1CHETRF(1) LAPACK routine (version 3.1) CHETRF(1)
2
6 CHETRF - the factorization of a complex Hermitian matrix A using the
7 Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE CHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 INTEGER IPIV( * )
17 COMPLEX A( LDA, * ), WORK( * )
19
21 CHETRF 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
26 where U (or L) is a product of permutation and unit upper (lower) tri‐
28 angular matrices, and D is Hermitian and block diagonal with 1-by-1 and
29 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 array, dimension (LDA,N)
43 On entry, the Hermitian 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 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 INFO (output) INTEGER
75 = 0: successful exit
76 < 0: if INFO = -i, the i-th argument had an illegal value
77 > 0: if INFO = i, D(i,i) is exactly zero. The factorization
78 has been completed, but the block diagonal matrix D is exactly
79 singular, and division by zero will occur if it is used to
80 solve a system of equations.
81
83 If UPLO = 'U', then A = U*D*U', where
84 U = P(n)*U(n)* ... *P(k)U(k)* ...,
85 i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1
86 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
87 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
88 by IPIV(k), and U(k) is a unit upper triangular matrix, such that if
89 the diagonal block D(k) is of order s (s = 1 or 2), then
90 ( I v 0 ) k-s
92 U(k) = ( 0 I 0 ) s
93 ( 0 0 I ) n-k
94 k-s s n-k
95 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
97 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
98 A(k,k), and v overwrites A(1:k-2,k-1:k).
99 If UPLO = 'L', then A = L*D*L', where
101 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
102 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
103 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
104 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
105 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
106 the diagonal block D(k) is of order s (s = 1 or 2), then
107 ( I 0 0 ) k-1
109 L(k) = ( 0 I 0 ) s
110 ( 0 v I ) n-k-s+1
111 k-1 s n-k-s+1
112 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
114 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
115 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
116 LAPACK routine (version 3.1) November 2006 CHETRF(1)