Abstract: One embodiment of the present invention provides a system that solves an overdetermined system of interval linear equations. During operation, the system receives a representation of the overdetermined system of interval linear equations Ax=b, wherein A is a matrix with m rows corresponding to m equations, and n columns corresponding to n variables, and wherein x includes n variable components, b includes m scalar components, and m>n. Next, the system performs a Gaussian Elimination operation to transform Ax=b into the form [ T W ] ? x = [ u v ] , wherein T is a square upper triangular matrix of order n, u is a vector with n components, v is a vector with m?n components, and W is a matrix with m?n rows and n columns, wherein W is zero except in the last column, which is represented as a column vector z with m?n components. Next, the system performs an interval intersection operation based on the equations zixn=vi (i=1, . . . , m?n) and Tnnx=un to solve for xn.

Abstract: One embodiment of the present invention provides a system that facilitates performing exception-free arithmetic operations within a computer system. During execution of a computer program, the system receives an instruction to perform an arithmetic operation that involves manipulating floating-point values. If the arithmetic operation manipulates a floating-point value representing {+0}, the arithmetic operation is performed in a manner consistent with {+0} representing a set containing a single value “?0”, wherein “?0” is the limit of a sequence of values that approaches zero only from above. Similarly, if the arithmetic operation manipulates a floating-point value representing {?0}, the arithmetic operation is performed in a manner consistent with {?0} representing a set containing a single value “+0”, wherein “+0” is the limit of a sequence of values that approaches zero only from below.

Abstract: One embodiment of the present invention provides a system that computes interval parameter bounds from fallible measurements. During operation, the system receives a set of measurements z1, . . . , zn, wherein an observation model describes each zi as a function of a p-element vector parameter x=(x1, . . . , xp). Next, the system forms a system of nonlinear equations zi?h(x)=0 (i=1, . . . , n) based on the observation model. Finally, the system solves the system of nonlinear equations to determine interval parameter bounds on x.

Abstract: One embodiment of the present invention provides a system that solves a global optimization problem specified by a function ƒ and a set of equality constraints q1(x)=0 (i=1, . . . , r), wherein ƒ is a scalar function of a vector x=(x1, x2, x3, . . . xn). During operation, the system receives a representation of the function ƒ and the set of equality constraints and stores the representation in a memory within a computer system. Next, the system and performs an interval global optimization process to compute guaranteed bounds on a globally minimum value of the function ƒ(x) subject to the set of equality constraints. Performing this interval global optimization process involves, applying term consistency to the set of equality constraints over a subbox X, and excluding portions of the subbox X that violate the set of equality constraints.

Abstract: A system that solves a global inequality constrained optimization problem specified by a function ƒ and a set of inequality constraints pi(x)?0(i=1, . . . , m), wherein ƒ and pi are scalar functions of a vector x=(x1, x2, x3, . . . xn). The system performs an interval inequality constrained global optimization process to compute guaranteed bounds on a globally minimum value of the function ƒ(x) subject to the set of inequality constraints. The system applies term consistency and box consistency to a set of relations associated with the global inequality constrained optimization problem over a subbox X, and excludes any portion of the subbox X that violates the set of relations. The system also performs an interval Newton step on the subbox X to produce a resulting subbox Y. The system integrates the sub-parts of the process with branch tests designed to increase the overall speed of the process.

Abstract: One embodiment of the present invention provides a system that solves an unconstrained interval global optimization problem specified by a function ƒ, wherein ƒ is a scalar function of a vector x=(x1, x2, x3, . . . xn). The system operates by receiving a representation of the function ƒ, and then performing an interval global optimization process to compute guaranteed bounds on a globally minimum value ƒ* of the function ƒ(x) and the location or locations x* of the global minimum. While performing the interval global optimization process, the system deletes all of part of a subbox X for which ƒ(x)>ƒ_bar, wherein ƒ_bar is the least upper bound on ƒ* that has been so far found. This is called the “ƒ_bar test”. The system applies term consistency to the ƒ_bar test over the subbox X to increase that portion of the subbox X that can be proved to violate the ƒ_bar test.

