Claims
- 1. A method for using a computer system to find the roots of a system of nonlinear equations within an interval vector X, wherein the system of non-linear equations is specified by a vector function, f, the method comprising:
receiving a representation of the interval vector X, wherein for each dimension, i, the representation of Xi includes a first floating-point number, ai, representing the left endpoint of Xi and a second floating-point number, bi, representing the right endpoint of Xi; performing an interval Newton step on X to produce a resulting interval vector, X′, wherein the point of expansion of the interval Newton step is a point, x, within the interval X, and wherein performing the interval Newton step involves evaluating f(x) to produce an interval result f1(x); evaluating a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within f1(x), J(x, X) is regular, wherein J(x, X) is the Jacobian of the function f evaluated with respect to x over the interval X, and X is contained within X′; and if the first termination condition is TRUE, terminating the interval Newton method and recording X′ as a final bound.
- 2. The method of claim 1, wherein if no termination condition is satisfied, the method further comprises returning to perform an interval Newton step on the interval X′.
- 3. The method of claim 1, further comprising:
evaluating a second termination condition; wherein the second termination condition is TRUE if a function of the width of the interval X′ is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval X′ is less than a pre-specified value, εF; and if the second termination condition is TRUE, terminating the interval Newton method and recording X′ as a final bound.
- 4. The method of claim 3, wherein the second termination condition is evaluated and the method possibly terminates before the first termination condition is evaluated.
- 5. The method of claim 3, wherein the second termination condition is evaluated only if J(x, X) is not proved to be regular.
- 6. The method of claim 5, further comprising determining if J(x, X) is regular by computing a pre-conditioned Jacobian, M(x, X)=BJ(x, X), wherein B is an approximate inverse of the center of J(x, X), and then attempting to solve M(x, X)(y−x)=r(x), where r(x)=−Bf(x).
- 7. The method of claim 1, further comprising determining whether J(x, X) is regular by attempting to solve the relation M(x, X)(y−x)=r(x) using Gaussian elimination.
- 8. The method of claim 2, wherein returning to perform an interval Newton step on the interval X′ can involve splitting the interval X′.
- 9. 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 find the roots of a system of nonlinear equations within an interval vector X, wherein the system of non-linear equations is specified by a vector function, f, the method comprising:
receiving a representation of the interval vector X, wherein for each dimension, i, the representation of X, includes a first floating-point number, ai, representing the left endpoint of Xi and a second floating-point number, bi, representing the right endpoint of Xi; x, within the interval X, and wherein performing the interval Newton step involves evaluating f(x) to produce an interval result f′(x); evaluating a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within f1(x), J(x, X) is regular, wherein J(x, X) is the Jacobian of the functions f evaluated with respect to x over the interval X, and X is contained within X′; and if the first termination condition is TRUE, terminating the interval Newton method and recording X′ as a final bound.
- 10. The computer-readable storage medium of claim 9, wherein if no termination condition is satisfied, the method further comprises returning to perform an interval Newton step on the interval X′.
- 11. The computer-readable storage medium of claim 9, wherein the method further comprises:
evaluating a second termination condition; wherein the second termination condition is TRUE if a function of the width of the interval X′ is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval X′ is less than a pre-specified value, εF; and if the second termination condition is TRUE, terminating the interval Newton method and recording X′ as a final bound.
- 12. The computer-readable storage medium of claim 11, wherein the second termination condition is evaluated and the method possibly terminates before the first termination condition is evaluated.
- 13. The computer-readable storage medium of claim 11, wherein the second termination condition is evaluated only if J(x, X) is not proved to be regular.
- 14. The computer-readable storage medium of claim 13, wherein the method further comprises determining if J(x, X) is regular by computing a pre-conditioned Jacobian, M(x, X)=BJ(x, X), wherein B is an approximate inverse of the center of J(x, X), and then attempting to solve M(x, X)(y−x)=r(x), where r(x)=−Bf(x).
- 15. The computer-readable storage medium of claim 9, wherein the method further comprises determining whether J(x, X) is regular by attempting to solve the relation M(x, X)(y−x)=r(x) using Gaussian elimination.
- 16. The computer-readable storage medium of claim 10, wherein returning to perform an interval Newton step on the interval X′ can involve splitting the interval X′.
- 17. An apparatus that finds the roots of a system of nonlinear equations within an interval vector X, wherein the system of non-linear equations is specified by a vector function, f, the apparatus comprising:
a receiving mechanism that is configured to receive a representation of the interval vector X, wherein for each dimension, i, the representation of Xi includes a first floating-point number, ai, representing the left endpoint of Xi and a second floating-point number, bi, representing the right endpoint of Xi; an interval Newton mechanism that is configured to perform an interval Newton step on X to produce a resulting interval vector, X′, wherein the point of expansion of the interval Newton step is a point, x, within the interval X, and wherein performing the interval Newton step involves evaluating f(x) to produce an interval result f1(x); a termination mechanism that is configured to evaluate a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within f1(x), J(x, X) is regular, wherein J(x, X) is the Jacobian of the function f evaluated with respect to x within the interval X, and X is contained within X′; and wherein if the first termination condition is TRUE, the termination mechanism is configured to terminate the interval Newton method and to record X′ as a final bound.
- 18. The apparatus of claim 17, wherein if no termination condition is satisfied, the apparatus is configured to return to perform an interval Newton step on the interval X′.
- 19. The apparatus of claim 17, wherein the termination mechanism is additionally configured to:
evaluate a second termination condition; wherein the second termination condition is TRUE if a function of the width of the interval X′ is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval X′ is less than a pre-specified value, εF; and wherein if the second termination condition is TRUE, the termination mechanism is configured to terminate the interval Newton method and to record X′ as a final bound.
- 20. The apparatus of claim 19, wherein the termination mechanism is configured to evaluate the second termination condition, and possibly to terminate, before evaluating the first termination condition.
- 21. The apparatus of claim 19, wherein the termination mechanism is configured to evaluate the second termination condition only if J(x, X) is not proved to be regular.
- 22. The apparatus of claim 21, the termination mechanism is configured to evaluate determine if J(x, X) is regular by computing a preconditioned Jacobian, M(x, X)=BJ(x, X), wherein B is an approximate inverse of the center of J(x, X), and then attempting to solve M(x, X)(y−x)=r(x), where r(x)=−Bf(x).
- 23. The apparatus of claim 17, the termination mechanism is configured to determine whether J(x, X) is regular by attempting to solve the relation M(x, X)(y−x)=r(x) using Gaussian elimination.
- 24. The apparatus of claim 18, wherein while returning to perform an interval Newton step on the interval X′, the apparatus is configured to split the interval X′.
RELATED APPLICATION
[0001] The subject matter of this application is related to the subject matter in a co-pending non-provisional application by the same inventors as the instant application and filed on the same day as the instant application entitled, “Termination Criteria for the One-Dimensional Interval Version of Newton's Method,” having Ser. No. 09/927,270, and filing date Aug. 9, 2001 (Attorney Docket No. SUN-P6282).