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MATH226 Numerical Methods for Applied Mathematics

创建日期：2024-06-17 17:38:23

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标签： MATH226

Numerical Methods for Applied Mathematics

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MATH226 Numerical Methods for Applied Mathematics

1 Programming in Maple
Make sure Maple has been configured properly before trying any of the
examples below for yourself. Configuration instructions are in the module
handbook.
1.1 Getting started
• A Maple program is a sequence of statements, each of which tells Maple to
perform some action(s). For example typing
restart
and pressing return causes Maple to clear any currently stored data. It is a
good idea to issue a restart command immediately before starting a new
problem (including at the beginning of the Maple worksheet).
• Pressing shift+return moves the cursor down one line without executing.
This is useful when writing complicated statements.
• The result of a statement that ends with a colon is not displayed. Otherwise,
results will be displayed if they do not generate an excessive amount of
output (e.g. a huge matrix). This behaviour can be altered by changing the
printlevel and rtablesize parameters (see §1.12 and §1.9, respectively).
• The black square brackets to the left of the window indicate the scope of
execution groups. If two statements are inside the same execution group,
then they need to be separated by a colon or a semicolon.
• Maple is case sensitive; for example int and Int are different commands.
• Maple has a comprehensive online help system. To obtain information about
a command, enter a question mark, followed by the command, e.g. ?evalf.
It is usually easiest to scroll down to the examples at the foot of the help
page.
• Maple very rarely crashes, but this does sometimes happen. When you start
a new worksheet, save it immediately after putting restart on the first
line. By default, a saved worksheet will save again automatically every three
minutes. Do not turn off autosave.
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Numerical Methods for Applied Mathematics MATH226 2021–22
• If a calculation is taking too long, you can try to stop it by pressing the
interrupt button in the toolbar. This may take a few seconds to work, but
it may not work at all. This is one reason why you should never deactivate
autosave.
1.2 Exact vs approximate results
Unless told to do otherwise, Maple tries to produce exact results, which can be
problematic. Suppose we wish to determine whether the inequality∫ 1
0
x
1 + x5 dx > 0.4
holds. The statement
int( x / ( 1 + x^5 ) , x = 0..1 )
tells Maple to evaluate the integral, but the result is rather complicated, and doesn’t
really help. The evalf command tells Maple to calculate an approximate numerical
value, so the result of
evalf( Int( x / ( 1 + x^5 ) , x = 0..1 ) )
is much easier to interpret. Complicated exact results can cause Maple to slow
down and appear to freeze if they are reused in later calculations.
Use evalf to avoid complicated exact results.
Remark: evalf( int( ... ) ) with a lower case ‘i’ tells Maple to evaluate
the integral exactly (if it can) and then convert the result to a decimal. See
problem 2.1.
1.3 Comments
Maple ignores material from a # symbol until the end of the line. This is used to
insert explanatory notes for statements whose effect is not obvious.
# Create a 2 x 2 matrix
Matrix( 2 )
If a comment is inserted after a statement, then a colon or semicolon must be
included in between.
a := 1 ; # Would generate an error without the semicolon
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MATH226 2021–22 Numerical Methods for Applied Mathematics
1.4 Variables
Variables are used to store data. To assign a value to a variable, use the assignment
operator :=.
a := 27 :
b := 4 :
c := a + b :
c
31
Assignments are not equations. The expression on the right is computed, then
the result is stored in the variable on the left. This means the right-hand side can
reference the variable that is about to be assigned.
a := 27 :
a := a + 1 :
a
28
Maple allows sequences of assignments to be made in the same statement.
a , b := 15 , 21 :
a
15
b
21
This provides a very useful shortcut for swapping variable values.
a := 15 :
b := -6 :
a , b := b , a :
a
-6
b
15
Often, Maple does not need to distinguish between different types of variable such
as integer or complex, but there are situations where this is important.
a := 1 :
b := 1.5 :
type( a , integer )
true
3
Numerical Methods for Applied Mathematics MATH226 2021–22
type( b , integer )
false
Variables in Maple can possess more than one type.
type( a , integer )
true
type( a , numeric )
true
type( a , complex )
true
In this way, Maple differs from languages such as C and Fortran.
See sections 2.11 and 2.12 in Understanding Maple for more about variables.
1.5 Conditional statements
A conditional (if) statement causes Maple to test a condition, then carry out
different operations depending on whether the condition is true or false. The
simplest conditional statement instructs Maple to perform a set of actions only if a
