Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model

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1 IMA Journa of Numerca Anayss (2011) 1, do:10.109/manum/drq04 Advance Access pubcaton on May 0, 2011 Souton of a nonsymmetrc agebrac Rccat equaton from a one-dmensona mutstate transport mode TIEXIANG LI Department of Mathematcs, Southeast Unversty, Nanng , Peope s Repubc of Chna tx@seu.edu.cn ERIC KING-WAH CHU Schoo of Mathematca Scences, Budng 28, Monash Unversty, Vctora 800, Austraa Correspondng author: erc.chu@sc.monash.edu.au JONG JUANG Department of Apped Mathematcs, Natona Chao Tung Unversty, Hsnchu 00, Tawan uang@math.nctu.edu.tw AND WEN-WEI LIN Centre of Mathematca Modeng and Scentfc Computng, Centre of Theoretca Scences and Department of Mathematcs, Natona Tawan Unversty, No. 1, Sec. 4, Roosevet Road, Tape Tawan wwn@math.nctu.edu.tw [Receved on 18 January 2010; revsed on 9 September 2010] For the steady-state souton of a dfferenta equaton from a one-dmensona mutstate mode n transport theory, we sha derve and study a nonsymmetrc agebrac Rccat equaton B X F F + X + X B + X = 0, where F ± (I F)D ± and B ± B D ± wth postve dagona matrces D ± and possby ow-ranked matrces F and B. We prove the exstence of the mnma postve souton X under a set of physcay reasonabe assumptons and study ts numerca computaton by fxed-pont teraton, Newton s method and the doubng agorthm. We sha aso study severa speca cases. For exampe when B and F are ow ranked then X = Γ ( 4 =1 U V T ) wth ow-ranked U and V that can be computed usng more effcent teratve processes. Numerca exampes w be gven to ustrate our theoretca resuts. Keywords: agebrac Rccat equaton; doubng agorthm; fxed-pont teraton; Newton s method; refecton; transport theory. 1. Introducton Transport theory has been an actve area of research, assocated wth masters ke R. E. Beman and S. Chandrasekhar (see the references n Juang (1995)). A one-dmensona mode was studed frst n Juang (1995), startng a seres of numerca studes, for exampe, n Lu (2005), Ba et a. (2008), c The author Pubshed by Oxford Unversty Press on behaf of the Insttute of Mathematcs and ts Appcatons. A rghts reserved.

2 1454 T. LI ET AL. FIG. 1. The one-dmensona rod. Bn et a. (2008), Ln et a. (2008), and Mehrmann & Xu (2008), n the past 15 years. We sha study a dfferent one-dmensona mutstate mode from Beman et a. (197) and Beman & Wng (1975, equaton (1.7), p. 15) that s generazed sghty n ths paper. We start from a smpe one-dmensona rod or ne segment that extends from 0 to x and denote a generc pont n the rod by z, as n Fg. 1. Partces move to the rght and eft aong ths rod wthout codng wth one another whe nteractng wth the rod tsef wthout affectng t. We frst assume that a partces are of the same type and have the same speed. The obectve s to obtan nformaton about the densty of the beam of partces as a functon of the poston z. We further assume that the probabty of a partce at z (movng n ether drecton) nteractng wth the rod whe movng a dstance of Δ s gven by the expresson σ (z)δ + o(δ), (1.1) where σ ( ) > 0 s the macroscopc cross-secton and o( ) denotes hgher-order terms. As a resut of ths nteracton, an expected average of f (z) and b(z) new partces emerge at the pont z n the same (forward) and opposte (backward) drectons, respectvey, as the orgna partce. Partces traveng to the eft of z = 0 and the rght of z = x are ost to the system. Partces nected at the eft and rght ends, together wth the new partces generated through the coson process, make up the tota partce popuaton of the system. Intay, et us assume a tme-ndependent state, where the expected partce popuaton s statonary and ndependent of the tme at whch the system s observed. We defne u(z) and v(z) to be the expected numbers of rght- and eft-movng partces, respectvey, passng through the pont z each second. (The adectve expected s sometmes negected but s necessary due to the stochastc nature of the probem.) From the defnton n (1.1) the probabty of partces passng through from z to z + Δ wthout nteractng wth the rod s 1 σ (z)δ + o(δ). The expected contrbuton to u(z + Δ) from ths type of occurrence s [1 σ (z)δ + o(δ)]u(z) = [(1 σ (z)δ)]u(z) + o(δ). (1.2) However, some rght- (or eft-) movng partces passng through z w nteract wth the rod before reachng z +Δ. Each such event w produce an expected number f (z) (or b(z)) of partces proceedng n the drecton of nterest. The expected contrbutons to u(z + Δ) from these types of occurrences are σ (z) f (z)u(z)δ + o(δ), σ (z)b(z)v(z)δ + o(δ). (1.) Other events can take pace but the contrbutons are o(δ) and nsgnfcant. Hence, summng the contrbutons n (1.2) and (1.) yeds u(z + Δ) = [(1 σ (z)δ)]u(z) + σ (z) f (z)u(z)δ + σ (z)b(z)v(z)δ + o(δ). (1.4) Takng the mt Δ 0 n (1.4) eads to the foowng dfferenta equaton for u: du dz = σ (z){[ f (z) 1]u(z) + b(z)v(z)}, u(0) = 0. (1.5)

