February 11, 2018 Sunday

Bedtime Story 

Note F of Ada Lovelace - 2 

Tonight we are continuing with the Note F of Lady Lovelace where she takes on the vexing issue of multiple punched cards required for mathematical operations.

“As a mere example of the degree to which the combined systems of cycle and backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes.

Suppose it be required to eliminate nine variables from 10 simple equations of the form –

                    ax₀ + bx₁ + cx₂ + dx₃… = p         (1)

                  a¹x₀ + b¹x₁ + c¹x₂ + d¹x₃… = p’      (2)

and so on               

 

We should explain, before proceeding, that it is not our object to consider this problem with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution.

The first would be the elimination of the first unknown quantity x₀ between the first two equations.

This would be obtained by the form-

(a¹a-aa¹)x₀ + (a¹b-ab¹)x₁ + (a¹c-ac¹)x₂ +
+ (a¹d-ad¹)x₃ + · · · · · · · · · · · · · · · · · · · · · · · · = a¹p-ap¹,   

For which the operations 10 (x, x, -) would be needed.

The second step would be the elimination of x₀ between the second and third equations, for which the operations would be precisely the same.  

We should then have had altogether the following operations:-

10 (x, x, -), 10 (x, x, -) = 20(x, x, -)

Continuing in the same manner, the total number of operations for the complete elimination of x₀ between all the successive pairs of equations would be -           

9 . 10 (x, x, -) = 90 (x, x, -)

We should then be left with nine simple equations of nine variables from which to eliminate the next variable x₁, for which the total of the processes would be –

8 . 9 (x, x, -) = 72 (x, x, -)

We should then be left with eight simple equations of eight variables from which to eliminate x₂, for which the processes would be -    

7 . 8 (x, x, -) = 56 (x, x, -)

And so on.

The total operations for the elimination of all the variables would thus be –

9.10 + 8.9 + 7.8 + 6.7 + 5.6 + 4.5 + 3.4 + 2.3 + 1.2 = 330

So that three Operation-cards would perform the office of 330 such cards.”

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Good night mon ami and my fellow cousin ape.