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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 INTERNAL REPORT NO. 63 
 
 A BRIEF DESCRIPTION OF THE ROUTINES IN THE ILLIAC 
 ACTIVE PROGRAM LIBRARY 
 
 Robert T. Gregory 
 March 16, 1955 
 
 PART I 
 
 This work has been supported in part by 
 the Office of Naval Research under 
 Contract NR Okk 001 
 
A BRIEF DESCRIPTION OF THE ROUTINES IN THE 
 ILLIAC ACTIVE PROGRAM LIBRARY 
 
 A 1 - 63 Floating Decimal Arithmetic Routine 
 
 ThiF is an interpretive routine which manipulates numbers 
 in the form A x 10 x ', where l/lO < |a| < 1 and -6k < p < 6k. It is 
 especially effective for problems with scaling difficulties. In effect, 
 it converts the Illiac into a medium-speed floating decimal computer. 
 
 A 3 - 12 5 Convert _a Number from Floating Decimal Representation to 
 Normal Machine Form 
 
 This routine converts numbers of the form A x lCr (see A l) 
 into standard Illiac form. It is used with the floating decimal auxiliaries 
 (see A 6) in general. 
 
 A k - 87 1.7 Precision Floating Binary Arithmetic with Floating Decimal 
 Conversion 
 
 This interpretive routine manipulates numbers in the form 
 B x 2 where -512 < q < 512 and |B| < 1. However numbers are input and 
 output in the form A x Kr where -1 < A < 1 and |p| : 153 • The modulus 
 of B has 68 binary digits instead of the usual 39 digits. A h is slower 
 than A 1. 
 
 A 5 - 138 Complex Number Arithmetic 
 
 This interpretive routine manipulates complex numbers in 
 floating decimal form, i.e., numbers in the form (A + iB) x 10 (see A 1 
 above ) . 
 
 A 6 - 15^ Floating Decimal Routine and Auxiliaries 
 
 This makes available on a single tape the following routines: 
 A 1, A 3, RA 1, SA 2, SA 3, TA 1, and TA 2. Thus one can find the square 
 root, exponential, logarithm, sine, and arctangent of an argument using 
 floating decimal. 
 
 -1- 
 
C 1 -48 Post Mortem Version of the Decimal Order Input 
 
 When code-checking a new routine one usually finds that 
 the routine "dies" due to some coding error. One can perform a post 
 mortem on such a routine by reading it into the memory again using C 1 
 as the input routine instead of the Decimal Order Input. Whenever a 
 discrepancy is found there is printed a line of information giving the 
 address of the discrepancy, word on tape, word in memory. It then 
 corrects the content of the memory to agree with the tape and continues 
 to read the problem into the memory so it can be started again. 
 
 C 3 - 9^ Post Mortem for Fractions and their Locations 
 
 This routine will go to ten specified consecutive memory 
 locations and print the content (except when zero) of each as a signed 
 12-place decimal fraction preceded by its decimal location in the memory. 
 As many groups of ten locations can be examined as desired. 
 
 C h - 97 Post Mortem for Integers and their Locations 
 
 Same as C 3 except that the content of each location is 
 printed as a signed 12-place decimal integer. 
 
 C 5 - 103 Post Mortem for Sets of Order Pairs and their Locations 
 Same as C 3 and C h except that the content of each 
 location is printed as a pair of orders (a zero left shift order is 
 not printed) . 
 
 C6 - lk-6 Address Search Routine 
 
 If a coding error is such that an incorrect address is written 
 in an order (e.g., a control transfer) the order containing that address can 
 be located with this routine. C 6 locates and prints all words which have a 
 given address (other than zero) in either address portion of the word. 
 
 -2- 
 
D 1 - 95 Check Point Routine 
 
 This routine is possibly the most useful aid in the library 
 for one interested in code-checking a new routine. It enables the pro- 
 grammer to specify certain check points in his routine at which the 
 ordinary operation of the routine is interrupted long enough to print 
 out information desired for checking on the proper operation of the 
 routine. There is a variety of types of information which can be called 
 for at each check point. 
 
 D k - 70 Control Transfe r Check Routine 
 
 This routine is another aid in finding errors in a program. 
 It tabulates the last p control transfers obeyed by the program. Thus 
 it can be used to trace the last actions of a program leading to a 
 hang-up on a zero left shift. 
 
 E 2 - I58 Integration by Simpson's Rule 
 
 Given "a set of n values of a function f , tabulated at equal 
 intervals," this routine computes the approximation 
 
 r 1 1 
 
 JL f(x)dx = -=i ( f_ + kf. + 2f + ...+ 2f _ + kf . + f ) 
 
 -A) 3n 1 2 n-2 n-1 n 
 
 where n is even. A simple transformation permits one to use a different 
 range of integration. 
 
 F 1 - 11^ Solution of a System of Ordinary Differential Equations 
 
 This subroutine will integrate a set of n simultaneous first 
 order ordinary differential equations in which each derivative is expressed 
 explicitly in terms of the dependent variables. An n order ordinary 
 differential equation can be solved by expressing it as n first order 
 equations. The Runge-Kutta method, as modified by Gill, is used. 
 
 F 2 - 115 Solution of a System of Differential Equations by Milne's 
 Iterative Method 
 Same as F 1 except Milne's method is used. 
 
 -3- 
 
F 3 - 129 Integration of n Simultaneous Second Order Differential Equations 
 with Initial Conditions Specified 
 This routine will integrate n second order differential equations 
 of the type : 
 
 y± = f i^0' ' • • y n-l> y 6> • • ' ' y n-l } I - 0, . . . , n - 1, 
 
 where initial values of y. are specified. 
 