condition is true.
if condition then
statement[s]
end if
Note that this is a single statement (which contains other statements). The line
breaks are produced by pressing shift+return, and the whole structure must be
executed together.
Any statement that evaluates to true or false can be used in an if statement.
cold_today := true :
if cold_today then
"Wear your hat!" :
end if
"Wear your hat!"
In cases where a conditional contains a sequence of statements (as opposed to a
single statement), these must be separated by colons or semicolons. Otherwise
Maple has no way to know where one statement ends and the next begins.
4
MATH226 2021–22 Numerical Methods for Applied Mathematics
a := 1 :
b := a :
increase_vars := true :
if increase_vars then
a := a + 1 : # <--- The colon on this line is crucial!
b := b + 2 :
end if
Boolean operators including and, not and or can be used to construct more
complex conditions.
if a > 0 and frac( a ) = 0 then
"a is a positive integer." :
end if
Here, the message is displayed if a is a positive integer. Note the frac command,
which returns the fractional part of a number (i.e. the digits after the decimal
point). Using else instructs Maple to take different actions (rather than none at
all) if the condition is false.
if condition then
statement[s]
else
alternative statement[s]
end if
Finally, we can check additional conditions if the first turns out to be false.
if condition then
statement[s] # Execute if condition is true
elif second condition then
alternative statement[s] # Execute if first condition is false,
# but second condition is true.
else
more alternative statement[s] # Execute if all above conditions
# are false.
end if
5
Numerical Methods for Applied Mathematics MATH226 2021–22
In the next example, the message "a is negative" will only be printed if both
conditions (a > 0 and a = 0 are false).
if a > 0 then
"a is positive" :
elif a = 0 then
"a is zero" :
else
"a is negative" :
end if
See section 7.1 in Understanding Maple for more about conditional state-
ments.
1.6 do loops
A do loop causes Maple to repeatedly execute the same statements. The next
example causes Maple to display the approximate value of pi five times.
from 1 to 5 do
evalf( Pi ) :
end do
3.141592654
3.141592654
3.141592654
3.141592654
3.141592654
Often we need to use the step number itself during the iteration. This is achieved
using a loop with an index variable.
for index from start to finish do
statement[s]
end do
Here, index , start, and finish are integers. Initially, index is set to equal start.
After each iteration, index is increased by 1. If the result exceeds finish, the loop
terminates. Otherwise another iteration is performed. If start exceeds finish, the
statements inside the do loop are not executed at all. We can compute 10! as
follows.
6
MATH226 2021–22 Numerical Methods for Applied Mathematics
p := 1 :
for j from 2 to 10 do
p := p * j : # Updates the value of p
end do :
p
3628800
As another example, we can compute the sum 1 + 12 +
1
3 + · · ·+ 1n as follows.
n := 100 : # (Or any other natural number)
s := 0 :
for j from 1 to n do
s := s + evalf( 1 / j ) :
end do :
s
5.187377520
Increments other than 1 are possible.
for index from start by increment to finish do
statement[s]
end do
If increment is negative, the loop will terminate when index reaches a value that is
smaller than finish, and will not execute at all if finish exceeds start.
for j from 5 by -1 to 1 do
j ;
end do
5
4
3
2
1
To increment a variable through noninteger values, calculate a step size before the
loop starts. In the next example, x varies from 0 to 3 in steps of size 3/7.
nsteps := 7 : # Number of steps
dx := evalf( 3 / nsteps ) : # Step size
7
Numerical Methods for Applied Mathematics MATH226 2021–22
for j from 0 to nsteps do
x := j * dx ;
end do
x := 0.
x := 0.4285714286
x := 0.8571428572
x := 1.285714286
x := 1.714285714
x := 2.142857143
x := 2.571428572
x := 3.000000000
Using a floating point (decimal) index can lead to a disastrous situation in which
one too few steps is performed because the last value for x exceeds the upper limit
due to rounding error.
dx := evalf( 3 / nsteps ) : # Step size
for x from 0 to 3 by dx do
x ;
end do
x := 0.
x := 0.4285714286
x := 0.8571428572
x := 1.285714286
x := 1.714285714
x := 2.142857144
x := 2.571428573
x
3.000000002
Never use a decimal as a do loop index.
Remark: In Maple, it is possible to use an exact fraction as a loop index (e.g.
try the above example without evalf). However, very few other programming
languages allow this, so we will always use integers.
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MATH226 2021–22 Numerical Methods for Applied Mathematics
1.7 Break statements
Sometimes it is not possible to predict how many iterations will be needed for a
particular task. For these cases we can use a do loop with no final value for the
index (or no index at all). In such cases the loop must be terminated by a break
statement.

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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 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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