1 SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
2 $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, UPLO
12 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
17 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
25 * compute the solution to a complex system of linear equations
26 * A * X = B,
27 * where A is an N-by-N Hermitian positive definite band matrix and X
28 * and B are N-by-NRHS matrices.
29 *
30 * Error bounds on the solution and a condition estimate are also
31 * provided.
32 *
33 * Description
34 * ===========
35 *
36 * The following steps are performed:
37 *
38 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
39 * the system:
40 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
41 * Whether or not the system will be equilibrated depends on the
42 * scaling of the matrix A, but if equilibration is used, A is
43 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
44 *
45 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
46 * factor the matrix A (after equilibration if FACT = 'E') as
47 * A = U**H * U, if UPLO = 'U', or
48 * A = L * L**H, if UPLO = 'L',
49 * where U is an upper triangular band matrix, and L is a lower
50 * triangular band matrix.
51 *
52 * 3. If the leading i-by-i principal minor is not positive definite,
53 * then the routine returns with INFO = i. Otherwise, the factored
54 * form of A is used to estimate the condition number of the matrix
55 * A. If the reciprocal of the condition number is less than machine
56 * precision, INFO = N+1 is returned as a warning, but the routine
57 * still goes on to solve for X and compute error bounds as
58 * described below.
59 *
60 * 4. The system of equations is solved for X using the factored form
61 * of A.
62 *
63 * 5. Iterative refinement is applied to improve the computed solution
64 * matrix and calculate error bounds and backward error estimates
65 * for it.
66 *
67 * 6. If equilibration was used, the matrix X is premultiplied by
68 * diag(S) so that it solves the original system before
69 * equilibration.
70 *
71 * Arguments
72 * =========
73 *
74 * FACT (input) CHARACTER*1
75 * Specifies whether or not the factored form of the matrix A is
76 * supplied on entry, and if not, whether the matrix A should be
77 * equilibrated before it is factored.
78 * = 'F': On entry, AFB contains the factored form of A.
79 * If EQUED = 'Y', the matrix A has been equilibrated
80 * with scaling factors given by S. AB and AFB will not
81 * be modified.
82 * = 'N': The matrix A will be copied to AFB and factored.
83 * = 'E': The matrix A will be equilibrated if necessary, then
84 * copied to AFB and factored.
85 *
86 * UPLO (input) CHARACTER*1
87 * = 'U': Upper triangle of A is stored;
88 * = 'L': Lower triangle of A is stored.
89 *
90 * N (input) INTEGER
91 * The number of linear equations, i.e., the order of the
92 * matrix A. N >= 0.
93 *
94 * KD (input) INTEGER
95 * The number of superdiagonals of the matrix A if UPLO = 'U',
96 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
97 *
98 * NRHS (input) INTEGER
99 * The number of right-hand sides, i.e., the number of columns
100 * of the matrices B and X. NRHS >= 0.
101 *
102 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
103 * On entry, the upper or lower triangle of the Hermitian band
104 * matrix A, stored in the first KD+1 rows of the array, except
105 * if FACT = 'F' and EQUED = 'Y', then A must contain the
106 * equilibrated matrix diag(S)*A*diag(S). The j-th column of A
107 * is stored in the j-th column of the array AB as follows:
108 * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
109 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
110 * See below for further details.
111 *
112 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
113 * diag(S)*A*diag(S).
114 *
115 * LDAB (input) INTEGER
116 * The leading dimension of the array A. LDAB >= KD+1.
117 *
118 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
119 * If FACT = 'F', then AFB is an input argument and on entry
120 * contains the triangular factor U or L from the Cholesky
121 * factorization A = U**H *U or A = L*L**H of the band matrix
122 * A, in the same storage format as A (see AB). If EQUED = 'Y',
123 * then AFB is the factored form of the equilibrated matrix A.
124 *
125 * If FACT = 'N', then AFB is an output argument and on exit
126 * returns the triangular factor U or L from the Cholesky
127 * factorization A = U**H *U or A = L*L**H.
