1 SUBROUTINE ZHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
2 $ LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER FACT, UPLO
11 INTEGER INFO, LDB, LDX, N, NRHS
12 DOUBLE PRECISION RCOND
13 * ..
14 * .. Array Arguments ..
15 INTEGER IPIV( * )
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
18 $ X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
25 * A = L*D*L**H to compute the solution to a complex system of linear
26 * equations A * X = B, where A is an N-by-N Hermitian matrix stored
27 * in packed format and X and B are N-by-NRHS matrices.
28 *
29 * Error bounds on the solution and a condition estimate are also
30 * provided.
31 *
32 * Description
33 * ===========
34 *
35 * The following steps are performed:
36 *
37 * 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
38 * A = U * D * U**H, if UPLO = 'U', or
39 * A = L * D * L**H, if UPLO = 'L',
40 * where U (or L) is a product of permutation and unit upper (lower)
41 * triangular matrices and D is Hermitian and block diagonal with
42 * 1-by-1 and 2-by-2 diagonal blocks.
43 *
44 * 2. If some D(i,i)=0, so that D is exactly singular, then the routine
45 * returns with INFO = i. Otherwise, the factored form of A is used
46 * to estimate the condition number of the matrix A. If the
47 * reciprocal of the condition number is less than machine precision,
48 * INFO = N+1 is returned as a warning, but the routine still goes on
49 * to solve for X and compute error bounds as described below.
50 *
51 * 3. The system of equations is solved for X using the factored form
52 * of A.
53 *
54 * 4. Iterative refinement is applied to improve the computed solution
55 * matrix and calculate error bounds and backward error estimates
56 * for it.
57 *
58 * Arguments
59 * =========
60 *
61 * FACT (input) CHARACTER*1
62 * Specifies whether or not the factored form of A has been
63 * supplied on entry.
64 * = 'F': On entry, AFP and IPIV contain the factored form of
65 * A. AFP and IPIV will not be modified.
66 * = 'N': The matrix A will be copied to AFP and factored.
67 *
68 * UPLO (input) CHARACTER*1
69 * = 'U': Upper triangle of A is stored;
70 * = 'L': Lower triangle of A is stored.
71 *
72 * N (input) INTEGER
73 * The number of linear equations, i.e., the order of the
74 * matrix A. N >= 0.
75 *
76 * NRHS (input) INTEGER
77 * The number of right hand sides, i.e., the number of columns
78 * of the matrices B and X. NRHS >= 0.
79 *
80 * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
81 * The upper or lower triangle of the Hermitian matrix A, packed
82 * columnwise in a linear array. The j-th column of A is stored
83 * in the array AP as follows:
84 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
85 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
86 * See below for further details.
87 *
88 * AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
89 * If FACT = 'F', then AFP is an input argument and on entry
90 * contains the block diagonal matrix D and the multipliers used
91 * to obtain the factor U or L from the factorization
92 * A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
93 * a packed triangular matrix in the same storage format as A.
94 *
95 * If FACT = 'N', then AFP is an output argument and on exit
96 * contains the block diagonal matrix D and the multipliers used
97 * to obtain the factor U or L from the factorization
98 * A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
99 * a packed triangular matrix in the same storage format as A.
100 *
101 * IPIV (input or output) INTEGER array, dimension (N)
102 * If FACT = 'F', then IPIV is an input argument and on entry
103 * contains details of the interchanges and the block structure
104 * of D, as determined by ZHPTRF.
105 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
106 * interchanged and D(k,k) is a 1-by-1 diagonal block.
107 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
108 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
109 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
110 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
111 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
112 *
113 * If FACT = 'N', then IPIV is an output argument and on exit
114 * contains details of the interchanges and the block structure
115 * of D, as determined by ZHPTRF.
116 *
117 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
118 * The N-by-NRHS right hand side matrix B.
119 *
120 * LDB (input) INTEGER
121 * The leading dimension of the array B. LDB >= max(1,N).
122 *
123 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
124 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
125 *
126 * LDX (input) INTEGER
127 * The leading dimension of the array X. LDX >= max(1,N).
128 *
129 * RCOND (output) DOUBLE PRECISION
130 * The estimate of the reciprocal condition number of the matrix
131 * A. If RCOND is less than the machine precision (in
132 * particular, if RCOND = 0), the matrix is singular to working
133 * precision. This condition is indicated by a return code of
134 * INFO > 0.
