1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
|
SUBROUTINE ZSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
* @precisions normal z -> s d c
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZSYSV computes the solution to a complex system of linear equations
* A * X = B,
* where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
* matrices.
*
* The diagonal pivoting method is used to factor A as
* A = U * D * U**T, if UPLO = 'U', or
* A = L * D * L**T, if UPLO = 'L',
* where U (or L) is a product of permutation and unit upper (lower)
* triangular matrices, and D is symmetric and block diagonal with
* 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then
* used to solve the system of equations A * X = B.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input/output) COMPLEX*16 array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
*
* On exit, if INFO = 0, the block diagonal matrix D and the
* multipliers used to obtain the factor U or L from the
* factorization A = U*D*U**T or A = L*D*L**T as computed by
* ZSYTRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (output) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D, as
* determined by ZSYTRF. If IPIV(k) > 0, then rows and columns
* k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
* diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
* then rows and columns k-1 and -IPIV(k) were interchanged and
* D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
* IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
* -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
* diagonal block.
*
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The length of WORK. LWORK >= 1, and for best performance
* LWORK >= max(1,N*NB), where NB is the optimal blocksize for
* ZSYTRF.
* for LWORK < N, TRS will be done with Level BLAS 2
* for LWORK >= N, TRS will be done with Level BLAS 3
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, D(i,i) is exactly zero. The factorization
* has been completed, but the block diagonal matrix D is
* exactly singular, so the solution could not be computed.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LWKOPT, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZSYTRF, ZSYTRS, ZSYTRS2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
LWKOPT = 1
ELSE
CALL ZSYTRF( UPLO, N, A, LDA, IPIV, WORK, -1, INFO )
LWKOPT = WORK(1)
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZSYSV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL ZSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
IF ( LWORK.LT.N ) THEN
*
* Solve with TRS ( Use Level BLAS 2)
*
CALL ZSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
ELSE
*
* Solve with TRS2 ( Use Level BLAS 3)
*
CALL ZSYTRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
*
END IF
*
END IF
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZSYSV
*
END
|