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MATH 317 ASSIGNMENT
创建日期：2024-06-04 14:11:17
所属分类：数学mathematics
标签： MATH 317
ASSIGNMENT

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MATH 317 ASSIGNMENT 1

DUE SATURDAY SEPTEMBER 30, 22:00 EDT
1. For x > 0 large, consider the computation of e−x by truncating the Taylor series
e−x = 1− x+ x
2
2
− x
3
3!
+ . . . .
Suppose that we are given a tolerance parameter ε > 0, and we truncate the series once
the current term becomes smaller than ε in absolute value. As the series is alternating,
in exact arithmetic we would have the error of the truncated the series bounded by ε.
(a) Explain why you would expect cancellation of digits under floating point arithmetic.
With the help of a calculator or a computer, produce a concrete and illustrative
example where such a cancellation occurs.
(b) Come up with a method to compute e−x without cancellation of digits. Show off
the performance of your method by a concrete example.
2. (a) Perform a detailed round-off error analysis on the pairwise summation algorithm.
(b) Perform the same analysis on the “pairwise product” algorithm, which is the ana-
logue of the pairwise summation algorithm for computing products.
3. Design a paper-and-pencil method that computes cubic roots digit by digit, and argue
convincingly that it works correctly. The method (but not necessarily the design prin-
ciples) should be simple enough that it can be taught to a 6th grader. Show off your
method by illustrative examples.
4. (a) Let us call the functions tanx, arctanx, ex, and log x the basic functions. Then
reduce the evaluation of sinx, cosx, arcsinx, arccosx, and xa (a ∈ R) into basic
functions and elementary arithmetic operations. Here by elementary arithmetic
operations we understand addition, subtraction, multiplication, division, and square
root extraction
√
x for x > 0, and all variables are real.
(b) Reduce the argument of arctanx into [0, b], where b < 1 is to be chosen by you
(Generally, b ≈ 12 would be considered satisfactory). In other words, express arctanx
with x ∈ R, in terms of arctan y with 0 ≤ y ≤ b.
5. Perform a detailed error analysis on an algorithm that computes log y by employing the
Gregory series
log
1 + x
1− x = 2
(
x+
x3
3
+
x5
5
+ . . .
)
.
You can assume −12 ≤ x ≤ 12 in the analysis. Then describe a procedure to reduce the
argument into −12 ≤ x ≤ 12 .
Date: Fall 2023.
1
2 DUE SATURDAY SEPTEMBER 30, 22:00 EDT
Homework policy
You are welcome to consult each other provided (1) you list all people and sources who
aided you, or whom you aided and (2) you write-up the solutions independently, in your
own language. If you seek help from other people, you should be seeking general advice, not
specific solutions, and must disclose this help. This applies especially to internet knowledge
bases such as MathStackExchange and ChatGPT.
Similarly, if you consult books and papers outside your notes, you should be looking for
better understanding of or different points of view on the material, not solutions to the
problems.
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EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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