128 *
129 * If FACT = 'E', then AFB is an output argument and on exit
130 * returns the triangular factor U or L from the Cholesky
131 * factorization A = U**H *U or A = L*L**H of the equilibrated
132 * matrix A (see the description of A for the form of the
133 * equilibrated matrix).
134 *
135 * LDAFB (input) INTEGER
136 * The leading dimension of the array AFB. LDAFB >= KD+1.
137 *
138 * EQUED (input or output) CHARACTER*1
139 * Specifies the form of equilibration that was done.
140 * = 'N': No equilibration (always true if FACT = 'N').
141 * = 'Y': Equilibration was done, i.e., A has been replaced by
142 * diag(S) * A * diag(S).
143 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
144 * output argument.
145 *
146 * S (input or output) DOUBLE PRECISION array, dimension (N)
147 * The scale factors for A; not accessed if EQUED = 'N'. S is
148 * an input argument if FACT = 'F'; otherwise, S is an output
149 * argument. If FACT = 'F' and EQUED = 'Y', each element of S
150 * must be positive.
151 *
152 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
153 * On entry, the N-by-NRHS right hand side matrix B.
154 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
155 * B is overwritten by diag(S) * B.
156 *
157 * LDB (input) INTEGER
158 * The leading dimension of the array B. LDB >= max(1,N).
159 *
160 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
161 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
162 * the original system of equations. Note that if EQUED = 'Y',
163 * A and B are modified on exit, and the solution to the
164 * equilibrated system is inv(diag(S))*X.
165 *
166 * LDX (input) INTEGER
167 * The leading dimension of the array X. LDX >= max(1,N).
168 *
169 * RCOND (output) DOUBLE PRECISION
170 * The estimate of the reciprocal condition number of the matrix
171 * A after equilibration (if done). If RCOND is less than the
172 * machine precision (in particular, if RCOND = 0), the matrix
173 * is singular to working precision. This condition is
174 * indicated by a return code of INFO > 0.
175 *
176 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
177 * The estimated forward error bound for each solution vector
178 * X(j) (the j-th column of the solution matrix X).
179 * If XTRUE is the true solution corresponding to X(j), FERR(j)
180 * is an estimated upper bound for the magnitude of the largest
181 * element in (X(j) - XTRUE) divided by the magnitude of the
182 * largest element in X(j). The estimate is as reliable as
183 * the estimate for RCOND, and is almost always a slight
184 * overestimate of the true error.
185 *
186 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
187 * The componentwise relative backward error of each solution
188 * vector X(j) (i.e., the smallest relative change in
189 * any element of A or B that makes X(j) an exact solution).
190 *
191 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
192 *
193 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
194 *
195 * INFO (output) INTEGER
196 * = 0: successful exit
197 * < 0: if INFO = -i, the i-th argument had an illegal value
198 * > 0: if INFO = i, and i is
199 * <= N: the leading minor of order i of A is
200 * not positive definite, so the factorization
201 * could not be completed, and the solution has not
202 * been computed. RCOND = 0 is returned.
203 * = N+1: U is nonsingular, but RCOND is less than machine
204 * precision, meaning that the matrix is singular
205 * to working precision. Nevertheless, the
206 * solution and error bounds are computed because
207 * there are a number of situations where the
208 * computed solution can be more accurate than the
209 * value of RCOND would suggest.
210 *
211 * Further Details
212 * ===============
213 *
214 * The band storage scheme is illustrated by the following example, when
215 * N = 6, KD = 2, and UPLO = 'U':
216 *
217 * Two-dimensional storage of the Hermitian matrix A:
218 *
219 * a11 a12 a13
220 * a22 a23 a24
221 * a33 a34 a35
222 * a44 a45 a46
223 * a55 a56
224 * (aij=conjg(aji)) a66
225 *
226 * Band storage of the upper triangle of A:
227 *
228 * * * a13 a24 a35 a46
229 * * a12 a23 a34 a45 a56
230 * a11 a22 a33 a44 a55 a66
231 *
232 * Similarly, if UPLO = 'L' the format of A is as follows:
233 *
234 * a11 a22 a33 a44 a55 a66
235 * a21 a32 a43 a54 a65 *
236 * a31 a42 a53 a64 * *
237 *
238 * Array elements marked * are not used by the routine.
