Method and apparatus for solving overdetermined systems of interval linear equations

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 formwherein 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. If xn is not the empty interval, the system performs a back substitution operation using xn and Tx=u to solve for the remaining components (xn−1, . . . , x1) of x.

BACKGROUND

1. Field of the Invention

The present invention relates to techniques for performing arithmetic operations involving interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus for solving overdetermined systems of interval linear equations within a computer system.

2. Related Art

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.)

In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.

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 232, 264or 2128possible 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.

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.

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<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. Also note that the infimum and the supremum can be represented by floating point numbers.)

One commonly performed computational operation is to solve a system of linear equations Ax=b, wherein A is an (n×n) matrix and b is a (n×1) column vector. Such a system is said to be “consistent” if there is a unique (n×1) vector x for which the system Ax=b is satisfied. In many cases, a system of linear equations is “overdetermined,” which means that there are more equations than unknowns. In an overdetermined system Ax=b, the number of rows in A and elements in b is m, which is greater than n (the number of columns in A and elements in x).

In the point (non-interval) case, there is no generally reliable way to decide if an overdetermined system is consistent or not. Instead a least squares solution is generally sought. However, in an overdetermined system of linear equations with interval coefficients, the additional equations can potentially help in bounding the set of solutions.

Hence, what is needed is a method and an apparatus for solving an overdetermined system of interval linear equations.

SUMMARY

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

[TW]⁢x=[uv],
wherein T is a square upper triangular matrix of order n, u is an interval vector with n components, v is an interval 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 Tnnxn=unto solve for

xn=unTnn⁢⋂i=1m-n⁢vizi.
If xnis not the empty interval, the system also performs a back substitution operation using xnand Tx=u to solve for the remaining components (xn−1, . . . , x1) of x.

In a variation on this embodiment, before performing the Gaussian Elimination operation, the system uses a preconditioning matrix B to precondition the system of interval linear equations Ax=b to generate a modified system BAx=Bb that can be solved with reduced interval width.

In a further variation, the system generates the preconditioning matrix B by: (1) determining a non-interval matrix Ac, which is the approximate center of the interval matrix A; (2) augmenting the m by n matrix Acto produce an n×n partitioned matrix

C=[Ac′0Ac″I],
wherein Ac′ is an n by n matrix, Ac″ is an m−n by n matrix, I is the identity matrix of order m−n, and 0 is an n by m−n matrix of zeros; and (3) calculating the approximate inverse of the partitioned matrix C to produce the preconditioning matrix B.

In a variation on this embodiment, the system linearizes an initial system of nonlinear equations to form the system of interval linear equations Ax=b.

In a variation on this embodiment, while performing the Gaussian Elimination operation, the system performs column interchanges in the system of interval linear equations Ax=b.

In a variation on this embodiment if xnis determined to be the empty interval during the interval intersection operation, the system indicates that the overdetermined system of interval linear equations Ax=b is inconsistent.

In a variation on this embodiment, if Ax=b is determined to be inconsistent, the system selects equations to remove from Ax=b to make a resulting system of equations consistent, and then removes the selected equations.

In a variation on this embodiment, if Ax=b is determined to be inconsistent, the system determines that at least one of the following is true: (1) a theory underlying Ax=b is false; (2) an error model underlying Ax=b is false; and (3) measurement error was involved in generating Ax=b.

DETAILED DESCRIPTION

Computer System

FIG. 1illustrates a computer system100in accordance with an embodiment of the present invention. As illustrated inFIG. 1, computer system100includes processor102, which is coupled to a memory112and a to peripheral bus110through bridge106. Bridge106can generally include any type of circuitry for coupling components of computer system100together.

Processor102can 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. Processor102includes an arithmetic unit104, which is capable of performing computational operations using floating-point numbers.

Processor102communicates with storage device108through bridge106and peripheral bus110. Storage device108can 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.

Processor102communicates with memory112through bridge106. Memory112can include any type of memory that can store code and data for execution by processor102. As illustrated inFIG. 1, memory112contains computational code for intervals114. Computational code114contains instructions for the interval operations to be performed on individual operands, or interval values115, which are also stored within memory112. This computational code114and these interval values115are described in more detail below with reference toFIGS. 2-5.

Note that although the present invention is described in the context of computer system100illustrated inFIG. 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 system100illustrated inFIG. 1.

Compiling and Using Interval Code

FIG. 2illustrates the process of compiling and using code for interval computations in accordance with an embodiment of the present invention. The system starts with source code202, which specifies a number of computational operations involving intervals. Source code202passes through compiler204, which converts source code202into executable code form206for interval computations. Processor102retrieves executable code206and uses it to control the operation of arithmetic unit104.

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.

Arithmetic Unit for Intervals

FIG. 3illustrates arithmetic unit104for 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. Arithmetic unit104receives intervals302and312as inputs and produces interval322as an output.