135 *
136 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
137 * The estimated forward error bound for each solution vector
138 * X(j) (the j-th column of the solution matrix X).
139 * If XTRUE is the true solution corresponding to X(j), FERR(j)
140 * is an estimated upper bound for the magnitude of the largest
141 * element in (X(j) - XTRUE) divided by the magnitude of the
142 * largest element in X(j). The estimate is as reliable as
143 * the estimate for RCOND, and is almost always a slight
144 * overestimate of the true error.
145 *
146 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
147 * The componentwise relative backward error of each solution
148 * vector X(j) (i.e., the smallest relative change in
149 * any element of A or B that makes X(j) an exact solution).
150 *
151 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
152 *
153 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
154 *
155 * INFO (output) INTEGER
156 * = 0: successful exit
157 * < 0: if INFO = -i, the i-th argument had an illegal value
158 * > 0: if INFO = i, and i is
159 * <= N: D(i,i) is exactly zero. The factorization
160 * has been completed but the factor D is exactly
161 * singular, so the solution and error bounds could
162 * not be computed. RCOND = 0 is returned.
163 * = N+1: D is nonsingular, but RCOND is less than machine
164 * precision, meaning that the matrix is singular
165 * to working precision. Nevertheless, the
166 * solution and error bounds are computed because
167 * there are a number of situations where the
168 * computed solution can be more accurate than the
169 * value of RCOND would suggest.
170 *
171 * Further Details
172 * ===============
173 *
174 * The packed storage scheme is illustrated by the following example
175 * when N = 4, UPLO = 'U':
176 *
177 * Two-dimensional storage of the Hermitian matrix A:
178 *
179 * a11 a12 a13 a14
180 * a22 a23 a24
181 * a33 a34 (aij = conjg(aji))
182 * a44
183 *
184 * Packed storage of the upper triangle of A:
185 *
186 * AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191 DOUBLE PRECISION ZERO
192 PARAMETER ( ZERO = 0.0D+0 )
193 * ..
194 * .. Local Scalars ..
195 LOGICAL NOFACT
196 DOUBLE PRECISION ANORM
197 * ..
198 * .. External Functions ..
199 LOGICAL LSAME
200 DOUBLE PRECISION DLAMCH, ZLANHP
201 EXTERNAL LSAME, DLAMCH, ZLANHP
202 * ..
203 * .. External Subroutines ..
204 EXTERNAL XERBLA, ZCOPY, ZHPCON, ZHPRFS, ZHPTRF, ZHPTRS,
205 $ ZLACPY
206 * ..
207 * .. Intrinsic Functions ..
208 INTRINSIC MAX
209 * ..
210 * .. Executable Statements ..
211 *
212 * Test the input parameters.
213 *
214 INFO = 0
215 NOFACT = LSAME( FACT, 'N' )
216 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
217 INFO = -1
218 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
219 $ THEN
220 INFO = -2
221 ELSE IF( N.LT.0 ) THEN
222 INFO = -3
223 ELSE IF( NRHS.LT.0 ) THEN
224 INFO = -4
225 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
226 INFO = -9
227 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
228 INFO = -11
229 END IF
230 IF( INFO.NE.0 ) THEN
231 CALL XERBLA( 'ZHPSVX', -INFO )
232 RETURN
233 END IF
234 *
235 IF( NOFACT ) THEN
236 *
237 * Compute the factorization A = U*D*U**H or A = L*D*L**H.
238 *
239 CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
240 CALL ZHPTRF( UPLO, N, AFP, IPIV, INFO )
241 *
242 * Return if INFO is non-zero.
243 *
244 IF( INFO.GT.0 )THEN
245 RCOND = ZERO
246 RETURN
247 END IF
248 END IF
249 *
250 * Compute the norm of the matrix A.
251 *
252 ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
253 *
254 * Compute the reciprocal of the condition number of A.
255 *
256 CALL ZHPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, INFO )
257 *
258 * Compute the solution vectors X.
259 *
260 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
261 CALL ZHPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
262 *
263 * Use iterative refinement to improve the computed solutions and
264 * compute error bounds and backward error estimates for them.
265 *
266 CALL ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
267 $ BERR, WORK, RWORK, INFO )
268 *
269 * Set INFO = N+1 if the matrix is singular to working precision.
