Claims
- 1. A method for using a computer system to solve a system of nonlinear equations specified by a vector function, f, wherein f(x)=0 represents a set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒ(x)=0, wherein x is a vector (x1, x2, x3, . . . xn), the method comprising:
receiving a representation of a subbox X=(X1, X2, . . . , Xn), 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; storing the representation in a computer memory; applying term consistency to the set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn (x)=0, over X, and excluding portions of X that violate any of these nonlinear equations; applying box consistency to the set of nonlinear equations over X, and excluding portions of X that violate any of the nonlinear equations; and performing an interval Newton step on X to produce a resulting subbox Y, wherein the point of expansion of the interval Newton step is a point x within X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 2. The method of claim 1, wherein performing the interval Newton step involves:
computing J(x,X), wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X; and determining if J(x,X) is regular as a by product of solving for the subbox Y that contains values of y that satisfy M(x,X)(y−x)=r(x), where M(x,X)=BJ(x,X), r(x)=−Bf(x), and B is an approximate inverse of the center of J(x,X).
- 3. The method of claim 2, further comprising:
applying term consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and excluding portions of X that violate the preconditioned set of nonlinear equations.
- 4. The method of claim 2, further comprising:
applying box consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and excluding portions of X that violate the preconditioned set of nonlinear equations.
- 5. The method of claim 1, wherein applying term consistency to the set of nonlinear equations involves:
for each nonlinear equation ƒi(x)=0 in the system of equations f(x)=0, symbolically manipulating ƒi(x)=0 to solve for an invertible term, g(x′j), thereby producing a modified equation g(x′j)=h(x), wherein g(x′j) can be analytically inverted to produce an inverse function g−1(y); substituting the subbox X into the modified equation to produce the equation g(X′j)=h(X); solving for X′j=g−1(h(X)); and intersecting X′j with the vector element Xj to produce a new subbox X+; wherein the new subbox X+ contains all solutions of the system of equations f(x)=0 within the subbox X, and wherein the width of the new subbox X+ is less than or equal to the width of the subbox X.
- 6. The method of claim 1, further comprising:
evaluating a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within fI(x), J(x,X) is regular, wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X, and the solution Y of M(x,X) (y−x)=r contains X; and if the first termination condition is TRUE, terminating and recording X as a final bound.
- 7. The method of claim 6, 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 subbox X is less than a pre-specified value, εX, and the width of the function f over the subbox X is less than a pre-specified value, εF; and if the second termination condition is TRUE, terminating and recording X as a final bound.
- 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 nonlinear equations specified by a vector function, f, wherein f(x)=0 represents a set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn(x)=0, wherein x is a vector (x1, x2, x3, . . . xn), the method comprising:
receiving a representation of a subbox X=(X1, X2, . . . , Xn), 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; storing the representation in a computer memory; applying term consistency to the set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn(x)=0, over X, and excluding portions of X that violate any of these nonlinear equations; applying box consistency to the set of nonlinear equations over X, and excluding portions of X that violate any of the nonlinear equations; and performing an interval Newton step on X to produce a resulting subbox Y, wherein the point of expansion of the interval Newton step is a point x within X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 9. The computer-readable storage medium of claim 8, wherein performing the interval Newton step involves:
computing J(x,X), wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X; and determining if J(x,X) is regular as a byproduct of solving for the subbox Y that contains values of y that satisfy M(x,X)(y−x)=r(x), where M(x,X)=BJ(x,X), r(x)=−Bf(x), and B is an approximate inverse of the center of J(x,X).
- 10. The computer-readable storage medium of claim 9, wherein the method further comprises:
applying term consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and excluding portions of X that violate the preconditioned set of nonlinear equations.
- 11. The computer-readable storage medium of claim 9, wherein the method further comprises:
applying box consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and excluding portions of X that violate the preconditioned set of nonlinear equations.
- 12. The computer-readable storage medium of claim 8, wherein applying term consistency to the set of nonlinear equations involves:
for each nonlinear equation ƒi(x)=0 in the system of equations f(x)=0, symbolically manipulating ƒi(x)=0 to solve for an invertible term, g(x′j), thereby producing a modified equation g(x′j)=h(x), wherein g(x′j) can be analytically inverted to produce an inverse function g−1(y); substituting the subbox X into the modified equation to produce the equation g(X′j)=h(X); solving for X′j=g−1(h(X)); and intersecting X′j with the vector element Xj to produce a new subbox X+; wherein the new subbox X+ contains all solutions of the system of equations f(x)=0 within the subbox X, and wherein the width of the new subbox X+ is less than or equal to the width of the subbox X.
