Polynomials are attractive because they are well understood and they have signi64257cant simplicity and structure in that they are vector spaces and rings Additionally degreetwo polynomials conic sections that are also known as quadrics show up in m ID: 23004 Download Pdf

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Polynomials are attractive because they are well understood and they have signi64257cant simplicity and structure in that they are vector spaces and rings Additionally degreetwo polynomials conic sections that are also known as quadrics show up in m

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JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 354 INTRODUCTION Multivariate polynomials show up in many applications. Polynomials are attractive because they are well understood and they have signiﬁcant simplicity and structure in that they are vector spaces and rings. Additionally, degree-two polynomials (conic sections that are also known as quadrics) show up in many engineering applications, including multilateration. This article begins with a discussion on ﬁnding roots of univariate polynomials via eigenvalues/eigenvectors of companion

matrices. Next, we brieﬂy introduce the concept of a ring to enable us to discuss ring representation of polynomials that are a generalization of companion matrices. We then discuss how multivariate polynomial systems can be solved with ring representations. Following that, we outline an algo rithm attributable to Emiris that is used to ﬁnd ring representations. Finally, we give an example of the methods applied to a trilateration quadric-intersection problem. Solving Polynomial Equations Using Linear Algebra Michael Peretzian Williams engineering problems, such as

multilateration. Typically, uadric intersection is a common class of nonlinear systems of equations. Quadrics, which are the class of all degree-two polynomials in three or more variables, appear in many numerical methods are used to solve such problems. Unfortunately, these methods require an initial guess and, although rates of convergence are well understood, convergence is not necessarily certain. The method discussed in this article transforms the problem of simultaneously solving a system of polynomials into a linear algebra problem that, unlike other root-ﬁnding methods, does not

require an initial guess. Additionally, iterative methods only give one solution at a time. The method outlined here gives all solutions (including complex ones).

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JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 355 COMPANION MATRICES: FINDING ROOTS OF UNIVARIATE POLYNOMIALS AS AN EIGENVALUE/EIGENVECTOR PROBLEM Recall that by deﬁnition an matrix has eigenvectors and eigenvalues if Mv Such a matrix can have at most eigenvalues/eigenvectors. Solving for is equiva lent to solving det( xI ) = 0. This calculation yields a polynomial in called the

“characteristic polynomial,” the roots of which are the eigenvalues . Once the values are obtained, the eigenvectors can be calculated by solving ( = 0. Hence, it is not surprising that eigenvalues/eigenvectors provide a method for solving polynomials. Next, we consider the univariate polynomial C x = 0 ) = where = 1. The matrix C C 0 1 0 0 0 1 MM OM is known as the “companion matrix” for . Recall that the Vandermonde matrix is deﬁned as V x x x x x x x n n ( , , , 1 2 1 2 1 1 MM LM As it turns out, the roots of the characteristic polynomial of [i.e., det( xI 0] are precisely the roots

of the polynomial , and the eigenvectors are Vandermonde vectors of the roots. Indeed, if is a root of [i.e., 0] and is a Vandermonde vector such that = [1 ] then Mv C C 0 1 0 0 0 1 MM OM jj i j j j ( ) Hence, the eigenvalues of are the roots of . Likewise, the right eigenvectors of are the columns of the Vandermonde matrix of the roots of The most important property of companion matrices in this article can be stated as follows: Given a polynomial , the companion matrix deﬁnes a matrix such that the characteristic polynomial of is [i.e., det( xI ) = )]. To give a better understanding

of the above statement, consider the following con crete example. Let ) = ( – 2)( – 3)( – 5) = – (2 + 3 + 5) + (2 3 + 2 5 + 3 5) – 2 3 5 = – 10 + 31 – 30 We can verify (for this case) that the eigenvalue/eigenvector statements made above hold [i.e., det( xI = )]. The companion matrix for ) is

