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 M 0 x=r, wherein M 0 =BA and r=Bb. Next, the system widens the matrix M 0  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 M 0 x=r.

Description:
BACKGROUND  
         [0001]    1. Field of the Invention  
           [0002]    The present invention relates to performing arithmetic operations on interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus that uses interval arithmetic to bound the solution set of a system of linear equations.  
           [0003]    2. Related Art  
           [0004]    Rapid advances in computing technology make it possible to perform trillions of computational operations each second. This tremendous computational speed makes it practical to perform computationally intensive tasks as diverse as predicting the weather and optimizing the design of an aircraft engine. Such computational tasks are typically performed using machine-representable floating-point numbers to approximate values of real numbers. (For example, see the Institute of Electrical and Electronics Engineers (IEEE) standard 754 for binary floating-point numbers.)  
           [0005]    In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.  
           [0006]    One limitation is that machine-representable floating-point numbers have a fixed-size word length, which limits their accuracy. Note that a floating-point number is typically encoded using a 32, 64 or 128-bit binary number, which means that there are only 2 32 , 2 64  or 2 128  possible symbols that can be used to specify a floating-point number. Hence, most real number values can only be approximated with a corresponding floating-point number. This creates estimation errors that can be magnified through even a few computations, thereby adversely affecting the accuracy of a computation.  
           [0007]    A related limitation is that floating-point numbers contain no information about their accuracy. Most measured data values include some amount of error that arises from the measurement process itself. This error can often be quantified as an accuracy parameter, which can subsequently be used to determine the accuracy of a computation. However, floating-point numbers are not designed to keep track of accuracy information, whether from input data measurement errors or machine rounding errors. Hence, it is not possible to determine the accuracy of a computation by merely examining the floating-point number that results from the computation.  
           [0008]    Interval arithmetic has been developed to solve the above-described problems. Interval arithmetic represents numbers as intervals specified by a first (left) endpoint and a second (right) endpoint. For example, the interval [a, b], where a&lt;b, is a closed, bounded subset of the real numbers, R, which includes a and b as well as all real numbers between a and b. Arithmetic operations on interval operands (interval arithmetic) are defined so that interval results always contain the entire set of possible values. The result is a mathematical system for rigorously bounding numerical errors from all sources, including measurement data errors, machine rounding errors and their interactions. (Note that the first endpoint normally contains the “infimum”, which is the largest number that is less than or equal to each of a given set of real numbers. Similarly, the second endpoint normally contains the “supremum”, which is the smallest number that is greater than or equal to each of the given set of real numbers.)  
           [0009]    One commonly performed computational operation is to find the solution of a system of interval linear equations. What is needed is a method and an apparatus that uses interval arithmetic to efficiently compute narrow bounds on the solution set of a system of linear equations.  
         SUMMARY  
         [0010]    One embodiment of the present invention provides a system that bounds the solution set of a system 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 M 0 x=r, wherein M 0 =BA and r=Bb. Next, the system widens the matrix M 0  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 M 0 x=r.  
           [0011]    In a variation on this embodiment, the system computes the matrix B by computing an approximate center A C  of the matrix A, and then forming B by computing the approximate inverse of A C , B (AC) −1 .  
           [0012]    In a variation on this embodiment, the system additionally assures that sup(r i )≧0 by changing the sign of r i  and x i  if necessary.  
           [0013]    In a variation on this embodiment, the system uses M and r to compute the hull h by forming P as an inverse of the left endpoint of M. The system also forms c i =1/(2P ii −1) for i=1, . . . , n and forms z i =(inf(r i )+sup(r i ))P ii −e i   T Psup(r), wherein e i   T is a unit vector in which the i-th element is 1 and other elements are 0. The system then forms h by: setting inf(h i )=c i z i  if z i &gt;0; setting inf(h i )=z i  if z i ≦0; and setting sup(h)=Psup(r).  
           [0014]    In a variation on this embodiment, the system determines whether or not M is regular. If the inverse of inf(M) exists and is denoted by P, and if inf(M ii )&gt;0 for all i, then M, M 0  and A are all regular if and only if P≧I. If not, the system terminates the process of computing the hull h. 
       
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0015]    [0015]FIG. 1 illustrates a computer system in accordance with an embodiment of the present invention.  
         [0016]    [0016]FIG. 2 illustrates the process of compiling and using code for interval computations in accordance with an embodiment of the present invention.  
