Cincinnati Observatory
d2P
THE EQU ATION dr~ = P·F (where F is a pre-determined function of r), arises in the computation of the wave equations for atoms in various stages of ioniza-tion. In that case it is necessary to replace F by (F
+
E),where
E
is such a constant as will cause P to vanish when r approaches infinity. In practice the correct value, of E must be determined by trials, and hence it is necessary to run through this kind of a solution many times.From the calculus of finite differences we have the fol-lowing relationships:
d2P
( 6.r)2 - . :1r~
=
(6.r)2 P (F+
E)=
. f6.iip = f
+
6.iifJ12 - 6.ivf1240+
The ultimate objective of our computations is to obtain a table of numerical values of P (r) which satisfy these two conditions and which may be illustrated by the following arrangement of the intermediate results:
r P f::.ip f::.iip f f::.if f::.iif f::. iiif
0.0 1.000 000 -0.000 012 0.000 000 -146
-0.072 712 +927 + 3
0.1 0.927 288 +0.000 915 0.000 927 -143
-0.071 797 +784 +4
0.2 0.855 491 +0.001 699 0.G01 711 -139
-0.070 098 +645 + 9
0.3 0.785 393 +0.002 345 0.002 356 -130
-0.067 753 +515 +8
0.4 0.717 640 +0.002 861 0.002 871 -122
-0.064 892 +393 +12
0.5 0.625 748 +0.003 255 0.003 255 -110
This illustration represents the solution of the simplified d2P
equation (1;:2
=
Pr, where P(O)=
1, and P (00)=
0; the problem is to find dP / dr at r=
0 such that the condition P ( 00)=
0 will be fulfilled.The only numbers which can be entered directly into the table are in the
f
column, when they are computed according to the first equation above. vVhen these are dif-39
ferenced, it becomes possible to compute
6.
iip and build up the6.
ip and P columns by addition. It is necessary to proceed step by step and by successive approximations.In the solution of the original, more general, equation it was possible to employ a Type 601 MUltiplying Punch equipped with sign control and a net balance summary counter. The board wiring may be illustrated schemati-cally:
Reading hrushes
Multiplier Multiplicand
LH Counter (5) = 0 Pickup
Sum. Counter
Punch
The first position in the punched field receives an X punch.
The group multiplier switch is wired oFF and ON at the same time. The OFF switch permits the multiplier to reset on every card. The column in which the X is punched is wired to read as if it were the group mUltiplier master card indication. This has the effect that when any card is punched and thei1 fed through the machine a second time it will be skipped out as a master card, without punching.
The field A is always cross footed into the LHC and it transfers to the SC with sign control. The only multiplier is (6.r) 2 F, which is prepunched into a set of salmon cards.
The multiplicand is wired reversed from the positions where the punched field is "reflected" through the center of the card. All cards have their index numbers pre-punched. The P's are manila cards, the
6.
ip'S are green cards, and the6.
iif 112's are blue cards. These must be obtained from a previous approximation and have the above mentioned X punch.40
2. The blue card is allowed to fall into the stacker.
3. The top card from the blue pile is picked up and
8. The second manila card is placed behind the other cards being held in one hand. other cards being held in one hand.
11. The last salmon card is placed (in the direct posi-tion) ahead of all the cards being held in one hand, and
12. This deck is now placed in the feed hopper to begin the next cycle.
The operator may be illiterate, so long as he is not color blind! The work proceeds at the rate of thirty seconds per step in the table, which is nearly the speed at which the cards can pass through the machine.
The SC does not reset except under control of the class selector. The selector transfer is obtained only from a
pre-SCI E N T I F I e COM PUT A T 1·0 N
punched Xon the salmon card when it is fed in the re-versed position. This automatically clears the counter at the end of each cycle of operations.
I f we undertake to apply these 'principles to the solution of the first order differential equation, (6r
)~p = f,
it Now,f
may consist of the algebraic sum of any number of cards, and if the higher order difference cards are already available from a previous approximation,it is only necessary to include one control card in each control group and to use two counters in order to build up the table of numerical values of P. All the cards representing quan-tities on the ith line are entered into both counters. The control change causes an intermediate (progressive) total This gives the value of the integral, P ( i), on the ith line.As the next control group starts through the machine, the first card is the control card, and this rolls the second that Dr. Herget has a big point in using the human element in his cycle. We can all make use of the tricks that Mr.
g(x) is a function of one variable only. The equation is linear and of the second order. Now a more general equa-tion of the second order in the normal forn) may be written
d
2
S = g(xJ S) S
+
p(x)dx2
This equation is non-linear. You might conceivably treat it in the same manner as the speaker has suggested, except that you would obviously run into the difficulty of .com-puting the quanties Yn. We might construct some sUltable program beforehand and use it to estimate the Yn. I wonder if either of you gentlemen have tried such a method. .
Dr. Thomas: This is exactly the thing that Hartree dId in his so-called "Self-consistent Field" computations.
There are two ways you can do it. vVith the notation:
where
He used V and any numerical approximation to t/ln to get V nJ then solved the differential equation to get t/lnJ these to get new V and VnJ and so on until you come out with what you put in. A somewhat different trick was one we tried a few years ago. Instead of assuming Vn we assumed V and put o/n to get Vn continuously as t/ln was being com-puted. I don't think you gain anything by that, except that every answer you get is a solution of the differential equation.
Dr. Caldwell: It might be possible to do that kind of thing with the 602 provided the functions were not too complicated.
Dr. Thomas: It was the double integral that I had in mind. You can do two of them simultaneously on the 405 as well as the constant. You go all through an integration to get preliminary values for t/ln. These must be normalized. These integrals must be obtained to get an
"energy" to put in on the right-hand side before repeating the integration.