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M. P. WILLIAMS JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 356 0 1 0 0 30 31 10 Since the roots of are (2, 3, 5) by construction, the eigenvectors of should be given by the Vandermonde matrix 11 23 49 25 (2, 3, 5) = We

observe that 0 1 0 0 30 31 10 30 − + 31 4 1 0 1 0 0 30 31 10 − + 30 3 3 1 9 10 27 0 1 0 0 30 31 10 25 25 30 5 3 1 2 5 1 − + 25 125 25 Hence, the eigenvalues of are given by 2, 3, 5, the roots of ) as claimed. Also, we note that is the characteristic polynomial for [i.e., det( xI )]. Indeed, ( ) + de t( ) d et M I − = 1 0 0 1 30 31 10 = de 31 10 0 1 30 10 de = 31 30 de ( ) 10 31 30 10 31 30 3 2 − + = + = Hence, det( xI ) as claimed. In this section, we have outlined univariate examples of the companion matrix and Vandermonde matrix. Indeed, we have demonstrated

that ﬁnding the roots of a univariate polynomial is an eigenvalue/eigenvector problem. In a later section, we discuss the u-resultant algorithm, which is used to construct a generalization of the companion matrix. These generalizations, known herein as “ring representations, can be used to solve polynomial systems, including quadric intersection. RINGS AND RING REPRESENTATIONS Previously, we discussed transforming univariate polynomial root ﬁnding into an eigenvalue/eigenvector problem. For the remainder of this article, we will generalize the method above to simultaneously solve

systems of multivariate polynomial equa tions. More precisely, we want to ﬁnd coordinates ( , , ) such that for poly nomials , ,

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SOLVING POLYNOMIAL EQUATIONS USING LINEAR ALGEBRA JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 357 f x x x f x x x f x x x 1 1 2 1 1 2 ( , , , ( , , , ( , , , 0. As with many multivariate generalizations, the univariate case does not require understanding of the deeper structure inherent to the problem. The last section is no exception. A brief discussion regarding the algebraic concept of a ring is necessary to advance

the generalization. Rings are an algebraic structure having both a multiplication and addition but not necessarily a division. The most common rings are integers, univariate polynomials, and multivariate polynomials. For any , , in a ring , the following properties hold. Addition: A. Closure, are in B. Associativity, ( C. Identity, D. Inverse, ( ( E. Commutativity, Multiplication: A. Closure, ab is in B. Associativity, ( ab bc C. Identity, (optional) D. Inverse, –1 –1 1 (optional) E. Commutativity, ab ba (optional) Algebra, the ﬁeld of mathematics in which rings are studied, commonly

looks at an operation called adjoining , or including new elements. If is a ring, then adjoining is accomplished by including some new element j and creating the ring ] in the following fashion: + . . for all , , , . in . A common instance of this operation is adjoining the imaginary unit to the real numbers to obtain the complex numbers. Because matrices are well understood and easily implemented in code, ring rep resentations (a special family of matrices particular to a ring) are a common way of handling algebraic applications. In the same way that logarithms map multiplication to addition

(log( ab ) = log( log( )) and Fourier transforms map convolution to mul tiplication ( ] = ]), ring representations map ring multiplication and ring addition to matrix multiplication and addition. There are many ways to view a matrix. It can be, and typically is, viewed as a transfor mation whose eigenvalues could be adjoined to a ring if they were not present. Alterna tively, because 0 (where is the characteristic polynomial of ), the matrix can itself be viewed as an element that can be adjoined because matrices are roots of their characteristic polynomial. Let be any ring. The mapping to

adjoin a matrix to the ring obtaining the ring ] is as follows: ] = for all , , ., Each element in ] is a ring representation of the element in ]. For a concrete example, consider the complex numbers, which form a ring as they satisfy all of the properties above. To cast the complex numbers as ring representa tions over the real numbers, consider the characteristic polynomial of the complex numbers, i.e., 0. In order to form a ring representation, we are interested in adjoining a matrix whose eigenvalues are the roots of . The com panion matrix of this polynomial, as discussed above, will

