1ZLAHEF(1) LAPACK routine (version 3.1) ZLAHEF(1)
2
6 ZLAHEF - a partial factorization of a complex Hermitian matrix A using
7 the Bunch-Kaufman diagonal pivoting method
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10 SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, KB, LDA, LDW, N, NB
15 INTEGER IPIV( * )
17 COMPLEX*16 A( LDA, * ), W( LDW, * )
19
21 ZLAHEF computes a partial factorization of a complex Hermitian matrix A
22 using the Bunch-Kaufman diagonal pivoting method. The partial factor‐
23 ization has the form:
24 A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
26 ( 0 U22 ) ( 0 D ) ( U12' U22' )
27 A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L'
29 ( L21 I ) ( 0 A22 ) ( 0 I )
30 where the order of D is at most NB. The actual order is returned in the
32 argument KB, and is either NB or NB-1, or N if N <= NB. Note that U'
33 denotes the conjugate transpose of U.
34 ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code
36 (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
37 A22 (if UPLO = 'L').
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41 UPLO (input) CHARACTER*1
42 Specifies whether the upper or lower triangular part of the
43 Hermitian matrix A is stored:
44 = 'U': Upper triangular
45 = 'L': Lower triangular
46 N (input) INTEGER
48 The order of the matrix A. N >= 0.
49 NB (input) INTEGER
51 The maximum number of columns of the matrix A that should be
52 factored. NB should be at least 2 to allow for 2-by-2 pivot
53 blocks.
54 KB (output) INTEGER
56 The number of columns of A that were actually factored. KB is
57 either NB-1 or NB, or N if N <= NB.
58 A (input/output) COMPLEX*16 array, dimension (LDA,N)
60 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
61 n-by-n upper triangular part of A contains the upper triangular
62 part of the matrix A, and the strictly lower triangular part of
63 A is not referenced. If UPLO = 'L', the leading n-by-n lower
64 triangular part of A contains the lower triangular part of the
65 matrix A, and the strictly upper triangular part of A is not
66 referenced. On exit, A contains details of the partial factor‐
67 ization.
68 LDA (input) INTEGER
70 The leading dimension of the array A. LDA >= max(1,N).
71 IPIV (output) INTEGER array, dimension (N)
73 Details of the interchanges and the block structure of D. If
74 UPLO = 'U', only the last KB elements of IPIV are set; if UPLO
75 = 'L', only the first KB elements are set.
76 If IPIV(k) > 0, then rows and columns k and IPIV(k) were inter‐
78 changed and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U'
79 and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
80 -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diag‐
81 onal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
82 rows and columns k+1 and -IPIV(k) were interchanged and
83 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
84 W (workspace) COMPLEX*16 array, dimension (LDW,NB)
86 LDW (input) INTEGER
88 The leading dimension of the array W. LDW >= max(1,N).
89 INFO (output) INTEGER
91 = 0: successful exit
92 > 0: if INFO = k, D(k,k) is exactly zero. The factorization
93 has been completed, but the block diagonal matrix D is exactly
94 singular.
95 LAPACK routine (version 3.1) November 2006 ZLAHEF(1)