1DSYTRF(1) LAPACK routine (version 3.2) DSYTRF(1)
2
6 DSYTRF - computes the factorization of a real symmetric matrix A using
7 the Bunch-Kaufman diagonal pivoting method
8
10 SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 INTEGER IPIV( * )
17 DOUBLE PRECISION A( LDA, * ), WORK( * )
19
21 DSYTRF computes the factorization of a real 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 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
63 (MAX(1,LWORK))
64 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
65 LWORK (input) INTEGER
67 The length of WORK. LWORK >=1. For best performance LWORK >=
68 N*NB, where NB is the block size returned by ILAENV. If LWORK
69 = -1, then a workspace query is assumed; the routine only cal‐
70 culates the optimal size of the WORK array, returns this value
71 as the first entry of the WORK array, and no error message
72 related to LWORK is issued by XERBLA.
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
91 U(k) = ( 0 I 0 ) s
92 ( 0 0 I ) n-k
93 k-s s n-k
94 If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s =
95 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
96 A(k,k), and v overwrites A(1:k-2,k-1:k).
97 If UPLO = 'L', then A = L*D*L', where
98 L = P(1)*L(1)* ... *P(k)*L(k)* ...,
99 i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n
100 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
101 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined
102 by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
103 the diagonal block D(k) is of order s (s = 1 or 2), then
104 ( I 0 0 ) k-1
105 L(k) = ( 0 I 0 ) s
106 ( 0 v I ) n-k-s+1
107 k-1 s n-k-s+1
108 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s =
109 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and
110 A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
111 LAPACK routine (version 3.2) November 2008 DSYTRF(1)