provide just such a matrix. The companion matrix is

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M. P. WILLIAMS JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 358 0 1 1 0 Indeed, de t( ) d et de J I − = 0 1 1 0 = + = , . We can now deﬁne a ring representation of the complex numbers as rep for all , . More concretely, we have re p u u i u u u u ( ) 0 1 0 1 1 0 + = This construction preserves addition. Indeed, re p u u i re p v v i u u u u ( ) ( ) 0 1 0 1 0 1 1 0 + + + = 0 1 1 0 0 0 1 1 1 1 0 0 v v u v u v u v u v + + − + ( ) = + + + re p u v u v i (( ) ( )) 0 0 1 1 Also, the

construction preserves multiplication as re p u u i re p v v i u u u u v v ( ) ( ) 0 1 0 1 0 1 1 0 ++ 1 0 0 0 1 1 0 1 1 0 0 1 −+ v v u v u v u v u v u v v u v u re p u v u v u 0 0 0 1 0 0 1 1 0 1 (( ) ( = ++ =+ u v re p u u i v v 1 0 0 1 0 1 )) (( )( )) Remarkably, this process can be generalized to include the roots of several multivari ate polynomials. MOTIVATION FOR USING RING REPRESENTATIONS As stated in the Companion Matrices section, one interpretation of the companion matrix is as an answer to the inverse eigenvalue problem. Namely, given a polynomial , ﬁnd a matrix such

that is the characteristic polynomial of . We will show that ring representations are a multivariate generalization of companion matrices. In this section, we will motivate discussion of Emiris’ algorithm for ﬁnding ring representa tions with an example. Consider the following system of equations: 10 xy 16 0. Let now be the real numbers. These above equations are elements in the poly nomial ring , ]. If we could ﬁnd ring representations and for and , respec tively, in a ring where 0, the following would be true by the properties of ring representations: 10 XY 16 0.

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SOLVING POLYNOMIAL EQUATIONS USING LINEAR ALGEBRA JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 359 Furthermore, because the ring deﬁned by the equations is commutative, and would necessarily commute. As a result, if and are nondefective (have distinct eigenvalues), then they are simultaneously diagonalizable (they can be diagonalized with the same basis or, equivalently, they share all of their eigenvectors). Conse quently, the diagonal of the representations must be the coordinates of the roots. If is the matrix of eigenvectors for and , then the

statements above can be expressed symbolically as follows: V X Y I V X V V Y V V I + + 1 2 1 2 1 2 10 10 ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 10 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x y x y x y 10 0 0 10 + + + x y 10 + Similarly, V X XY Y I x x y y x x + + = + + 1 2 1 1 2 2 2 1 16 + + + + + x x y y x x y y 3 3 4 4 16 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Indeed, it turns out for this example that ring representations for and are as follows: 0 1 0 0 4 0 0 0 0 0 0 3 2 0 0 0 0 0 6 0 0 3 It is easy to verify f X Y I 2 2 10 = + 0 0 1 0 0

0 0 1 6 0 0 1 0 3 2 0 0 7 2 0 0 3 2 0 12 0 0 6 0 0 1 0 3 2 0 0 3 8 0 12 0 0 10 0 0 0 0 1 0 0 0 0 10 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 1 4 0 0 1 0 0 0 1 0 3 2 0 10 0 0 0 0 1 0 0 0 0 10 0 0 0 10 0 1 0 0

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M. P. WILLIAMS JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4 2010 360 f X Y I 2 2 10 = + 0 0 1 0 0 0 0 1 6 0 0 1 0 3 2 0 0 7 2 0 0 3 2 0 12 0 0 6 0 0 1 0 3 2 0 0 3 8 0 12 0 0 10 0 0 0 0 1 0 0 0 0 10 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 1 4 0 0 1 0 0 0 1 0 3 2 0 10 0 0 0 0 1 0 0 0 0 10 0 0 0 10 0 1 0 0 Similar calculations can be done for .