- 13. The computer-readable storage medium of claim 8, wherein the method further comprises:
evaluating a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within fI(x), J(x,X) is regular, wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X, and the solution Y of M(x,X) (y−x)=r contains X; and if the first termination condition is TRUE, terminating and recording X as a final bound.
- 14. The computer-readable storage medium of claim 13, 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 subbox X is less than a pre-specified value, εX, and the width of the function f over the subbox X is less than a pre-specified value, εF; and if the second termination condition is TRUE, terminating and recording X as a final bound.
- 15. An apparatus that solves a system of nonlinear equations specified by a vector function, f, wherein f(x)=0 represents a set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn(x)=0, wherein x is a vector (x1, x2, x3, . . . xn), the apparatus comprising:
a receiving mechanism that is configured to receive a representation of a subbox X=(X1, X2, . . . , Xn), 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; a computer memory for storing the representation; a term consistency mechanism that is configured to apply term consistency to the set of nonlinear equations, ƒ1(x)=0, f2(x)=0, ƒ3(x)=0, . . . , ƒn(x)=0, over X, and to exclude portions of X that violate any of these nonlinear equations; a box consistency mechanism that is configured to apply box consistency to the set of nonlinear equations over X, and to exclude portions of X that violate any of the nonlinear equations; and an interval Newton mechanism that is configured to perform an interval Newton step on X to produce a resulting subbox Y, wherein the point of expansion of the interval Newton step is a point x within X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 16. The apparatus of claim 15, wherein the interval Newton mechanism is configured to:
compute J(x,X), wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X; and determine if J(x,X) is regular as a byproduct of solving for the subbox Y that contain the values of y that satisfy M(x,X)(y−x)=r(x), where M(x,X)=BJ(x,X), r(x)=−Bf(x), and B is an approximate inverse of the center of J(x,X).
- 17. The apparatus of claim 16, wherein the term consistency mechanism is configured to:
apply term consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and to exclude portions of X that violate the preconditioned set of nonlinear equations.
- 18. The apparatus of claim 16, wherein the box consistency mechanism is configured to:
apply box consistency to the preconditioned set of nonlinear equations Bf(x)=0 over the subbox X; and to exclude portions of X that violate the preconditioned set of nonlinear equations.
- 19. The apparatus of claim 15, wherein for each nonlinear equation ƒi,(x)=0 in the system of equations f(x)=0, the term consistency mechanism is configured to:
symbolically manipulate ƒi(x)=0 to solve for an invertible term, g(x′j), thereby producing a modified equation g(x′j)=h(x), wherein g(x′j) can be analytically inverted to produce an inverse function g−1(y); substitute the subbox X into the modified equation to produce the equation g(X′j)=h(X); solve for X′j=g−1(h(X)); and to intersect X′j with the vector element Xj to produce a new subbox X+; wherein the new subbox X+ contains all solutions of the system of equations f(x)=0 within the subbox X, and wherein the width of the new subbox X+ is less than or equal to the width of the subbox X.
- 20. The apparatus of claim 15, further comprising a termination mechanism that is configured to:
evaluate a first termination condition, wherein the first termination condition is TRUE if,
zero is contained within fI(x), J(x,X) is regular, wherein J(x,X) is the Jacobian of the function f evaluated as a function of x over the subbox X, and the solution Y of M(x,X) (y−x)=r contains X; and to terminate and record X as a final bound if the first termination condition is TRUE.
- 21. The apparatus of claim 20, 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 subbox X is less than a pre-specified value, εX, and the width of the function f over the subbox X is less than a pre-specified value, εF; and to terminate and record X as a final bound if the second termination condition is TRUE.
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 entitled, “Solving Systems Of Nonlinear Equations Using Interval Arithmetic And Term Consistency”, having Ser. No. 09/991,477, and a filing date of Nov. 16, 2001 (Attorney Docket No. SUN-P6429-SPL).