3 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL 1455 A smar partce countng process aso produces the foowng dfferenta equaton for v: dv dz = σ (z){b(z)u(z) + [ f (z) 1]v(z)}, v(x) = 1. (1.6) For a mutstate mode we aow n dfferent states (e.g., speed, energy, type or any other features other than drecton that dstngush between the partces). The macroscopc cross-secton for the th state s σ (z) > 0 and the probabty n (1.1) resutng n the emsson of partces n the th state then reads σ (z)δ + o(δ). Smary, we have functons u (z) and v (z) ( = 1,..., n) representng the expected number of partces n state, movng to the rght and eft, respectvey, past the pont z each second. We defne the matrx functons F(z) = [ f (z)], B(z) = [b (z)], F(z) = [ f (z)] and B(z) [ b (z)], where f (z) = σ (z)[ f (z) δ ], b (z) = σ (z)b (z), (, = 1,..., n), δ s the Kronecker δ, and f (z) and b (z) are the expected numbers of partces traveng, respectvey, n the forward and backward drectons, respectvey, after the coson of a partce of state emttng partces of state. A smar argument to that eadng to (1.5) and (1.6) then produces du dz = F(z)u + B(z)v, dv dz = B(z)u + F(z)v, wth u = [u 1,..., u n ] T, v = [v 1,..., v n ] T and the conventon M = [m ] (capta etters for matrces and the correspondng ower-case etters wth ndces for eements), and the nta condtons u (0) = 0 and v (x) = δ (correspondng to the nta necton of a partce of state from the rght). From the above dscusson we expect B, F > 0 to satsfy ( f + b ) < 1. (1.7) We w aow the crtca case of equaty n (1.7) (the pure scatterng case n Beman & Wng (1975, equaton (4.1), p. 55)) ater. To carry out the nvarant mbeddng procedure the functons R and T are ntroduced, where r (x) s the expected number of partces emergent each second at z = x n state from a rod of ength x when the ony nput s one partce per second n state at the rght end z = x, and t (x) s defned smary except the emergence s at the other end z = 0. Consder a rod of ength x +Δ wth the sub-rod of ength x mbedded. Assumng that the refectng response functon R(z) = [r (z)] s known, the transmsson response functon T (z) = [t (z)] can be defned through a dfferenta equaton derved from a partce countng process. Countng a the sgnfcant events as enumerated n Wng (1962), and aowng dfferent macroscopc cross-sectons σ ± (z) > 0 for sources from the eft and the rght, the foowng equaton for R(x) [r (x)] can be derved: dr(x) dx = B (x) R(x)F (x) F + (x)r(x) + R(x)B + (x)r(x), R(0) = 0, (1.8) where B ± B D ±, F ± (I F)D ±, D ± dag{σ k ± } > 0, and B and F are possby ow ranked. (In Wng (1962) the sgns of the near terms on the rght-hand sde of (1.8) were postve. We change these

