Abstract: One embodiment of the present invention provides a system for performing a minimum/maximum computation for an interval operation. The system operates by receiving at least four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number. Next, the system computes a minimum/maximum of the at least four floating-point numbers, wherein if the at least four floating-point numbers include one or two default NaN (not-a-number) values and the remaining values are not default NaN values, the default NaN values are ignored in computing the minimum/maximum.

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 for performing a minimum computation for an interval multiplication operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number. The system then computes a minimum of the four floating-point numbers, wherein if the four floating-point numbers include one or more default NaN (not-a-number) values, the system sets the minimum to negative infinity. One embodiment of the present invention provides a system for performing a minimum computation for an interval division operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number.

Abstract: A system for performing dependent interval subtraction, wherein a first interval is subtracted from a third interval to produce a resulting interval, given knowledge that the third interval is the sum of the first interval and a second interval. If the left endpoint of the third interval is negative infinity, the left endpoint of the resulting interval becomes negative infinity. Otherwise, the left endpoint of the resulting interval is computed by subtracting a left endpoint of the first interval from a left endpoint of the third interval, rounded down to a nearest floating-point number. Similarly, if the right endpoint of the third interval is positive infinity, the right endpoint of the resulting interval becomes positive infinity. Otherwise, the right endpoint of the resulting interval is computed by subtracting a right endpoint of the first interval from a right endpoint of the third interval, rounded up to a nearest floating-point number.

Abstract: One embodiment of the present invention provides a system for performing a minimum computation for an interval multiplication operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number. The system then computes a minimum of the four floating-point numbers, wherein if the four floating-point numbers include one or more default NaN (not-a-number) values, the system sets the minimum to negative infinity. One embodiment of the present invention provides a system for performing a minimum computation for an interval division operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number.

Abstract: A system is presented that selectively enables expression folding during compilation of a program, wherein the compilation converts the program from source code into executable code. The system operates by forming an expression tree for an expression within the source code which includes an assignment operator. If the assignment operator is a first assignment operator that is a value assignment, only the computed value can be used in subsequent expressions, thereby disabling expression folding during the compilation process. On the other hand, if the assignment operator is a second assignment operator that specifies an expression assignment, the entire expression can be used in place of the variable on the left of the expression assignment, thereby enabling expression folding during the compilation process. The expression can include a mathematical interval. The expression folding can involve substituting a first expression for a variable within a second expression, and then simplifying the result.

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 for compiling computer code to perform a subtraction operation between a first interval and a third interval to produce a resulting interval. The system operates by receiving source code within a compiler. The system next determines if a subtraction operation within the source code is a dependent subtraction operation, wherein the third interval is the sum of the first interval and a second interval. If so, the system produces executable code for the subtraction operation that computes a left endpoint for the resulting interval and a right endpoint for the resulting interval. If the left endpoint of the third interval is negative infinity, the left endpoint of the resulting interval is assigned to be negative infinity.

Abstract: One embodiment of the present invention provides a system for performing a division operation between arithmetic intervals within a computer system. The system operates by receiving interval operands, including a first interval and a second interval, wherein the first interval is to be divided by the second interval to produce a resulting interval. Next, the system uses the operand values to create a mask. The system uses this mask to perform a multi-way branch, so that an execution flow of a program performing the division operation is directed to code that is tailored to compute the resulting interval for specific relationships between the interval operands and zero. In one embodiment of the present invention, creating the mask additionally involves, determining whether the first and/or second intervals are empty, and modifying the mask so that the multi-way branch directs the execution flow of the program to the appropriate code for this case.

Abstract: One embodiment of the present invention provides a system for representing intervals within a computer system to facilitate efficient and sharp arithmetic interval operations. The system operates by receiving a representation of two intervals. These representations include a first floating-point number representing a first endpoint of the interval and a second floating-point number representing a second endpoint of the interval. Next, the system performs an interval arithmetic operation using the interval operands to produce an interval result. In performing this arithmetic operation, if the first endpoint is negative infinity and the second endpoint is finite, the system treats the first endpoint as a negative overflow toward negative infinity. On the other hand, if the second endpoint is positive infinity and the first endpoint is finite, the system treats the first endpoint as a positive overflow toward positive infinity.

Abstract: One embodiment of the present invention provides a system that facilitates performing a mask-driven multiplication operation between arithmetic intervals within a computer system. The system first receives interval operands, including a first interval and a second interval, to be multiplied together to produce a resulting interval. Next, the system uses the operand values to create a mask. The system uses this mask to perform a multi-way branch to the code for the interval operands. In one embodiment of the present invention, creating the mask additionally involves: determining whether the first interval and/or second intervals are empty, and modifying the mask so the multi-way branch directs the execution flow of the program to appropriate code for this case. In one embodiment of the present invention, if the first interval is empty or if the second interval is empty, the multi-way branch directs the execution flow of the program to code that sets the resulting interval to be empty.

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 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 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 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). During operation, the system receives a representation of the function ƒ and the set of inequality constraints, and stores 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 a globally minimum value of the function ƒ(x) subject to the set of inequality constraints. During this process, the system applies term 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.

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.

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.

Abstract: One embodiment of the present invention provides a system that receives a representation of the function f 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 f(x) over a subbox X. This interval global optimization process applies term consistency to a set of relations associated with the function f 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 f 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 that solves a global optimization problem specified by a function | and a set of equality constraints q1(x)=0 (i=1, . . . , r), wherein | 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: 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.