Filter Diagonalization Method which Constructs an Approximation of Orthonormal Basis of the Invariant Subspace from the Filtered Vectors

1 Av=λBv [a, b] subspace subspace B- subspace B- [a, b] B- Filter Diagonalization Method which Constructs an Approximation of Orthonormal Basis of the Invariant Subspace from the Filtered Vectors Hiroshi Murakami 1 By the filter diagonalization method, for a real symmetric definite generalized eigenproblem Av=λBv of large size, only those eigenpairs are solved whose eigenvalues are in the interval [a, b]. The filter diagonalization method applys the subspace method to the subspace spanned by the set of filter s output vectors to give approximations of eigenpairs. The filter we use is a linear combination of resolvents with complex shifts. Since the set of filter s output vectors usually has many rank deficiencies, the direct application of the subspace method to the set is not appropriate. Therefore, previously the set of B-orthonormal basis for the subspace method has been constructed by the use of SVD of the filter s output vectors in B-metric with the cut-off of singular values below a threshold. Our new method makes use the property of the filter operator and both information of input and output vectors of the filter, and constructs a set of B-orthonormal basis so that the set spans an approximation of the invariant subspace whose eigenvalues are in the interval [a, b]. Some results of numerical of experiments are shown which solved real symmetric definite generalized eigenproblems for the cases both of coefficient matrices are banded. 1. Av=λBv [a, b] 6) 11),14) subspace subspace B- B- A B- [a, b] B-. Vol.11-HPC9 No.1 11/3/15 N GEVP Av=λBv [a, b] [a, b] F τ R(τ) (A τb) 1 B 1 Department of Mathematics and Information Sciences, Tokyo Metropolitan University 1 c 11 Information Processing Society of Japan

F = c I + n γpr(τp) λ v p=1 Fv = f(λ)v f(λ) = c + n γp/(λ τp) λ p=1 λ F f(λ) n τ p γ p c F f(λ) 4 ),5) Butterworth Chebyshev inverse Chebyshev elliptic 1),14) 7),8) Butterworth.1 λ [a, b] t [ 1, 1] λ t t 1 passband 1 < µ t stopbands 1 < t < µ transitionbands f(λ) = g(t) g(t) g min g(t) g max g(t) 1 g min tight 4 3 µ g min g max n n g(t) 3 µ g min n g max g max g(t) 3 g min g max n µ µ g(t) g(t) f(λ) = g(t) λ [a, b] F 1) 14) 3. Subspace N m B N m X X F N m Y Y [a, b] subspace Y GEVP Y N m Y B r( m) Z Z B- r Z T AZ Y B GEVP [a, b] Z subspace Z subspace Vol.11-HPC9 No.1 11/3/15 1 f(λ) 4. Subspace F GEVP Av=λBv c 11 Information Processing Society of Japan

(λ, v) v Fv=f(λ)v F Y F (ρ, v) v B- F (ρ, v) ρ [a, b] f(λ) v Z Z GEVP Av=λBv subspace 4.1 F F Y (ρ, v) Ritz-Galerkin Fv=ρv v=y u X T B m GEVP αu=ρβu α=x T BFY β=x T BY F BF=F T B α=y T BY α β=x T BFX=Y T BX=β T β 4 f(λ) β c n inverse Chebyshev elliptic β Y α β αu=ρβu 4. αu=ρβu Y α β 1 β α Rutishauser Jacobi 1) β β=q T DQ u Qw G QαQ T Gw=ρDw 1 ε MAC ɛ ɛ D G D w Ĝŵ=ρ Dŵ ŵ D1/ z H D 1/ Ĝ D 1/ EVP Hz=ρz Jacobi (ρ, z) z ŵ w u v=y u Jacobi z u β- v (i) = (1/ ρ (i) ) Y u (i) v (i) B- 5. (λ, v) v B- r (A λb)v B 1 - r T B 1 r λ Wilkinson CPU intel Core i7-6k 3.4GHz 8MB 4 Hyperthread P67chipset); 16Gbytes dual channel; 4 4GB DDR3 16MHz PC38 DIMM intel Fortran/OpenMP v1. for intel64 -fast -openmp IEEE754 64-bit OS Fedora14 for intel64 5.1 1 n=1 elliptic 1 A B N=1 5 h=1 p q h A p,q= p pq Bp,q= 1 +q p+q 1 + δp,q. p, q=1,,..., N δ p,q Kronecker [5,1] 11 X m=15 B- n=1 elliptic g min = 3[dB] g max = 15.14[dB] µ=1.4 c 9.7 1 16 Y subspace Z SVD 1 Vol.11-HPC9 No.1 11/3/15 B-SVD Y 1 9 13 13 subspace 13 [5,1] 111 111 3 3 c 11 Information Processing Society of Japan

Vol.11-HPC9 No.1 11/3/15 SIGMA 1.8 LOG 1 SIGMA -1.6.4. -14-16 4 6 8 1 1 14 16 4 6 8 1 1 14 16 1 m=15 1 9 4 1 ρ m=15.5 IT IT LOG 1 DELTA LOG 1 DELTA -1-1 5 6 7 8 9 1 5 6 7 8 9 1 3 1 [5,1] m=15 5 1 [5,1] m=15 IT IT 1 1 11 1 6 7 8 IT Rayleigh 1 IT 1 [5,1] 11 1 1 1 1 14. 16.9 4 44. 6.3 4 c 11 Information Processing Society of Japan