4 1456 T. LI ET AL. sgns to make the resutng nonsymmetrc agebrac Rccat equaton (NARE) n (1.9) more consstent wth notaton n other papers on NAREs. Note that the {σ k } were ndependent of drecton n Beman et a. (197) and Wng (1962).) After the determnaton of R, the functon T can be derved from the smper equaton dt (x) = T (x)[ F(x) + B(x)R(x)], T (0) = I. dx For the steady-state souton for a partcuar x, (1.8) eads to wth R repaced by the usua varabe X for NAREs. B X F F + X + X B + X = 0, (1.9) 2. Exstence of a souton Some reevant defntons are as foows. For any matrces Â, B R m n we wrte  B or  > B f ther eements satsfy â b or â > b, respectvey, for a and. A rea square matrx  s caed a Z-matrx f a of ts off-dagona eements are nonpostve. It s cear that any Z-matrx  can be wrtten as s I B wth B 0. A Z-matrx  s caed an M-matrx f s ρ( B), where ρ( ) s the spectra radus, and t s a snguar M-matrx f s = ρ( B) and a nonsnguar M-matrx f s > ρ( B). We have the foowng usefu resuts from Berman & Pemmons (1994) and Guo & Hgham (2007, Theorem 1.1). LEMMA 2.1 For a Z-matrx  the foowng statements are equvaent: (a)  s a nonsnguar M-matrx; (b)  1 0; (c) Âv > 0 for some vector v > 0. THEOREM 2.2 Let us consder the NARE XĈ X X D ÂX + B = 0, (2.1) where Â, B, Ĉ and D are rea matrces of szes m m, m n, n m and n n, respectvey. Assume that [ ] D Ĉ M = (2.2) B  s a nonsnguar M-matrx or an rreducbe snguar M-matrx. Then the NARE has a mnma nonnegatve souton S. If M s rreducbe, then S > 0, and  SĈ and D Ĉ S are rreducbe M-matrces. If M s a nonsnguar M-matrx, then  SĈ and D Ĉ S are nonsnguar M-matrces. If M s an rreducbe snguar M-matrx wth postve eft and rght nu vectors [u T 1, ut 2 ]T and [v T 1, vt 2 ]T (where u 1, v 1 R n and u 2, v 2 R m ) satsfyng then u T 1 v 1 u T 2 v 2, M S = I n ( SĈ) + ( D Ĉ S) T I m s a nonsnguar M-matrx. If M s an rreducbe snguar M-matrx wth u T 1 v 1 = u T 2 v 2, then M S s an rreducbe snguar M-matrx.

5 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL 1457 We have the foowng exstence resut. THEOREM 2. Under assumpton (1.7), the unque mnma non-negatve souton X of (1.9) exsts. Proof. From Theorem 2.2 we need to show that the Z-matrx [ (I F)D B D M = + ] { [ ]} [ F B D B D (I F)D + = I B F D + ] (2.) s a nonsnguar M-matrx. Note that A s an M-matrx f and ony f A T s an M-matrx. Appyng Lemma 2.1, we need to fnd a vector v > 0 such that M T v > 0, whch s trva from (1.7). 2.1 NARE as an egenvaue probem The NARE (1.9) can be reformuated as the foowng egenvaue probem: [ ] [ ] [ I I F B H = S, H + ] {[ ] [ ]} [ I F B D X X B F + = + I B F D + ]. (2.4) From (1.7) t s easy to see that the egenvaues of H are shfted sghty from ±σk, spttng equay on opposte sdes of the magnary axs. Usng the Gerschgorn theorem, wth D(a, r) {x C: x a r}, the egenvaues are n [ k D(σ k, ασ k )] [ k D( σ k +, ασ k + )], dvded equay on opposte sdes of the magnary axs, wth α ( F + B 1 ) < 1 from (1.7). REMARK 2.4 For the crtca case wth α = 1 a smpe appcaton of the Gerschgorn theorem mpes that the matrces H n (2.4) and M n (2.) may be snguar. However, the potenta snguarty may be detected or excuded by appyng the extensons of the Gerschgorn theorem n Horn & Johnson (1985, Secton 6.2). Consder a of the Gerschgorn dsks of H T contanng the orgn. At east one of the correspondng nequates shoud not be satsfed wth equaty. In other words, we may have to excude the utra-crtca case that a of the frst or ast n rows have ther correspondng off-dagona row sums equa to unty. Note that, even f H or M are snguar, the exstence resut n Theorem 2.2 st hods provded that M s rreducbe. Wth the addtona requrement for the nu vectors as n Theorem 2.2, Newton s method n Secton 4.1 w be quadratcay convergent. 2.2 NARE as a nonnear equaton To compute the mnma non-negatve souton X for the NARE (1.9), consder t n component form as foows: (σ + σ + )x = (B D ) + x (F D ) + (F D + ) x + x (B D + )x. (Here x and x denote the th row and th coumn, respectvey, of X, and ( ) denotes the eement (, ) of a matrx.) Equvaenty, we have X = φ(x) Γ (B D + X F D + F D + X + X B D + X), Γ [(σ + σ + ) 1 ], (2.5) wth X beng a Hadamard product. Note that Theorem 2., (2.5) and the assumpton B > 0 mpy that X > 0.

