AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES

Size: px
Start display at page:

Download "AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES"

Transcription

1 MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-578(XX)- AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI Abstract. We propose a new matrx geometrc mean satsfyng the ten propertes gven by Ando, L and Mathas [Lnear Alg. Appl. 4]. Ths mean s the lmt of a sequence whch converges superlnearly wth convergence of order 3 whereas the mean ntroduced by Ando, L and Mathas s the lmt of a sequence havng order of convergence. Ths maes ths new mean very easly computable. We provde a geometrc nterpretaton and a generalzaton whch ncludes as specal cases our mean and the Ando-L-Mathas mean.. Introducton In several contexts, t s natural to generalze the geometrc mean of two postve real numbers a#b := ab to real symmetrc postve defnte n n matrces as (.) A#B := A(A B) / = A / (A / BA / ) / A /. Several papers, e.g. [3, 4, 9], and a chapter of the boo [], are devoted to studyng the geometry of the cone of postve defnte matrces P n endowed wth the Remannan metrc defned by ds = A / daa /, where B =,j b,j denotes the Frobenus norm. The dstance nduced by ths metrc s (.) d(a,b) = log(a / BA / ) It turns out that on ths manfold the geodesc jonng X and Y has equaton γ(t) = X / (X / Y X / ) t X / = X(X Y ) t =: X # t Y, t [,], and thus A#B s the mdpont of the geodesc jonng A and B. An analyss of numercal methods for computng the geometrc mean of two matrces s carred out n [7]. It s less clear how to defne the geometrc mean of more than two matrces. In the semnal wor [], Ando, L and Mathas lst ten propertes that a good matrx geometrc mean should satsfy, and show that several natural approaches based on generalzaton of formulas worng for the scalar case, or for the case of two matrces, do not wor well. They propose a new defnton of mean of Receved by the edtor??? and, n revsed form,???. Mathematcs Subject Classfcaton. Prmary 65F3; Secondary 5A48, 47A645. Key words and phrases. Matrx geometrc mean, geometrc mean, postve defnte matrx. c XXXX Amercan Mathematcal Socety

2 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI matrces satsfyng all the requested propertes. We refer to ths mean as to the Ando-L-Mathas mean, or shortly ALM-mean. The ALM-mean s the lmt of a recursve teraton process where at each step of the teraton geometrc means of matrces must be computed. One of the man drawbac of ths teraton s ts lnear convergence. In fact, the large number of teratons needed to approxmate each geometrc mean at all the recursve steps maes t qute expensve to actually compute the ALM-mean wth ths algorthm. Moreover, no other algorthms endowed wth a hgher effcency are nown. A class of geometrc means satsfyng the Ando, L, Mathas requrements has been ntroduced n [8]. These means are defned n terms of the soluton of certan matrx equatons. Ths approach provdes nterestng theoretcal propertes concernng the means but no effectve tools for ther computaton. In ths paper, we propose a new matrx geometrc mean satsfyng the ten propertes of Ando, L and Mathas. Lewse the ALM-mean, our mean s defned as the lmt of an teraton process wth the relevant dfference that convergence s superlnear wth order of convergence at least three. Ths property maes t much less expensve to compute ths geometrc mean snce the number of teratons requred to reach a hgh accuracy s dropped down to just a few ones. The teraton on whch our mean s based has a smple geometrcal nterpretaton. In the case = 3, gven the postve defnte matrces A,A,A 3, we generate three matrx sequences A (m),a (m),a (m) 3 startng from A () = A, =,,3. At the step m+, the matrx A (m+) s chosen along the geodesc whch connects A (m) wth the mdpont of the geodesc connectng A (m) to A (m) 3 at dstance /3 from A (m). The matrces A (m+) and A (m+) 3 are smlarly defned. In the case of the Eucldean geometry, just one step of the teraton provdes the value of the lmt,.e., the centrod of the trangle wth vertces A (m),a (m), A (m) 3. In fact, n a trangle the medans ntersect each other at /3 of ther length. In the dfferent geometry of the cone of postve defnte matrces the geodescs whch play the role of the medans, mght even not ntersect each other. In the case of matrces A,A,...,A, the matrx A (m+) s chosen along the geodesc whch connects A (m) wth the geometrc mean of the remanng matrces, at dstance /(+) from A (m). In the customary geometry, ths pont s the common ntersecton pont of all the medans of the -dmensonal smplex formed by all the matrces A (m), =,...,. We prove that the sequences {A (m) } m, =,...,, converge to a common lmt Ā wth order of convergence at least 3. The lmt Ā s our defnton of geometrc mean of A,...,A. It s nterestng to pont out that our mean and the ALM-mean of matrces can be vewed as two specfc nstances of a class of more general means dependng on parameters s [,], =,...,. All the means of ths class satsfy the requrements of Ando, L and Mathas, moreover, the ALM-mean s obtaned wth s = (/,,,...,), for s = (s ), whle our mean s obtaned wth s = (( )/,( )/( ),...,/). The new mean s the only one n ths class for whch the matrx sequences generated at each recursve step converge superlnearly. The artcle s structured as follows. After ths ntroducton, n Secton we present the ten Ando L Mathas propertes and brefly descrbe the ALM-mean; then, n Secton 3, we propose our new defnton of a matrx geometrc mean and

