1ZSYTRF(1) LAPACK routine (version 3.1) ZSYTRF(1)
2
6 ZSYTRF - the factorization of a complex symmetric matrix A using the
7 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
26 where U (or L) is a product of permutation and unit upper (lower) tri‐
28 angular matrices, and D is symmetric and block diagonal with with
29 1-by-1 and 2-by-2 diagonal blocks.
30 This is the blocked version of the algorithm, calling Level 3 BLAS.
32
35 UPLO (input) CHARACTER*1
36 = 'U': Upper triangle of A is stored;
37 = 'L': Lower triangle of A is stored.
38 N (input) INTEGER
40 The order of the matrix A. N >= 0.
41 A (input/output) COMPLEX*16 array, dimension (LDA,N)
43 On entry, the symmetric matrix A. If UPLO = 'U', the leading
44 N-by-N upper triangular part of A contains the upper triangular
45 part of the matrix A, and the strictly lower triangular part of
46 A is not referenced. If UPLO = 'L', the leading N-by-N lower
47 triangular part of A contains the lower triangular part of the
48 matrix A, and the strictly upper triangular part of A is not
49 referenced.
50 On exit, the block diagonal matrix D and the multipliers used
52 to obtain the factor U or L (see below for further details).
53 LDA (input) INTEGER
55 The leading dimension of the array A. LDA >= max(1,N).
56 IPIV (output) INTEGER array, dimension (N)
58 Details of the interchanges and the block structure of D. If
59 IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
60 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
61 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
62 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
63 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
64 rows and columns k+1 and -IPIV(k) were interchanged and
65 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
66 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
68 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
69 LWORK (input) INTEGER
71 The length of WORK. LWORK >=1. For best performance LWORK >=
72 N*NB, where NB is the block size returned by ILAENV.
73 If LWORK = -1, then a workspace query is assumed; the routine
75 only calculates the optimal size of the WORK array, returns
76 this value as the first entry of the WORK array, and no error
77 message related to LWORK is issued by XERBLA.
78 INFO (output) INTEGER
80 = 0: successful exit
81 < 0: if INFO = -i, the i-th argument had an illegal value
82 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
83 has been completed, but the block diagonal matrix D is exactly
84 singular, and division by zero will occur if it is used to
85 solve a system of equations.
86
88 If UPLO = 'U', then A = U*D*U', where
89 U = P(n)*U(n)* ... *P(k)U(k)* ...,
90 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
91 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
92 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
93 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
94 the diagonal block D(k) is of order s (s = 1 or 2), then
95 ( I v 0 ) k-s
97 U(k) = ( 0 I 0 ) s
98 ( 0 0 I ) n-k
99 k-s s n-k
100 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
102 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
103 A(k,k), and v overwrites A(1:k-2,k-1:k).
104 If UPLO = 'L', then A = L*D*L', where
106 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
107 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
108 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
109 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
110 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
111 the diagonal block D(k) is of order s (s = 1 or 2), then
112 ( I 0 0 ) k-1
114 L(k) = ( 0 I 0 ) s
115 ( 0 v I ) n-k-s+1
116 k-1 s n-k-s+1
117 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
119 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
120 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
121 LAPACK routine (version 3.1) November 2006 ZSYTRF(1)