In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.
Background
editWhen successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent.
Definition
editWe call an n × n matrix T a convergent matrix if
(1) |
Example
editLet
Computing successive powers of T, we obtain
and, in general,
Since
and
T is a convergent matrix. Note that ρ(T) = 1/4, where ρ(T) represents the spectral radius of T, since 1/4 is the only eigenvalue of T.
Characterizations
editLet T be an n × n matrix. The following properties are equivalent to T being a convergent matrix:
Iterative methods
editA general iterative method involves a process that converts the system of linear equations
(2) |
into an equivalent system of the form
(3) |
for some matrix T and vector c. After the initial vector x(0) is selected, the sequence of approximate solution vectors is generated by computing
(4) |
for each k ≥ 0.[8][9] For any initial vector x(0) ∈ , the sequence defined by (4), for each k ≥ 0 and c ≠ 0, converges to the unique solution of (3) if and only if ρ(T) < 1, that is, T is a convergent matrix.[10][11]
Regular splitting
editA matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations (2) above, with A non-singular, the matrix A can be split, that is, written as a difference
(5) |
so that (2) can be re-written as (4) above. The expression (5) is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0, that is, B−1 and C have only nonnegative entries. If the splitting (5) is a regular splitting of the matrix A and A−1 ≥ 0, then ρ(T) < 1 and T is a convergent matrix. Hence the method (4) converges.[12][13]
Semi-convergent matrix
editWe call an n × n matrix T a semi-convergent matrix if the limit
(6) |
exists.[14] If A is possibly singular but (2) is consistent, that is, b is in the range of A, then the sequence defined by (4) converges to a solution to (2) for every x(0) ∈ if and only if T is semi-convergent. In this case, the splitting (5) is called a semi-convergent splitting of A.[15]
See also
editNotes
edit- ^ Burden & Faires (1993, p. 404)
- ^ Isaacson & Keller (1994, p. 14)
- ^ Varga (1962, p. 13)
- ^ Burden & Faires (1993, p. 404)
- ^ Isaacson & Keller (1994, pp. 14, 63)
- ^ Varga (1960, p. 122)
- ^ Varga (1962, p. 13)
- ^ Burden & Faires (1993, p. 406)
- ^ Varga (1962, p. 61)
- ^ Burden & Faires (1993, p. 412)
- ^ Isaacson & Keller (1994, pp. 62–63)
- ^ Varga (1960, pp. 122–123)
- ^ Varga (1962, p. 89)
- ^ Meyer & Plemmons (1977, p. 699)
- ^ Meyer & Plemmons (1977, p. 700)
References
edit- Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3.
- Isaacson, Eugene; Keller, Herbert Bishop (1994), Analysis of Numerical Methods, New York: Dover, ISBN 0-486-68029-0.
- Carl D. Meyer, Jr.; R. J. Plemmons (Sep 1977). "Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems". SIAM Journal on Numerical Analysis. 14 (4): 699–705. doi:10.1137/0714047.
- Varga, Richard S. (1960). "Factorization and Normalized Iterative Methods". In Langer, Rudolph E. (ed.). Boundary Problems in Differential Equations. Madison: University of Wisconsin Press. pp. 121–142. LCCN 60-60003.
- Varga, Richard S. (1962), Matrix Iterative Analysis, New Jersey: Prentice–Hall, LCCN 62-21277.