**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)^{-1}B | ||||||

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)^{-1}B n = 5 | ||||||

15^{ } | (I - A)^{-1}B n = 6 | ||||||

16^{ } | (I - A)^{-1}B n = 10 | ||||||

17^{ } | (I - A)^{-1}B n = 11 | ||||||

18^{ } | (I - A)^{-1}B n = 20 | ||||||

19^{ } | (I - A)^{-1}B 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 |