1DSYSVX(1) LAPACK driver routine (version 3.2) DSYSVX(1)
2
6 DSYSVX - uses the diagonal pivoting factorization to compute the solu‐
7 tion to a real system of linear equations A * X = B,
8
10 SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
11 X, LDX, RCOND, FERR, BERR, WORK, LWORK, IWORK, INFO
12 )
13 CHARACTER FACT, UPLO
15 INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
17 DOUBLE PRECISION RCOND
19 INTEGER IPIV( * ), IWORK( * )
21 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
23 BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
24
26 DSYSVX uses the diagonal pivoting factorization to compute the solution
27 to a real system of linear equations A * X = B, where A is an N-by-N
28 symmetric matrix and X and B are N-by-NRHS matrices.
29 Error bounds on the solution and a condition estimate are also pro‐
30 vided.
31
33 The following steps are performed:
34 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
35 The form of the factorization is
36 A = U * D * U**T, if UPLO = 'U', or
37 A = L * D * L**T, if UPLO = 'L',
38 where U (or L) is a product of permutation and unit upper (lower)
39 triangular matrices, and D is symmetric and block diagonal with
40 1-by-1 and 2-by-2 diagonal blocks.
41 2. If some D(i,i)=0, so that D is exactly singular, then the routine
42 returns with INFO = i. Otherwise, the factored form of A is used
43 to estimate the condition number of the matrix A. If the
44 reciprocal of the condition number is less than machine precision,
45 INFO = N+1 is returned as a warning, but the routine still goes on
46 to solve for X and compute error bounds as described below. 3. The
47 system of equations is solved for X using the factored form
48 of A.
49 4. Iterative refinement is applied to improve the computed solution
50 matrix and calculate error bounds and backward error estimates
51 for it.
52
54 FACT (input) CHARACTER*1
55 Specifies whether or not the factored form of A has been sup‐
56 plied on entry. = 'F': On entry, AF and IPIV contain the fac‐
57 tored form of A. AF and IPIV will not be modified. = 'N':
58 The matrix A will be copied to AF and factored.
59 UPLO (input) CHARACTER*1
61 = 'U': Upper triangle of A is stored;
62 = 'L': Lower triangle of A is stored.
63 N (input) INTEGER
65 The number of linear equations, i.e., the order of the matrix
66 A. N >= 0.
67 NRHS (input) INTEGER
69 The number of right hand sides, i.e., the number of columns of
70 the matrices B and X. NRHS >= 0.
71 A (input) DOUBLE PRECISION array, dimension (LDA,N)
73 The symmetric matrix A. If UPLO = 'U', the leading N-by-N
74 upper triangular part of A contains the upper triangular part
75 of the matrix A, and the strictly lower triangular part of A is
76 not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
77 gular part of A contains the lower triangular part of the
78 matrix A, and the strictly upper triangular part of A is not
79 referenced.
80 LDA (input) INTEGER
82 The leading dimension of the array A. LDA >= max(1,N).
83 AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
85 If FACT = 'F', then AF is an input argument and on entry con‐
86 tains the block diagonal matrix D and the multipliers used to
87 obtain the factor U or L from the factorization A = U*D*U**T or
88 A = L*D*L**T as computed by DSYTRF. If FACT = 'N', then AF is
89 an output argument and on exit returns the block diagonal
90 matrix D and the multipliers used to obtain the factor U or L
91 from the factorization A = U*D*U**T or A = L*D*L**T.
92 LDAF (input) INTEGER
94 The leading dimension of the array AF. LDAF >= max(1,N).
95 IPIV (input or output) INTEGER array, dimension (N)
97 If FACT = 'F', then IPIV is an input argument and on entry con‐
98 tains details of the interchanges and the block structure of D,
99 as determined by DSYTRF. If IPIV(k) > 0, then rows and columns
100 k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal
101 block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows
102 and columns k-1 and -IPIV(k) were interchanged and
103 D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
104 IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)
105 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal
106 block. If FACT = 'N', then IPIV is an output argument and on
107 exit contains details of the interchanges and the block struc‐
108 ture of D, as determined by DSYTRF.
109 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
111 The N-by-NRHS right hand side matrix B.
112 LDB (input) INTEGER
114 The leading dimension of the array B. LDB >= max(1,N).
115 X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
117 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
118 LDX (input) INTEGER
120 The leading dimension of the array X. LDX >= max(1,N).
121 RCOND (output) DOUBLE PRECISION
123 The estimate of the reciprocal condition number of the matrix
124 A. If RCOND is less than the machine precision (in particular,
125 if RCOND = 0), the matrix is singular to working precision.
126 This condition is indicated by a return code of INFO > 0.
127 FERR (output) DOUBLE PRECISION array, dimension (NRHS)
129 The estimated forward error bound for each solution vector X(j)
130 (the j-th column of the solution matrix X). If XTRUE is the
131 true solution corresponding to X(j), FERR(j) is an estimated
132 upper bound for the magnitude of the largest element in (X(j) -
133 XTRUE) divided by the magnitude of the largest element in X(j).
134 The estimate is as reliable as the estimate for RCOND, and is
135 almost always a slight overestimate of the true error.
136 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
138 The componentwise relative backward error of each solution vec‐
139 tor X(j) (i.e., the smallest relative change in any element of
140 A or B that makes X(j) an exact solution).
141 WORK (workspace/output) DOUBLE PRECISION array, dimension
143 (MAX(1,LWORK))
144 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
145 LWORK (input) INTEGER
147 The length of WORK. LWORK >= max(1,3*N), and for best perfor‐
148 mance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where NB is
149 the optimal blocksize for DSYTRF. If LWORK = -1, then a
150 workspace query is assumed; the routine only calculates the
151 optimal size of the WORK array, returns this value as the first
152 entry of the WORK array, and no error message related to LWORK
153 is issued by XERBLA.
154 IWORK (workspace) INTEGER array, dimension (N)
156 INFO (output) INTEGER
158 = 0: successful exit
159 < 0: if INFO = -i, the i-th argument had an illegal value
160 > 0: if INFO = i, and i is
161 <= N: D(i,i) is exactly zero. The factorization has been com‐
162 pleted but the factor D is exactly singular, so the solution
163 and error bounds could not be computed. RCOND = 0 is returned.
164 = N+1: D is nonsingular, but RCOND is less than machine preci‐
165 sion, meaning that the matrix is singular to working precision.
166 Nevertheless, the solution and error bounds are computed
167 because there are a number of situations where the computed
168 solution can be more accurate than the value of RCOND would
169 suggest.
170 LAPACK driver routine (version 3.N2o)vember 2008 DSYSVX(1)