Abstract: One embodiment of the present invention provides a system that solves a global optimization problem specified by a function ƒ and a set of inequality constraints pi(x)?0 (i=1, . . . , m), wherein ƒ and pi are scalar functions of a vector x=(x1, x2, x3, . . . xn). The system operates by receiving a representation of the function ƒ and the set of inequality constraints, and then storing the representation in a memory within the computer system. Next, the system performs an interval inequality constrained global optimization process to compute guaranteed bounds on the minimum value of the function ƒ(x) subject to the set of inequality constraints. While performing the interval global optimization process, the system applies term consistency at various places in the process over a subbox X, and excludes any portion of the subbox X that violates term consistency.

Abstract: One embodiment of the present invention provides a system that bounds the solution set of a system of nonlinear equations specified by the set of linear equations Ax=b, wherein A is an interval matrix and b is an interval vector. During operation, the system preconditions the set of linear equations Ax=b by multiplying through by a matrix B to produce a preconditioned set of linear equations M0x=r, wherein M0=BA and r=Bb. Next, the system widens the matrix M0 to produce a widened matrix, M, wherein the midpoints of the elements of M form the identity matrix. Finally, the system uses M and r to compute the hull h of the system Mx=r, which bounds the solution set of the system M0x=r.

Abstract: One embodiment of the present invention provides a system that solves a problem involving an interval parameter p through an interval solution process. During operation, the system receives a representation of the problem, wherein the problem includes a number of variables x1, x2, x3, . . . xn and at least one interval parameter p. The system stores the representation in a computer memory, and then performs the interval solution process on the problem. During this interval solution process, the system splits the problem into sub-problems by splitting the interval parameter p into subintervals, and creating separate sub-problems for each subinterval. The system then performs the interval solution process on the sub-problems. By splitting the interval parameter p, the system can achieve a tighter bound on the solution set of the problem. The decision to split on any parameter p is made in exactly the same way it would be made if p were a variable of the problem.

Abstract: One embodiment of the present invention provides a system that solves a global optimization problem specified by a function ƒ and a set of equality constraints qi(x)=0 (i=1, . . . , r), wherein ƒ is a scalar function of a vector x=(x1, x2, x3, . . . xn). During operation, the system receives a representation of the function ƒ and the set of equality constraints and stores the representation in a memory. Next, the system performs an interval equality constrained global optimization process to compute guaranteed bounds on a globally minimum value of the function ƒ(x) subject to the set of equality constraints. During this process, the system applies term consistency to a set of relations associated with the interval equality constrained global optimization problem over a subbox X, and excludes any portion of the subbox X that violates the set of relations. It also applies box consistency to the set of relations, and excludes any portion of the subbox X that violates the set of relations.

Abstract: One embodiment of the present invention provides a system that performs a procedure to solve a system of linear inequalities. During operation, the system receives a representation of the system of linear inequalities Ax?b, wherein Ax?b can be a linearized form of a system of nonlinear equations. Within this representation, A is an interval matrix with m rows corresponding to m inequalities, and with n columns corresponding to n variables, the vector x includes n variable components, and the vector b includes m scalar interval components. The system solves the system of linear inequalities Ax?b by performing a Gaussian elimination process using only positive multipliers so as not to change the sense of any inequality.

Abstract: One embodiment of the present invention provides a system that receives a representation of the function ƒ and stores the representation in a memory. Next, the system performs an interval global optimization process to compute guaranteed bounds on a globally minimum value of the function ƒ(x) over a subbox X. This interval global optimization process applies term consistency to a set of relations associated with the function ƒ over the subbox X, and excludes any portion of the subbox X that violates any member of the set of relations. It also applies box consistency to the set of relations associated with the function ƒ over the subbox X, and excludes any portion of the subbox X that violates the set of relations.

Abstract: One embodiment of the present invention provides a system for finding the roots of a system of nonlinear equations within an interval vector X=(X1, . . . , Xn), wherein the system of non-linear equations is specified by a vector function ƒ=(ƒ1, . . . , ƒn). The system operates by receiving a representation of the interval vector X (which is also called a box), wherein for each dimension, i, the representation of Xi includes a first floating-point number, ?i, representing the left endpoint of Xi, and a second floating-point number, bi, representing the right endpoint of Xi. Next, the system performs 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 ƒ(x) to produce an interval result ƒ1(x).

