A note on the modified Hermitian and skew-hermitian splitting methods for non-hermitian positive definite linear systems

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1 A ote o the modified Hermitia ad skew-hermitia splittig methods for o-hermitia positive defiite liear systems Shi-Liag Wu Tig-Zhu Huag School of Applied Mathematics Uiversity of Electroic Sciece ad Techology of Chia Chegdu Sichua 654 People s Republic of Chia wushiliag999@6.com or tigzhuhuag@6.com Abstract: Comparig the lopsided Hermitia/skew-Hermitia splittig (L) method ad Hermitia/skew- Hermitia splittig () method a ew criterio for choosig the above two methods is preseted which is better tha that of Li Huag ad Liu [Modified Hermitia ad skew-hermitia splittig methods for o- Hermitia positive-defiite liear systems Numer. Li. Alg. Appl. 4 (7): 7-35]. Key-Words: o-hermitia matrix; splittig; skew-hermitia matrix; Hermitia matrix; iteratio Itroductio The solutios of may problems i scietific computig are evetually tured ito the solutios of the large liear systems that is Ax= b () where A C is a large sparse o-hermitia positive defiite matrix ad x b C. To solve () iteratively the efficiet splittig of the coefficiet matrix A are usually required. For example the classic Jacobi ad Gauss-Seidel iteratio split the matrix A ito its diagoal ad off-diagoal parts. Oe ca see [3 5-8] for a comprehesive survey. Recetly a Hermitia ad skew-hermitia splittig [] gais people s attetio that is A = H + S where H = ( A+ A ) ad S = ( A A ). I [] Bai Golub ad Ng developed the iteratio method by the above splittig of A for o- Hermitia positive defiite system: Give a iitial guess x () for k = ( k ) util x coverges compute ( k+ ) ( k) ( αi + H) x = ( αi S) x + b () ( k+ ) ( k+ ) ( αi + Sx ) = ( αi Hx ) + b where H = ( A+ A ) ad S = ( A A ) are the Hermitia ad skew-hermitia parts of A respectively ad α is a give positive costat. I matrix-vector form the iteratio method () ca be equivaletly rewritte as ( k+ ) ( k) x = M( α) x + G( α) b k = where M ( α) = ( αi + S) ( αi H)( αi + H) ( αi S) G( α) = α( αi + S) ( αi + H). Here M ( α ) is the iteratio matrix of the iteratio method. To describe the covergece property of the iteratio the followig theorem was established i []. Theorem Let A C be a positive defiite matrix H = ( A+ A ) ad S = ( A A ) be respectively its Hermitia ad skew-hermitia parts ad α be a positive costat. The the spectral radius ρ( M ( α )) of the iteratio matrix M ( α) of the method is bouded by α λi γα ( ) = λi λ( H ) α + λi with λ λ λ beig the eigevalues of H. Thus it holds that ρ( M ( α)) γ( α) < for all α >. The optimal parameter α is α λ α = arg mi = λλ α λ λ λ α + λ ad λ λ γ ( α) =. (3) λ + λ Recetly Li Huag ad Liu [] preseted the lopsided Hermitia ad skew-hermitia splittig ISSN: Issue 5 Volume 7 May 8

