AD THE MAXIMUM AND MINIMUM OF A POSITIVE DEFINITE QUADRATIC POLYNOMIAL ON A SPHERE ARE CONVEX FUNCTIONS OF THE RADIUS

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1 AD THE MAXMUM AND MNMUM OF A POSTVE DEFNTE QUADRATC POLYNOMAL ON A SPHERE ARE CONVEX FUNCTONS OF THE RADUS George E. Forsythe Stanford University Stanford, California July 1969 i DiStribllttd,.,'to foster, serve and promote the nation's economic development and technological advancement.' U.S. DEPARTMENT OF COMMERCE/Natlonal Bureau of Standards l This document has been approved for public release and sale.

2 «CS 144 Ct.; ^ THE MAXMUM AND MNMUM OF A POSTVE DEFNTE ^ QUADRATC POLYNOMAL ON A SPHERE ARE, CONVEX FUNCTONS OF THE RADUS BY GEORGE E. FORSYTHE TECHNCAL REPORT NO. CS 144 JULY 1969 y ' 1 ' ' i N G H C Ü S f COMPUTER SCENCE DEPARTMENT School of Humanities and Sciences STANFORD UNVERSTY v>

3 mmi. i p THE MAXMUM AND MNMUM OF A POSTVE DEFNTE QUADRATC POLYNOMAL ON A SPHERE ARE CON^TX FUNCTONS OF THE RADUS by George E. Poraythe July 1969 Reporduction in whole or in part is permitted for any purpose of the United States Government. Computer Science Department Stanford University

4 Abstract t is proved that in euclidean n-space the maximum M(p) and minimum m(p) of a fixed positive definite quadratic polynomial Q on spheres with fixed center are both convex functions of the radius p of the sphere. n the proof, which uses elementary calculus and a result of Forsythe ar.d Golub, m"(p) and M" (p) are shown to exist and lie in the interval [2>u, X ], where ^. are the eigenvalues of the quadratic form of Q. iience m"(p) > 0 and M"(p) > 0.

5 mmmmmtmmm < uununary Let A be a given symmetric, nonsinrular matrix of real elements and order n. Lc-c b be a given column vector of a real elements. For each real column n-vector x, the nonhomop;oneous quadratic polynomial Q(x) = (x-b) T A(x-b) (T denotes transpose) is a real number. Let ^i < \> < < A be the (necessarily) real eigenvalues of A. Let m(p) be the minimum of Q(x) on the sphere S = {x: x x = p }, and let M(p) be the maximum r of Q(x) on S. M. J. D. Powell asked the author whether m(p) is a convex function of p when A is positive definite. An affirmative answer is given by the theorem: (1) Theorem. f A is positive definite (i.e., if 0 < X ), then both m(p) and M(p) are convex functions of p, for all p > 0. Theorem (l) will follow from the following result: (2) Theorem. Let A be any nonsingular matrix. Then for p > 0, the second derivatives m"(p) and M"(p) both exist, and b) m M (p) > 2X 1 jmd M"(p) > 2?^. Equality occurs in (3) if and only if Ab = A,b. Moreover, (U) m"(p) <2X n and r(p) < 2A n and equality occurs in (k) if and only if Ab = X b. j

6 Heviev ol' Previous Work The proof of Theorem (2) is based on techniques developed in Forcythe and Golub [l], which dealt only with the cnse p = 1. The relevant results of [l] are now summarized and extended to general p. Jet {u,,...,u } he an orthonormal real set of eifjenvectors of A, with An. = X.u, (i - i,...,n). Let b = ^ b.u.. For any vector x in i' at whicli Q,(x) is stationary with respect to.'-, there is n real nijjnber X wi + h (5) A(x-b) = Ax (oj x x = p Letting x -... J] x u., we find from (5) t hat (7) x,. i x.b. 1 1 X^^-X so that (6) becomet n X 2 b? 1=1 (A.-xr For each ^iven vntne of p > 0, equation (6) determines from 2 to 2n real values of X. For each X so determined, equation (5) determines om or more vectors x (if all b. / 0, then x is unique). For any x, wc nave (9) Q(x X ) = f(x), whe-rc

7 ' ^^f J o JL y^s r (10) fw - r r {\'*y Now Q(x) is stationary with respect to S at any x. For given p, let A = A (p) and A = A (p) be the sinallest resp. largest values of X satisfying equation (8). Theorem (l+,l) of [l] states that f(a and f(a ) are the minimum resp. maximum values of Q(x) on C. r( D Much of [l] was devoted to the singular cases where come b. = 0. For the present investigation, where we are interested only in the values of Q(x), we simply omit from the sums (8) and (lö) all terms with b. = 0, and reduce n, if necessary. Having done that, it is then clear from (8) that, for any p, (11) A L <^1 and \ <A R This concludes the necessary summary of [l]. As a digression, the author notes that the main theorems (2.7) and (U. l) of [l] were proved in [l] by studying f(>,) and g{\) for complex values of X. n late 1965, Professor W. Kahan f unpublished] showed us how to prove those theorems more simply, using only real values of >. Proof of Theorem (2). With the above apparatus our problem is reduced to an exercise in the differential calculus. For each p > 0 we determine a unique Lagrange multiplier > = >v(p) from (8) eituer the minimal A or maximal A, For ease of exposition, suppose ^(p) = A. Then the function (12) m (p) = f(a(p)) 3