The eigenvalues for and that should yield the coordinates of the roots of the polynomial system are = ( , and 8 1 = 2, 3). Indeed, ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + − = 8 2 10 8 8 2 2 2 1 6 0 + + + − = ( ) ( ) ( ) ( ) ( ) mm 3 1 0 0 1 1 3 2 3 1 6 0 EMIRIS’ ALGORITHM We have established that, if we have a way of determining the ring representa tions for and , we can solve the system of equations. Emiris’ algorithm, a method to accomplish just that, is discussed notionally here. For a full proof and description see Ref. 1 or 2. Given polynomials ,

, each of degree , , , respectively, the algo rithm is as follows: 1. Introduce an extra polynomial , where , are arbitrarily chosen constants with at least one being non-zero 2. Consider 0 , , , }, B x x x i i i D i i += {| 12 01 01 12 KK where = 3. Consider S b B x S b B x { | di vi de s } { | di vi de but doe sn ot { | di vi de but b x S b B b n n S b B x b x , d on ot { | di vi de but d d n n x x S b B b , , , d on ot 1 2 4. Find a linear transformation on the monomials in such that

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SOLVING POLYNOMIAL EQUATIONS USING LINEAR ALGEBRA JOHNS HOPKINS APL TECHNICAL DIGEST,

VOLUME 28, NUMBER 4 2010 361 B x x x B x S x S x S x ( , , , ) ( ( ) ( ) ( ) 1 2 = = M B x M S x S x S x ( ) ( ) ( ) ( ) S x f x S x x f S x 0 0 1 1 ( ) ( ) ( ( )/ ) ( ( ( vv vv )/ ) ( x f 5. Consider Mv p p M B p S p S p S p ( , , , ) ( ( ) ( ) ( ) 1 2 = = S p f p S p 0 0 1 1 ( ) ( ) ( ( )/ vv ) ( ( ( )/ ) ( f p S p p f n n vv S p f p 0 0 ( ) ( ) vv 6. Break into blocks as follows: M M M M M S p S p ( ) ( ) ( ) 00 01 10 11 S p S p f p ( ) ( ) ( ) ( ( vv 0 0 p p f p S p p f n n )/ ) ( ( ( )/ ) ( 1 1 vv S p f p 0 0 ( ) ( ) vv 7. Block triangularlize with the following transformation: I

M M M M M 01 11 00 01 10 11 S p S p S p ( ) ( ) ( ) M M S p S p S p 10 11 ( ) ( ) ( ) I M f p 01 11 ( ) 0 0 ( ) ( ) ( ) vv p f p S 8. As a result of the above, %% M M u u u M M M ( , , , 1 2 00 01 11 10 9. Solve the following eigenvector problem: vv MS p f p S p u u p u p S p n n 0 0 0 0 1 1 0 1 ( ) ( ) ( ) = = + + ,, , , p p 10. Since , , , , , .] (because it will be derived numerically), extracting the root coordinates can be done by dividing the eigenvector by and collecting the 2nd through th entries The roots of the polynomial system are the coordinates ( , ). APPLICATION One application

for Emiris’ algorithm is trilateration. Trilateration is achieved by measuring the time difference of arrival between pairs of sensors. The so-called “TDOA” (time difference of arrival) equation is || S1|| || S2|| = TDOA * C , where S1 and S2 are sensors, is an emitter position, TDOA is the time difference of arrival, and C is the emission propagation rate. It is well known that this can be repre sented as a polynomial (indeed, it is hyperbolic). Thus, the polynomial that describes the level surface above is given by

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M. P. WILLIAMS JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME

28, NUMBER 4 2010 362 TDOA * − = − + 1 2 TDOA * = [(S1 S2)/2 (S1 S2)/2] [(S1 S2)/2 (S1 S2)/2] SS yx TDOA * yv yv =+ = =+ + −= () () 12 12 44 yv yv yv yv yv yy yv TT += += + ++ TT vy yy vv y v y v yv = −+ −+ = 44 44 −= −+ = 22 42 22 yv yv yv yv yv () () yv yy yv vv yv yy TT += −+ += 42 22 242 () 42 22 vv yv vy yy vv yv vI yv TT TT TT += () += The equation above becomes Let As an example, let’s consider the problem of locating a gunshot within a city. Suppose there are three acoustic sensors (S1, S2, and S3) at (9, 39),