6 1458 T. LI ET AL. We have the foowng theorem for X and the fxed-pont teraton. THEOREM 2.5 Let X (0) = 0 and X (k+1) = φ(x (k) ). Then, under assumpton (1.7), we have the foowng () the terates satsfy X X (k+1) X (k) Γ B D > 0; () X (k) X as k. Proof. It s easy to show that X (1) = Γ B D > 0 from (2.5). For () consder the dfference between X = φ(x ) and X (k) = φ(x (k 1) ) as foows: X X (k) = Γ [(X X (k 1) )F D + F D + (X X (k 1) ) + (X X (k 1) )B D + X + X (k 1) B D + (X X (k 1) )]. Inducton w then compete the argument for X X (k) 0 n (). Smary, nducton on the dfference between X (k+1) = φ(x (k) ) and X (k) = φ(x (k 1) ) eads to X (k+1) X (k) 0 n (). For () convergence s mped by () wth the mt X m k X (k) = X because () mpes that X X > 0 and X s mnma.. Low-ranked B and F When the fu-ranked decompostons F = F 1 F2 T and B = B 1 B2 T are of rank m and p, respectvey, (2.5) mpes that wth the auxary varabes X = Γ (B 1 B T 2 D + Z 1 F T 2 D + F 1 Z T 2 + Z Z T 4 ) (.1) Z 1 X F 1, Z 2 X T D + F 2, Z X B 1, Z 4 X T D + B 2. (.2) Substtutng X n (.1) nto (.2), we have 2(m + p)n nonnear equatons for the 2(m + p)n unknowns n Z ( = 1,..., 4) as foows: Z 1 = [Γ (B 1 B2 T D + Z 1 F2 T D + F 1 Z2 T + Z Z4 T)]F 1, Z 2 = [Γ T (D B 2 B1 T + D F 2 Z1 T + Z 2F1 T + Z 4Z T)]D+ F 2, Z = [Γ (B 1 B2 T D + Z 1 F2 T D + F 1 Z2 T + Z Z4 T)]B (.) 1, Z 4 = [Γ T (D B 2 B1 T + D F 2 Z1 T + Z 2F1 T + Z 4Z T)]D+ B 2, (cf. the 2n equatons n 2n unknowns n Juang (1995) for a smper NARE wth m = p = 1). Smary, X can be retreved usng (.1) after the Z are obtaned. It s obvous from Theorem 2.5 and (.) that Z > 0 ( = 1,..., 4) and X > 0. The convergence of varous teratve schemes for the set of nonnear equatons (.) can be shown. Frst et R ( = 1,..., 4) be the th rght-hand sde n (.). Startng from Z (0) = 0 ( = 1,..., 4), we sha consder the foowng teratve methods. (1) Smpe teraton (SI): for, = 1,..., 4 we have = R (..., Z (k ),...), k = k (, ). (.4) The rght-hand sdes R ony nvove the prevous terates Z (k).

7 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL 1459 (2) Modfed smpe teraton (MSI): for, = 1,..., 4 we have = R (..., Z ( k ),...), k = { k + 1 f <, k otherwse. (.5) The rght-hand sdes R nvove Z (k), as we as f they have been computed. () Nonnear bock Jacob method (NBJ): for, = 1,..., 4 we have = R (..., Z ( k ),...), k = { k + 1 f =, k otherwse. The rght-hand sdes R nvove Z (k), as we as wth = and the correspondng terms moved to the eft-hand sdes. (4) Nonnear bock Gauss Sede method (NBGS): for, = 1,..., 4 we have { = R (..., Z (ǩ ) k + 1 f,,...), ǩ = (.7) k otherwse. The rght-hand sdes R nvove Z (k) and (when avaabe), as we as wth = and the correspondng terms moved to the eft-hand sdes. From the above formuae we obvousy have the foowng nequates for the ndces: (.6) k = k k ǩ, k = k k ǩ (, = 1,..., 4). (.8) The foowng smpe emma w be repeatedy apped n the proof of Theorem.2. LEMMA.1 Let U = [u ], V = [v ] and W = [w ] R n q be non-negatve, and et Γ = [ τ ] R n n be postve. Consder the near system and ts th row has the form where U [ Γ (U V T )]W = R, (.9) u (I P ) = r, (.10) (P ) s = n τ v s w, (.11) =1 for = 1,..., n and s, = 1,..., q. In addton, assume that u (I P ) = r, (.12) where P s constructed as n (.11) wth v s repaced by v s = (V ) s, r s the th row of R and R R > 0, U > 0, V V. (.1) Then I P and I P are nonsnguar M-matrces and P P, (I P ) 1 (I P ) 1 0. (.14)