2 (L) iteratio method based o the followig splittig: A= H + S = ( α I + S) ( α I H). The L iteratio method: Give a iitial guess () ( k ) x for k = util x coverges compute ( k+ ) ( k) Hx = Sx + b (4) ( k+ ) ( k+ ) ( αi + Sx ) = ( αi Hx ) + b where α is a give o-zero positive costat. Note that whe the matrix A is positive defiite H must be a positive defiite matrix with S beig skew-hermitia ad A + A = H + H. Sice S is skew-hermitia it is ot difficult to see that α I + S is also osigular. The above L iteratio method (4) ca be equivaletly trasformed ito the followig matrix-vector form: ( k+ ) ( k) x = M( α) x + G( α) b k = (5) where M( α) = ( αi + S) ( αi H) H ( S) G( α) = α( αi + S) H. Here M ( α) is the iteratio matrix of the L method. I fact (5) may also result from the splittig A = M N of the coefficiet matrix A with M = α H( αi + S) N = α ( αi H)( S). The followig theorem established i [] describes the covergece property of the L iteratio. Theorem Let A H ad S be defied as those i Theorem ad α be a o-zero costat. The the spectral radius ρ( M ( α)) of the iteratio matrix M( α) of the L iteratio is bouded by α λi δα ( ) = λi λ( H ) α + λ i where λ λ λ are the eigevalues of H ad is the imum sigular value of S. Moreover we have (i) If λ whe λ α > or α < λ the boud of δ ( α ) < i.e. the L iteratio coverges; (ii) If λ < < λ whe λ λ < α < or α < λ λ the boud of δ ( α ) < i.e. the L iteratio coverges; (iii) If λ whe λ < α < λ the boud of δ ( α ) < i.e. the L iteratio coverges. The optimal parameter α is obtaied at λλ α = λ + λ ad ( λ λ ) δα ( ) =. (6) 4 λλ + ( λ λ ) Theorem shows that if the coefficiet matrix A is positive defiite the L iteratio (5) coverges to the uique solutio of the liear systems () for a loose restrictio o the choice of α. Moreover the upper boud of the cotractio factor of the L iteratio is depedet o the choice of α the spectrum of the Hermitia part H ad the imum sigular value of the skew-hermitia part S that is but is idepedet of the rest sigular values of S as well as the eigevectors of the matrices A H ad S. Remark Sice S is the skew-hermitia matrix the the eigevalues of S are complex umber without the real parts. Let μ( S) = { iμ iμ iμ} μi R ( i = ) ad let μ = { μ μ }. It is easy to kow that = μ. That is to say of Theorem ca be replaced by μ which shows that the upper boud of the cotractio factor of the L iteratio is depedet o the choice of α the spectrum of the Hermitia part H ad the imum module of the eigevalues of the skew- Hermitia part S but is idepedet of the module of the rest eigevalue of S as well as the eigevectors of the matrices A H ad S. Mai results We ow give the followig mai result which is a ew criterio for choosig betwee the two methods. ISSN: Issue 5 Volume 7 May 8

3 Theorem 3 Let λ λ α ad δ ( α ) be defied as those i Theorem respectively. Let α ad γ ( α ) be defied as those i Theorem respectively. Case : If λ = λ the δ ( α ) γ ( α) = ; Case : If λ λ ad λλ ( λ λ ) λλ < + the the followig iequality holds: ( ) < δ( α ). γ α Proof. It is easy to kow that Case holds from (3) ad (6). We eed oly to prove that Case holds. Supposig γ ( α) < δ( α ). From λ > λ we have λ λ ( λ λ ) < λ + λ 4 λλ + ( λ λ ) which implies ( λ + λ ) <. λ + λ 4 λλ + ( λ λ ) Applyig squarig operatio o both sides ad combiig the coefficiets of we get 4 4 λλ < [( λ+ λ) ( λ λ) ]. By the simple maipulatios we obtai λλ < λλ( λ+ λ) which completes the proof. Apparetly we have the followig theorem. Theorem 4 Uder coditios of Theorem 3 If λ λ ad the γ ( α) δ( α ) λλ λ λ λλ < ( + ) < holds. Remark It is easy to kow that of Theorem 3 ad 4 ca be replaced by μ. Moreover Theorem.5 i [] is ivalid whe λ = λ. I fact δ ( α ) = γ α. i this case ( ) Corollary Uder coditios of Theorem 3 if λ λ ad λ λ < the γ ( α) < δ( α ). Example We cosider the three-dimesioal covectio-diffusio equatio ( uxx + uyy + uzz ) + q( ux + uy + uz ) = f( x y z) o the uit cube Ω = [] [] [] with costat coefficiet q ad subject to Dirichlet-type boudary coditios. Discretizig this equatio with seve-poit fiite differece ad assumig the umbers ( ) of grid poits i all three directios are the same we defie h= ( + ) as the step size r = qh/( q > ) is the mesh Reyolds umber. From [] whe the problem size becomes reasoably large i.e. h is reasoably small we kow for the cetred differece scheme that λ = 6( + cos π h) λ = 6( cos πh) 3 π h = 6rcosπ h 3 qh. From Theorem 3 we kow that if 3 3 8π h 8π h = π < q (7) ( + πh) 8πh that is q > π the the method may be a good choice. From Theorem 4 we fid that if 3 3 8π h 8π h q < (8) 9 ( + π h) the L method may be a good choice. I the sequel we ivestigate the spectral radius of the matrix M ( α ) ad M ( α) with the differet value of q. All the matrices tested are 5 5 uless otherwise metioed i Example i.e. = 8. From the above discussig we kow that if = 8 the 3 8π h To cofirm the results of (7) ad (8) here we test q is equal to.5 5 ad respectively. I figures ()-(4) we show that the spectral radius ρ( M ( α )) ad ρ( M ( α)) of the iteratio matrices of both L ad method with the defferet value of α ad the boud δ ( α ). 3 Two examples ISSN: Issue 5 Volume 7 May 8

