Claims
- 1. A method for using a computer system to solve a system of equations in fixed-point form, comprising:
receiving the system of equations in fixed-point form; storing the system of equations in a computer memory; reducing the dimension of the system of equations, when possible, by eliminating variables in the system of equations to produce a reduced system of equations; and performing interval intersections based on the Fixed Point theorem to reduce the size of a box containing solutions to the reduced system of equations.
- 2. The method of claim 1, further comprising applying interval techniques to find solutions to the system of equations, when such solutions exist.
- 3. The method of claim 2, wherein applying interval techniques involves:
applying term consistency over a subbox X; and excluding any portion of the subbox X that violates term consistency.
- 4. The method of claim 1, wherein reducing the dimension of the system of equations involves:
identifying cycles in the system of equations; identifying a generating set of variables that covers all cycles in the system of equations; and forming the reduced system of equations by expressing the system of equations in terms of the generating set.
- 5. The method of claim 4, wherein reducing the dimension of the system of equations also involves reordering the system of equations so that equations that are part of the same irreducible block are contiguous.
- 6. The method of claim 1, wherein performing the interval intersections involves performing component-wise interval intersections for equations in the reduced system of equations.
- 7. The method of claim 6, wherein performing the interval intersections involves updating interval arguments as soon as updated values for the interval arguments are available.
- 8. A computer-readable storage medium storing instructions that when executed by a computer cause the computer to perform a method for using a computer system to solve a system of equations in fixed-point form, the method comprising:
receiving the system of equations in fixed-point form; storing the system of equations in a computer memory; reducing the dimension of the system of equations, when possible, by eliminating variables in the system of equations to produce a reduced system of equations; and performing interval intersections based on the Fixed Point theorem to reduce the size of a box containing solutions to the reduced system of equations.
- 9. The computer-readable storage medium of claim 8, wherein the method further comprises applying interval techniques to find solutions to the system of equations, when such solutions exist.
- 10. The computer-readable storage medium of claim 9, wherein applying interval techniques involves:
applying term consistency over a subbox X; and excluding any portion of the subbox X that violates term consistency.
- 11. The computer-readable storage medium of claim 8, wherein reducing the dimension of the system of equations involves:
identifying cycles in the system of equations; identifying a generating set of variables that covers all cycles in the system of equations; and forming the reduced system of equations by expressing the system of equations in terms of the generating set.
- 12. The computer-readable storage medium of claim 11, wherein reducing the dimension of the system of equations also involves reordering the system of equations so that equations that are part of the same irreducible block are contiguous.
- 13. The computer-readable storage medium of claim 8, wherein performing the interval intersections involves performing component-wise interval intersections for equations in the reduced system of equations.
- 14. The computer-readable storage medium of claim 13, wherein performing the interval intersections involves updating interval arguments as soon as updated values for the interval arguments are available.
- 15. An apparatus that uses a computer system to solve a system of equations in fixed-point form, comprising:
a memory for storing the system of equations in fixed-point form; a reduction mechanism that is configured to reduce the dimension of the system of equations, when possible, by eliminating variables in the system of equations to produce a reduced system of equations; and an intersection mechanism that is configured to perform interval intersections based on the Fixed Point theorem to reduce the size of a box containing solutions to the reduced system of equations.
- 16. The apparatus of claim 15, further comprising an interval mechanism that is configured to apply interval techniques to find solutions to the system of equations, when such solutions exist.
- 17. The apparatus of claim 16, wherein the interval mechanism is configured to:
apply term consistency over a subbox X; and to exclude any portion of the subbox X that violates term consistency.
- 18. The apparatus of claim 15, wherein the reduction mechanism is configured to:
identify cycles in the system of equations; identify a generating set of variables that covers all cycles in the system of equations; and to form the reduced system of equations by expressing the system of equations in terms of the generating set.
- 19. The apparatus of claim 18, wherein the reduction mechanism is additionally configured to reorder the system of equations so that equations that are part of the same irreducible block are contiguous.
- 20. The apparatus of claim 15, wherein the intersection mechanism is configured to perform component-wise interval intersections for equations in the reduced system of equations.
- 21. The apparatus of claim 20, wherein the intersection mechanism is configured to update interval arguments as soon as updated values for the interval arguments are available.
RELATED APPLICATION
[0001] This application hereby claims priority under 35 U.S.C. §119 to U.S. Provisional Patent Application No. 60/338,884, filed on Dec. 7, 2001, entitled “Sparse Systems in Fixed Point Form,” by inventors G. William Walster and Ramon E. Moore.
Provisional Applications (1)
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Number |
Date |
Country |
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60338884 |
Dec 2001 |
US |