8 1460 T. LI ET AL. Proof. From (.11) and (.1) we have that u, r > 0 and I P s a Z-matrx. Consequenty, the transpose of (.12) and Lemma 2.1 mpy that (the transpose of) I P s a nonsnguar M-matrx wth a non-negatve nverse and (I P )v > 0 for some vector v > 0. From (.11) P s near n V, wth V V mpyng that P P, I P I P and (I P )v (I P )v > 0. Lemma 2.1 then mpes that I P s a nonsnguar M-matrx wth a non-negatve nverse and (.14) foows. Wth the addtona subscrpts I = S, M, J, G for the four dfferent methods (.4) (.7), respectvey (and gnorng them when the resut hods for a the methods), we have the foowng resuts that are smar to those n Guo & Ln (2010). THEOREM.2 We sha assume that (1.7) hods and we have the spttng F = F 1 F T 2 and B = B 1 B T 2, wth the fu-ranked B 1, B 2, F 1, F 2 0. We have for = 1,..., 4 and k = 0, 1,... that the foowng hods () the terates satsfy Z () Z (k) Z as k ; () 0 Z (k) (k) (k) S, Z M, Z G, ; (v) 0 Z (k) (k) (k) S, Z J, Z G,. Z (k) > 0, except Z (0) = 0; Proof. For NBJ and NBGS the formuae n (.) (gnorng the superscrpts (k +1) on the eft-hand sdes and (k) on the rght for Z as n (.6) and (.7)) are equvaent to Z 1 [Γ (Z 1 F T 2 )]D F 1 = R 1 [Γ (B 1 B T 2 D + F 1 Z T 2 + Z Z T 4 )]F 1, (.15) Z 2 [Γ T (Z 2 F T 1 )]D+ F 2 = R 2 [Γ T (D B 2 B T 1 + D F 2 Z T 1 + Z 4Z T )]D+ F 2, (.16) Z [Γ (Z Z T 4 )]B 1 = R [Γ (B 1 B T 2 D + Z 1 F T 2 D + F 1 Z T 2 )]B 1, (.17) Z 4 [Γ T (Z 4 Z T )]D+ B 2 = R 4 [Γ T (D B 2 B T 1 + D F 2 Z T 1 + Z 2F T 1 )]D+ B 2. (.18) The operators on the eft-hand sde of (.15) (.18) are of smar form and we need to nvert them wth known rght-hand sdes R. For the generc term U [ Γ (U V T )]W for Γ = [ τ ] R n n and U = [u ], V = [v ], W = [w ] R n q (wth q = m or p), the (, ) component equas ( q n ) {U [ Γ (U V T )]W } = u u s τ v s w, (.19) s=1 =1 mpyng that the th row n (.15) (.18) has the generc form (.10) wth P R q q ( = 1,..., n) as n (.11). Note that u and r n (.10) are the th rows of U and the rght-hand sde R n (.15) (.18) or Tabe 1 beow, respectvey. We sha prove () by nducton. For the k = 0 case n (), except for Z 4 n NBGS, t s easy to see from (.) and (.4) (.7) that Z (1) are we defned and Z Z (1) > Z (0) = 0 ( = 1,..., 4), (.20) where the mts (ndcated by ( ) ) are guaranteed to exst by Theorems 2. or 4.1 together wth (.2). Note from Tabe 1 and Z (0) = 0 that the P are constant n (.10) for Z 1 and Z 2 n NBJ and NBGS, and P = 0 for Z and Z 4 n NBJ as we as Z n NBGS (because the correspondng V s n Tabe 1 vansh).