In the embodiment illustrated inFIG. 3, interval302includes a first floating-point number304representing a first endpoint of interval302, and a second floating-point number306representing a second endpoint of interval302. Similarly, interval312includes a first floating-point number314representing a first endpoint of interval312, and a second floating-point number316representing a second endpoint of interval312. Also, the resulting interval322includes a first floating-point number324representing a first endpoint of interval322, and a second floating-point number326representing a second endpoint of interval322.

Note that arithmetic unit104includes circuitry for performing the interval operations that are outlined inFIG. 5. This circuitry enables the interval operations to be performed efficiently.

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, compiler204converts interval operations into a executable code that can be executed using standard computational hardware that is not specially designed for interval operations.

FIG. 4is 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 number304and second floating-point number306(step402). Next, the system performs an arithmetic operation using the representation of the interval to produce a result (step404). The possibilities for this arithmetic operation are described in more detail below with reference toFIG. 5.

Interval Operations

FIG. 5illustrates 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,

xdenotes the lower bound of X, and

xdenotes the upper bound of X.

The interval X is a closed subset of the extended (including −∞ and +∞) system of real numbers R* (see line1ofFIG. 5). Similarly the interval Y also has two endpoints and is a closed subset of the extended real numbers R* (see line2ofFIG. 5).

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∪{−∞})∪{+∞}=[−∞,+∞].

We also define R** by replacing the unsigned zero, {0}, from R* with the interval [−0,+0].
R**=R*−{0}∪[−0,+0]=[−∞,+∞], because 0=[−0,+0].

In the equations that appear inFIG. 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.

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.

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.

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.

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*.

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.

Solving an Overdetermined System of Interval Linear Equations

Given the real (n×n) matrix A and the (n×1) column vector b, the linear system of equations
Ax=b(1)
is consistent if there is a unique (n×1) vector x for which the system in (1) is satisfied. If the number of rows in A and elements in b is m≠n, then the system is said to be either under- or overdetermined depending on whether m<n or n<m. In the overdetermined case, if m-n equations are not linearly dependent, there is no solution vector x that satisfies the system. In the underdetermined case there is no unique solution.

In the point (non-interval) case, there is no generally reliable way to decide if an overdetermined system is consistent or not. Instead a least squares solution is generally sought. In the interval case, it is possible to delete inconsistent cases and bound the set of solutions to the remaining consistent equations.

We now consider the problem of solving overdetermined systems of equations in which the coefficients are intervals. That is, we consider a system of the form
AIx=bI(2)
where AIis an interval matrix of m rows and n columns with m>n. The interval vector bIhas m components. Such a system might arise directly or by linearizing an overdetermined system of nonlinear equations. (Note that within this specification and in the following claims, we sometimes drop the superscript “I” when referring the interval matrices or vectors.)

The solution set of (2) is the set of vectors x for which there exists a real matrix AεAIand a real vector bεbIsuch that (1) is satisfied. In general, the system in (2) is inconsistent if its solution set is empty. However, we assume that there exists at least one AεAIand bεbIsuch that (1) is inconsistent. Moreover, we also assume that the data in AIand bIare fallible. That is, there exists at least one AεAIand bεbIsuch that (1) is inconsistent. Our goal is to implicitly exclude at least some of these cases. For example, the redundancy resulting from the fact that there are more equations than variables might be deliberately introduced to sharpen the interval bound on the set of solutions to (2). In a following section, we show how this sharpening is accomplished.

We shall simplify the system using Gaussian elimination. In the point case, it is good practice to avoid forming normal equations from the original system. Instead, one performs elimination using normal operation matrices to triangularize the coefficient matrix. After this first phase, the normal equations of this simpler system can be formed and solved. Our procedure begins with a phase similar to the first phase just described. However, we do not quite complete the usual procedure. We have no motivation to use normal operations because we do not form the normal equations. This is just as well because interval normal matrices do not exist.

When using interval Gaussian elimination, it is generally necessary to precondition the system to avoid excessive widening of intervals due to dependence. In the following section, we show how preconditioning can be done in the present case where AIis not square.

Preconditioning can be done in the same way it is done when AIis square. Let Acdenote the center of the interval matrix AI. Partition Acas

Ac=[Ac′Ac″](3)
where Ac′ is an n by n matrix and Ac″ is an m−n by n matrix. Note that Ac′ need only be an approximation for the center of AI. Define the partitioned matrix

C=[Ac′0Ac″I](4)
where I denotes the identity matrix of order m−n, and the block denoted by 0 is an n×m−n matrix of zeros.

