Recursions and Their Stability

  • W. Richard Kolk
  • Robert A. Lerman


Recursion formulae are algorithms in which an initial value is used to produce a new value, which in turn is inserted into the algorithm to produce again a new value, the process being repeated as many times as is desired. Jacobi’s iteration is just such a recursion, whose algorithm is the basis for solving a set of linear algebraic equations. We consider the simplest case of two such equations in two unknowns,
$$\begin{array}{*{20}{c}} {{{a}_{{11}}} \cdot {{x}_{1}} + {{a}_{{12}}} + {{x}_{2}} = {{b}_{1}}} \\ {{{a}_{{21}}} \cdot {{x}_{1}} + {{a}_{{22}}} \cdot {{x}_{2}} = {{b}_{2}}} \\ \end{array}$$


Nonlinear System Dynamics Recursion Formula Cube Root Stable Node Stable Focus 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shields, Paul C, Elementary Linear Algebra, Worth, New York (1980).Google Scholar
  2. 2.
    Gelfand, S. I., Gerver, M. L., Kirillov, A. A., Konstantinov, N. N., and Kushnirenko, A. G., Learn Limits Through Problems, Gordon & Breach, New York (1969).MATHGoogle Scholar
  3. 3.
    Smith, W. Allen, Elementary Numerical Analysis, Reston Englewood Cliffs, New Jersey (1986).MATHGoogle Scholar
  4. 4.
    Gleick, James, Chaos, Making a New Science, Viking, New York (1987).MATHGoogle Scholar
  5. 5.
    Campbell, David K., Nonlinear science, from paradigms to practicalities, Los Alamos Science 15 (special issue) pp. 218–262 (1987).Google Scholar
  6. 6.
    Spiegel, Murray, Applied Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1981).MATHGoogle Scholar
  7. 7.
    Wylie, C. Ray, Differential Equations, McGraw-Hill, New York (1979).MATHGoogle Scholar
  8. 8.
    Gille, J.-C, Pelegrin, M. J., and DeCaulne, P., Feedback Control Systems, McGraw-Hill, New York (1959).Google Scholar

Copyright information

© Van Nostrand Reinhold 1992

Authors and Affiliations

  • W. Richard Kolk
    • 1
  • Robert A. Lerman
    • 2
  1. 1.PortlandUSA
  2. 2.West HartfordUSA

Personalised recommendations