9 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL 1461 TABLE 1 Constructon of P n (.10) or (.19) for NBJ and NBGS Method U Γ V W R Both 1 Γ D F 2 F 1 R 1 2 Γ T D + F 1 F 2 R 2 Γ Z (k) 4 B 1 R NBJ NBGS 4 Γ T D + Z (k) B 2 R 4 4 Γ T D + B 2 R 4 For Z 4 n NBGS the teraton has the foowng form that s smar to (.10): where P (1) s near n V = Z (1), whch has been proved to satsfy u (1) [I P (1) ] = r (0), (.21) Z Z (1) > 0. (.22) From (1.7) and (.20) we have u > 0 (from methods other than NBGS) and r r (0) > 0. Wth (.21) n pace of (.10), or u (1), P (1) and r (0) n pace of u, P and r, respectvey, Lemma.1 then mpes that I P (1) s nonsnguar and (I P ) 1 [I P (1) ] 1 0. Consequenty, u (1) and thus Z (1) 4 are we defned, and we have u = r (I P ) 1 r (0) [I P (1) ] 1 = u (1) Z4 (1) Z 4 > Z (0) 4 = 0. We have proved the k = 0 case n (). Assumng that () hods up to some vaue of k, we sha prove the (k + 1) case. The concuson can be easy drawn for SI and MSI by consderng the dfferences between (.4) and (.5) for successve vaues of k as we as the mtng case when k. For NBJ and NBGS n (.6) and (.7) for Z 1 and Z 2, we have that P (s) = P = P (for a s) are constant as V = F 2, F 1 from Tabe 1. The mtng case n (.12) or a trva appcaton of Lemma.1 mpy that [I P (s) ] 1 0 (for a s). For NBJ and NBGS for Z and Z 4, the teratons take the foowng generc form, as n (.10): u (s+1) [I P (s+1) ] = r (s) (s = 0, 1,..., k + 1), (.2) wth r (s) and P (s+1) dependent on V = Z (s) or Z (s+1) (for NBGS for the teraton for Z 4, where Z (k+2) when the teratons n (.7) are executed n the ntended order = 1,..., 4). From the nducton hypothess and the nearty of R wth respect to Z (for a ) n (.15) (.18), we have U > 0, V V > 0 and R R (s) > 0. Wth (.2) n pace of (.10), or u (s+1), P (s+1) and r (s) n pace of u, P and r, respectvey, Lemma.1 mpes that I P (s+1) (s = 0, 1,..., k + 1) are nonsnguar M-matrces wth non-negatve nverses, P P (k+2) P (k+1) and (I P ) 1

10 1462 T. LI ET AL. [I P (k+2) ] 1 [I P (k+1) ] 1 0. From successve vaues of s = k, k + 1 and the mtng case k, approprate dfferences yed u u (k+2) = r [I P = ( r r (k+1) )(I P ] 1 r (k+1) [I P (k+2) ] 1 ) 1 + r (k+1) [I P (k+2) ] 1 [P P (k+2) ](I P ) 1 0. Smary, we have u (k+2) u (k+1) = r (k+1) [I P (k+2) ] 1 r (k) [I P (k+1) t] 1 0. Thus a Z Z (k+2) ( = 1,..., 4) and the nducton for () s compete. For () a smar argument as n the proof of () n Theorem 2.5 can be apped. For () and (v) we note that the terates Z (k) are ncreasng towards ther respectve mts Z, and R and R preserve the order of postvty of ther arguments. We sha prove the nequates agan by nducton. The nta cases for k = 0 are obvous. Assume that the resuts hod for some vaue of k. From (.8), for = 1,..., 4 and k = 0, 1,... we have S, = R (..., Z (k) S,,...) R (..., Z (k) S, = R (..., Z (k) S,,...) R (..., Z (k) M, = R (..., Z ( k ) M,,...) R (..., Z ( k ) M,,...) R (..., Z ( k ) M, J,,...) R (..., Z ( k ) J, G,,...) R (..., Z (ǩ ) G,,...) = Z (k+1) M,,,...) = Z (k+1) J,,,...) = G,. For the rght-most nequates n (v) consder the teratons n the genera form (.2). We then have u (k+1) J, snce P (k+1) J, = r (k) J, P (k+1) G, [I P(k+1) J, and r (k) J, ] 1 r (k) G, [I P(k+1) J, ] 1 r (k) G, [I P(k+1) G, ] 1 = u (k+1) G, (k) (k+1) r G,. Thus Z J, G,, and nducton s compete. REMARK. The assumpton that B, F 0 ( = 1, 2) s ust a convenent suffcent condton for Theorem.2. There are many other weaker but more tedous suffcent condtons that we can wrte down. For exampe, by carefu appcaton of (.15) (.18) n the proof, we can make the aternatve assumpton, wth W 1 (Γ D ) B and W 2 (D + Γ D ) B, that the foowng matrces are postve: 4. Genera case F 2 F T 1 ; W 1 B 1, W 1 F 1, B T 2 W 1 B 1 ; F T 2 W 2, B T 2 W 2 B 1. For the genera case wth B and F beng fu ranked, the NARE (1.9), namey, B X F F + X + X B + X = 0, or the equvaent (2.5), can be soved by fxed-pont teraton (as n Theorem 2.5), Newton s method (Lu, 2005; Guo & Hgham, 2007; Ln et a., 2008) or doubng (Guo et a., 2006; Chang et a., 2009). The exstence of the unque mnma postve souton X of (1.9) s guaranteed by Theorem 2..

