Abstract: A SQRT calculator capable of calculation with a minimal error is provided. The integer calculation unit selects a largest integer from a set of integers with a square of each of the integers smaller than an input datum. The transformation unit transforms the selected integer from the integer calculation unit by multiplying it by 2 and shifts a decimal point of the resulting number to the right by 1 place, thereby adding a certain number less than 10 to the decimal point shifted number to calculate a transformation value. The calculation unit shifts a decimal point of the number less than 10 to the left by 2 places and multiplies the transformation value by the resulting value, thereby subtracting the multiplied value from the input datum and choosing a largest number less than 10 with the subtracted value being in a desired range as a second decimal number of the square root. Thus, the SQRT calculator is capable of calculation with minimal error and, furthermore, has a minimum size of hardware.

Abstract: A Givens rotation computation technique is provided that makes use of polynomial approximations of an expression that contains a square root function. The polynomial approximation uses polynomial coefficients that are specifically adapted to respective ones of a number of subintervals within the range of possible values of the input variable of the expression. The technique may be used in data communications devices such as those in wireless local area networks. An example is the application of the Givens rotations technique in a decision feedback equalizer.

Abstract: A transfer function is formed starting from a technical system described by a system of parameter-dependent differential algebraic equations. Subsequently, information about the number of turns per unit length and about the monotonicity behavior of this transfer function is derived. It is established by using this information whether a Hopf bifurcation point is present. In the case of the presence of a Hopf bifurcation point, the latter is determined.

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 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 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 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.

Abstract: One embodiment of the present invention provides a system for solving a system of equations in fixed-point form. During operation, the system receives a representation of the equations in fixed-point form and stores the representation a computer memory. Next, the system reduces the dimension of the system of equations, when possible, by eliminating variables in the system of equations to produce a reduced system of equations. The system then performs interval intersections based on the Fixed Point theorem to reduce the size of a box containing solutions to the reduced system of equations. In a variation on this embodiment, the system additionally applies interval techniques to find solutions to the system of equations, when such solutions exist.

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 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)).

Abstract: In order to provide an approximate solution having high accuracy to a given partial differential equation made up of one of a Poisson equation, diffusion equation or other partial differential equation similar in form to a Poisson or diffusion equation, the given equation being applied on a plurality of grid points dispersed at irregular intervals, a program is generated in which not only the dependent variable of the original equation is used, but in addition first order derivatives thereof also are input independently as additional dependent variables, the program thereby serving to execute and solve discretized equations using discretized expressions made up of high accuracy second and third order derivative terms of a dependent variable of the given partial differential equation.

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 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)).

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=[S1L, S1U], S1L≦S1+1,L. Next, the system establishes interval roots for F(x) from the interval entries in list L.

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 f(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 f(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 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 f=(f1, . . . , fn). 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, ai, 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 f(x) to produce an interval result f1(x).

Abstract: One embodiment of the present invention provides a system for finding zeros of a function, f, 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 f(x) to produce an interval result fI(x). If fI(x) contains zero, the system evaluates f(a) to produce an interval result fI(a). It also evaluates f(b) to produce an interval result fI(b).

Abstract: One embodiment of the present invention provides a system for efficiently perform a modification (cmod) operation in solving a system of linear algebraic equations involving a sparse coefficient matrix. The system operates by identifying supernodes in the sparse matrix, wherein each supernode comprises a set of contiguous columns having a substantially similar pattern of non-zero elements. In solving the equation, the system performs a cmod operation between a source supernode and a destination supernode. As part of this cmod operation, the system determines a subset of the source supernode that will be used in creating an update for the destination supernode. The system partitions the subset into a plurality of tiles, each tile being a rectangular shape of fixed dimensions chosen so as to substantially optimize a computational performance of the cmod operation on a particular computer architecture.

Abstract: In a preconditioning process for an iteration method to solve simultaneous linear equations through multilevel block incomplete factorization of a coefficient matrix, a set of variable numbers of variables to be removed is determined at each level of the factorization such that a block matrix comprising coefficients of the variables can be diagonal dominant. The approximate inverse matrix of the block matrix is obtained in iterative computation, and non-zero elements of a coefficient matrix at a coarse level are reduced.

Abstract: A hybrid system for efficiently performing a cmod operation in solving a system of linear algebraic equations involving a sparse coefficient matrix. The system operates by identifying supernodes in the sparse matrix, wherein each supernode comprises a set of contiguous columns having a substantially similar pattern of non-zero elements. In solving the equation, the system performs a column modification (CMOD) operation between a source supernode and a destination supernode. As part of this CMOD operation, the system determines dimensions of the source supernode and the destination supernode. If a result of a function on the dimensions is lower than a threshold value, the system performs the CMOD operation between the source supernode and the destination supernode using a kernel that is written in an architecture-independent high-level language.

Abstract: A simulation device comprises an equation generating unit for generating a simultaneous linear equation by application of the implicit integration formula and the Newton iteration method to the description data of an electronic circuit to be simulated, a plurality of block ILU factorization units for performing incomplete LU factorization processing in parallel on each block in a coefficient matrix of the generated simultaneous linear equation, a plurality of fill-in adding units for adding a plurality of fills-in generated by the incomplete LU factorization to a combined portion of coefficient matrices, in parallel, a plurality of line collection ILU factorization units for ILU-factorizing each of several line collections on the combined portion where the fills-in are added, and a convergent solution judging unit for repeating a series of the above processing until convergence of a solution in the simultaneous linear equation generated by the equation generating unit is reached.

Abstract: The reliability of solutions of simultaneous algebraic equations with integral coefficients is improved and high-speed information processing is realized. A method processes information wherein, in a model in which restrictions among parameters are given by simultaneous algebraic equations with integral coefficients, when the simultaneous equations have only a finite number of solutions, zero points of a polynomial set F with integral coefficients, which represents the simultaneous equations and is described in a memory, are represented by rational expressions based on zero points of a one-variable polynomial regarding one variable so that the information is represented using the representation of rational expression. The method processes information by choosing a term order, calculates a Grobner basis, calculates a minimum polynomial f1, and controls a digital processor to find solutions represented by rational expressions.

Abstract: Electron repulsion integrals are classified according to atomic nucleus coordinates, etc., coefficients are generated and are stored in a data memory, multiplication with addition operation is executed according to a product sum procedure of auxiliary integrals of recursive order 1 or less, and the result is stored in the data memory. Next, density matrix element is stored in the data memory, a multiplication with addition operation procedure of an electron repulsion integral of recursive order 2 not containing any procedure of recursive order 1 or less is generated, and an instruction memory is updated. Multiplication with addition operation is executed while data is read from the data memory, and the result is stored in the data memory. At the termination of the product sum procedure, calculation of electron repulsion integral gRstu is complete and the Fock matrix element value is updated.