270 *
271 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
272 $ INFO = N + 1
273 *
274 RETURN
275 *
276 * End of ZHPSVX
277 *
278 END
2 $ LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER FACT, UPLO
11 INTEGER INFO, LDB, LDX, N, NRHS
12 DOUBLE PRECISION RCOND
13 * ..
14 * .. Array Arguments ..
15 INTEGER IPIV( * )
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
18 $ X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
25 * A = L*D*L**H to compute the solution to a complex system of linear
26 * equations A * X = B, where A is an N-by-N Hermitian matrix stored
27 * in packed format and X and B are N-by-NRHS matrices.
28 *
29 * Error bounds on the solution and a condition estimate are also
30 * provided.
31 *
32 * Description
33 * ===========
34 *
35 * The following steps are performed:
36 *
37 * 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
38 * A = U * D * U**H, if UPLO = 'U', or
39 * A = L * D * L**H, if UPLO = 'L',
40 * where U (or L) is a product of permutation and unit upper (lower)
41 * triangular matrices and D is Hermitian and block diagonal with
42 * 1-by-1 and 2-by-2 diagonal blocks.
43 *
44 * 2. If some D(i,i)=0, so that D is exactly singular, then the routine
45 * returns with INFO = i. Otherwise, the factored form of A is used
46 * to estimate the condition number of the matrix A. If the
47 * reciprocal of the condition number is less than machine precision,
48 * INFO = N+1 is returned as a warning, but the routine still goes on
49 * to solve for X and compute error bounds as described below.
50 *
51 * 3. The system of equations is solved for X using the factored form
52 * of A.
53 *
54 * 4. Iterative refinement is applied to improve the computed solution
55 * matrix and calculate error bounds and backward error estimates
56 * for it.
57 *
58 * Arguments
59 * =========
60 *
61 * FACT (input) CHARACTER*1
62 * Specifies whether or not the factored form of A has been
63 * supplied on entry.
64 * = 'F': On entry, AFP and IPIV contain the factored form of
65 * A. AFP and IPIV will not be modified.
66 * = 'N': The matrix A will be copied to AFP and factored.
67 *
68 * UPLO (input) CHARACTER*1
69 * = 'U': Upper triangle of A is stored;
70 * = 'L': Lower triangle of A is stored.
71 *
72 * N (input) INTEGER
73 * The number of linear equations, i.e., the order of the
74 * matrix A. N >= 0.
75 *
76 * NRHS (input) INTEGER
77 * The number of right hand sides, i.e., the number of columns
78 * of the matrices B and X. NRHS >= 0.
79 *
80 * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
81 * The upper or lower triangle of the Hermitian matrix A, packed
82 * columnwise in a linear array. The j-th column of A is stored
83 * in the array AP as follows:
84 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
85 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
86 * See below for further details.
87 *
88 * AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
89 * If FACT = 'F', then AFP is an input argument and on entry
90 * contains the block diagonal matrix D and the multipliers used
91 * to obtain the factor U or L from the factorization
92 * A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
93 * a packed triangular matrix in the same storage format as A.
94 *
95 * If FACT = 'N', then AFP is an output argument and on exit
96 * contains the block diagonal matrix D and the multipliers used
97 * to obtain the factor U or L from the factorization
98 * A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
99 * a packed triangular matrix in the same storage format as A.
100 *
101 * IPIV (input or output) INTEGER array, dimension (N)
102 * If FACT = 'F', then IPIV is an input argument and on entry
103 * contains details of the interchanges and the block structure
104 * of D, as determined by ZHPTRF.
105 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
106 * interchanged and D(k,k) is a 1-by-1 diagonal block.
107 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
108 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
109 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
110 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
111 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
112 *
113 * If FACT = 'N', then IPIV is an output argument and on exit
114 * contains details of the interchanges and the block structure
115 * of D, as determined by ZHPTRF.
116 *
117 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
118 * The N-by-NRHS right hand side matrix B.
119 *
120 * LDB (input) INTEGER
121 * The leading dimension of the array B. LDB >= max(1,N).
122 *
123 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
124 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
125 *
126 * LDX (input) INTEGER
127 * The leading dimension of the array X. LDX >= max(1,N).
128 *
129 * RCOND (output) DOUBLE PRECISION
130 * The estimate of the reciprocal condition number of the matrix
131 * A. If RCOND is less than the machine precision (in
132 * particular, if RCOND = 0), the matrix is singular to working
133 * precision. This condition is indicated by a return code of
134 * INFO > 0.