3 AN EFFECTIVE MATRIX GEOMETRIC MEAN 3 prove some of ts propertes by gvng also a geometrcal nterpretaton; n Secton 4 we provde a generalzaton whch ncludes the ALM-mean and our mean as two specal cases. Fnally, n Secton 5 we present some numercal experments of explct computatons nvolvng ths means concernng some problems from Physcs. It turns out that, n the case of sx matrces, the speed up reached by our approach wth respect to the ALM-mean s by a factor greater than. We also expermentally demonstrate that the ALM-mean s dfferent, even though very close, from our mean. Fnally, for = 3 we provde a pctoral descrpton of the parametrc famly of geometrc means.. Known results Throughout ths secton we use the postve semdefnte orderng defned by A B f A B s postve semdefnte. We denote by A the transpose conjugate of A... Ando L Mathas propertes for a matrx geometrc mean. Ando, L and Mathas [] proposed the followng lst of propertes that a good geometrc mean G( ) of three matrces should satsfy. P: Consstency wth scalars. If A, B, C commute then G(A,B,C) = (ABC) /3. P: Jont homogenety. G(αA,βB,γC) = (αβγ) /3 G(A,B,C). P3: Permutaton nvarance. For any permutaton π(a,b,c) of A, B, C t holds G(A,B,C) = G(π(A,B,C)). P4: Monotoncty. If A A, B B, C C, then G(A,B,C) G(A,B,C ). P5: Contnuty from above. If A n, B n, C n are monotonc decreasng sequences convergng to A, B, C, respectvely, then G(A n,b n,c n ) converges to G(A, B, C). P6: Congruence nvarance. For any nonsngular S, G(S AS,S BS,S CS) = S G(A,B,C)S. P7: Jont concavty. If A = λa + ( λ)a, B = λb + ( λ)b, C = λc +( λ)c, then G(A,B,C) λg(a,b,c )+( λ)g(a,b,c ). P8: Self-dualty. G(A,B,C) = G(A,B,C ). P9: Determnant dentty. detg(a,b,c) = (det Adet B det C) /3. P: Arthmetc geometrc harmonc mean nequalty: A + B + C 3 ( A + B + C ) G(A,B,C). 3 Moreover, t s proved n [] that P5 and P are consequences of the others. Notce that all these propertes can be easly generalzed to the mean of any number of matrces. We wll call a geometrc mean of three or more matrces any map G( ) satsfyng P P or ther analogous for a number 3 of entres... The Ando L Mathas mean. Here and hereafter, we use the followng notaton. We denote by G (A,B) the usual geometrc mean A#B and, gven the -tuple (A,...,A ), we defne Z (A,...,A ) = (A,...,A,A +,...,A ), =,...,, that s, the -tuple where the -th term has been dropped out.

4 4 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI In [], Ando, L and Mathas note that the prevously proposed defntons of means of more than two matrces do not satsfy all the propertes P P, and propose a new defnton that fulflls all of them. Ther mean s defned nductvely on the number of arguments. Gven A,...,A postve defnte, and gven the defnton of a geometrc mean G ( ) of matrces, they set A () = A, =,...,, and defne for r (.) A (r+) := G (Z (A (r),...,a(r) )), =,...,. For = 3, the teraton reads A (r+) G (B (r),c (r) ) B (r+) = G (A (r),c (r) ). C (r+) G (A (r),b (r) ) Ando, L and Mathas show that the sequences (A (r) ) r= converge to the same matrx Ã, and fnally defne G (A,...,A ) = Ã. In the followng, we shall denote by G( ) the Ando L Mathas mean, droppng the subscrpt when not essental. An addtonal property of the Ando L Mathas mean whch wll turn out to be mportant n the convergence proof s the followng. Recall that ρ(x) denotes the spectral radus of X, and let R(A,B) := max(ρ(a B),ρ(B A)). Ths functon s a multplcatve metrc, that s, we have R(A,B) wth equalty ff A = B, and R(A,C) R(A,B)R(B,C). The addtonal property s P: For each, and for each par of sequences (A,...,A ), (B,...,B ) t holds ( / R (G(A,...,A ),G(B,...,B )) R(A,B )). = 3. A new matrx geometrc mean 3.. Defnton. We are gong to defne for each a new mean Ḡ( ) of matrces satsfyng P P. Let Ḡ(A,B) = A#B, and suppose that the mean has already been defned for up to matrces. Let us denote shortly T (r) = Ḡ (Z (Ā(r),...,Ā(r) )) and defne Ā(r+) for =,..., as (3.) Ā (r+) := Ḡ(Ā(r),T (r),t (r),...,t (r) ), } {{ } tmes wth Ā() = A for all. Notce that apparently ths needs the mean Ḡ( ) to be already defned; n fact, n the specal case n whch of the arguments are concdent, the propertes P and P6 alone allow one to determne the common value of any geometrc mean: G(X,Y,Y,...,Y ) =X / G(I,X / Y X /,...,X / Y X / )X / =X / (X / Y X / ) X / = X # Y.

