Douglas Aircraft Company, Incorporated
MAN Y MET HOD S exist for solving simultaneous equations with punched card accounting machines. The one presented here takes advantage of the speed and flexibility of the 604 electronic calculator. A 10th order matrix can be inverted in one hour by use of this method, which com-pares with approximately eight hours through use of relay multipliers. Furthermore, the method is extremely simple.a The basic reduction cycle consists of: sort (650 cards per minute), reproduce (100 cards per minute), sort (650 cards per minute), and calculate (100 cards per minute).
This cycle must be repeated a number of times equal to the number of equations.
THEORY
Several variations of the basic elimination method can be used with the machine procedure outlined. The one de-scribed requires no back solution and is well suited to ma-chine methods. It is well known and will be described very briefly.
The equations may be expressed in matrix notation as AX
=
C. C and X may have, of course, any number of columns. If A-I is desired, C becomes 1 and X becomes A-I(see reference 1).
The object of the calculation is to operate on the matrices A and C, considered as equations, so as to reduce A to a unit matrix, thus reducing C to X.
Let M be the augmented matrix composed of A and C.
Choose any row, k, of M and form M' such that
, mkj
m k j = -mkk
, m k j . k
mij = mij - mik - , ~
=F •
mkk
The kth column of M' is zero, excepting the kth row which is unity. Therefore, no cards are made for the kth column of M'.
Form M" from M' using the above equations, but a dif-ferent row for k. If this process is repeated until each row has been used and all the columns of A eliminated, the columns of C will have been reduced to X.
aThe value of the determinant of the matrix of coefficients can be obtained as a by-product of the process. See reference 1.
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For best accuracy and to insure that all numbers stay within bounds, the e1emeI).ts of M should be close to unity, and, if possible, the principal diagonal elements of A should be larger than the other elements.
A column of check sums (negative sums of each row) appended to M provides an easy and complete check on the work. These check sums can be calculated by machine, but if they are manually calculated and written as a part of M they provide an excellent check on the key punching. Also, experience has shown that the agreement of the final check sums with X is an index to the accuracy of X.
MACHINE PROCEDURE
Layout
The following card fields are necessary:
A. Row (of M) B. Column (of M)
C. Order (initially n and reduced by one each cycle until it has become zero)
D. Common 12 punch E. Pivotal column 11 punch F. Pivotal row 12 punch G. Next pivotal row 12 punch H. Product or quotient 1. Cross-foot or dividend
J.
Multiplicand or divisor Procedure, Using Rows in Order1. Start with M punched in fields (A), (B), (C) and amounts in (H).
2. Sort to column. Emit 11 in (E) of column 1.
3. Place column 1 in front and sort to row. Emit 12 in (G) of row 1.
4. Reproduce cards. Emit 12 in (D) of all cards. Re-produce (A) to (A), (B) to (B), and (G) to (F).
Reproduce (H) to (1) except that (H) of the piv-otal column cards is gang punched in (J). Emit 11 in (E) of the first card of each gang punched group (column 2 in this case). It is advisable to pass blanks on the punch side for the 11 in ( E) masters. See note (3).
S E M I N A R
5. Sort the new cards to column [row 1 with 12 in (F) should automatically be the first card of each column] . 6. Calculate on the 604. On the 12 in (F) masters, tabulated controlling on row. All rows should sum to zero except the pivotal row which should sum to -1.
2. In (6), using a standard 40-program 604, the follow-ing limits on size of numbers seem to exist:
Address John Lowe, Douglas Aircraft Company, Inc., Engineer-ing Department, Santa Monica, California.
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The following procedure provides eight-decimal accuracy when x<1.
a. As terms are calculated on the 604, they are checked, and if > 1. , an 11 is punched (not shown on schedule be-comes zero, and eight-decimal accuracy is obtained at all times.
1. WILLIAM EDMUND MILNE, Numerical Calculus (Princeton Uni-versity Press, 1949), p. 26.
2. FRANK M. VERZUH, "The Solution of Simultaneous Linear Equations with the Aid of the 602 Calculating Punch," M athe-matical Tables and Other Aids to Computation. III, No. 27
(J uly 1949), pp. 453-462.
DISCUSSION
Mr. Turner: What do you do if B22 happens to be very small ?
Mr. Lowe: In the manner I described, you would actu-ally pick the starting rows and column in sequence-that is, the first column in the first row and the second column in the second row, and so forth. It isn't necessary to do that.
You can pick anyone you want. In picking, pick the one that would give you the most advantageous numbers. In particular, we usually try to pick the one that gives the smallest quotients in doing this division.
Mr. Wolanski: We have a method that is similar to this, but we always use the element that is greatest; we cannot say the first row or the first column. In the first column we use an element which is the largest; when we do eliminate and get B 2H and B 31 equals zero, we start in on our second
column, and we pick the element that is the largest.
Mr. Lowe: Our method for finding out if the numbers get too big is simply to punch out a few more numbers than we can use the next time and then sight-check the cards.
Mr. Bell: In our handling of this problem we try to re-move judgment from 'the operation which the operator performs. We don't want him to have to look at it and evaluate and decide which term to use. So, in handling matrices-usually, in groups-we simply start up from the main diagonal. Perhaps. ten per cent of the problems will go bad. We take that ten per cent and start down the main diagonal, and maybe ten per cent of those go bad. Well, then we have ljlOOth left over, of the total working vol-ume, and those we actually evaluate and select proper big terms in order to make it behave. But by doing that the mass of the work is h~ndled in a routine way.