         [0017]    [0017]FIG. 3 illustrates an arithmetic unit for interval computations in accordance with an embodiment of the present invention.  
         [0018]    [0018]FIG. 4 is a flow chart illustrating the process of performing an interval computation in accordance with an embodiment of the present invention.  
         [0019]    [0019]FIG. 5 illustrates four different interval operations in accordance with an embodiment of the present invention.  
         [0020]    [0020]FIG. 6 illustrates the process of bounding the solution set of a system of linear equations in accordance with an embodiment of the present invention. 
     
    
     DETAILED DESCRIPTION  
       [0021]    The following description is presented to enable any person skilled in the art to make and use the invention, and is provided in the context of a particular application and its requirements. Various modifications to the disclosed embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the spirit and scope of the present invention. Thus, the present invention is not limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and features disclosed herein.  
         [0022]    The data structures and code described in this detailed description are typically stored on a computer readable storage medium, which may be any device or medium that can store code and/or data for use by a computer system. This includes, but is not limited to, magnetic and optical storage devices such as disk drives, magnetic tape, CDs (compact discs) and DVDs (digital versatile discs or digital video discs), and computer instruction signals embodied in a transmission medium (with or without a carrier wave upon which the signals are modulated). For example, the transmission medium may include a communications network, such as the Internet.  
         [0023]    Computer System  
         [0024]    [0024]FIG. 1 illustrates a computer system  100  in accordance with an embodiment of the present invention. As illustrated in FIG. 1, computer system  100  includes processor  102 , which is coupled to a memory  112  and a to peripheral bus  110  through bridge  106 . Bridge  106  can generally include any type of circuitry for coupling components of computer system  100  together.  
         [0025]    Processor  102  can include any type of processor, including, but not limited to, a microprocessor, a mainframe computer, a digital signal processor, a personal organizer, a device controller and a computational engine within an appliance. Processor  102  includes an arithmetic unit  104 , which is capable of performing computational operations using floating-point numbers.  
         [0026]    Processor  102  communicates with storage device  108  through bridge  106  and peripheral bus  110 . Storage device  108  can include any type of non-volatile storage device that can be coupled to a computer system. This includes, but is not limited to, magnetic, optical, and magneto-optical storage devices, as well as storage devices based on flash memory and/or battery-backed up memory.  
         [0027]    Processor  102  communicates with memory  112  through bridge  106 . Memory  112  can include any type of memory that can store code and data for execution by processor  102 . As illustrated in FIG. 1, memory  112  contains computational code for intervals  114 . Computational code  114  contains instructions for the interval operations to be performed on individual operands, or interval values  115 , which are also stored within memory  112 . This computational code  114  and these interval values  115  are described in more detail below with reference to FIGS.  2 - 5 .  
         [0028]    Note that although the present invention is described in the context of computer system  100  illustrated in FIG. 1, the present invention can generally operate on any type of computing device that can perform computations involving floating-point numbers. Hence, the present invention is not limited to the computer system  100  illustrated in FIG. 1.  
         [0029]    Compiling and Using Interval Code  
         [0030]    [0030]FIG. 2 illustrates the process of compiling and using code for interval computations in accordance with an embodiment of the present invention. The system starts with source code  202 , which specifies a number of computational operations involving intervals. Source code  202  passes through compiler  204 , which converts source code  202  into executable code form  206  for interval computations. Processor  102  retrieves executable code  206  and uses it to control the operation of arithmetic unit  104 .  
         [0031]    Processor  102  also retrieves interval values  115  from memory  112  and passes these interval values  115  through arithmetic unit  104  to produce results  212 . Results  212  can also include interval values.  
         [0032]    Note that the term “compilation” as used in this specification is to be construed broadly to include pre-compilation and just-in-time compilation, as well as use of an interpreter that interprets instructions at run-time. Hence, the term “compiler” as used in the specification and the claims refers to pre-compilers, just-in-time compilers and interpreters.  
         [0033]    Arithmetic Unit for Intervals  
         [0034]    [0034]FIG. 3 illustrates arithmetic unit  104  for interval computations in more detail accordance with an embodiment of the present invention. Details regarding the construction of such an arithmetic unit are well known in the art. For example, see U.S. Pat. Nos. 5,687,106 and 6,044,454, which are hereby incorporated by reference in order to provide details on the construction of such an arithmetic unit. Arithmetic unit  104  receives intervals  302  and  312  as inputs and produces interval  322  as an output.  