5 AN EFFECTIVE MATRIX GEOMETRIC MEAN 5 Thus we can use ths smpler expresson drectly n (3.) and set (3.) Ā (r+) = Ā(r) # T (r). In sectons 3.3 and 3.4, we are gong to prove that the sequences (Ā(r) ) r= converge to a common lmt Ā wth order of convergence at least three, and ths wll enable us to defne Ḡ(A,...,A ) := Ā. In the followng, we wll drop the ndex from Ḡ( ) when t can be easly nferred from the context. 3.. Geometrcal nterpretaton. In [3], an nterestng geometrcal nterpretaton of the Ando L Mathas mean s proposed for = 3. We propose an nterpretaton of the new mean Ḡ( ) n the same sprt. For = 3, the teraton defnng Ḡ( ) reads Ā (r+) Ā (r) # B (r+) ( B (r) # C (r) ) 3 = B (r) # (Ā(r) # C (r) ) C (r+) 3 C (r) # (Ā(r) # B. (r) ) 3 We can nterpret ths teraton as a geometrcal constructon n the followng way. To fnd e.g. Ā (r+), the algorthm s: () Draw the geodesc jonng B (r) and C (r), and tae ts mdpont M (r) ; () Draw the geodesc jonng Ā(r) and M (r), and tae the pont lyng at /3 of ts length: ths s Ā(r+). If we execute the same algorthm on the Eucldean plane, replacng the word geodesc wth straght lne segment, t turns out that Ā(), B(), and C () concde n the centrod of the trangle wth vertces A, B, C. Thus, unle the Eucldean counterpart of the Ando L Mathas mean, ths process converges n one step on the plane. Roughly speang, when A, B and C are very close to each other, we can approxmate (n some ntutve sense) the geometry on the Remannan manfold P n wth the geometry on the Eucldean plane: snce ths constructon to fnd the centrod of a plane trangle converges faster than the Ando L Mathas one, we can expect that also the convergence speed of the resultng algorthm s faster. Ths s ndeed what wll result after a more accurate convergence analyss Global convergence and propertes P P. In order to prove that the teraton (3.) s convergent (and thus that Ḡ( ) s well defned), we are gong to adapt a part of the proof of Theorem 3. of [] (namely, Argument ). Theorem 3.. Let A,...A be postve defnte. () All the sequences (Ā(r) ) r= converge for r to a common lmt Ā; () the functon Ḡ(A,...,A ) satsfes P P. Proof. We wor by nducton on. For =, our mean concdes wth the ALMmean, so all the requred wor has been done n []. Let us now suppose that the thess holds for all. We have Ā (r+) (Ā(r) ) + ( )T (r) = Ā (r), where the frst nequalty follows from P for the ALM-mean G ( ) (remember that n the specal case n whch of the arguments concde, G ( ) = Ḡ( )),

6 6 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI and the second from P for Ḡ ( ). Thus, (3.3) = Ā (r+) = Ā (r) A. Therefore, the sequence (Ā(r),...,Ā(r) ) r= s bounded, and there must be a convergng subsequence, say, convergng to (Ā,...,Ā). Moreover, for each p,q {,...,} we have R(Ā(r+) p,ā(r+) q ) R(Ā(r) p,ā(r) q R(Ā(r) p,ā(r) q ) / R(T (r) p / ) (R(Ā(r) q,t q (r) = ),Ā(r) p ) ) = R(Ā(r) p,ā(r) q ) /, where the frst nequalty follows from P n the specal case, and the second from P n the nductve hypothess. Passng at the lmt of the convergng subsequence, one can verfy that R(Āp,Āq) R(Āp,Āq) /, from whch we get R(Āp,Āq), that s, Ā p = Āq, because of the propertes of R,.e., the lmt of the subsequence s n the form (Ā,Ā,...,Ā). Suppose there s another subsequence convergng to ( B, B,..., B); then, by (3.3), we have both Ā B and B Ā, that s, Ā = B. Therefore, the sequence has only one lmt pont, thus t s convergent. Ths proves the frst pont of the theorem. We now turn to show that P holds for our mean Ḡ( ). Consder -tuples B (r) A,...,A and B,...,B, and let be defned as Ā(r) but startng the teraton from the -tuple (B ) nstead of (A ). We have for each R(Ā(r+), B(r+) ) R(Ā(r), R(Ā(r) (r), B (r) / B ) R(Ḡ(Z(Ā(r) ) /,...,Ā(r) )),Ḡ(Z ( R(Ā(r) (r) j, B j ) j B (r),..., B (r) ))) Ths yelds = j= R(Ā(r) (r) j, B j ) /. = R(Ā(r+), (r+) B ) = R(Ā(r) (r), B ); channg together these nequaltes for successve values of r and passng to the lmt, we get R(G(A,...,A ),G(B,...,B )) R(A,B ), whch s P. The other propertes P P4 and P6 P9 (remember that P5 and P are consequences of these) are not dffcult to prove. All the proofs are qute smlar, and can be establshed by nducton, usng also the fact that snce they hold for the ALM-mean, they can be appled to the mean Ḡ( ) appearng n (3.) (snce we just proved that all possble geometrc means tae the same value f appled wth =