11 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL Newton s method Consderng the NARE (1.9), et R(X) denote the eft-hand sde of the equaton. At the (k+1)th teraton wth X (k) beng an approxmate souton and X (k+1) = X (k) + δx (k+1), Newton s method requres the souton of the Syvester equaton (F + X (k) B + )δx (k+1) + δx (k+1) (F B + X (k) ) = R(X (k) ). (4.1) The convergence of Newton s method s guaranteed by the foowng theorem quoted from Guo & Hgham (2007, Theorem 2.). THEOREM 4.1 Let S be the mnma postve souton of (1.9). Then, under assumpton (1.7), for the Newton teraton (4.1) wth X (0) = 0, the sequence {X (k) } s we defned, X (k) X (k+1) S for a k 0, and m k X (k) = S. The proof makes use of seected resuts from Theorem 2.2. In partcuar, when vectorzed the above Syvester operator can be wrtten as the matrx operator M S (wth m = n) as n Theorem Doubng We sha quote the doubng agorthm for the genera NARE (2.1), wth the matrx M n (2.2) beng a nonsnguar M-matrx, from Guo et a. (2006). Note that, per teraton, the doubng agorthm s faster than Newton s method, as concuded n Guo et a. (2006), Guo (2007) and Tabe 2, and we refer the reader to the detas n these references. For the genera NARE XĈ X X D ÂX + B = 0, wth the correspondng matrx M n (2.2) beng a nonsnguar M-matrx, we frst transform Â, B, Ĉ and D to E γ = I 2γ Vγ 1, G γ = 2γ Dγ 1 γ, F γ = I 2γ Wγ 1, H γ = 2γ Wγ 1 B Dγ 1, ĈW 1 wth the parameter γ max{â,..., â nn ; d 11,..., d nn } and A γ = Â + γ I, D γ = D + γ I, W γ = A γ B Dγ 1 Ĉ, V γ = D γ Ĉ A 1 γ B. The doubng agorthm can then be summarzed as foows: E 0 = E γ, F 0 = F γ, G 0 = G γ, H 0 = H γ, E k+1 = E k (I G k H k ) 1 E k, F k+1 = F k (I H k G k ) 1 F k, G k+1 = G k + E k (I G k H k ) 1 G k F k, H k+1 = H k + F k (I H k G k ) 1 H k E k. TABLE 2 Operaton counts per teraton B, F Method Fops Low-ranked NBGS 8(m + p + 2)n 2 Genera Fxed-pont teraton 4n Newton s method 6n Doubng 10 2 n

12 1464 T. LI ET AL. The terates are we defned wth I H k G k and I G k H k beng nonsnguar M-matrces for a k, and H k X and G k Y (the souton to the adont agebrac Rccat equaton to (2.1)) from beow quadratcay as k (see Guo et a., 2006, Theorem 5.1). Note that, f D ± = D, B = Ĉ = B D and  = D = (I F)D, then we can have the computaton as E k = F k and G k = H k for a k. Some savng n computaton can aso be made n Newton s method as the Syvester equatons n the teraton become Lyapunov equatons. 5. Numerca exampes For comparson, we sha summarze the operaton counts per teraton of varous teratve methods n Tabe 2. We sha show ony the domnant terms, assumng that n m, p. For ow-ranked B and F, ony the fastest method NBGS s consdered. The sow fxed-pont teraton method s aso ncuded for comparson. REMARK 5.1 For the smper NARE consdered n Lu (2005), Ba et a. (2008), Ln et a. (2008) and Mehrmann & Xu (2008), the assocated structure gave rse to teratve souton processes (anaogous to our SI, MSI, NBJ and NBGS methods) of O(n) computatona compexty. The fast Newton method n Bn et a. (2008) s of O(n 2 ) compexty and s uncompettve. However, our NARE n (1.9), for both the ow-ranked and the genera cases, has very dfferent structures. It s key that faster souton methods can be found but we do not antcpate methods of ess compexty than the O(n 2 ) NBGS method (for the ow-ranked case) and the O(n ) doubng method (for the genera case). We sha consder two randomy generated exampes for n = 64, 128, 256, 512, 1024 and Exampe 1 has B and F beng fu ranked, and Exampe 2 has B and F of rank 10. For the exampes the assumptons n Theorems 2. and.2 are satsfed. The numerca computaton has been carred out usng MATLAB R2008b on a aptop wth precson eps = (MathWorks, 2002). For Exampe 1, fxed-pont teraton, Newton s method and the doubng agorthm have been compared for varous vaues of n. The teratons have been run unt convergence wth toerance to = The resuts are summarzed n Tabe, wth t n denotng the CPU tme, r n t n /t n/2 and #It beng the number of teratons requred, for partcuar vaues of n. The terates are aso potted n Fg. 2 for n = Note that the resduas n Fgs 2 and are potted usng a ogarthmc scae. Tabe and Fg. 2 seem to ndcate that the doubng agorthm performs better than Newton s method n CPU-tme and the fxed-pont teraton method s the sowest, as predcted n Tabe 2. The ratos r n ustrate the O(n ) compexty of the methods. The graphs n Fg. 2 ustrate the quadratc convergence of the doubng agorthm and Newton s method, wth the fxed-pont teraton method obvousy convergng neary. Newton s method s two to three tmes faster than the doubng method n terms of number of teratons, but the atter has an advantage n operaton count per teraton by a factor of.6, resutng n ts better effcency n terms of CPU tme. Note that the cputme command n MATLAB (MathWorks, 2002) s not an exact refecton of CPU tme consumed and shoud be used as a rough gude ony. Aso, users have no contro over some parts of the agorthms, such as the nverson of the Syvester operators by the MATLAB command yap (MathWorks, 2002) n Newton s method. For Exampe 2, ony the fastest teraton method NBGS has been tested aganst the doubng method and the resuts are summarzed n Tabe 4 (for n = 64, 128, 256, 512, 1024, 2048) and Fg. (for n = 1024), wth to = The O(n 2 ) compexty of NBGS and the O(n ) compexty of the doubng method are ustrated n the ratos r n n Tabe 4. The near convergence of NBGS and the quadratc convergence of the doubng method can be seen ceary n Fg.. NBGS usuay requres ess teratons than the doubng method and s aso more effcent n terms of CPU tme because of ts superor