4 L L Fig. The spectral radius ad boud with q= L Fig 4. The spectral radius ad boud with q= From figures ()-(4) it is ot difficult to fid that the spectral radius of the iteratio matrix L method is much smaller tha that of the method with q be ad.5 o the other had the spectral radius of the method seems to be better tha that of L method whe q are equal to 5 ad. I the meawhile here the distributio of the eigevalues of the iteratio matrix is depicted i figures (5)-(8) with α = which correspod to the differet value of q Fig. The spectral radius ad boud with q= L Fig 5. The distributio of eigevalues with q= Fig 3. The spectral radius ad boud with q= Fig 6. The distributio of eigevalues with q=.5 ISSN: Issue 5 Volume 7 May 8

5 Fig 7. The distributio of eigevalues with q=5 From figures (5)-(8) we fid that the distributio of the eigevalues of the iteratio matrix becomes clustered whe q becomes large. It is ot difficult to fid that the spectral radius becomes large with the icreasig of q (the matrix becomes domiat). Next we study the ad L iteatio method. We try to use the ad L iteratio method to solve the systems of liear equatio () which raises from the disccretized three dimesio covectio-diffusio equatios Fig 8. The distributio of eigevalues with q = For the simplicity we set up the tested problem so that the right had side fuctio is equal to throughout the uit cube domai. All tests are started from the zero vector performed i Matlab 6.5 ad termiated whe the curret iterate satisfies r k 6 < r where r = b Ax ad rk is the residual of the k-th ad L iteratio. Some results are preseted to illustrate the behavior of the covergece of the ad L method respectively which are listed i Tables 4 correspodig to Figures (9)-(). The purpose of these experimets is just to ivestigate the ifluece of the eigevalue distributio o the covergece behavior of ad L method respectively. α (IT) L(IT) Table. Iteratio umber (IT) of ad L withq=. α (IT) L(IT) Table. Iteratio umber (IT) of ad L withq=.5. α (IT) L(IT) Table 3. Iteratio umber (IT) of ad L with q= 5. α (IT) L(IT) ISSN: Issue 5 Volume 7 May 8