7 AN EFFECTIVE MATRIX GEOMETRIC MEAN 7 equal arguments). For the sae of brevty, we provde only the proof for three of these propertes. P: We need to prove that f the A commute then Ḡ(A,...,A ) = (A A ) /. Usng the nductve hypothess, we have T () = j Ā. Usng the fact that P holds for the ALM-mean, we have = A / j Ā () A j = = A /, as needed. So, from the second teraton on, we have Ā(r) = Ā(r) Ā (r) P4: Let T (r) = = A/. and Ā (r) be defned as T (r) = = and Ā(r) but startng from A A. Usng monotoncty n the nductve case and n the ALM-mean, we have for each s and for each and thus T (r+) Ā (r+) T (r+) Ā (r+). Passng to the lmt for r, we obtan P4. P7: Suppose A = λa (r) + ( λ)a, and let T, (resp. T (r) ) and Ā (r), (resp. Ā (r) ) be defned as T (r) and Ā(r) but startng from A (resp. A ). Suppose that for some r t holds Ā(r) λā (r) +( λ)ā (r) for all. Then by jont concavty and monotoncty n the nductve case we have T (r+) =Ḡ(Z(Ā(r),...,Ā(r) )) Ḡ(Z(λĀ (r) + ( λ)ā (r),...,λā (r) + ( λ)ā (r) )) λt (r) + ( λ)t (r), and by jont concavty and monotoncty of the Ando L Mathas mean we have Ā (r+) =Ā(r) ( # λā (r) λā (r+) T (r) + ( λ)ā (r) + ( λ)ā (r+). ) ( # λt (r) Passng to the lmt for r, we obtan P7. ) + ( λ)t (r) 3.4. Cubc convergence. In ths secton, we wll use the bg-o notaton n the norm sense, that s, we wll wrte X = Y +O(ε h ) to denote that there are unversal postve constants ε < and θ such that for each < ε < ε t holds X Y θε h. The usual arthmetc rules nvolvng ths notaton hold. In the followng, these constants may depend on, but not on the specfc choce of the matrces nvolved n the formulas.

8 8 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI Theorem 3.. Let < ε <, M and Ā() = A, =,...,, be postve defnte n n, and E := M A I. Suppose that E ε for all =,...,. Then, for the matrces Ā() defned n (3.) the followng hold. C: We have (3.4) M Ā () I = T + O(ε 3 ) where T := j= E j 4 (E E j ).,j= C: There are postve constants θ, σ and ε < (all of whch may depend on ) such that for all ε ε t holds M Ā () I θε 3 for a sutable matrx M satsfyng M M I σε. C3: The teraton (3.) converges at least cubcally. C4: We have (3.5) M Ḡ(A,...,A ) I = O(ε 3 ). Proof. Let us frst fnd a local expanson of a generc pont on the geodesc A# t B: let M A = I + F and M B = I + F wth F δ, F δ, < δ <. Then we have M (A# t B) =M A(A B) t = (I + F ) ( (I + F ) (I + F ) ) t (3.6) =(I + F ) ( (I F + F + O(δ 3 ))(I + F ) ) t =(I + F ) ( I + F F F F + F + O(δ 3 ) ) t =(I + F ) ( I + t(f F F F + F ) ) t(t ) + (F F ) + O(δ 3 ) t(t ) =I + ( t)f + tf + (F F ) + O(δ 3 ), where we made use of the matrx seres expanson (I + X) t = I + tx + t(t ) X + O(X 3 ). Now, we are gong to prove the theorem by nducton on n the followng way. Let C denote the asserton C of the theorem (for =,...4) for a gven value of. We show that () C holds; () C = C ; (3) C = C3,C4 ; (4) C4 = C +. It s clear that these clams mply that the results C C4 hold for all by nducton; we wll now turn to prove them one by one. () Ths s smply equaton (3.6) for t =. () It s obvous that T = O(ε); thus, choosng M := M(I + T ) one has (3.7) Ā () = M(I + T + O(ε 3 )) = M (I + (I + T ) O(ε 3 )) = M (I + O(ε 3 )).

9 AN EFFECTIVE MATRIX GEOMETRIC MEAN 9 (3.8) Usng explct constants n the bg-o estmates, we get M Ā () I θε 3, M M I σε for sutable constants θ and σ. (3) Suppose ε s small enough to have θε 3 ε. We shall apply C wth ntal matrces Ā(), wth ε = θε 3 n leu of ε and M n leu of M, gettng M Ā () I θε 3, M M I σε. Repeatng agan for all the steps of our teratve process, we get for all s =,,... Ms Ā (s) I θε 3 s = ε s, M s M s+ I σεs (3.9) wth ε s+ := θε 3 s and M := M. For smplcty s sae, we ntroduce the notaton d(x,y ) := X Y I for any two n n symmetrc postve defnte matrces X and Y. It wll be useful to notce that X Y X X Y I X d(x,y ) and d(x,z) = (X Y I)(Y Z I) + X Y I + Y Z I d(x,y )d(y,z) + d(x,y ) + d(y,z). Wth ths notaton, we can restate (3.8) as d(m s,ā(s) ) ε s, d(m s,m s+ ) σε s. We wll now prove by nducton that, for ε smaller than a fxed constant, t holds (3.) d(m s,m s+t ) ( ) t σε s. Frst of all, t holds for all t ε s+t = θ 3t ε 3t, whch, for ε smaller than mn(/8,θ ), mples εs+t ε s ε 3 t s ε t s Now, usng (3.9), and supposng addtonally ε σ, we have d(m s,m s+t+ ) d(m s,m s+t )d(m s+t,m s+t+ ) + d(m s,m s+t ) + d(m s+t,m s+t+ ) ( ) ( t σε s + σε s σε s+t + ε ) s+t ε s ( ) ( t σε s + σε s ε ) s+t ε s Thus, we have for each t ( t ) σε s + σε s t+ = ( ) t+ σε s M t M M M M t I σ M ε, t+.

