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AME 150 L

Program Control

Subroutines and Functions

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Program Control - Loops

• Powerful numerical tool - SERIES– Polynomials or Power Series

– Infinite Series of Generalized Functions

2 30 1 2 3

0

( ) ...

( )

NN

Ni

ii

f x c c x c x c x c x

f x c x

0

( ) ( )k kk

g x c x

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Arrays of Variables• The use of indexed or subscripted variables

can be treated in one of two waysa) Create different variables

c0 C0, c1 C1, … , cN CN

b) Use a formal array structureREAL, dimension(0:12) :: Cthen c0 C(0), c1 C(1), … , cN C(N)

Size of an array is either a fixed number, a parameter, (or an argument or dynamic)

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Declaring Arrays

• Two equivalent methodsa) REAL, dimension(-2:5) :: a,b,cthe three arrays a,b,s all have the same dimension (-2:5)

b) REAL :: a(-2:5), B(15), c(0:5)makes it possible to use one declaration for arrays of different sizes

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Recursion• re·cur·sion (r-kûrzhn)

n. Mathematics – An expression, such as a polynomial, each term of which is

determined by application of a formula to preceding terms.

– A formula that generates the successive terms of a recursion.

• [Late Latin recursi, recursin- a running back, from Latin recursus, past participle of recurrere, to run back; see recur.]

• other definition 1, definition 2

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Recursion (Examples)

• Calculate xn

x = {set value of x}

x_to_n = 1

…

x_to_n = x * x_to_n

1st time x_to_n=x,

2nd time x_to_n=x*x,

etc.

• Calculate (-1)n

m1n = 1

…

m1n = - m1n

every time statement is executed, m1n changes sign +1, -1, +1

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Loops• To execute the same set of instructions

repeatedly

• Most common structure DO … END DO– Loop can be counted (original DO loop)– Loop can be conditional (DO WHILE)– Loop can be Infinite (but broken by a condition

or a test)

• DO may have a label

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Counted DO Loop• Syntax:

[label:]DO ctr = init, fin [, incr]

…Fortran statements

END DO [label]

If label is used, it is followed by one colon :

ctr is a variable, and must be declared(It is preferred that ctr be an integer)

If incr is omitted, it is assumed to be +1

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Counted DO (continued)• The DO loop is executed exactly

largest of [(fin-init)/incr, 0*] times

• DO loops can count backwards, but increment must be negative

• DO loops may have ctr as a REAL variable, but practice is discouraged

• DO index (ctr) can be used as an index of an array, or in normal expressions

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Trivial example of a Loop

• Sum of Integers, Program fragmentINTEGER :: n, i, sum

n = {read in some value for n}

sum = 0 !Initialize variable

loop: DO I = 1, n

sum = sum + I

END DO loop

WRITE(*,*)n,sum

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Parts of a Loop

• Initialization – Both Loop Index and calculations

• Incrementing– At completion of loop, increment (or 1) is

added to Loop Index

• Testing– Whenever Loop Index <= Final value, repeat

loop with new value of Loop Index

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Parts of a Loop (continued)

• Body of Loop– The statements that are repeatedly executed for

different values of the Loop Index

• Special cases– CYCLE - go to increment & test immediately– EXIT - exit loop immediately

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Power Series

INTEGER :: i, N, sum=0

REAL :: x, c(0:20)…read in x, N, and all c's (NOTE N<=20)

DO i = 0 , N

sum = sum + c(i) * x**i

END DO

0

( )N

ii

i

f x c x

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Special Power Series - ex

• This series is a special case of the preceding example, where ci = 1/i!

• n! is notation for n factorial (or the factorial function)

• n! = 1*2*3*…*(n-1)*n

0 !

ix

i

xe

i

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The Factorial Function !

• Normal definition -- repeated productProgram FragmentINTEGER :: I, n, nfact

n= {get some value for n}

nfact=1 !initialize factorial

DO I=1,n

nfact = nfact * I

END DO

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More Factorial Function !• Recursive definition

1! = 1n! = n*(n-1)! [=n*(n-1)*(n-2)*…*2*1]

• Hence2!=2*1!=2*1=2; 3!=3*2!=3*2=6; 4!=4*3!=4*6=24; 5!=5*4!=5*24=1206!= 6*5!=6*120=720 7!=7*6!=7*720=5040 … factorial grows fast!

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Factorial (Continued)

• Since n!=n*(n-1)!, and 1!=11! = 1* 0! 0! = 1

and

0! = 0 (-1)! =1 (-1)! is infinite (1/0)

and for every negative integer

(-n)! = (-n)*(-n+1)! (-n)! =and believe it or not,

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The Gamma Function

Factorial is related to an integral called the Gamma Function

and hence, n! is defined for non-integer values of n [a real exclamation point]

0

! ( 1) n tn n t e dt