6 Table 4. Iteratio umber (IT) of ad L withq = L L Fig 9. Iteratio umber with the differetα ad q= L Fig. Iteratio umber with the differetα ad q= L Fig. Iteratio umber with the differetα ad q= Fig. Iteratio umber with the differetα ad q= From tables -4 ad figures (9)-() it is show that the L performs very good for a wide rage of the parameter α whe q satisfies (8). I the meawhile whe q satisfies (7) the method is more efficiet tha L method. Example We cosider the two-dimesioal covectio-diffusio equatio ( uxx + uyy ) + τ ( ux + uy ) = g( x y) o the uit square [] [] with costat coefficiet τ ad subject to Dirichlet-type boudary coditios. Whe the five-poit cetered fiite differece discretizatio is used to it it is easy to get the system of the liear system () with the coefficiet matrix A = T I + T I ad T= tridiag( r + r) where r= τh/ is the mesh Reyolds umber ad deotes the Kroecker product symbol ad h = (+ m) is used i the discretizatio o both disrectios ad the atural lexicograghic orderig is employed to the ukows. From [4] it is easy to get that for the cetred differece scheme λ = 4( + cos π h) 8 λ = 4( cos πh) π h = 4rcosπh τh. By simple computatios it is easy to get that if τ > π (9) the the method may be a good choice. Ad if 3 3 8π h 8π h τ < () 9 ( + π h) the L method may be a good choice. Be similar to Example we cosider the spectral radius the distributio of the eigevalues of the iteratio matrix ad iteratio umber i Example ISSN: Issue 5 Volume 7 May 8

7 whe the ad L method are applied to solve liear systems () respectively. I the sequel we ivestigate the spectral radius of the matrix M ( α ) ad M ( α) with the differet value of τ. All the matrices tested are uless otherwise metioed i example i.e. m= From the above discussig it easy to get that if m= the 3 8π h To explai the results of (9) ad () here we test τ is also equal to.5 5 ad respectively. I figures (3)-(6) we show that the spectral radius ρ( M ( α)) ad ρ( M ( α)) of the iteratio matrices of both L ad method with the differet value ofτ ad the boud δ ( α ). From the followig figures (3)-(6) it is also to fid that the spectral radius of the iteratio matrix method is much larger tha that of L method with τ be ad.5 o the other had the spectral radius of the method seems to be better tha that of L method whe τ becomes large L Fig 3. The spectral radius ad boud with τ = I the meawhile here figures (7)-() describe the distributio of the eigevalues of the iteratio matrix with α = which correspod to the differet value of τ. From figures (7)-() it is easy to kow that the distributio of the eigevalues of the iteratio matrix becomes clustered whe τ becomes large. Now we ivestigate the perform of the ad L method respectively which is applied to solve the systems of liear equatio () raisig from the disccretized two-dimesioal covectio-diffusio equatio. For coveiece we set up Example so that the right had side fuctio is also equal to throughout the uit square domai. All tests are also started from the zero vector ad the curret iterate satisfies r k 6 <. r We preset some results i Tables 5 8 to make out the behavior of the covergece of the L Fig 5. The spectral radius ad boud with τ = L 4 3 L Fig 6. The spectral radius ad boud with τ = Fig 4. The spectral radius ad boud with τ =.5 ISSN: Issue 5 Volume 7 May 8

8 Fig 7. The distributio of eigevalues with τ = ad L method respectively. Figures ()- (4) depict the behavior of the ad L method with the differet value of τ which is applied to solve () respectively. The purpose of these experimets is just to further reflect ad cofirm the ifluece of the eigevalue distributio o the covergece behavior of ad L method i Example respectively. The solutios of the liear systems i each iteratio are computed exactly (hh) = ad τ= Fig. The distributio of eigevalues with τ = L Fig. Iteratio umber with the differet α ad τ = L Fig 8. The distributio of eigevalues with τ = Fig. Iteratio umber with the differetα ad τ =.5. L Fig 9. The distributio of eigevalues with τ = ISSN: Issue 5 Volume 7 May 8

