SCHUR-MONOTONIC HSS APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES

H1 =
U1
ˆ1 U
T1 ˆ1 T
≈ U1 T1 ≡ U1
T12
T14
T15
,
˜ij , T ˆij ). Note that a where Tij ’s are temporary matrices (also below, including any T tolerance is used in this approximate factorization. Then we can rewrite A as D U T T U1 T14 U1 T15 1 1 12 D1 (1) (1) (1) T T T12 A25 A24 A22 U1 A≈ (1)T (1) (1) , T T T14 I U1 A24 A44 A45 T T (1)T (1)T (1) T15 U1 A25 A45 A55 where A22 (1) T A24 (1)T A25 (b) Node 2.
Fig. 1.1. HSS tree.
an HSS matrix [9] also gives a superfast procedure to compute an approximate HSS factorization of the matrix. In fact, for a general SPD matrix A (not in HSS form) we can find an HSS approximation and in the meantime keep the positive definiteness of the matrix. Depending on the partition, HSS tree, and tolerance we may have different approximations. 2. Schur-monotonic HSS factorization. Given a matrix A and a partition sequence {mi } we compute a factorization A ≈ RT R where R is in postordering HSS

T D2 T T T T34 R2 U2 T T T T35 R2 U2
D1
I
U1 B1 V2T D2
U1 R1 T34 U2 R2 T34 (2) A44 (2)T A45
U1 R1 T35 U2 R2 T35 . (2) A45 (2) A55
(d) Node 4. Similarly compute
T A22 = D2 D2 , (1)
and write
T D1 T T T12 U1 A≈ T T T14 U1 T T T15 U1
T D2 T H2
I
D1
U1 T12 D2
U1 T14 U1 T15 H2 (1) (1) A45 A44 T − H2 H2 (1)T (1) A45 A55
SCHUR-MONOTONIC HSS APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES
S. CHANDRASEKARAN, M. GU, X. S. LI, J. XIA Abstract. Given a symmetric positive definite (SPD) matrix A, we compute an approximate factorization A ≈ RT R with any desired accuracy, where R is a hierarchically semi-separable (HSS) matrix. The factorization is both stable and efficient. The approximation RT R to A is an HSS matrix and is guaranteed to be positive definite.
A25
(1)
≈ U2
B2
T2
≡ U2
B2
T24 U1 T15 U2 T25 (2) . A45 (2) A55
T25
.
Now A becomes
T D1 T T T12 U1 A≈ T T T14 U1 T T U1 T15 (2) (2)
D1
T D2 T T T24 U2 T T U2 T25 (1)
,
(1) (1) A24 A25 . Now compress the second off-diagonal block row but ignoring any basis U, V (i.e. U1 here) T T12
−T where H2 = D2
H2
≡
T T12
A24
(1)
3
D1
U1 B1 V2T D2
U1 R1 T34 U2 R2 T34 D4
U1 R1 T35 U2 R2 T35 . H4 (2) T A55 − H4 H4
1
form. For simplicity we assume that R is associated with a complete tree such as in Fig. 1.1. We first demonstrate the procedure of constructing a 4 × 4 block HSS form (2.1) for R and {mi }: T T T T T D1 U1 B1 U2 U1 R1 B3 R4 U4 U1 R1 B3 R5 U5 T T T T D2 U2 R2 B3 R4 U4 U2 R2 B3 R5 U5 (2.1) R= T , D4 U4 B4 U5 D5 First we partition the matrix H into a A11 AT 12 A= AT 14 AT 15 4 × 4 block form A12 A22 AT 24 AT 25 A14 A24 A44 AT 45 A15 A25 , A45 A55
−T T A44 = D4 D4 , H4 = D4 A45 , T D1 T U2 B2 U1 A≈ T T T T34 R1 U1 T T T T35 R1 U1 (2) (2)


T D2 T T T T34 R2 U2 T T T T35 R2 U2 T D4 T H4
I
I
(1)
U1 T12 D2
U1 T14 U2 T24 (2) A44 (2)T A45
A45 A44 A45 A44 T T2 . − T2 = (1)T (1) (2)T (2) A45 A55 A45 A55 (c) Node 3. Node 3 is in upper level (level 1) with children node 1 and 2. A has two block rows/columns in terms of level 1. The off-diagonal block row corresponding to node 3 can be obtained by merging appropriate blocks of the off-diagonal block rows of node 1 and 2. We identify and compress it (ignoring any U basis) where T14 T24 Now A becomes T D1 T U2 B2 U1 A≈ T T T T34 R1 U1 T T T T35 R1 U1 T15 T25 = R1 R2 T34 T35 .
T A11 = D1 D1
and write A=
T D1 T H1
I

D1
Байду номын сангаас
A22 AT 24 AT 25
A24 A44 AT 45
H1 A25 , T A45 − H1 H1 A55
−T A12 A14 A15 . Now compress the first off-diagonal block row where H1 ≡ D1 in level 2 (bottom level) by a (rank revealing) QR factorization
2
(1)
A24 (1) A44 (1)T A45
(1)
(1) A25 A22 (1) A45 = AT 24 T (1) A 25 A55
A24 A44 AT 45
A25 T A45 − T1 T1 . A55
Now compute a Cholesky factorization
1. Introduction. For a given SPD matrix A [4] presents a fast and stable procedure to compute a sequentially semi-separable (SSS) approximation A ≈ RT R, where R is an upper triangular matrix in SSS form. See [1, 2, 3] for more details about SSS matrices. Here we show an algorithm that computes a factorization RT R where R is an HSS matrix. HSS structured are proposed by Chandrasekaran, Gu, et al. [6, 5] and they are generalizations of SSS structures and and H-matrices (see, e.g. [7, 8]). The generalizations of HSS matrices and HSS operations are included in [9]. Specifically [9] proposes a postordering HSS form T T T T V5 V4 U1 R1 B3 W5 U1 R1 B3 W4 D1 U1 B1 V2T T T T T V5 V4 U2 R2 B3 W5 D2 U2 R2 B3 W4 U2 B2 V1T (1.1) T T T T T . U4 R4 B6 W1 D4 U4 B4 V5 V1 U4 R4 B6 W2 V2 T T T T D5 U5 B5 V4T V2 V1 U5 R5 B6 W2 U5 R5 B6 W1 The postordering HSS form is a simplified version of the original HSS representation and more directly reflect the tree nature of the structure. Such a form is thus associated with an HSS tree. Fig. 1.1 shows a complete HSS tree for (1.1). Given