Machine Solutions
of Linear Differential Equations
Applications to a
Dynamic Economic Model

Kenneth E. Iverson

Preface
Synopsis
 
1.  A Dynamic Economic Model
Notes
 
2.  Problem Preparation for an Automatic Computer
A.  General Aspects of Problem Preparation
1.  Introduction
2.  Choice of Numerical Method
3.  High Accuracy Operation
4.  Simplication of Programming
B.  Mark IV Programming
Codes
Notes
 
3.  Linear Differential Equations
A.  Introduction
B.  The Complementary Function
C.  Reduction to Canonical Form
1.  Principal Vectors and Latent Roots
2.  The Adjoint Matrix
3.  Principal Vectors Derived from the Adjoint
4.  Principal Vectors of a General Matrix
D.  The Singular Case
E.  The Particular Integral
F.  The Economic Model
Notes
 
4.  Latent Roots and Principal Vectors
A.  Bounds on the Roots
B.  The Power Method
1.  Single Dominant Root
2.  Smaller Roots
3.  Multiple Roots
C.  Methods Employing the Characteristic Polynomial
1.  The Method of Bingham
2.  The Method of Frame
D.  Error Analysis
E.  Improving an Approximate Solution
F.  Comparison of Methods
Notes
 
5.  Zeros of a Polynomial
A.  Methods Depending on an Initial Approximation
1.  Synthetic Division
2.  Change of Origin
3.  The Birge-Vieta Process
4.  The Quadratic Factor Method
5.  Higher Order Processes
6.  Factors of Higher Degree
7.  Solution of the Quartic
8.  Solution of the Sextic
B.  Initial Approximations
1.  Method of Search
2.  Graeffe’s Method
C.  A General Method of Solution
Notes
 
6.  Machine Programs
A.  Summary of Calculations
B.  Matrix Subroutines
1.  Introduction
2.  Floating Vector Operation
3.  Matrix Programs
C.  Matrix Inversions
1.  A New Machine Method
2.  Inversion of a Complex Matrix
D.  The Method of Frame
E.  Solution of the Characteristic Equation
1.  The Graeffe Process
2.  The Quadratic Factor Method
F.  The Principal Vectors
Notes
 
7.  Results and Conclusions
A.  Tables of Results
B.  Conclusions
Notes
 
Appendices
I Tables of Flow Coefficients “A”
II Tables of Capital Coefficients “B”
III Tables of (I - S - μB)-1
IV Tables of (I - A)-1B
V Tables of Modal Matrix S and Latent Roots
VI Tables of S-1
VII  Tables of Industry Classification
 
List of Tables
0 List of Matrix Subroutines
1 Matrix Subroutines
2 Flow Coefficients “A” n = 6
3 Flow Coefficients “A” n = 11
4 Flow Coefficients “A” n = 21
5 Capital Coefficients “B” n = 6
6 Capital Coefficients “B” n = 11
7 Capital Coefficients “B” n = 21
8  (I - A - μB)-1 n = 5
9  (I - A - μB)-1 n = 6
10  (I - A - μB)-1 n = 10
11  (I - A - μB)-1 n = 11
12  (I - A - μB)-1 n = 20
13  (I - A - μB)-1 n = 21
14  (I - A)-1B n = 5
15  (I - A)-1B n = 6
16  (I - A)-1B n = 10
17  (I - A)-1B n = 11
18  (I - A)-1B n = 20
19  (I - A)-1B n = 21
20 Modal Matrix “S” and Latent Roots n = 5
21 Modal Matrix “S” and Latent Roots n = 6
22 Modal Matrix “S” and Latent Roots n = 10
23 Modal Matrix “S” and Latent Roots n = 11
24 Modal Matrix “S” and Latent Roots n = 20
25  S-1 n = 5
26  S-1 n = 6
27   Industry Classification
 
Bibiography