         [0035]    In the embodiment illustrated in FIG. 3, interval  302  includes a first floating-point number  304  representing a first endpoint of interval  302 , and a second floating-point number  306  representing a second endpoint of interval  302 . Similarly, interval  312  includes a first floating-point number  314  representing a first endpoint of interval  312 , and a second floating-point number  316  representing a second endpoint of interval  312 . Also, the resulting interval  322  includes a first floating-point number  324  representing a first endpoint of interval  322 , and a second floating-point number  326  representing a second endpoint of interval  322 .  
         [0036]    Note that arithmetic unit  104  includes circuitry for performing the interval operations that are outlined in FIG. 5. This circuitry enables the interval operations to be performed efficiently.  
         [0037]    However, note that the present invention can also be applied to computing devices that do not include special-purpose hardware for performing interval operations. In such computing devices, compiler  204  converts interval operations into a executable code that can be executed using standard computational hardware that is not specially designed for interval operations.  
         [0038]    [0038]FIG. 4 is a flow chart illustrating the process of performing an interval computation in accordance with an embodiment of the present invention. The system starts by receiving a representation of an interval, such as first floating-point number  304  and second floating-point number  306  (step  402 ). Next, the system performs an arithmetic operation using the representation of the interval to produce a result (step  404 ). The possibilities for this arithmetic operation are described in more detail below with reference to FIG. 5.  
         [0039]    Interval Operations  
         [0040]    [0040]FIG. 5 illustrates four different interval operations in accordance with an embodiment of the present invention. These interval operations operate on the intervals X and Y. The interval X includes two endpoints,  
         [0041]    [0041] x  denotes the lower bound of X, and  
         [0042]    {overscore (x)} denotes the upper bound of X.  
         [0043]    The interval X is a closed subset of the extended (including −∞ and +∞) real numbers R* (see line  1  of FIG. 5). Similarly the interval Y also has two endpoints and is a closed subset of the extended real numbers R* (see line  2  of FIG. 5).  
         [0044]    Note that an interval is a point or degenerate interval if X=[x, x]. Also note that the left endpoint of an interior interval is always less than or equal to the right endpoint. The set of extended real numbers, R* is the set of real numbers, R, extended with the two ideal points negative infinity and positive infinity:  
         
       R*=R∪{−∞}∪{+∞}.  
     
         [0045]    In the equations that appear in FIG. 5, the up arrows and down arrows indicate the direction of rounding in the next and subsequent operations. Directed rounding (up or down) is applied if the result of a floating-point operation is not machine-representable.  
         [0046]    The addition operation X+Y adds the left endpoint of X to the left endpoint of Y and rounds down to the nearest floating-point number to produce a resulting left endpoint, and adds the right endpoint of X to the right endpoint of Y and rounds up to the nearest floating-point number to produce a resulting right endpoint.  
         [0047]    Similarly, the subtraction operation X−Y subtracts the right endpoint of Y from the left endpoint of X and rounds down to produce a resulting left endpoint, and subtracts the left endpoint of Y from the right endpoint of X and rounds up to produce a resulting right endpoint.  
         [0048]    The multiplication operation selects the minimum value of four different terms (rounded down) to produce the resulting left endpoint. These terms are: the left endpoint of X multiplied by the left endpoint of Y; the left endpoint of X multiplied by the right endpoint of Y; the right endpoint of X multiplied by the left endpoint of Y; and the right endpoint of X multiplied by the right endpoint of Y. This multiplication operation additionally selects the maximum of the same four terms (rounded up) to produce the resulting right endpoint.  
         [0049]    Similarly, the division operation selects the minimum of four different terms (rounded down) to produce the resulting left endpoint. These terms are: the left endpoint of X divided by the left endpoint of Y; the left endpoint of X divided by the right endpoint of Y; the right endpoint of X divided by the left endpoint of Y; and the right endpoint of X divided by the right endpoint of Y. This division operation additionally selects the maximum of the same four terms (rounded up) to produce the resulting right endpoint. For the special case where the interval Y includes zero, X/Y is an exterior interval that is nevertheless contained in the interval R*.  
         [0050]    Note that the result of any of these interval operations is the empty interval if either of the intervals, X or Y, are the empty interval. Also note, that in one embodiment of the present invention, extended interval operations never cause undefined outcomes, which are referred to as “exceptions” in the IEEE 754 standard.  