9 Fig 3. Iteratio umber with the differetα ad τ = L Fig 4. Iteratio umber with the differetα ad τ = α (IT) L(IT) Table 5. Iteratio umber (IT) of ad L withτ =. α (IT) L(IT) Table 6. Iteratio umber (IT) of ad L withτ =.5. α (IT) L(IT) Table 7. Iteratio umber (IT) of ad L withτ = 5. α (IT) L(IT) Table 8. Iteratio umber (IT) of ad L withτ =. Remark 3 From (7)-() we fid that if the problem size becomes more ad more large that is h is reasoably small q orτ must be very small whe the L method is used to apply. That is Theorem 4 is too restrictive to be useful. I fact for the case that q or τ is very small may other efficiet methods like the geeralized cojugate gradiet method should be much more efficiet ad practical tha the L method itroduced by Li et al []. However how to make the L method more efficiet may be further studied. Remark 4 By observig Examples ad q orτ may be chose to be idepedet of h whe the method is applied to solve (). Namely eve if the problem size becomes reasoably large q orτ chose is still idepedet of the value of h ivolved whe the method is applied to solve (). Remark 5 I fact a upper boud may ot truly reflect the covergece behavior of a iteratio method from Theorem 3 ad Theorem 4. However by comparig the method ad the L method it is ot difficult to fid that the method may be much more efficiet ad practical tha the L method. There exist two mai aspects. Firstly from Theorem ad Theorem it is ot difficult to fid that the covergece domai of L may be smaller tha that of. Secodly from () ad (4) uder coditios of Theorem ad Theorem we ca fid that L may be less efficiet tha as oly a sub-system of liear equatios of the coefficiet matrix H is solved rather tha that of the better-coditioed coefficiet matrix α I + H. Ideed let λ ( H) ad λ ( H) mi respectively be the imum ad miimum ISSN: Issue 5 Volume 7 May 8

10 eigevalue of H. κ( H ) deotes the spectral coditio umber of H. It is easy to obtai that α + λ ( H) λ ( H) κα ( I + H) = < = κ( H) α + λmi ( H) λmi ( H) with α >. 4 Coclusio I this ote we have compared the ad L method to solve o-hermitia positive defiite liear systems. A ew criterio for choosig the above two methods has bee preseted. Numerical tests show that the L method performs very well if the Hermitia part of the coefficiet matrix is domiat however as the skew-hermitia part becomes domiat the performace of the L method becomes ot as good as the method. Therefore we eed to improve the L method such as itroduce aother parameter β i the first equatio i (4) which we will study i the future. [5] D.D. Hamide Computers ad liear algebra Wseas Trasactios o Mathematics Vol.3 Issue [6] M.T. Darivishi The best values of parameters i Accelerated successive overrelaxatio methods Wseas Trasactios o Mathematics Vol.3 Issue 3 4 pp [7] Y.M. Li A mootoe iterative method for bipolar juctio trasistor circuit simulatio Wseas Trasactios o Mathematics Vol. Issue 4 pp [8] S.J. Zhag E. Blosch X. Zhao Cojugate gradiet liear solvers for computatioal fluid dyamics Wseas Trasactios o Mathematics Vol.6 Issue 4 pp Ackowledgemets we would like to thak the aoymous referees whose may helpful suggestios have substatially improved this paper. This research was supported by 973 Programs (8CB37) NSFC (773) the Scietific ad Techological Key Project of the Chiese Educatio Miistry (798) the PhD. Programs Fud of Chiese Uiversities (764) Sichua Provice Project for Applied Basic Research (8JY5) ad the Project for Academic Leader ad Group of UESTC. Refereces: [] Z.-Z Bai G.H. Golub M.K. Ng Hermitia ad skew-hermitia splittig methods for o- Hermitia positive defiite liear systems SIAM J. Matrix Aal. Appl. Vol.4 No.3 3 pp [] L. Li T.Z. Huag X.P Liu Modified Hermitia ad skew-hermitia splittig methods for o-hermitia positive-defiite liear systems Numer. Li. Alg. Appl. 4 7 pp [3] Y. Saad Iterative Methods for Sparse Liear Systemsk PWS Publishig Compay Bosto 996. [4] Z.-Z Bai G.H. Golub C.K. Li Optimal parameter i hermitia ad skew-hermitia splittig method for certai two-by-two block matrices SIAM J. Sci. Comput. Vol.8 No. 6 pp ISSN: Issue 5 Volume 7 May 8