(65, 10), and (64, 71), respectively, in a local Cartesian coordinate system with meters as the dis tance unit. Suppose further that a gunshot is heard by S1 at 19:19:57.0875, by S2 at 19:19:57.1719, and by S3 at 19:19:57.1797, and the speed of sound ( ) is 341 m/s. Also suppose the unknown emitter location is at (27, 42) and that it emitted at 19:19:57:0340 (see Fig. 1). The time difference of arrival between S1 and S2 is 0.0844 and between S1 and S3 it is 0.0078. Hence, we wish to solve the system ( ) ( ) ( ) ( ) x y x y − + − + − = 9 3 9 6 5 1 0.0844 341 0.0078 341.

− + − + − = 9 3 9 6 4 7 ) ( ) ( ) ( Equivalently, we solve 207485 6417 19846 0765 31843 375 537 028 . . 812 36 7219 2480777 5687 88508 52 x x y y 23 37529 9994 549 4318 880 49 1818 x x y y + + 0. To apply Emiris’ algorithm, we pick = + + y = 8 + 28 + 80 and calcu late the ring representation = + + Y = 8 + 28 + 80 –48618.3988 –2483454.438 –73249499.44 84.5799 –1549.6064 –43682.6736 80 –112.9542 64660.2744 1768850.5654 28 2548.0324 62895.7242 1969279.3923

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SOLVING POLYNOMIAL EQUATIONS USING LINEAR ALGEBRA JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 4

2010 363 100 80 60 40 20 –20 160 140 120 100 80 60 40 20 –20 –40 Sensor 3 Sensor 2 Sensor 1 Emitter (27, 42) 19:19:57:1797 19:19:57:1719 19:19:57:0875 19:19:57:034 Known: S1 = (9, 39) Known: S2 = (65, 10) Known: S3 = (64, 71) Unknown: = (27, 42) Figure 1. Emitter and sensor locations. 44.1845 –100.0836 –4422.1484 25.9777 254.0739 6600.2544 27 42 1134 xy 70.1086 39.2354 2750.7347 Sensor 3 Sensor 2 Sensor 1 Known: S1 = (9, 39) Known: S2 = (65, 10) Known: S3 = (64, 71) Found: = (27, 42) Emitter (27, 42) 100 80 60 40 20 –20 160 140 120 100 80 60 40 20 –20 –40 Figure 2. Emitter and sensor

locations and eigenvector solutions. Michael Peretzian Williams received his B.S. in Mathematics in 1998 from the University of South Carolina. He earned his M.S. in Mathematics in 2001, again from the University of South Carolina, focusing on Number Theory. He received his Ph.D. in Mathematics in 2004 from North Carolina State University, working in Lie Algebras. From 2005 to 2007, Dr. Williams worked as a researcher for Northrop Grumman, developing algorithms for Dempster–Shafer fusion. He has been with APL’s Global Engagement Department since 2007, and his current work is in upstream data

fusion. His e-mail address is michael.williams@jhuapl.edu. P Michael P. Williams The eigenvectors of are 25.9777 254.0739 6600.2544 70.1086 39.2354 2750.7347 27 42 1134 44.1845 –100.0836 –4422.1484 The second and third rows are the solutions, and we have recovered the emitter loca tion at (27, 42), as seen in Fig. 2. CONCLUSION We have discussed some eigenvector methods for ﬁnding the roots of multi- variate polynomials. Unlike iterative, numerical methods typically applied to this problem, the methods outlined in this article possess the numerical stability of numer ical linear

algebra, do not require a good initial guess of the solution, and give all solu tions simultaneously. Furthermore, if the initial guess is poor enough, the methods outlined herein may converge more quickly than iterative methods. REFERENCES Emiris, I. Z., “On the Complexity of Sparse Elimination, J. Complexity 12 , 134–166 (1996). Cox, D., Little, ., and O’Shea, D., Using Algebraic Geometry , Graduate Texts in Mathematics Series, Vol. 185, Springer, New York, 2nd Ed., pp. 122–128 (2005).

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