1ZSYTRF(1) LAPACK routine (version 3.2) ZSYTRF(1)
2
6 ZSYTRF - computes the factorization of a complex symmetric matrix A
7 using the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE ZSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 INTEGER IPIV( * )
17 COMPLEX*16 A( LDA, * ), WORK( * )
19
21 ZSYTRF computes the factorization of a complex symmetric matrix A using
22 the Bunch-Kaufman diagonal pivoting method. The form of the factoriza‐
23 tion is
24 A = U*D*U**T or A = L*D*L**T
25 where U (or L) is a product of permutation and unit upper (lower) tri‐
26 angular matrices, and D is symmetric and block diagonal with with
27 1-by-1 and 2-by-2 diagonal blocks.
28 This is the blocked version of the algorithm, calling Level 3 BLAS.
29
31 UPLO (input) CHARACTER*1
32 = 'U': Upper triangle of A is stored;
33 = 'L': Lower triangle of A is stored.
34 N (input) INTEGER
36 The order of the matrix A. N >= 0.
37 A (input/output) COMPLEX*16 array, dimension (LDA,N)
39 On entry, the symmetric matrix A. If UPLO = 'U', the leading
40 N-by-N upper triangular part of A contains the upper triangular
41 part of the matrix A, and the strictly lower triangular part of
42 A is not referenced. If UPLO = 'L', the leading N-by-N lower
43 triangular part of A contains the lower triangular part of the
44 matrix A, and the strictly upper triangular part of A is not
45 referenced. On exit, the block diagonal matrix D and the mul‐
46 tipliers used to obtain the factor U or L (see below for fur‐
47 ther details).
48 LDA (input) INTEGER
50 The leading dimension of the array A. LDA >= max(1,N).
51 IPIV (output) INTEGER array, dimension (N)
53 Details of the interchanges and the block structure of D. If
54 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
55 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
56 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
57 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
58 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
59 rows and columns k+1 and -IPIV(k) were interchanged and
60 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
61 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
63 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
64 LWORK (input) INTEGER
66 The length of WORK. LWORK >=1. For best performance LWORK >=
67 N*NB, where NB is the block size returned by ILAENV. If LWORK
68 = -1, then a workspace query is assumed; the routine only cal‐
69 culates the optimal size of the WORK array, returns this value
70 as the first entry of the WORK array, and no error message
71 related to LWORK is issued by XERBLA.
72 INFO (output) INTEGER
74 = 0: successful exit
75 < 0: if INFO = -i, the i-th argument had an illegal value
76 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
77 has been completed, but the block diagonal matrix D is exactly
78 singular, and division by zero will occur if it is used to
79 solve a system of equations.
80
82 If UPLO = 'U', then A = U*D*U', where
83 U = P(n)*U(n)* ... *P(k)U(k)* ...,
84 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
85 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
86 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
87 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
88 the diagonal block D(k) is of order s (s = 1 or 2), then
89 ( I v 0 ) k-s
90 U(k) = ( 0 I 0 ) s
91 ( 0 0 I ) n-k
92 k-s s n-k
93 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
94 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
95 A(k,k), and v overwrites A(1:k-2,k-1:k).
96 If UPLO = 'L', then A = L*D*L', where
97 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
98 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
99 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
100 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
101 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
102 the diagonal block D(k) is of order s (s = 1 or 2), then
103 ( I 0 0 ) k-1
104 L(k) = ( 0 I 0 ) s
105 ( 0 v I ) n-k-s+1
106 k-1 s n-k-s+1
107 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
108 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
109 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
110 LAPACK routine (version 3.2) November 2008 ZSYTRF(1)