Month: October 2024

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Chapter 1

What Is Numerical Analysis?
ALGORITHMS
The objective of numerical analysis is to solve complex numerical problems using only the simple operations of arithmetic, to develop and evaluate methods for computing numerical results from given data. The methods of computation are called algorithms.

Our efforts will be focused on the search for algorithms. For some problems no satisfactory algorithm has yet been found, while for others there are several and we must choose among them. There are various reasons for choosing one algorithm over another, two obvious criteria being speed and accuracy. Speed is clearly an advantage, though for problems of modest size this advantage is almost eliminated by the power of the computer. For larger scale problems speed is still a major factor, and a slow algorithm may have to be rejected as impractical. However, other things being equal, the faster method surely gets the nod.

EXAMPLE 1.1. Find the square root of 2 to four decimal places.
More than one algorithm, using only the four basic operations of arithmetic, exists. The favorite is without much doubt
$latex
x_1=1 \quad x_{n+1}=\frac{1}{2}\left(x_n+\frac{2}{x_n}\right)
$
from which a few mental calculations quickly manage
$latex
x_2=\frac{3}{2} \quad x_3=\frac{17}{12} \quad x_4=\frac{1}{2}\left(\frac{17}{12}+\frac{24}{17}\right)
$
or, rounded to four decimal places,
$latex
x_2=1.5000 \quad x_3=1.4167 \quad x_4=1.4142
$
the last being correct to all four places. This numerical algorithm has a long history, and it will be encountered again in Chapter 25 as a special case of the problem of finding roots of equations.

ERROR
The numerical optimist asks how accurate are the computed results; the numerical pessimist asks how much error has been introduced. The two questions are, of course, one and the same. Only rarely will the given data be exact, since it often originates in measurement processes. So there is probably error in the input information. And usually the algorithm itself introduces error, perhaps unavoidable roundoffs. The output information will then contain error from both of these sources.

EXAMPLE 1.2. Suppose the number .1492 is correct to the four decimal places given. In other words, it is an approximation to a true value that lies somewhere in the interval between .14915 and .14925 . The error is then at most five units in the fifth place, or half a unit in the fourth. In such a case the approximation is said to have four significant digits. Similarly, 14.92 has two correct decimal places and four significant digits provided its error does not exceed .005 .

EXAMPLE 1.3. The number .10664 is said to be rounded to four decimal places when abbreviated to .1066 , while .10666 would be rounded to .1067 . In both cases the error made by rounding is no more than .00005 , assuming the given figures are correct. The first is an example of rounding down, the second of rounding up. A borderline case such as .10665 is usually rounded to the nearest even digit, here to .1066 . This is to avoid long-range prejudice between the ups and downs.

EXAMPLE 1.4. When 1.492 is multiplied by 1.066 , the product is 1.590472 . Computers work to a fixed "word
1
2
WHAT IS NUMERICAL ANALYSIS?
[CHAP. 1
length," all numbers being tailored to that length. Assuming a fictitious four-digit machine, the above product
Posted on

Chapter 1

What Is Numerical Analysis?
ALGORITHMS
The objective of numerical analysis is to solve complex numerical problems using only the simple operations of arithmetic, to develop and evaluate methods for computing numerical results from given data. The methods of computation are called algorithms.

Our efforts will be focused on the search for algorithms. For some problems no satisfactory algorithm has yet been found, while for others there are several and we must choose among them. There are various reasons for choosing one algorithm over another, two obvious criteria being speed and accuracy. Speed is clearly an advantage, though for problems of modest size this advantage is almost eliminated by the power of the computer. For larger scale problems speed is still a major factor, and a slow algorithm may have to be rejected as impractical. However, other things being equal, the faster method surely gets the nod.

EXAMPLE 1.1. Find the square root of 2 to four decimal places.
More than one algorithm, using only the four basic operations of arithmetic, exists. The favorite is without much doubt
$latex
x_1=1 \quad x_{n+1}=\frac{1}{2}\left(x_n+\frac{2}{x_n}\right)
$
from which a few mental calculations quickly manage
$latex
x_2=\frac{3}{2} \quad x_3=\frac{17}{12} \quad x_4=\frac{1}{2}\left(\frac{17}{12}+\frac{24}{17}\right)
$
or, rounded to four decimal places,
$larex
x_2=1.5000 \quad x_3=1.4167 \quad x_4=1.4142
$
the last being correct to all four places. This numerical algorithm has a long history, and it will be encountered again in Chapter 25 as a special case of the problem of finding roots of equations.

ERROR
The numerical optimist asks how accurate are the computed results; the numerical pessimist asks how much error has been introduced. The two questions are, of course, one and the same. Only rarely will the given data be exact, since it often originates in measurement processes. So there is probably error in the input information. And usually the algorithm itself introduces error, perhaps unavoidable roundoffs. The output information will then contain error from both of these sources.

EXAMPLE 1.2. Suppose the number .1492 is correct to the four decimal places given. In other words, it is an approximation to a true value that lies somewhere in the interval between .14915 and .14925 . The error is then at most five units in the fifth place, or half a unit in the fourth. In such a case the approximation is said to have four significant digits. Similarly, 14.92 has two correct decimal places and four significant digits provided its error does not exceed .005 .

EXAMPLE 1.3. The number .10664 is said to be rounded to four decimal places when abbreviated to .1066 , while .10666 would be rounded to .1067 . In both cases the error made by rounding is no more than .00005 , assuming the given figures are correct. The first is an example of rounding down, the second of rounding up. A borderline case such as .10665 is usually rounded to the nearest even digit, here to .1066 . This is to avoid long-range prejudice between the ups and downs.