135 *
136 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
137 * The estimated forward error bound for each solution vector
138 * X(j) (the j-th column of the solution matrix X).
139 * If XTRUE is the true solution corresponding to X(j), FERR(j)
140 * is an estimated upper bound for the magnitude of the largest
141 * element in (X(j) - XTRUE) divided by the magnitude of the
142 * largest element in X(j). The estimate is as reliable as
143 * the estimate for RCOND, and is almost always a slight
144 * overestimate of the true error.
145 *
146 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
147 * The componentwise relative backward error of each solution
148 * vector X(j) (i.e., the smallest relative change in
149 * any element of A or B that makes X(j) an exact solution).
150 *
151 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
152 *
153 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
154 *
155 * INFO (output) INTEGER
156 * = 0: successful exit
157 * < 0: if INFO = -i, the i-th argument had an illegal value
158 * > 0: if INFO = i, and i is
159 * <= N: D(i,i) is exactly zero. The factorization
160 * has been completed but the factor D is exactly
161 * singular, so the solution and error bounds could
162 * not be computed. RCOND = 0 is returned.
163 * = N+1: D is nonsingular, but RCOND is less than machine
164 * precision, meaning that the matrix is singular
165 * to working precision. Nevertheless, the
166 * solution and error bounds are computed because
167 * there are a number of situations where the
168 * computed solution can be more accurate than the
169 * value of RCOND would suggest.
170 *
171 * Further Details
172 * ===============
173 *
174 * The packed storage scheme is illustrated by the following example
175 * when N = 4, UPLO = 'U':
176 *
177 * Two-dimensional storage of the Hermitian matrix A:
178 *
179 * a11 a12 a13 a14
180 * a22 a23 a24
181 * a33 a34 (aij = conjg(aji))
182 * a44
183 *
184 * Packed storage of the upper triangle of A:
185 *
186 * AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191 DOUBLE PRECISION ZERO
192 PARAMETER ( ZERO = 0.0D+0 )
193 * ..
194 * .. Local Scalars ..
195 LOGICAL NOFACT
196 DOUBLE PRECISION ANORM
197 * ..
198 * .. External Functions ..
199 LOGICAL LSAME
200 DOUBLE PRECISION DLAMCH, ZLANHP
201 EXTERNAL LSAME, DLAMCH, ZLANHP
202 * ..
203 * .. External Subroutines ..
204 EXTERNAL XERBLA, ZCOPY, ZHPCON, ZHPRFS, ZHPTRF, ZHPTRS,
205 $ ZLACPY
206 * ..
207 * .. Intrinsic Functions ..
208 INTRINSIC MAX
209 * ..
210 * .. Executable Statements ..
211 *
212 * Test the input parameters.
213 *
214 INFO = 0
215 NOFACT = LSAME( FACT, 'N' )
216 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
217 INFO = -1
218 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
219 $ THEN
220 INFO = -2
221 ELSE IF( N.LT.0 ) THEN
222 INFO = -3
223 ELSE IF( NRHS.LT.0 ) THEN
224 INFO = -4
225 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
226 INFO = -9
227 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
228 INFO = -11
229 END IF
230 IF( INFO.NE.0 ) THEN
231 CALL XERBLA( 'ZHPSVX', -INFO )
232 RETURN
233 END IF
234 *
235 IF( NOFACT ) THEN
236 *
237 * Compute the factorization A = U*D*U**H or A = L*D*L**H.
238 *
239 CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
240 CALL ZHPTRF( UPLO, N, AFP, IPIV, INFO )
241 *
242 * Return if INFO is non-zero.
243 *
244 IF( INFO.GT.0 )THEN
245 RCOND = ZERO
246 RETURN
247 END IF
248 END IF
249 *
250 * Compute the norm of the matrix A.
251 *
252 ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
253 *
254 * Compute the reciprocal of the condition number of A.
255 *
256 CALL ZHPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, INFO )
257 *
258 * Compute the solution vectors X.
259 *
260 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
261 CALL ZHPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
262 *
263 * Use iterative refinement to improve the computed solutions and
264 * compute error bounds and backward error estimates for them.
265 *
266 CALL ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
267 $ BERR, WORK, RWORK, INFO )
268 *
269 * Set INFO = N+1 if the matrix is singular to working precision.
270 *
271 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
272 $ INFO = N + 1
273 *
274 RETURN
275 *
276 * End of ZHPSVX
277 *
278 END