10 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI whch mples M t M for all t. By a smlar argument, (3.) M s+t M s M s d(m s+t,m s ) σ M ε s, (3.) Due to the bounds already mposed on ε, the sequence ε s tends monotoncally to zero wth cubc convergence rate; thus (M t ) s a Cauchy sequence and therefore converges. In the followng, let M be ts lmt. The convergence rate s cubc, snce passng to the lmt (3.) we get M M s σ M ε s. Now, usng the other relaton n (3.8), we get Ā(s) M Ā (s) M s + M M s M d(m s,ā(s) ) + σ M ε s (σ + ) M ε s, that s, Ā (s) converges wth cubc convergence rate to M. Thus C3 s proved. By (3.9), (3.), and (3.8), we have d(m,ā(t) ) d(m,m t )d(m t,ā(t) ) + d(m,m t ) + d(m t,ā(t) ) σε ε t + σε + ε t (4σ + )ε = O(ε 3 ), whch s C4. (4) Usng C4 and (3.6) wth F = E +, F = M Ḡ(A,...,A ) = T + O(ε 3 ), δ = ε, we have ( M Ā () + =M A + # Ḡ(A,...,A ) + =I + + E T Observe that ( + ) ( E + ) ) E + O(ε 3 ). T = S + P ( )Q where S = = E, Q = = E, P =,j=, j E E j. Snce S = P +Q and S + = S +E +, Q + = Q +E+, P + = P +E + S + S E +, from (3.) one fnds that M Ā () + =I + + S + =I + T + + O(ε 3 ). = ( + ) Q + + ( + ) P + + O(ε 3 ) Snce the expresson we found s symmetrc wth respect to the E, t follows that Ā() j has the same expanson for any j. Observe that Theorems 3. and 3. mply that the teraton (3.) s globally convergent wth order of convergence at least 3. It s worth to pont out that, n the case where the matrces A, =,...,A, commute each other, the teraton (3.) converges n just one step,.e., Ā () = Ā

11 AN EFFECTIVE MATRIX GEOMETRIC MEAN for any. In the noncommutatve general case, one has det(ā(s) ) = det(ā) for any and for any s,.e., the determnant converges n one sngle step to the determnant of the matrx mean. Our mean s dfferent from the ALM-mean, as we wll show wth some numercal experments n Secton 5. In the next secton 4 we prove that our mean and the ALM-mean belong to a general class of matrx geometrc means, whch depends on a set of parameters. 4. A new class of matrx geometrc means In ths secton we ntroduce a new class of matrx means dependng on a set of parameters s,...,s and show that the ALM-mean and our mean are two specfc nstances of ths class. For the sae of smplcty, we descrbe ths generalzaton n the case of = 3 matrces A,B,C. The case > 3 s outlned. Here, the dstance between two matrces s defned n (.). For = 3, the algorthm presented n Secton 3 replaces the trple A,B,C wth A,B,C where A s chosen n the geodesc connectng A wth the mdpont of the geodesc connectng B and C, at dstance /3 from A, and smlarly s made for B and C. In our generalzaton we use two parameters s,t [,]. We consder the pont P t = B# t C n the geodesc connectng B to C at dstance t from B. Then we consder the geodesc connectng A to P t and defne A the matrx on ths geodesc at dstance s from A. That s, we set A = A# s (B# t C). Smlarly we do wth B and C. Ths transformaton s recursvely repeated so that the matrx sequences A (r), B (r), C (r) are generated by means of (4.) A (r+) = A (r) # s (B (r) # t C (r) ), B (r+) = B (r) # s (C (r) # t A (r) ), C (r+) = C (r) # s (A (r) # t B (r) ), r =,,... startng wth A () = A, B () = B, C () = C. By followng the same arguments of Secton 3 t can be shown that the three sequences have a common lmt G s,t for any s,t [,]. Moreover, for s =, t = / one obtans the ALM-mean,.e., G = G,, whle for s = /3, t = / the lmt concdes wth our mean,.e., Ḡ = G 3,. Moreover, t s possble to prove that for any s,t [,] the lmt satsfes the condtons P P so that t can be consdered a good geometrc mean. Concernng the convergence speed of the sequence generated by (4.) we may perform a more accurate analyss. Assume that A = M(I + E ), B = M(I + E ), C = M(I + E 3 ), where E ε <, =,,3. Then, applyng (3.6) n (4.) yelds A B C. = M(I + ( s)e + s( t)e + ste 3 + st(t ) H + s(s ) (H + th ) ). = M(I + ( s)e + s( t)e 3 + ste + st(t ) H3 + s(s ) (H + th 3 ) ). = M(I + ( s)e3 + s( t)e + ste + st(t ) H + s(s ) (H 3 + th ) ) where. = denotes equalty up to O(ε 3 ) terms, wth H = E E, H = E E 3, H 3 = E 3 E. Whence we have A = M(I + E ), B = M(I + E ), C =