13 RICCATI EQUATION FROM A ONE-DIMENSIONAL MULTISTATE TRANSPORT MODEL 1465 FIG. 2. Resduas for Exampe 1 (n = 1024). FIG.. Resduas for Exampe 2 (n = 1024). TABLE CPU tmes and teraton numbers for Exampe 1 Fxed-pont teraton Newton Doubng n t n r n #It t n r n #It t n r n #It

14 1466 T. LI ET AL. TABLE 4 CPU tmes and teraton numbers for Exampe 2 wth NBGS and doubng and the methods NBGS Doubng n t n r n #It t n r n #It operaton count per teraton. For fxed ranks of B and F the number of teratons requred decreases wth respect to n, as ndcated n the fourth coumn of Tabe Concudng remarks For the one-dmensona mutstate mode n transport theory we need to sove a dfferenta equaton to obtan the refecton functon R. For the steady-state souton we have derved an NARE from the dfferenta equaton. We have proved the exstence and unqueness of the mnma postve souton of the NARE. When B and F are ow ranked the NBGS method of O(n 2 ) compexty soves the NARE effcenty. For the genera case the doubng agorthm seems to be more effcent than Newton s method. The numerca resuts support our theoretca fndngs. For future work we need to mprove on the effcency of the numerca agorthms for arge vaues of n. Fnay, there are other smar modes and probems n transport theory (Beman & Wng, 1975) worthy of nvestgaton. Fundng The Centre of Mathematca Modeng and Scentfc Computng and the Natona Centre of Theoretca Scences, Natona Chao Tung Unversty, Repubc of Chna (to E.C.); The Natona Scence Foundaton, Repubc of Chna (to T.L.). REFERENCES BAI, Z.-Z., GAO, Y.-H. & LU, L.-Z. (2008) Fast teratve schemes for nonsymmetrc agebrac Rccat equatons arsng from transport theory. SIAM J. Sc. Comput., 0, BELLMAN, R. E., UENO, S. & VASUDEVAN, R. (197) On the matrx Rccat equaton of transport processes. J. Math. Ana. App., 44, BELLMAN, R. E. & WING, G. M. (1975) An Introducton to Invarant Imbeddng. New York: Wey. BERMAN, A. & PLEMMONS, A. J. (1994) Nonnegatve Matrces n the Mathematca Scences. Phadepha, PA: SIAM. BINI, D. A., IANNAZZO, B. & POLONI, F. (2008) A fast Newton s method for a nonsymmetrc agebrac Rccat equaton. SIAM J. Matrx Ana. App., 0, CHIANG, C.-Y., CHU, E. K.-W., GUO, C.-H., HUANG, T.-M., LIN, W.-W. & XU, S.-F. (2009) Convergence anayss of the doubng agorthm for severa nonnear matrx equatons n the crtca case. SIAM J. Matrx Ana. App., 1, GUO, C.-H. (2007) A new cass of nonsymmetrc agebrac Rccat equatons. Lnear Agebra App., 426,

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