1ZHETRF(1) LAPACK routine (version 3.2) ZHETRF(1)
2
6 ZHETRF - computes the factorization of a complex Hermitian matrix A
7 using the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE ZHETRF( 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 ZHETRF computes the factorization of a complex Hermitian matrix A using
22 the Bunch-Kaufman diagonal pivoting method. The form of the factoriza‐
23 tion is
24 A = U*D*U**H or A = L*D*L**H
25 where U (or L) is a product of permutation and unit upper (lower) tri‐
26 angular matrices, and D is Hermitian and block diagonal with 1-by-1 and
27 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 Hermitian 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.
68 INFO (output) INTEGER
70 = 0: successful exit
71 < 0: if INFO = -i, the i-th argument had an illegal value
72 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
73 has been completed, but the block diagonal matrix D is exactly
74 singular, and division by zero will occur if it is used to
75 solve a system of equations.
76
78 If UPLO = 'U', then A = U*D*U', where
79 U = P(n)*U(n)* ... *P(k)U(k)* ...,
80 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
81 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
82 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
83 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
84 the diagonal block D(k) is of order s (s = 1 or 2), then
85 ( I v 0 ) k-s
86 U(k) = ( 0 I 0 ) s
87 ( 0 0 I ) n-k
88 k-s s n-k
89 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
90 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
91 A(k,k), and v overwrites A(1:k-2,k-1:k).
92 If UPLO = 'L', then A = L*D*L', where
93 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
94 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
95 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
96 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
97 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
98 the diagonal block D(k) is of order s (s = 1 or 2), then
99 ( I 0 0 ) k-1
100 L(k) = ( 0 I 0 ) s
101 ( 0 v I ) n-k-s+1
102 k-1 s n-k-s+1
103 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
104 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
105 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
106 LAPACK routine (version 3.2) November 2008 ZHETRF(1)