8 is determined from (10). Since f(x) and g(x) are; analytic for "K < "K, the function m(p) has derivatives of all order. We shall determine m"(p) by calculus. To simplify some expressions, we introduce the abbreviations 2 2 i i (15) a - ^ (p - 2, 5, k). Differentiating (10) and simplifying, we find; ilk) f = 2Aa 3 ; d2 (15) i = 2a + 6Aa, da ^ Now equation (8) states that, when A = A(p), (16) a 2 = P 2 Differentiating (8) twice with respect to p yields (17) a 3 = p ; (18) ^a?)+ 3(g) a u «i. dp ' ' 2 Solving (17) and (lb) in turn, we find (10) ax _ ^_, d P a-, (20) 0 d'a ] ^\ dp' Q 3 r-, L. a.. j

9 ^» F^PP^ P r Now, by the chain rule, dm df dp dx d^ dp and (21) d 2 m d 2 f /d^v 2 df df> dp 2 " :x 2 ^j ^ " dp 2 We now substitute into (2l) the expressions (U), (15), (19), and (20). We find that (22) r (p ) B 5 B ( 2 a 3 + 6^) ßl + 2>.a 5 ( J.. 3 JL^ dp 2 a. Hence ni"(p) ^ + C " a" (Xa 3 + ^^ ' by tä' simplifying, m"(p) i. f j L a 3 i-l(v^) 3 ' or (23) m"(p) = t 2 2 ^bi i-1 i\'-k)' Formula (25) is the end of our calculus exercise. n it, > is determined from solving (8). Note by (ll) that the factors (Xj-X) all -lave the sr'.me sign for i» 1, 2,..., n, whether "K m A or X A_. Hence ^ m"(p) is a weighted average with positive weights of the {X.}.

10 t followb that ~ m"(p) > }\ 1, with equality only when all "h. in (25) are dctunl to X., i.e., if b. «= 0 for ^. > A,. This proves (5), 1 1 x 1 and (1+) is proved analogously. This concludes the proof of Theorem (2). t would be desirable to have a simple geometrical proof. What if A is singular? f A is singular, that is, if some 'A. =0, the situation is somewhat more complicated, just as the case where some A.b. = 0 is complicated in [l]. Theorem (2) fails to hold for semidefinite matrices, because m"(p) may not exist for some p, as the following example shows; 2 T {2k) Example. For n = 2 let Q(x) S (x -l), where x s (x,xp) Then >(p) r i-p, 0 < p < 1, \o, 1 < p <«, so m"(i) does not exist. f A 1 B 0, the Lagrange multiplier remains at A «0 for all sufficiently large p. Theorem (l) can easily be extended to semidefinite matrices by continuity. We have (25) Theorem. f A is positive semidefinite (i.e., if 0 < A, ), then both m(p) and M(p) are convex functions of p for p > 0. n proof, we note that m(p) and M(p) are continuous functions of the elements of A. f A is semidefinite, it can be approximated by?. definite matrix A, for which m. and M. are convex, with A-AJ <. Letting - 0, we find that m lim m and M lim M- are convex. 6

11 HL m,. i Reference [l] George E. Forsythe and Gene H. Golub, "On the stationary values of a second-degrer polynomial on the unit sphere", J. Soc. ndust. Appl. Math., vol. 13 (1965)1 pp

12 i Unclassified SiTunlv ClüKHificulion DOCUMENT CONTROL DATA R&D isernriiy rlathillration of lltlo, body ol»tuttarl mid indfing HnnotU jn mwl hf tnffd tvhm hr ovmnll fpeil_l»_rl»t»jutä± OMioiN»tiNG tcivnt icntpottit mulhot) Computer Science Department Stanford University Stanford, California 9^305 ' «EPOBT TTLE 2«, neon r «ccunirv CL»»«iric»TPON UnclaDsifled >b OKOUP THE MAXMUM AND MNMUM OF A POSTVE DEFNTE QUADRATC POLYNOMAL ON A SPHERE ARE CONVEX FUNCTONS OF THE RADUS 4 DESCBPTVF NOTES rtvp* of'«porr «d inrlikivr '(( (» Manuscript for Piihl inat.inn (T^nhn^nl V -ort. ) "i AU THOWS ff im nam«, niidc//* iniri«f. mti nmmn George E. Forsythe t fl«repor T D* TE July 1969 CONTRACT OR GRANT NO 7«.»OT *l. NO OFf*Grs 9 9«. O*t GN A ' O A*fi REPORT NUMRCRtS^ ib. NO or»er» 1 h N000ll4-6T-A-O112-O0?9 PROJEC T NO NR CS 1^1* >t> OTHER RFPCH T NO'S) 'Anv nibmr nuinfe«rt that mas bt mulgntd hia raporl) none 10 DSTRBUTON STATEMENT Releasable withour limitations on dissemination. 11 SuPPLLMENTARV NOTES J SPONSORNG ML T ARy ACTVTY Office of Naval Research ABSTRACT t is proved that ir euclidean n-space the maximum M(p) and minimum m(p) of a fixed positive definite quadratic polynomial Q on spheres with fixed center are both convex functions of the radius p of the sphere. n the proof, which uses elementary calculus and a result of Forsythe and Golub, m"(p) and M"(p) are shown to exist and lie in the interval [2X [2^,2^],2A, where X^ X. are the eigenvalues of the quadratic form of Q. Hence" m"(p) > 0 and M"(p) > ^.1473 S/N (PAGf " Unclassified "^ecuritv Clastification

13 ^» ^^^» ^ n Unclassified 8>cufUy CluttftcHton HCV WO*0«noLt KOLC WT Maximum on a sphere Minimum on a sphere Quadratic function Convex DD :S?.AA73 < BA CK) (PAGE 2) Unclassified Steurity CUitiflcatlon