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FIT3139: Assignment 1 Computational Science

创建日期：2024-06-21 17:45:34

所属分类：数学mathematics

标签： FIT3139

FIT3139: Assignment 1 Computational Science

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FIT3139: Assignment 1 Computational Science

Part 1

(a) A common computation in many applied mathematical problems involves the evaluation of the function log(1 + x),
∗ which is often represented as log1p(x). Indeed most math libraries
∗Here log() is the natural logarithm function to the base e.
(including MATLAB and Python) define an in-built function log1p(x) for this purpose, over and beyond using the standard in-built logarithm function log(1 + x).
One way to compute this function is by using the following Taylor series expansion over infinite terms:
log1p(x) = log(1 + x) = x −x22+x33−x44+ · · · +(−1)n+1xnn+ · · · (1)
• Write your own function my log1p(x) that accepts a value x as input and evaluates the above series expansion until EITHER 1,000,000 terms in the expansion have been computed OR when the approximation to log(1 + x) has changed by less than 10−6.
• Your function should output the value it evaluated, the number of terms in the expansion it computed before it returned, the relative error compared to the the in-built log1p(x).
• Test and plot the accuracy of your function (in terms of the relative error) for varying values of x, including the values of x less than the machine epsilon of your floatingpoint representation.
What should you submit for this question?
Your submission should include your script that drives your function my log1p(x) for various values of x and plots the relative errors.
Your submission must also include a report. Your report should briefly comment on the values of x that lead to the most inaccurate results in terms of relative error, and identify/describe the advantages/limitations of using the above expansion. It should also specify the values of x for which you would use my log1p(x) over the in-built logarithm function log(1 + x), and vice versa. Note, this report should be separate from the script, and submitted in a PDF format, should be succinct and comprehensibly written.
(b) Consider another way of evaluating log1p(x). For any given x, one can choose another value y such that
1 + y
1 − y= 1 + x.
• Work out the expansion of log(1 + x) as log
1 + y
1 − y.

• Write another function my another log1p(x) that evaluates log1p(x) using this new expansion.†
• Test and plot the accuracy of this new function (in terms of relative error wrt the in-built log1p(x)).
What should you submit for this question?
Your submission should contain a script that drives my another log1p(x) for varying values of x and plots the relative errors. Your report should contain a fully worked out expansion †The termination of this series expansion is similar to what you have done in part 1(a).
for log(1 + x) (with intermediate steps that facilitated the expansion). Compare and contrast the results with those analysed in part 1(a). Your report should comment which of the two approaches lead to greater accuracy and why? Suggest any further observations you might have about the comparison between the these approaches.
Part 2
(a) Consider the dynamical system given by the following difference equation:
xt+1 = mxt + c
Find the steady states equilibrium (or equilibria) of the system and, if any, the conditions necessary for it to exist.
(b) Find an explicit solution to the system above. That is, a function f(x0, t) that describes the state of the system at time t, starting where the state at t = 0 is x0.
(c) The system above can be shown to show qualitatively different behaviours for different values of m and c as follows:
• Case 1: |m| < 1
• Case 2: |m| > 1
• Case 3: m = 1 and c = 0.
• Case 4: m = 1 and c 6= 0.
• Case 5: m = −1
For each case above, sketch a cobweb plot and explain the behaviour of the dynamical system.
You can draw the plots by hand or using a computer program, or both.
(d) A non-linear function f defines a dynamical system with the difference equation:
xt+1 = f(xt)
It is known that x
∗ = f(x
∗
). Using a Taylor expansion of f around x
∗
, linearise the system to determine conditions for local stability of x
∗
. You can use the findings of Part 2c to find this answer. Local stability means that the system will converge to x
∗ provided that it is sufficiently close to it. Local stability means that the system will converge to x
∗ provided that it is sufficiently close to it.
What should you submit for this question?
You will have to submit a report with your answers to all questions in part 2. If you used any program as part of your argument, please submit the code specifying in the name of the program the question it addresses.
Part 3
This part is about continuous time models that result in a system of Ordinary Differential Equations.
Consider a basic SEIR model with the following diagram:

Figure 1:
The compartments stand for Susceptible, Exposed, Infected and Recovered; with a transmission rate β, and recovery rate α. The parameter η is the per-capita rate of becoming infectious, and 1 η is approximately the length of the so-called latent period, where individuals do not show symptoms and are not contagious. We want to extend this model to have (i) some basic demographic features (that is, the population size N is not constant); and (ii) to account for asymptomatic transmission.

For all questions, You may use any of the tools studied in the lectures (or combinations of them); including inspecting the dynamics numerically for different parameter values, attempting to find steady state solutions or using phase-plane analysis when appropriate. There is no single way to analyse the models, so we will focus on how you justify what you report on the basis of the techniques of your choice ‡.

(a) Reformulate the SEIR system to include demography, assuming that the birth rate (or rate at which the number of susceptible individuals increase, for example due to migration) is constant, given by Λ, and a constant per-capita death rate µ.

• Draw the flow diagram describing the model.
• Specify the set of equations for the compartments SEIR.
• Using the equations you derived for SEIR, formulate an expression that shows how N changes in time. Use this equation to show that happens to the population size in the long-run.
• How does the long-term behaviour of the system change qualitatively when adding demography, as compared to the closed system that ignores Λ and µ?

(b) Let’s now say you can spread the disease without showing symptoms. An asymptomatic compartment is typically added, denoted A. Exposed individuals progress to the symptomatic infectious compartment with probability p, and to the asymptomatic infectious compartment with probability (1 − p). Assume that asymptomatic individuals are contagious, but transmit the disease at a reduced rate q. They also recover at a rate γ > α. Make sure Λ and µ are also included to reflect the same demographic assumptions of the previous model.

• Draw the flow diagram describing the model.
• Specify the set of equations the compartments SEIR.
• How does the long-term behaviour of the system changes qualitatively when adding asymptomatic transmission? Compare SEIR and SEIAR models for different values of p and q.

‡For all numerical explorations, please use N = 1000, E0 = 1. You are free to set other parameters to reasonable values. For doing this, it may help to think about the units of the quantities involved. For example, since I and E have units number of people, the units of β must be 1/(number of people × unit of time). When doing numerical exploration use several parameters to see how things vary and better support your arguments. A good idea is to use similar parameters when making comparisons, for example use the same β values when comparing SEIR and SEIAR.

What should you submit for this question?
You will have to submit a report with your answers to all questions in part 3. Make sure to include graphics and descriptions of the results in the report. Please submit all the code you used for supporting your report, specifying in the name of the program the question it addresses.

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 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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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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