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=(X 1 , . . . . , X n ), wherein the system of non-linear equations is specified by a vector function f=(f 1 , . . . , f n ). 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 X i  includes a first floating-point number, a i , representing the left endpoint of X i , and a second floating-point number, b i , representing the right endpoint of X i . 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 f 1 (x). The system then evaluates a first termination condition, wherein the first termination condition is TRUE if: zero is contained within f 1 (x), J(x, X) is regular (wherein J(x, X) is the Jacobian of the function f evaluated with respect to x over the box X); and X is contained within X′. If the first termination condition is TRUE, the system terminates the interval Newton method and records X′ as a final bound.

Description:
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 and filed on the same day as the instant application entitled, “Termination Criteria for the One-Dimensional Interval Version of Newton&#39;s Method,” having Ser. No. 09/927,270, and filing date Aug. 9, 2001 (Attorney Docket No. SUN-P6282). 
     
    
     
       BACKGROUND  
         [0002]    1. Field of the Invention  
           [0003]    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 for using a computer system to find the roots of a system of non-linear equations using the interval version of Newton&#39;s method.  
           [0004]    2. Related Art  
           [0005]    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.)  
           [0006]    In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.  
           [0007]    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.  
           [0008]    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.  
           [0009]    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.)  
           [0010]    One commonly performed computational operation is to find the roots of a nonlinear equation using Newton&#39;s method. The interval version of Newton&#39;s method works in the following manner. From the mean value theorem, 
             f ( x )− f ( x *)=( x−x *) f ′(ξ), 
           [0011]    where ξ is some generally unknown point between x and x*. If x* is a zero of f then f(x*)=0 and, from the previous equation, 
             x*=x−f ( x )/ f ′(ξ). 
           [0012]    Let X be an interval containing both x and x*. Since ξ is between x and x*, it follows that ξεX Moreover, it follows that f′(ξ)εf′(X). Hence, x*εN(x, X) where 
             N ( x,X )= x−f ( x )/ f ′( X ). 
           [0013]    Temporarily assume 0εf′(X) so that N(x, X) is a finite interval. Since any zero of f in X is also in N(x,X), the zero is in the intersection X∩N(x, X). Using this fact, we define a procedure for finding zero x*. Let X 0  be an interval containing x*. For n=0, 1, 2, . . . , define 
             X   n   =m ( X   n ) 
             N ( x   n   ,X   n )=x n   −f ( x   n )/ f ′( X   n ) 
             X   n+1   =X   n   ∩N ( x   n   ,X   n ). 
           [0014]    Wherein m(X) is the midpoint of the interval X. We call x n  the point of expansion for the Newton method. It is not necessary to choose x n  to be the midpoint of X n . The only requirement is that x n εX n  to assure that x*εN(x n ,X n ). However, it is convenient and efficient to choose x n =m(X n ).  
           [0015]    Roots of an interval equation can be intervals rather than points when the equation contains non-degenerate interval constants or parameters. Suppose the interval version of Newton&#39;s method to find the roots of a system of nonlinear equations has not yet satisfied the user-specified convergence tolerances. Then it is difficult to distinguish between the following three situations:  
           [0016]    a) the current interval is a tight enclosure of a single interval root;  
           [0017]    b) the current interval contains sufficiently distinct interval roots that they can be isolated with a reasonable amount of effort; and  
           [0018]    c) the current interval contains point and/or interval roots that are so close large with the existing wordlength.  
           [0019]    What is needed is a method and an apparatus for terminating the interval version of Newton&#39;s root finding method for a system of nonlinear equations before iterations lose their practical value in isolating meaningfully distinct interval roots.  
         SUMMARY  
         [0020]    One embodiment of the present invention provides a system for finding the roots of a system of nonlinear equations within an interval vector X=(X 1 , . . . , X n ), wherein the system of non-linear equations is specified by a vector function f=f 1 , . . . , f n ). 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 X i  includes a first floating-point number, a i , representing the left endpoint of X i , and a second floating-point number, b i , representing the right endpoint of X i .  
           [0021]    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 f 1 (x).  
           [0022]    The system then evaluates a first termination condition, wherein the first termination condition is TRUE if: zero is contained within f 1 (x), J(x, X) is regular (wherein J(x, X) is the Jacobian of the function f evaluated with respect to x over the box X); and X is contained within X′. If the first termination condition is TRUE, the system terminates the interval Newton method and records X′ as a final bound.  
           [0023]    In one embodiment of the present invention, if no termination condition is satisfied, the system returns to perform an interval Newton step on the box X′.  
           [0024]    In one embodiment of the present invention, the system also evaluates a second termination condition, wherein the second termination condition is TRUE if a function of the width of the interval X′ is less than a pre-specified value, ε X , and the magnitude of the function f over the interval X′ is less than a pre-specified value, ε F . If the second termination condition is TRUE, the system terminates the interval Newton method and records X′ as a final bound. (Note that the width of an interval X i =[a i , b i ] (denoted as w(X i )) is simply b i −a i . Also note that the magnitude of the function f over the interval X′ is the largest |f i (x)| for any f i  that is part of the function f and any xεX′.  
           [0025]    In a variation on this embodiment, the second termination condition is evaluated and the method possibly terminates before the first termination condition is evaluated.  
           [0026]    In another variation, the second termination condition is evaluated only if J(x, X) is not proved to be regular. Note that the system can determine if J(x, X) is regular by first 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 attempting to solve M(x, X)(y−x)=r(x), where r(x)=−Bf(x).  
           [0027]    In one embodiment of the present invention, the system determines whether J(x, X) is regular by attempting to invert the matrix formed by the left endpoints of the interval elements of M(x, X).  
           [0028]    In one embodiment of the present invention, returning to perform an interval Newton step on the interval X′ can involve splitting the interval X′. 
       