Vol.11-HPC9 No.1 11/3/15 1 SIGMA.8.6 LOG 1 SIGMA -1.4. -14-16 4 6 8 1 1 14 16 4 6 8 1 1 8 ρ m=15.5 6 m=15 1 9 IT IT LOG 1 DELTA LOG 1 DELTA -1-1 9 1 3 4 5 [,5] m=15 1 3 4 5 7 [,5] m=15 1 ρ 4 g min= 3[dB] ρ.51187 ρ.5 11 11 subspace [5,1] 5 IT Rayleigh 1 IT 5 c 11 Information Processing Society of Japan

1 9 1 8 1 11 1 1 134.9 183.8 4 4.4 53.7 5. n=16 elliptic A B N=3 1 5 h=1 p q h A p,q= p pq Bp,q= 1 +q p+q 1 + δp,q p, q=1,,..., N δ p,q Kronecker [,5] 11 X m=15 B- n=16 elliptic g min = 3[dB] g max = 14.7[dB] µ=1.1 c 5.3 1 15 Y subspace Z SVD B-SVD Y 6 1 9 118 118 subspace 118 [,5] 11 7 11 IT 118 Rayleigh 1 11 11 1 8 1 6 8 4 349.9 18.8. 1 14 β 1 ρ 8 g min = 3[dB] ρ.51187 ρ.5 11 11 subspace [,5] 9 IT Rayleigh 1 IT 4 1 1 4 1 9 11 4 89.3 168.5 5.3 3 n=4 Chebyshev 3 A B N=3 1 5 h=1 p q h A p,q= max(p, q) 1 B p,q= 1 p+q 1 + δ p,q p, q=1,,..., N δ p,q Kronecker [15,] 88 X m= B- n=4 Chebyshev g min= 3[dB] g max= 39.715[dB] µ= Chebyshev c Y subspace Z SVD 3 Vol.11-HPC9 No.1 11/3/15 B-SVD Y 1 1 3 1 1 subspace 1 [15,] 88 1 Rayleigh 88 11 88 IT 88 IT 1 Rayleigh 1 [15,] 88 IT 88 1 4 8.1 144.7 6 c 11 Information Processing Society of Japan

Vol.11-HPC9 No.1 11/3/15 SIGMA 1.8 LOG 1 SIGMA -1.6.4. -14-16 4 6 8 1 1 14 16 18 4 6 8 1 1 14 16 18 1 3 [15,] m= 1 3 ρ m=.5 IT IT LOG 1 DELTA LOG 1 DELTA -1-1 15 16 17 18 19 15 16 17 18 19 11 3 [15,] m= 13 3 [15,] m= 3 β 1. 1 9. 1 14 β 1 ρ g min= 3[dB] ρ.51187 ρ(88)=.511687 ρ(89)=.4754493 ρ(9)=.4313191 ρ(91)=.379976 ρ(9)=.845459 ρ(93)=.18536 88 ρ ρ.5 4 9 9 subspace [15,] 88 4 9 [15,] 88 13 IT 88 IT 7 c 11 Information Processing Society of Japan

9 Rayleigh 1 IT 1 3 1 4 11 4 66.4 134.3 6. [a, b] F m B- X X F Y Y Z Z subspace SVD Y B- Z 3) L- SVD subspace Z Y B GEVP A Z Z subspace Y [a, b] Z B- Z subspace GEVP [a, b] F GEVP (λ, v) F (ρ, v) ρ = f(λ) v f subspace [a, b] Z F Fv = ρv Y v (ρ, v) ρ f(λ) λ [a, b] [g min, 1] v B Z Vol.11-HPC9 No.1 11/3/15 1) Rutishauser, H.: The Jacobi method for real symmetric matrices, Numerische Mathematik, Vol.9 (1966), pp.1 1. ) Daniels,R.W.: Approximation Methods for Electronic Filter Design, McGraw-Hill, 1974. 3) Hansen, P.C. and O Leary, D.P.: The Use of the L-curve in the Regularization of Discrete Ill-posed Problems, SIAM J. Sci. Comput., Vol.14,No.6 (1993), pp.1487 153. 4) Daniel B.S.: Criteria for Combining Inverse and Rayleigh Quotient Iteration, SIAM J. Numer. Anal., Vol.5,No.6 (1988), pp.1369 1375. 5) Lutovac,M.D., To sić, D.Y. and Evans,B.L.: Filter Design for Signal Processing, 1.8, Prentice Hall, 1. 6) Toledo, S. and Rabani, E.: Very Large Electronic Structure Calculations Using an Out-of-Core Filter-Diagonalization Method, J. Comput. Phys., Vol.18, No.1 (), pp.56 69. 7) Sakurai,T. and Sugiura,H: A projection method for generalized eigenvalue problems using numerical integration, J. Comp. Appl. Math., Vol.159(3), pp.119 18. 8) Polizzi, E.: Density-matrix-based algorithm for solving eigenvalue problems, Phys.Rev.B, Vol.79, No.11(9), p.11511[6pages]. 9) :,, 7-HPC-11(6) (7), pp.31 36. 1) :,, Vol.49, No.SIG (ACS1) (8), pp.66 87. 11) Ikegami,T., Tadano,H., Umeda,H. and Sakurai,T.: Hierarchical parallel algorithm to solve large generalized eigenproblems, HPCS1 (1), pp.17 114. 1) :,, Vol.1-HPC4, No.3 (1). 13) :,, Vol.1-HPC5, No.1 (1). 14) :, (ACS31), Vol.3, No.3 (1), pp.1 1. 8 c 11 Information Processing Society of Japan