Define the preconditioning matrix B to be the approximate inverse of C, where

To precondition (2) we multiply by B. We obtain
MIx=rI(5)
Where MI=BAIis an m by n interval matrix and rI=BbIis an interval vector of m components. When computing MIand rI, we use interval arithmetic to bound rounding errors.
Elimination

We now perform elimination. We apply an interval version of Gaussian elimination to the system MIx=rIthereby transforming MIinto almost (see below) upper trapezoidal form. We assume that this procedure only fails when all possible pivot elements contain zero. Note that after preconditioning, no pivot selection is performed during the elimination to obtain a result with the form

[TIWI]⁢x=[uIvI](6)
where TIis a square upper triangular interval matrix of order n, and both uIand vIare interval vectors of n and m−n components, respectively. The submatrix WIis a matrix of m−n rows and n columns. It is zero except in the last column. Therefore, we can represent it in the form
WI=[0zI]
where 0 denotes an m−n by n−1 block of zeros, and zIis a vector of m−n intervals. We now have a set of equations
zixn=vi(i=1, . . . ,m−n).  (7)
Also,
Tnnxn=un(8)
Therefore, the unknown value xnis contained in the interval

xn=unTnn⁢⋂i=1m-n⁢vizi.(9)
Taking this intersection is what implicitly eliminates fallible data from AIand bI. It is this operation that allows us to get a sharper bound on the set of solutions to the original system (2) than might otherwise be obtained.

If the original system contains at least one consistent set of equations, the intersection in (9) must not be empty. Knowing xnwe can backsolve (6) for xn−1, . . . , x1. From (6), this takes the standard form of backsolving a triangular system TIx=uI. Sharpening xnusing (9) also produces sharper bounds xIon the other components of x when we backsolve.

Inconsistency

Now suppose the initial equations (2) are not consistent. Then the equations (7) might or might not be consistent: Widening of intervals due to dependence and roundoff can cause the intersection in (9) to be non-empty.

Nevertheless, suppose we find that the intersection in (9) is empty. This event proves that the original equations (2) are inconsistent. Proving inconsistency might be the signal that a theory is measurably false, which might be an extremely enlightening event. On the other hand, inconsistency might only mean that invalid measurements have been made.

If invalid measurements are suspected, it might be important to discover which equation (s) in (2) are inconsistent. We might know which equation (s) in the transformed system (6) must be eliminated to obtain consistency. However, an equation in (6) is generally a linear combination of all the original equations in (2). Therefore, to establish consistency in the original system, we generally cannot determine which of its equation (s) to remove.

We might be able to determine a likely removal candidate by using the following steps:1. Remove enough equations from (6) that the intersection in (9) is not empty.2. Solve (6) for xn−1, . . . , x1. This process cannot fail because we assume the elimination process to obtain (6) does not fail.3. Substitute the solution into the original system (2). Any equation (s) in (2) whose left and right members do not intersect can be discarded.
Summary of the Gaussian Elimination Operation

FIG. 6illustrates the process of performing a Gaussian Elimination operation on an overdetermined interval system of linear equations in accordance with an embodiment of the present invention. The system starts by receiving a representation of the overdetermined system of linear equations Ax=b (step602). In this representation, A is a matrix with m rows corresponding to m equations and n columns corresponding to n variables, x includes n variable components, b includes m scalar components, and m>n. The system then stores this representation in memory (step604).

Next, the system preconditions Ax=b to generate a modified system BAx=Bb that can be solved with reduced interval width (step606). This preconditioning process is described in more detail below with reference toFIG. 7.

The system then performs a Gaussian elimination operation on BAx=Bb to form

[TW]⁢x=[uv],
wherein T is a square upper triangular matrix of order n, u is an interval vector with n components, v is an interval vector with m−n components, and W is a matrix with m−n rows and n columns, and wherein W is zero except in the last column, which is represented as a column vector z with m−n components (step608).

Note that Gaussian elimination can fail. If so, the system simply terminates (step609).

If Gaussian elimination does not fail, the system performs an interval intersection operation based on the equations zixn=vi(i=1, . . . , m−n) and Tnnxn=unto solve for

Finally, if x, is not the empty interval, the system performs a back substitution operation using xnand Tx=u to solve for the remaining components (xn−1, . . . , x1) of x (step612).

FIG. 7illustrates the process of generating a preconditioning matrix in accordance with an embodiment of the present invention. The system starts by determining a non-interval matrix Ac, which is the approximate center of the interval matrix A (step702). Next, the system augments the m×n matrix Acto produce an n×n partitioned matrix

C=[Ac′0Ac″I],
wherein Ac′ is an n×n matrix, Ac″ is an m−n×n matrix, I is the identity matrix of order m−n, and 0 is an n×m−n matrix of zeros (step704). Finally, the system calculates the approximate inverse of the partitioned matrix C to produce the preconditioning matrix B (step706). If C happens to be singular, its elements can be perturbed until it is no longer so. This causes no difficulty because C is just used to compute the approximate inverse B.