EXAMPLE 1.4. When 1.492 is multiplied by 1.066 , the product is 1.590472 . Computers work to a fixed “word
1
2
WHAT IS NUMERICAL ANALYSIS?
[CHAP. 1
length,” all numbers being tailored to that length. Assuming a fictitious four-digit machine, the above product

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図書館

  • 2024~2028 “정신병 치료 및 요양”

North Pole Branch Library

>> 2028~ 2031~ 2034~ 2037~

  • (6) 2037~2043″봄 나들이”

Pio Pico

Little Tokyo

LA Central

  • (6) 2043~2049 UMT (stipend)
  • (6) 2049~2055″봄 나들이”

(9)(9)(9)(9)(9)

  • (9) Computational Science PhD, SDState (stipend)
  • – Economics MS, SDState
  • (9) CS MS >> Data Science & Engineering PhD, USD (stipend)
  • (9) Mathematics PhD, utexas (stipend)
  • – CS MS, utexas
  • (9) Economics PhD, utexas (stipend)
  • (9) 日本語 PhD, utexas (stipend)

condo, nice & cozy, nearby utexas (12月~05月)

  • Seattle, WA (05月)

가장 돈 많고 (>”$10B”)

가장 공부 많이 하고 (“PhDs”, utexas)

가장 오래 사는 {韓国人} (>2100)。

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stages

2034~ 소셜 시큐리티 70歳 수령

~2037 은퇴 I.II.III.

2037~2064 은퇴 연금 적립

(6)(6)(6)

(9)(9)(9)(9)(9)

~2100 JINU 2028~ 毎9年 × 8번 도전

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우선,

ㅇ

BBA, EA QBO CPA

  • CS AA, Foothill >> CS MS, USD / utexas
  • Economics AA, Foothill >> Economics MS, SDState / Economics PhD, utexas
  • English AA, Foothill >> English BA, UAF / English MA MFA, UAF
  • 日本語 AA, Foothill >> 日本語 BA, UAF / PhD, utexas

年 >$2万 저축 中、

REMOTE QBO job 확보

{CPA}.CsamiGroup.com

  • 연변 TM
  • 毎日 20통화

2028~2031~2034~2037 은퇴 I.II.III.

  • 2037~2064 {Solo 401(k)} 毎年$60,000 적립

(6)(6)(6)

(9)(9)(9)(9)(9) ~2100

d

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日本語 – 2024 10 12

コレステロールは気になるけれど、卵 は1日 1 個までと言われなくなった…何個食べても大丈夫なの?

콜레스테롤이 걱정되긴 하지만, 계란은 하루 1개까지만 먹으라는 말이 사라졌어요… 몇 개를 먹어도 괜찮은 건가요?

WordHiraganaMeaning in Korean
コレステロールころレスてろーる콜레스테롤
はは은/는
気にきに걱정되다
なるなる되다
けれどけれど그렇지만
卵たまご계란
はは은/는
1日いちにち하루
までまで까지
といわれなくなったといわれなくなった말하지 않게 되었다
………
何個なんこ몇 개
食べてもたべても먹어도
大丈夫だいじょうぶ괜찮다
なのなの인가요
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PASSED – “지난 세월” & SETTLEMENT

2009 Flagstaff, AZ

  • 지선 9歳, 지원 8歳, 지연 6歳

2010 家出

2011

2013 出家

  • 지선 13歳, 지원 12歳, 지연 10歳

2017 divorce

2018 Juneau, AK

2019 03 15 Mavis

2021 UAF 生活 始まり。

2023 LA wandering

2024 North Pole, AK LOT 장만

2026 지선 지원 지연 大卒

  • $3,500 REMOTE QBO
  • $3,000 {CPA}.CsamiGroup.com 毎年、月+$1,000

은퇴 I.II.III.

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“人生、日常 生活”

１日、$0.62; 月$1,600 BUDGET

  • ~2037 (6)(6)(6) ~2055 (9)(9)(9)(9)(9) ~2100~
  • North Pole, AK duplex (05月~12月)
  • CalStateLA (TA)
  • UMT (stipend)
  • CSULB (TA)
  • SDState (stipend/WRGP)
  • USD (stipend/WRGP)
  • Austin, TX condo nearby utexas (stipend)
  • –12月~05月
  • Seattle, WA (05月)

毎日4時間の学習 @McDonald’s

  • 05:00AM~09:00AM
  • Safeway
  • GREEN — BROWN, free
  • Barbershop, Mimi, Costco

daily COOK

REMOTE QBO (年$2万 저축 中、)

{CPA}.CsamiGroup.com

  • 연변 TM
  • １日、20통화

은퇴 연금 적립

JINU “미주장학재단” version READY

  • Credentialed Mathematics PhD
  • {言語-GLOBAL}.JinuAcademy.com

Forex Algorithmic Trading

  • Computational / Data Science PhDs
  • CS BS MS
  • Economics BA MS PhD
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~2034~

2024

2025

..

2028

..

2031

..

2034 소셜 시큐리티 70歳 수령

..

2037 (2028~2037)

  • (6)(6)(6)

2055

  • (9)(9)(9)(9)(9)
  • – SDState
  • – USD
  • – utexas (Mathematics PhD)
  • – utexas (Economics PhD)
  • – utexas (日本語 PhD)
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budget $20万 後、

($2,160) – duplex mortgage (20% downpay on $360,000)

$613 – SPIA

$1,050 – duplex rental income

($350) – 그로서리

($300) – 유틸리티

($400) – 재산세

($100) – 집보험

($100) – 쓰레기 수거비

($400) – 비지니스 경비

($100) – misc.

($2,247) – $613 = ~$1,600

$72,000 downpay

$100,000 SPIA

$100,000 SPIA

月$1,600 BUDGET 위해 $272,000 필요