Abstract: One embodiment of the present invention provides a system for finding zeros of a function, ƒ, within an interval, X, using the interval version of Newton's method. The system operates by receiving a representation of the interval X. This representation including a first floating-point number, a, representing the left endpoint of X, and a second floating-point number, b, representing the right endpoint of X. Next, the system performs an interval Newton step on X, wherein the point of expansion is the midpoint, x, of the interval X. Note that performing the interval Newton step involves evaluating ƒ(x) to produce an interval result ƒI(x). If ƒI(x) contains zero, the system evaluates ƒ(a) to produce an interval result ƒI(a). It also evaluates ƒ(b) to produce an interval result ƒI(b).

Abstract: One embodiment of the present invention provides a computer-based system for solving 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 system operates by 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. The system stores the representation in a computer memory. Next, the system applies term consistency to the set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn,(x)=0, over X, and excludes portions of X that violate the set of nonlinear equations. The system also applies box consistency to the set of nonlinear equations over X, and excludes portions of X that violate the set of nonlinear equations.

Abstract: One embodiment of the present invention provides a system for finding the roots of a polynomial or a quadratic equation with interval coefficients. The system operates by receiving a representation of a polynomial equation, which can be a quadratic equation of the form F(x)=Ax2+Bx+C=0, wherein A=[AL, AU], B=[BL, BU] and C=[CL, CU] are interval coefficients. Next, the system computes intervals containing roots of the functions F1(x), F2(x), F3(x) and F4(x), wherein F1(x)=ALx2+BLx+CL, F2(x)=AUx2+BUx+CU, F3(x)=ALx2+BUx+CL and F4(x)=AUx2+BLx+CU. The system then places the computed intervals into a list, L, and orders the computed intervals in L by their left endpoints, so that for a each entry, Si=[SiL, SiU], SiL?Si+1,L. Next, the system establishes interval roots for F(x) from the interval entries in list L.

Abstract: A computer-based system for solving 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 system operates by 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)=0 in the system of equations f(x)=0, each individual component function ƒi(x) can be written in the form ƒi(x)=g(x?j)?h(x) or g(x?j)=h(x), where g can be analytically inverted so that an explicit expression for x?j can be obtained: x?j=g?1(h(x)). Next, the system substitutes the interval vector element Xj into the modified equation to produce the equation g(X?j)=h(X), and solves for X?j=g?1(h(X)).

Abstract: One embodiment of the present invention provides a system for solving a nonlinear equation through interval arithmetic. During operation, the system receives a representation of the nonlinear equation ƒ(x)=0, as well as a representation of an initial interval, X, wherein this representation of X includes a first floating-point number, XL, for the left endpoint of X, and a second floating-point number, XU, for the right endpoint of X. Next, the system symbolically manipulates the nonlinear equation ƒ(x)=0 to solve for a first term, g1(x), thereby producing a modified equation g1(x)=h1(x), wherein the first term g1(x) can be analytically inverted to produce an inverse function g1−1(x). The system then plugs the initial interval X into the modified equation to produce the equation g1(X′)=h1(X), and solves for X′=g1−1[h1(X)].

Abstract: One embodiment of the present invention provides a system that computes interval parameter bounds from fallible measurements. During operation, the system receives a set of measurements z1, . . . , zn, wherein an observation model describes each z1 as a function of a p-element vector parameter x=(xi, . . . , xp). Next, the system forms a system of nonlinear equations zi−h(x)=0 (i=1, . . . , n) based on the observation model. Finally, the system solves the system of nonlinear equations to determine interval parameter bounds on x.

Abstract: One embodiment of the present invention provides a system that performs a procedure to solve a system of linear inequalities. During operation, the system receives a representation of the system of linear inequalities Ax≦b, wherein Ax≦b can be a linearized form of a system of nonlinear equations. Within this representation, A is an interval matrix with m rows corresponding to m inequalities, and with n columns corresponding to n variables, the vector x includes n variable components, and the vector b includes m scalar interval components. The system solves the system of linear inequalities Ax≦b by performing a Gaussian elimination process using only positive multipliers so as not to change the sense of any inequality.