         [0051]    Bounding the Solution Set of a System of Linear Equations  
         [0052]    [0052]FIG. 6 illustrates the process of bounding the solution set of a system of linear equations in accordance with an embodiment of the present invention. The system starts by receiving a representation of the system of linear equations Ax=b, wherein A is an interval matrix and b is an interval vector (step  606 ).  
         [0053]    The system then preconditions Ax=b to produce M 0 x=r, where M 0 =BA and r=Bb (step  608 ). The preconditioning matrix B can be formed by first computing an approximate center A C  of the matrix A, and then forming B, the approximate inverse of A C , B=(A C ) −1 . Note that we can write A=A C +Q[−1,1] where Q is a real matrix. Therefore, the preconditioned matrix is M=BA=I+BQ[−1,1]. That is, the center of M is the identity matrix. If A C  and B were computed exactly, the center of M 0  would be the identity matrix I. (Note that the system widens M 0  at step  612  below so that the center of the result, M, equals I.)  
         [0054]    If we denote M=[inf(M),sup(M)] and r=[inf(r), sup(r)], then for i,j=1, . . . , n, inf(M ij )=−sup(M ij ) (i≠j), and inf(M ii )+sup(M ii )=2.  
         [0055]    In the preceding discussion, we have ignored the fact that B does not exist if A C  is singular. Suppose we try to invert Ac using Gaussian elimination. If A C  is singular, this fails because a pivot element is zero. At this point, the system terminates. Otherwise, if A C  is not singular and B can be computed, the system continues.  
         [0056]    Next, the system assures that sup(r i )≧0 by changing the sign of r i  and x i  if necessary (step  611 ). Suppose we multiply the i-th equation of the system by −1 and simultaneously change the sign of x i . As noted above inf(M ij )=−sup(M ij ) for i≠j. Hence, the off-diagonal elements are unchanged. Moreover, the diagonal elements change sign twice so they have no net change. Thus, the coefficient matrix is unchanged while x i  and r i  change sign.  
         [0057]    We can assure that sup(r i )≧0 by changing the sign of r i  and x i  if necessary. Assume this is the case. If 0∈r i , we can change the sign of r i  and x i  if necessary and obtain −inf(r i )≦sup(r i ). Therefore, we can always assure that 0≦|inf(r i )|≦sup(r i ). Hereafter, we assume that the above relationship is satisfied for all i=1, . . . , n. This simplifies the procedure for finding the hull of M 0 x=r.  
         [0058]    Next, the system widens M 0  so that the center of the result, M, equals I (step  612 ). At this point, the system determines if M is regular (step  613 ).  
         [0059]    If so, the system forms the hull h from M and r. In doing so, the system computes P=(M L ) −1  as the inverse of the left endpoint M L  of M (step  614 ). The system then forms c i =1/(2P ii −1) for i=1, . . . , n (step  616 ), and also forms z i =(inf(r i )+sup(r i ))P ii −e i   T Psup(r) for i =1, . . . , n, wherein e i   T  is a unit vector in which the i-th element is 1 and other elements are 0 (step  618 ). Next, the system forms h by setting inf(h i )=c i z i  if z i &gt;0 for i=1, . . . , n, and by setting inf(h i )=z i  for i=1, . . . , n if z i ≦0 (step  620 ). The system also sets sup(h)=Psup(r) (step  622 ).  
         [0060]    If M was not regular at step  613 , the system uses the Gauss-Seidel process to compute the hull h (step  615 ) before terminating.  
         [0061]    The above-described procedure for finding the hull is valid only if M is regular. The following theorem enables us to verify regularity as a by-product of the computation of the hull.  
         [0062]    Theorem 1: Assume inf(M) is nonsingular so that P=(inf(M)) −1  exists. Also assume that inf(M ii )&gt;0 for all i=1, . . . , n. Then M is regular if and only if P≧I.  
         [0063]    If, using Theorem 1, we find that M is regular, we can compute the hull h using the above-described procedure. Note, however, that the hull of the preconditioned system Mx=r is generally larger than that of the original system Ax=b.  
         [0064]    The foregoing descriptions of embodiments of the present invention have been presented only for purposes of illustration and description. They are not intended to be exhaustive or to limit the present invention to the forms disclosed. Accordingly, many modifications and variations will be apparent to practitioners skilled in the art. Additionally, the above disclosure is not intended to limit the present invention. The scope of the present invention is defined by the appended claims.