AME 150 L Program Control Subroutines and Functions

AME 150 L Program Control Subroutines and Functions 11 - 2/4/2000 AME 150 L

Program Control - Loops • Powerful numerical tool - SERIES – Polynomials or Power Series – Infinite Series of Generalized Functions 11 - 2/4/2000 AME 150 L

Arrays of Variables • The use of indexed or subscripted variables can be treated in one of two ways a) Create different variables c 0 C 0, c 1 C 1, … , c. N CN b) Use a formal array structure REAL, dimension(0: 12) : : C then c 0 C(0), c 1 C(1), … , c. N C(N) Size of an array is either a fixed number, a parameter, (or an argument or dynamic) 11 - 2/4/2000 AME 150 L

Declaring Arrays • Two equivalent methods a) REAL, dimension(-2: 5) : : a, b, c the 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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

Recursion (Examples) • Calculate xn • Calculate (-1)n x = {set value of x} x_to_n = 1 m 1 n = 1 … m 1 n = - m 1 n … x_to_n = x * x_to_n 1 st time x_to_n=x, 2 nd time x_to_n=x*x, every time statement is executed, m 1 n changes sign +1, -1, +1 etc. 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

Trivial example of a Loop • Sum of Integers, Program fragment INTEGER : : 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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

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 11 - 2/4/2000 AME 150 L

Special Power Series - x e • 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 11 - 2/4/2000 AME 150 L

The Factorial Function ! • Normal definition -- repeated product Program Fragment INTEGER : : I, n, nfact n= {get some value for n} nfact=1 !initialize factorial DO I=1, n nfact = nfact * I END DO 11 - 2/4/2000 AME 150 L

More Factorial Function ! • Recursive definition 1! = 1 n! = n*(n-1)! [=n*(n-1)*(n-2)*…*2*1] • Hence 2!=2*1=2; 3!=3*2=6; 4!=4*3!=4*6=24; 5!=5*4!=5*24=120 6!= 6*5!=6*120=720 7!=7*6!=7*720=5040 … factorial grows fast! 11 - 2/4/2000 AME 150 L

Factorial (Continued) • Since n!=n*(n-1)!, and 1!=1 1! = 1* 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, 11 - 2/4/2000 AME 150 L

The Gamma Function G 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] 11 - 2/4/2000 AME 150 L