
[
1
1
]
. However, we do not know the coefficients a0, a1, · · · , a10.
Suppose that in an experiment, we collect the following data:
{xj, yj}500j=1.
Propose a method that we can use to learn the unknown coefficients {a0, a1, · · · , a10}
of the model. You don’t need to implement your method. You need to out-
line your method, to a point that if we have the data, we can implement
your method. The more details you are willing to provide, the better.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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