12 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI M(I + E 3), wth E E =. C(s,t) E 3 where E E E 3 + C(s,t) = st(t ) H H 3 H + s(s ) ( s)i s( t)i sti sti ( s)i s( t)i s( t)i sti ( s)i. (H th ) (H th 3 ) (H 3 th ) Observe that the bloc crculant matrx C(s,t) has egenvalues λ =, λ = ( 3 s)+ 3 s(t ), and λ 3 = λ, wth multplcty n, where =. Moreover, the par (s,t) = (/3,/) s the only one whch yelds λ = λ 3 =. In fact (/3, /) s the only par whch provdes superlnear convergence. For the ALMmean, where t = / and s = t holds λ = λ 3 = / whch s the rate of convergence of the ALM teraton []. In the case of > 3 matrces, gven the ( )-tuple (s,s,...,s ) we may recursvely defne G s,...,s (A,...,A ) as the common lmt of the sequences generated by A (r+) = A (r) # s G s,...,s (Z (A (r),...,a(r) )), =,...,. Observe that wth (s,...,s ) = (/,,,...,) one obtans the ALM-mean, whle wth (s,...,s ) = (( )/,( )/( ),...,/) one obtans the new mean ntroduced n Secton Numercal experments We have mplemented the two teratons convergng to the ALM mean and to the newly defned geometrc mean n Matlab, and run some numercal experments on a quad-xeon.8ghz computer. To compute matrx square roots we used Matlab s bult-n sqrtm functon, whle for p-th roots wth p > we used the rootm functon n Ncholas Hgham s Matrx Computaton Toolbox [6]. To counter the loss of symmetry due to the accumulaton of computatonal errors, we chose to dscard the magnary part of the computed roots. The experments have been performed on the same data set as the paper [9]. It conssts of fve sets each composed of four to sx 6 6 postve defnte matrces, correspondng to physcal data from elastcty experments conducted by Hearmon [5]. The matrces are composed of smaller dagonal blocs of szes up to 4 4, dependng on the symmetres of the nvolved materals. Two to three sgnfcatve dgts are reported for each experments. We have computed both the ALM mean and the newly defned mean of these sets; as a stoppng crteron for each computed mean, we chose max A (r+) A (r) < ε, where X := max,j X j, wth ε =. The CPU tmes, n seconds, are reported n Table. For four matrces, the speed gan s a factor of, and t ncreases even more for more than four matrces. We then focused on Hearmon s second data set (ammonum dhydrogen phosphate), composed of four matrces. In Table, we reported the number of outer ( = 4) teratons needed and the average number of teratons needed to reach

13 AN EFFECTIVE MATRIX GEOMETRIC MEAN 3 Data set (number of matrces) ALM mean New mean NaClO 3 (5) 3..3 Ammonum dhydrogen phosphate (4) Potassum dhydrogen phosphate (4) Quartz (6) Rochelle salt (4)..53 Table. CPU tmes n seconds for the Hearmon elastcty data ALM mean New mean Outer teratons 3 3 Avg. nner teratons 8.3 Matrx square roots (sqrtm) 55 7 Matrx p-th roots (rootm) 84 Table. Number of nner and outer teratons needed, and number of matrx roots needed convergence n the nner ( = 3) teratons (remember that the computaton of a mean of four matrces requres the computaton of three means of three matrces at each of ts steps). Moreover, we measured the number of square and p-th roots needed by the two algorthms, snce they are the most expensve operaton n the algorthm. From the results, t s evdent that the speed gan n the new mean s due not only to the reducton of the number of outer teratons, but also of the number of nner teratons needed to get convergence at each step of the nner mean calculatons. When the number of nvolved matrces becomes larger, these speedups add up at each level. Hearmon s elastcty data are not sutable to measure the accuracy of the algorthm, snce the results to be obtaned are not nown. To measure the accuracy of the computed results, we computed nstead G(A 4,I,I,I) A, whch should yeld zero n exact arthmetc (due to P), and ts analogue wth the new mean. We chose A to be the frst matrx n Hearmon s second data set. Moreover, n order to obtan results closer to machne precson, n ths experment we changed the stoppng crteron choosng ε = 3 Operaton G(A 4,I,I,I) A Ḡ(A 4,I,I,I) A Result 3.6E-3.8E-4 The results are well wthn the errors permtted by the stoppng crteron, and show that both algorthms can reach a satsfyng precson. The followng examples provde an expermental proof that our mean s dfferent from the ALM-mean. Consder the followng matrces [ a b A = b a ], B = [ a b b a ] [, C = c Observe that the trple (A,B,C) s transformed nto (B,A,C) under the map X S XS, for S = dag(, ). In ths way, any matrx mean G(A,B,C) ].

14 4 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI satsfyng condton P3 s such that G = S GS, that s, the off-dagonal entres of G are zero. Whence, G must be dagonal. Wth a =,b =,c = 4, for the ALM-mean G and our mean Ḡ one fnds that [ ] [ ] Ḡ =, G =, where we reported the frst dgts. Observe that the determnant of both the matrces s 6, that s, the geometrc mean of det A,det B,det C, moreover, ρ(ḡ) < ρ(g). For the matrces A = C = 3 5, B =, D = 3, 5, one has Ḡ = , G = Ther egenvalues are (6.58, 3.845,.39), and (6.85, ,.368), respectvely. Observe that, unle n the prevous example, t holds ρ(ḡ) > ρ(g). In order to llustrate the propertes of the set {G s,t : (s,t) (,] (,)}, where G s,t s the mean of three matrces defned n Secton 4, we consdered the ntervals [/5, ], [/5, 4/5] and dscretzed them nto two sets S, T of 5 equdstant ponts {/5 = s < s < < s 5 = }, {/5 = t < t < < t 5 = 4/5}, respectvely. For each par (s,t j ) S T,,j =,...,5, we computed G s,t j and the orthogonal projecton (x(,j),y(,j),z(,j)) of the matrx G s,t j G 3,, over a three dmensonal fxed randomly generated subspace. The set V = {(x(,j),y(,j),z(,j)) R 3,,j =,...,5} has been plotted wth the Matlab command mesh(x,y,z) whch connects each pont of coordnates (x(, j), y(, j), z(, j)) to ts four neghborhoods of coordnates (x( + δ,j + γ),y( + δ,j + γ),z( + δ,j + γ))) for δ,γ {, }. Fgure dsplays the set V from sx dfferent ponts of vew, where the matrces A,B and C of sze 3, have been randomly generated. The set appears to be a flat surface wth part of the edge tghtly folded on tself. The geometrc mean G 3, corresponds to the pont of coordnates (,,) whch s denoted by a small crcle and seems to be located n the central part of the fgure. These propertes, reported for only one trple (A,B,C), are mantaned wth very lght dfferences n all the plots that we have performed. The software concernng our experments can be delvered upon request.

