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 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 an interval vector 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; for each nonlinear equation ƒi(x)=g(x′j)−h(x)=0 in the system of equations f(x)=0, symbolically manipulating ƒi(x)=0 within the computer system to solve for any 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 interval vector 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 interval vector X+; wherein the new interval vector X+ contains all solutions of the system of equations f(x)=0 within the interval vector X, and wherein the width of the new interval vector X+ is less than or equal to the width of the interval vector X.
- 2. The method of claim 1, further comprised of performing an interval Newton step on X to produce a resulting interval vector, Y, wherein the point of expansion of the interval Newton step is a point, x, within the interval vector X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 3. The method of claim 2, 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 interval vector X, and Y contained within X; and if the first termination condition is TRUE, terminating and recording X=X∩Y as a final bound.
- 4. The method of claim 3, 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 solving for the interval vector Y that contains the value of y that satisfies M(x,X)(y-x)=r(x), where r(x)=−Bf(x).
- 5. The method of claim 4, further comprising applying term consistency to Bf(x)=0.
- 6. 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 vector Y.
- 7. The method of claim 6, wherein returning to perform the interval Newton step on the interval vector Y can involve splitting the interval vector X=Y∩X.
- 8. The method of claim 2, further comprising:
evaluating a second termination condition; wherein the second termination condition is TRUE if a function of the width of the interval vector X is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval vector 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.
- 9. The method of claim 1, wherein for each term, g(xj), that can be analytically inverted within the equation ƒi(x)=0, the method further comprises:
setting Xj=Xj+ in X; and repeating the process of symbolically manipulating, substituting, solving and intersecting to produce the new interval vector Xj+.
- 10. The method of claim 1, wherein symbolically manipulating ƒi(x)=0 involves selecting the invertible term g(xj) as the dominating term of the function ƒi(x)=0 within the interval vector X.
- 11. 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 ƒi(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 an interval vector 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; for each nonlinear equation ƒi(x)=g(x′j)−h(x)=0 in the system of equations f(x)=0, symbolically manipulating ƒi(x)=0 within the computer system to solve for any 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 interval vector 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 interval vector X+; wherein the new interval vector X+ contains all solutions of the system of equations f(x)=0 within the interval vector X, and wherein the width of the new interval vector X+ is less than or equal to the width of the interval vector X.
- 12. The computer-readable storage medium of claim 11, wherein the method further comprises performing an interval Newton step on X to produce a resulting interval vector, Y, wherein the point of expansion of the interval Newton step is a point, x, within the interval vector X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 13. The computer-readable storage medium of claim 12, 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 interval vector X, and Y is contained within X; and +P2 if the first termination condition is TRUE, terminating and recording X=X∩Y as a final bound.
- 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 solving for the interval vector Y that contains the value of y that satisfies M(x,X)(y−x)=r(x), where r(x)=−Bf(x).
- 15. The computer-readable storage medium of claim 14, wherein the method further comprises applying term consistency to Bf(x)=0.
- 16. The computer-readable storage medium of claim 11, wherein if no termination condition is satisfied, the method further comprises returning to perform an interval Newton step on the interval vector Y.
- 17. The computer-readable storage medium of claim 16, wherein returning to perform the interval Newton step on the interval vector Y can involve splitting the interval vector X=Y∩X.
- 18. The computer-readable storage medium of claim 12, 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 vector X is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval vector 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.
- 19. The computer-readable storage medium of claim 11, wherein for each term, g(xj), that can be analytically inverted within the equation ƒi(x)=0, the method further comprises:
setting Xj=Xj+ in X; and repeating the process of symbolically manipulating, substituting, solving and intersecting to produce the new interval vector Xj+.
- 20. The computer-readable storage medium of claim 11, wherein symbolically manipulating ƒi(x)=0 involves selecting the invertible term g(xj) as the dominating term of the function ƒi(x)=0 within the interval vector X.
- 21. An apparatus that uses a computer system to solve a system of nonlinear equations specified by a vector function, f, wherein f(x)=0 represents ƒi(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 an interval vector 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 symbolic manipulation mechanism, wherein for each nonlinear equation ƒ(x)=g(x′j)−h(x)=0 in the system of equations f(x)=0, the symbolic manipulation mechanism is configured to manipulate ƒi(x)=0 to solve for any 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); a solving mechanism that is configured to,
substitute the interval vector X into the modified equation to produce the equation g(X′j)=h(X), and to solve for X′j=g−1(h(X)); and an intersecting mechanism that is configured to intersect X′j with the vector element Xj to produce a new interval vector X+, wherein the new interval vector X+ contains all solutions of the system of equations f(x)=0 within the interval vector X, and wherein the width of the new interval vector X+ is less than or equal to the width of the interval vector X.
- 22. The apparatus of claim 21, further comprising an interval Newton mechanism that is configured to perform an interval Newton step on X to produce a resulting interval vector, Y, wherein the point of expansion of the interval Newton step is a point, x, within the interval vector X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x).
- 23. The apparatus of claim 22, 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 interval vector X, and Y is contained within X; and to wherein if the first termination condition is TRUE, the termination mechanism is configured to terminate and recording X=X∩Y as a final bound.
- 24. The apparatus of claim 23, wherein the termination mechanism is configured to determine 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 to solve for the interval vector Y that contains the value of y that satisfies M(x,X)(y−x)=r(x), where r(x)=−Bf(x).
- 25. The apparatus of claim 24, wherein the symbolic manipulation mechanism is additionally configured to apply term consistency to Bf(x)=0.
- 26. The apparatus of claim 21, wherein if no termination condition is satisfied, the apparatus is configured to return to perform an interval Newton step on the interval vector Y.
- 27. The apparatus of claim 26, wherein returning to perform the interval Newton step on the interval vector Y can involve splitting the interval vector X=Y∩X.
- 28. The apparatus of claim 22, wherein the termination mechanism that is configured to:
evaluate a second termination condition; wherein the second termination condition is TRUE if a function of the width of the interval vector X is less than a pre-specified value, εX, and the absolute value of the function, f, over the interval vector 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 and record X as a final bound.
- 29. The apparatus of claim 21, wherein for each term, g(xj), that can be analytically inverted within the equation ƒi(x)=0, the apparatus is configured to:
set Xj=Xj+ in X; and to repeat the process of symbolically manipulating, substituting, solving and intersecting to produce the new interval vector Xj+.
- 30. The apparatus of claim 21, wherein symbolically manipulating ƒi(x)=0 involves selecting the invertible term g(xj) as the dominating term of the function ƒi(x)=0 within the interval vector 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 entitled, “Solving a Nonlinear Equation Through Interval Arithmetic and Term Consistency,” having serial number 09/952,759, and filing date of Sep. 12, 2001 (Attorney Docket No. SUN-P6284-SPL).