 FA 1 - 122 Second Order Linear Differential Equation with Two Point Boundary 
 Conditions 
 This routine gives the solution to the differential equation 
 
 A(x)y" + B(x)y' + C(x)y = D(x), 
 subject to the boundary conditions 
 
 E(x)y* + F(x)y = G(x Q ) 
 
 H(x)y' + K(x)y = M(x n ) , 
 
 where x = x_ + nh and h = x. , -x. i = 0, . . » n - 1 = The calculations 
 
 n l+l i ' ' 
 
 are performed in floating point arithmetic. 
 
 G 1 - 78 Laplace ' s Equation - Liebmann Method 
 
 This routine gives the solution V(x,y), corresponding to a 
 given set of boundary values, to the difference equation V_ + V = 
 which approximates Laplace's equation 
 
 *J + *i = 0. 
 
 > 2 -> 2 
 
 ox dj 
 
 The solution is found for all interior grid points of an arbitrary closed 
 
 region with a square mesh. 
 
 G 2 - 98 Poisson's Equation - Liebmann-Frankel Method 
 
 This routine gives the solution V(x,y), corresponding to given 
 
 boundary values of V and given interior values of p, of the difference 
 
 2 
 
 equation V + V_ = oh which approximates Poisson's equation 
 
 b*+i? = e (x ' y) - 
 
 The solution is found for all interior grid points of an arbitrary closed 
 region with a square mesh of side h. 
 
 -k- 
 
H 1 - 71 Inverse Interpolation . A Real Root of f(x) = 
 
 Given two values of the argument, x and x p , this routine 
 will determine whether there is an odd number or an even number of real 
 roots of f(x) = between x and x . If the number is even (e.g. zero) 
 it places -1 in the accumulator. If the number is odd (e.g. one) it 
 places an approximation to one of the roots in the accumulator. Multiple 
 roots are treated as single roots. 
 
 H 2 - 72 A Search for the Real Roots of f(x) = 
 
 This routine examines the interval x n < x < x„ + nh, in n 
 steps of length h, for real roots of f(x) = using subroutine H 1. 
 The programmer must have sufficient information about f(x) to choose 
 x n , n, and h compatible with both economy of time and the finding of 
 all the roots . 
 
 H 3 - 80 Minimization of a Function of Two Variables 
 
 This routine will find an approximation to the values of 
 x, and x_ for which f (x. ,x p ) has a relative minimum. 
 
 H h - 84 Minimization of a Function of Four Variables 
 Same as H 3 except f = f(x , x , x , x. ) 
 
 H 5 - 85 Minimization of a Function of n Variables 
 
 Same as H 3 except f = f(x , x , . . . , x ) 
 
 H 6 - 86 Minimization of a Function of n Variables Treating One Variable 
 at a Time 
 A function poorly conditioned in having a very small gradient 
 with respect to some argument can deceive routines H 3, H ^, and H 5 in 
 that this coordinate of the minimum will be very poorly found. H 6 is a 
 "brute force' 1 approach to the minimum by varying one argument at a time. 
 This routine essentially is designed for finishing the job begun by one 
 of the routines mentioned above. 
 
 -5- 
 
HF 1 - 111 Zero of a Solution to a Differential Equation 
 
 This routine is used with F 1 to find a value of x for which 
 y. (x) = where y! = f.(y , y , . . . , y ) is the system of differential 
 equations under consideration. 
 
 I 1 - 67 Interpolation 
 
 Using Neville's method of successive linear interpolations 
 this routine interpolates in one or more tables stored in the machine. 
 
 It is assumed that functional values f(x.) are stored for the constant 
 
 1' 
 
 interval x. , - x. = h where i = 0, 1, . . . , n-1. 
 i+1 1 > f j 
 
 J 1 - 159 Roots of a Polynomial 
 
 This complete program calculates the roots of a polynomial 
 whose coefficients have been punched on tape in floating decimal form. 
 An n degree polynomial with complex coefficients may be punched and n 
 complex roots obtained. Real numbers are treated as complex numbers 
 whose imaginary parts are zero. 
 
 K 2 - 135 Product Moment Correlations , Means , Standard Deviations , 
 
 Variances and Covariances 
 
 The product moment correlation coefficient is a measure 
 of the degree of relation of two variables . It may be shown to range 
 between +1 and -1. This program computes the matrix of product moment 
 correlations between each pair of a set of variables. 
 
 K 3 - 137 Least Squares Polynomial 
 
 This routine finds the best polynomial of the form 
 
 n-1 
 
 F(x) = is§o a s xS 
 
 to fit a set of N weighted experimental points f (x. ), i = 0, . . . , N-1. 
 
 The criterion of excellence for the polynomial is that the sum of squares 
 
 of the deviations 
 
 N-1 p 
 
 M =.Z [ F(x.) - f(x.)r w(x.) 
 
 shall be a minimum with respect to arbitrary variations of the coefficients 
 a . The resulting polynomial is called the "least squares polynomial." 
 
 -6- 
 
LI- 112 Solution of a Set of Simultaneous Linear Algebraic Equations 
 This routine solves a set of n simultaneous linear algebraic 
 equations when provided with the augmented matrix of coefficients and 
 constants associated with the system. The method used is to reduce the 
 augmented matrix to triangular form and then use backward substitution 
 to obtain the solution. The number of equations n should not exceed 39 • 
 
 L 2 - 113 Automatic Linear Equation Solver 
 
 This routine is a complete program which combines L 1 with 
 the subroutines it needs to solve a system of linear equations and punch 
 out the solution. 
 
 L 3 - 100 Complete Linear Equation Solver 
 
 This routine solves the system described in L 1 by an iterative 
 process rather than by a direct process. The number of equations n should 
 not exceed 37* 
 
 RTGregory/hc 
 
 -7-