# Essays/MSLDE

Quotations from
Iverson, K.E., *Machine Solutions of Linear Differential Equations: Applications to a Dynamic Economic Model*,
Doctoral Thesis, Harvard University, 1954-01.

• The writer wishes to express his appreciation to Professor Howard H. Aiken
for his helpful guidance and encouragement,
and to Professor Wassily W. Leontief who proposed the problem and
offered many valuable suggests for its solution.

` ` ` ` — Preface

[Howard Aiken (1900-1973) was a major figure of the early digital era. He is best known for his first machine, the IBM Automatic Sequence Controlled Calculator or Harvard Mark I, conceived in 1937 and put into operation in 1944. But he also made significant contributions to the development of applications for the new machines and to the creation of a university curriculum for computer science. (from I. Bernard Cohen, *Howard Aiken: Portrait of a Computer Pioneer*, 1999).]

[Wassily Leontief won the Nobel Prize in Economics in 1973 "for the development of the input-output method and for its application to important economic problems". Three of his students also won the prize.]

• The writer also wishes to express his thanks to Miss Jacquelin Sanborn
who typed the plates for printing ... and to ... Mr. Robert Burns ...
all of whom assisted in the preparation of the manuscript.

` ` ` ` — Preface

[Jacquelin Sanborn was also acknowledged in *A Programming Language* for providing "much early and continuing guidance in matters of style, format, and typography". Robert Burns was acknowledged in same as among "unusually competent typists and draughtsmen" who provided assistance.]

• The speed of modern computers has reached a point where, for many applications,
the time required to program a problem may be a more important factor than the actual computing time.
Attempts are made, therefore, to simplify the work of programming in various ways.

` ` ` ` — Section 2A4, Simplification of Programming

• The calculator has a storage capacity of 10,000 orders,
each of which is identified by a “line-number” in the range 0000-9999. ...

The two hundred fast storage registers are referred to
by the numbers of the β-codes controlling them and are numbered from 0000 to 0199. ...

The function registers are ten in number (β0220-0229) and are designated as f0-9. ...

Provision is made for the storage of four thousand numbers in “slow storage” positions numbered 0000-3999. ...

The average time required for certain operations is given below in terms of machine cycles.
A machine cycle is 1.2 milliseconds, the time required for a single order.

` ` ` ` — Section 2B, Mark IV Programming

• All equations are to be understood as matrix or vector relations
unless otherwise specified.

` ` ` ` — Section 3, Linear Differential Equations

• Since the matrix is not symmetric, complex roots may be expected
to occur and indeed these complex roots are of particular interest
in the theory of the business cycle.
(The term of the general solution corresponding to a pair of complex roots is periodic.)

` ` ` ` — Section 4, Latent Roots and Principal Vectors

• Uniformity is desirable in automatic computation since exceptions
may require as much programming as the main process.

` ` ` ` — Section 5A4, The Quadratic Factor Method

• The Frame method [for generating the characteristic equation]
was first run using floating vector operation.
This proved satisfactory for the matrices of order ten and less
but the round-off accumulation in the higher order systems
was too severe.
The calculation was repeated, therefore,
using double-accuracy operation.
However, the approximate value of the solution obtained
by the floating vector method was useful in choosing
the decimal point in the double-accuracy calculation
so as to minimize the round-off error.
For a matrix of order twenty the floating vector program
takes two hours and the double-accuracy program six hours.

` ` ` ` — Section 6D, The Method of Frame

• The machine computations were begun during the final testing of Mark IV
when the machine was first coming into operation.
Hence conditions were far from ideal and the lack of experience
in programming and operation combined with machine failures to cause a considerable loss of time.
However it is under just such adverse conditions
that the relative ease of programming reruns on Mark IV proves especially valuable.
The reliability of the computer has now been greatly improved and at present writing
it has been operating with more than 85% good running time during the month of November 1953.

` ` ` ` — Section 7B, Conclusions