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0029]    [0029]FIG. 1 illustrates a computer system in accordance with an embodiment of the present invention.  
         [0030]    [0030]FIG. 2 illustrates the process of compiling and using code for interval computations in accordance with an embodiment of the present invention.  
         [0031]    [0031]FIG. 3 illustrates an arithmetic unit for interval computations in accordance with an embodiment of the present invention.  
         [0032]    [0032]FIG. 4 is a flow chart illustrating the process of performing an interval computation in accordance with an embodiment of the present invention.  
         [0033]    [0033]FIG. 5 illustrates four different interval operations in accordance with an embodiment of the present invention.  
         [0034]    [0034]FIG. 6 illustrates a process of solving for zeros of a function that specifies a system of nonlinear equations using the interval Newton method in accordance with an embodiment of the present invention.  
         [0035]    [0035]FIG. 7 illustrates another process of solving for zeros of a function that specifies a system of nonlinear equations using the interval Newton method in accordance with an embodiment of the present invention. 
     
    
     DETAILED DESCRIPTION  
       [0036]    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 intended to be limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and features disclosed herein.  
         [0037]    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.  
         [0038]    Computer System  
         [0039]    [0039]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  10  through bridge  106 . Bridge  106  can generally include any type of circuitry for coupling components of computer system  100  together.  
         [0040]    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.  
         [0041]    Processor  102  communicates with storage device  108  through bridge  106  and peripheral bus  1   10 . 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.  
         [0042]    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 .  
         [0043]    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.  
         [0044]    Compiling and Using Interval Code  
         [0045]    [0045]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 .  
         [0046]    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.  
         [0047]    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.  
         [0048]    Arithmetic Unit for Intervals  
         [0049]    [0049]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.  
         [0050]    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 .  
         [0051]    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.  
         [0052]    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.  
         [0053]    [0053]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.  
         [0054]    Interval Operations  
         [0055]    [0055]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,  
         [0056]    [0056] x  denotes the lower bound of X, and  
         [0057]    {overscore (x)} denotes the upper bound of X.  
         [0058]    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).  
         [0059]    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∪{−∞}∪{+∞}. 
     