15 AN EFFECTIVE MATRIX GEOMETRIC MEAN 5 4 x 4 3 x 4 4 x x 4.5 x x 4 x x x x 3 x x 4 x 4 4 x x 3 x x x 4 6 Fgure. Plot of the set V, the small crcle corresponds to G /3,/. Acnowledgments The authors wsh to than Bruno Iannazzo for the many nterestng dscussons on ssues related to matrx means, and an anonymous referee for the useful comments and suggestons to mprove the presentaton.

16 6 DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI References. T. Ando, Ch-Kwong L, and Roy Mathas, Geometrc means, Lnear Algebra Appl. 385 (4), MR MR63358 (5f:4749). Rajendra Bhata, Postve defnte matrces, Prnceton Seres n Appled Mathematcs, Prnceton Unversty Press, Prnceton, NJ, 7. MR MR8476 (7:55) 3. Rajendra Bhata and John Holbroo, Noncommutatve geometrc means, Math. Intellgencer 8 (6), no., MR MR893 (7g:473) 4., Remannan geometry and matrx geometrc means, Lnear Algebra Appl. 43 (6), no. -3, MR MR9895 (7c:53) 5. R. F. S. Hearmon, The elastc constants of pesoelectrc crystals, J. Appl. Phys. 3 (95), Ncholas J. Hgham, The Matrx Computaton Toolbox, 7. Bruno Iannazzo and Beatrce Men, The matrx geometrc mean and other matrx functons: a unfyng framewor, Tech. report, Dpartmento d Matematca, Unverstà d Psa, Yongdo Lm, On Ando-L-Mathas geometrc mean equatons, Lnear Algebra Appl. 48 (8), no. 8-9, MR MR Maher Moaher, On the averagng of symmetrc postve-defnte tensors, J. Elastcty 8 (6), no. 3, MR MR365 (7a:747) Dpartmento d Matematca, Unverstà d Psa, Largo B. Pontecorvo 5, 567 Psa, Italy E-mal address: bn, Scuola Normale Superore, Pazza de Cavaler 6, 566 Psa, Italy E-mal address:

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

More information

where the coordinates are related to those in the old frame as follows.

where the coordinates are related to those in the old frame as follows. Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

Loop Parallelization

Loop Parallelization - - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

On the Solution of Indefinite Systems Arising in Nonlinear Optimization

On the Solution of Indefinite Systems Arising in Nonlinear Optimization On the Soluton of Indefnte Systems Arsng n Nonlnear Optmzaton Slva Bonettn, Valera Ruggero and Federca Tnt Dpartmento d Matematca, Unverstà d Ferrara Abstract We consder the applcaton of the precondtoned

More information

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Research Note APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES * Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Generalizing the degree sequence problem

Generalizing the degree sequence problem Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

On some special nonlevel annuities and yield rates for annuities

On some special nonlevel annuities and yield rates for annuities On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

More information

Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering

Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering Out-of-Sample Extensons for LLE, Isomap, MDS, Egenmaps, and Spectral Clusterng Yoshua Bengo, Jean-Franços Paement, Pascal Vncent Olver Delalleau, Ncolas Le Roux and Mare Oumet Département d Informatque

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

TENSOR GAUGE FIELDS OF DEGREE THREE

TENSOR GAUGE FIELDS OF DEGREE THREE TENSOR GAUGE FIELDS OF DEGREE THREE E.M. CIOROIANU Department of Physcs, Unversty of Craova, A. I. Cuza 13, 2585, Craova, Romana, EU E-mal: manache@central.ucv.ro Receved February 2, 213 Startng from a

More information

Embedding lattices in the Kleene degrees

Embedding lattices in the Kleene degrees F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded

More information

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching) Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

A Performance Analysis of View Maintenance Techniques for Data Warehouses

A Performance Analysis of View Maintenance Techniques for Data Warehouses A Performance Analyss of Vew Mantenance Technques for Data Warehouses Xng Wang Dell Computer Corporaton Round Roc, Texas Le Gruenwald The nversty of Olahoma School of Computer Scence orman, OK 739 Guangtao

More information

The k-binomial Transforms and the Hankel Transform

The k-binomial Transforms and the Hankel Transform 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 9 (2006, Artcle 06.1.1 The k-bnomal Transforms and the Hankel Transform Mchael Z. Spvey Department of Mathematcs and Computer Scence Unversty of Puget

More information

Supplementary material: Assessing the relevance of node features for network structure

Supplementary material: Assessing the relevance of node features for network structure Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Matrix Multiplication I

Matrix Multiplication I Matrx Multplcaton I Yuval Flmus February 2, 2012 These notes are based on a lecture gven at the Toronto Student Semnar on February 2, 2012. The materal s taen mostly from the boo Algebrac Complexty Theory

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Conversion between the vector and raster data structures using Fuzzy Geographical Entities Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Do Hidden Variables. Improve Quantum Mechanics?

Do Hidden Variables. Improve Quantum Mechanics? Radboud Unverstet Njmegen Do Hdden Varables Improve Quantum Mechancs? Bachelor Thess Author: Denns Hendrkx Begeleder: Prof. dr. Klaas Landsman Abstract Snce the dawn of quantum mechancs physcst have contemplated

More information

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

More information

Calculating the high frequency transmission line parameters of power cables

Calculating the high frequency transmission line parameters of power cables < ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs

More information

General Auction Mechanism for Search Advertising

General Auction Mechanism for Search Advertising General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

More information

The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance

The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance JOURNAL OF RESEARCH of the Natonal Bureau of Standards - B. Mathem atca l Scence s Vol. 74B, No.3, July-September 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1 (4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

CHAPTER 7 VECTOR BUNDLES

CHAPTER 7 VECTOR BUNDLES CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

INSTITUT FÜR INFORMATIK

INSTITUT FÜR INFORMATIK INSTITUT FÜR INFORMATIK Schedulng jobs on unform processors revsted Klaus Jansen Chrstna Robene Bercht Nr. 1109 November 2011 ISSN 2192-6247 CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL Insttut für Informat

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

On fourth order simultaneously zero-finding method for multiple roots of complex polynomial equations 1

On fourth order simultaneously zero-finding method for multiple roots of complex polynomial equations 1 General Mathematcs Vol. 6, No. 3 (2008), 9 3 On fourth order smultaneously zero-fndng method for multple roots of complex polynomal euatons Nazr Ahmad Mr and Khald Ayub Abstract In ths paper, we present

More information

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

I. INTRODUCTION. 1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes

I. INTRODUCTION. 1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes he Knematc Analyss of a Symmetrcal hree-degree-of-freedom lanar arallel Manpulator Damen Chablat and hlppe Wenger Insttut de Recherche en Communcatons et Cybernétque de Nantes, rue de la Noë, 442 Nantes,

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

A Fast Incremental Spectral Clustering for Large Data Sets

A Fast Incremental Spectral Clustering for Large Data Sets 2011 12th Internatonal Conference on Parallel and Dstrbuted Computng, Applcatons and Technologes A Fast Incremental Spectral Clusterng for Large Data Sets Tengteng Kong 1,YeTan 1, Hong Shen 1,2 1 School

More information

Fast degree elevation and knot insertion for B-spline curves

Fast degree elevation and knot insertion for B-spline curves Computer Aded Geometrc Desgn 22 (2005) 183 197 www.elsever.com/locate/cagd Fast degree elevaton and knot nserton for B-splne curves Q-Xng Huang a,sh-mnhu a,, Ralph R. Martn b a Department of Computer Scence

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz

More information

Adaptive Fractal Image Coding in the Frequency Domain

Adaptive Fractal Image Coding in the Frequency Domain PROCEEDINGS OF INTERNATIONAL WORKSHOP ON IMAGE PROCESSING: THEORY, METHODOLOGY, SYSTEMS AND APPLICATIONS 2-22 JUNE,1994 BUDAPEST,HUNGARY Adaptve Fractal Image Codng n the Frequency Doman K AI UWE BARTHEL

More information

Stochastic epidemic models revisited: Analysis of some continuous performance measures

Stochastic epidemic models revisited: Analysis of some continuous performance measures Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,

More information

Binomial Link Functions. Lori Murray, Phil Munz

Binomial Link Functions. Lori Murray, Phil Munz Bnomal Lnk Functons Lor Murray, Phl Munz Bnomal Lnk Functons Logt Lnk functon: ( p) p ln 1 p Probt Lnk functon: ( p) 1 ( p) Complentary Log Log functon: ( p) ln( ln(1 p)) Motvatng Example A researcher

More information

Performance Analysis and Coding Strategy of ECOC SVMs

Performance Analysis and Coding Strategy of ECOC SVMs Internatonal Journal of Grd and Dstrbuted Computng Vol.7, No. (04), pp.67-76 http://dx.do.org/0.457/jgdc.04.7..07 Performance Analyss and Codng Strategy of ECOC SVMs Zhgang Yan, and Yuanxuan Yang, School

More information

Traffic State Estimation in the Traffic Management Center of Berlin

Traffic State Estimation in the Traffic Management Center of Berlin Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

More information

Optimal resource capacity management for stochastic networks

Optimal resource capacity management for stochastic networks Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, ton.deker@sye.gatech.edu

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

The descriptive complexity of the family of Banach spaces with the π-property

The descriptive complexity of the family of Banach spaces with the π-property Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s40065-014-0116-3 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the π-property Receved: 25 March 2014

More information

Software Alignment for Tracking Detectors

Software Alignment for Tracking Detectors Software Algnment for Trackng Detectors V. Blobel Insttut für Expermentalphysk, Unverstät Hamburg, Germany Abstract Trackng detectors n hgh energy physcs experments requre an accurate determnaton of a

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

More information