         [0060]    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.  
         [0061]    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.  
         [0062]    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.  
         [0063]    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.  
         [0064]    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*.  
         [0065]    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.  
         [0066]    Interval Version of Newton&#39;s Method for Systems of Nonlinear Equations  
         [0067]    [0067]FIG. 6 illustrates the process of solving for zeros of a function that specifies a system of nonlinear equations using the interval Newton method in accordance with an embodiment of the present invention. The system starts with a multi-variable function f(x)=0, wherein x is a vector (x 1 , x 2 , x 3 , . . . X n ), and wherein f(x)=0 represents a system of equations f 1 (x)=0, f 2 (x)=0, f 3 (x)=0, . . . , f n (x)=0.  
         [0068]    Next, the system receives a representation of an interval vector, X (step  602 ). In one embodiment of the present invention, for each dimension, i, the representation of X i  includes a first floating-point number, a i , representing the left endpoint of X i  in the i-th dimension, and a second floating-point number, b i , representing the right endpoint of X i .  
         [0069]    The system then performs a Newton step on X, wherein the point of expansion is x, to compute a resulting interval X′=N(x, X) (step  604 ).  
         [0070]    Next, the system evaluates termination criteria A and B, which relate to the size of the box X and the function f respectively (step  606 ). Criterion A is satisfied if the width of the interval X, w(X), is less than ε X  for some ε X &gt;0, wherein w(X) is be defined as the maximum width of any component X 1  of the interval X. Note that ε X  is user-specified and is an absolute criterion. Criterion A can alternatively be a relative criterion w(X)/|X|&lt;ε X  if the box X does not contain zero. Moreover, ε X  can be a vector, ε X , wherein there exists a separate component ε Xi  for each dimension in the interval vector X In this case, components containing zero can use absolute criteria, while other components use relative criteria.  
         [0071]    Criterion B is satisfied if ||f||&lt;ε F  for some user-specified ε F &gt;0, wherein ||f||=max(|f 1 (X)|, |f 2 (X)|, |f 3 (X)|, . . . , |f n (X)|). Note that as with ε X , element-specific values ε Fi  can be used, but they are always absolute.  
         [0072]    However defined, criteria A and B are satisfied, the system terminates and accepts X′ as a final bounding box for the zeros of f(step  610 ). Otherwise, if either criterion A or criterion B is not satisfied, the system proceeds to evaluate criterion C (step  612 ).  
         [0073]    Criterion C is satisfied if three conditions are satisfied. A first condition is satisfied if zero is contained within f 1 (x), wherein x is a point within the box X, and wherein f 1 (x) is a box that results from evaluating f(x). Note that performing the interval Newton step in step  604  involves evaluating f(x) to produce an interval result f 1 (x). Hence, f 1 (x) does not have to be recomputed in evaluating criterion C.  
         [0074]    A second condition is satisfied if M(x, X)=BJ(x, X) is regular. J(x, X) is the Jacobian (matrix of second order partial derivatives) of the vector function f with respect to the point x in the interval X. B is an approximate inverse of the center of J(x, X). Note that multiplying J(x, X) by B preconditions J(x, X) so it is easier to determine whether J(x, X) is regular. Hence, M(x, X) is referred to as the “preconditioned” Jacobian. Note that M(x, X) is regular if it is possible to invert M(x, X) using a technique such as Gaussian elimination.  
         [0075]    Finally, a third condition is satisfied if X=X′. This indicates that the interval Newton step (in step  604 ) failed to make progress.  
         [0076]    If criterion C is satisfied, the system terminates and accepts X′ as a final bounding box for the zeros of f(step  616 ).  
         [0077]    Otherwise, if criterion C is not satisfied, the system returns for another iteration. This may involve splitting X′ into multiple intervals to be separately solved if the Newton step has not made sufficient progress to assure convergence at a reasonable rate (step  618 ). The system then sets X=X′ (step  620 ) and returns to step  604  to perform another interval Newton step.  
         [0078]    The above-described process works well if tolerances ε X  and ε F  are chosen “relatively large”. In this case, processing stops early and computing effort is relatively small.  
         [0079]    Alternatively, the process illustrated in FIG. 7 seeks to produce to best (or near best) possible result for simple zeros. The system starts with a multi-variable function f(x)=0. Next, the system receives a representation of an interval vector X (step  702 ). The system then performs a Newton step on X, wherein the point of expansion is xεX, to compute a resulting interval X′=N(x, X) (step  704 ).  
         [0080]    Next, the system evaluates criterion C (step  705 ). If criterion C is satisfied, the system terminates and accepts X′ as a final bounding box for the zeros of f(step  708 ). Otherwise, if criterion C is not satisfied, the system proceeds to determine if M(x, X) is regular (step  709 ).  
         [0081]    If M(x, X) is regular, the system returns for another iteration. This may involve splitting X′ into multiple intervals to be separately solved if the Newton step has not made sufficient progress to assure convergence at a reasonable rate (step  717 ). The system also sets X=X′ (step  718 ) before returning to step  704  to perform another interval Newton step.  
         [0082]    If M(x, X) is not regular, the system evaluates termination criteria A and B (step  712 ). If criteria A and B are satisfied, the system terminates and accepts X′ as a final bounding box for the zeros of f(step  716 ). Otherwise, if either criterion A or criterion B is not satisfied, the system returns for another iteration (steps  717  and  718 ).  
         [0083]    The foregoing descriptions of embodiments of the present invention have been presented for purposes of illustration and description only. 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.