239 *
240 * =====================================================================
241 *
242 * .. Parameters ..
243 DOUBLE PRECISION ZERO, ONE
244 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
245 * ..
246 * .. Local Scalars ..
247 LOGICAL EQUIL, NOFACT, RCEQU, UPPER
248 INTEGER I, INFEQU, J, J1, J2
249 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
250 * ..
251 * .. External Functions ..
252 LOGICAL LSAME
253 DOUBLE PRECISION DLAMCH, ZLANHB
254 EXTERNAL LSAME, DLAMCH, ZLANHB
255 * ..
256 * .. External Subroutines ..
257 EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU,
258 $ ZPBRFS, ZPBTRF, ZPBTRS
259 * ..
260 * .. Intrinsic Functions ..
261 INTRINSIC MAX, MIN
262 * ..
263 * .. Executable Statements ..
264 *
265 INFO = 0
266 NOFACT = LSAME( FACT, 'N' )
267 EQUIL = LSAME( FACT, 'E' )
268 UPPER = LSAME( UPLO, 'U' )
269 IF( NOFACT .OR. EQUIL ) THEN
270 EQUED = 'N'
271 RCEQU = .FALSE.
272 ELSE
273 RCEQU = LSAME( EQUED, 'Y' )
274 SMLNUM = DLAMCH( 'Safe minimum' )
275 BIGNUM = ONE / SMLNUM
276 END IF
277 *
278 * Test the input parameters.
279 *
280 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
281 $ THEN
282 INFO = -1
283 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
284 INFO = -2
285 ELSE IF( N.LT.0 ) THEN
286 INFO = -3
287 ELSE IF( KD.LT.0 ) THEN
288 INFO = -4
289 ELSE IF( NRHS.LT.0 ) THEN
290 INFO = -5
291 ELSE IF( LDAB.LT.KD+1 ) THEN
292 INFO = -7
293 ELSE IF( LDAFB.LT.KD+1 ) THEN
294 INFO = -9
295 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
296 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
297 INFO = -10
298 ELSE
299 IF( RCEQU ) THEN
300 SMIN = BIGNUM
301 SMAX = ZERO
302 DO 10 J = 1, N
303 SMIN = MIN( SMIN, S( J ) )
304 SMAX = MAX( SMAX, S( J ) )
305 10 CONTINUE
306 IF( SMIN.LE.ZERO ) THEN
307 INFO = -11
308 ELSE IF( N.GT.0 ) THEN
309 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
310 ELSE
311 SCOND = ONE
312 END IF
313 END IF
314 IF( INFO.EQ.0 ) THEN
315 IF( LDB.LT.MAX( 1, N ) ) THEN
316 INFO = -13
317 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
318 INFO = -15
319 END IF
320 END IF
321 END IF
322 *
323 IF( INFO.NE.0 ) THEN
324 CALL XERBLA( 'ZPBSVX', -INFO )
325 RETURN
326 END IF
327 *
328 IF( EQUIL ) THEN
329 *
330 * Compute row and column scalings to equilibrate the matrix A.
331 *
332 CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
333 IF( INFEQU.EQ.0 ) THEN
334 *
335 * Equilibrate the matrix.
336 *
337 CALL ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
338 RCEQU = LSAME( EQUED, 'Y' )
339 END IF
340 END IF
341 *
342 * Scale the right-hand side.
343 *
344 IF( RCEQU ) THEN
345 DO 30 J = 1, NRHS
346 DO 20 I = 1, N
347 B( I, J ) = S( I )*B( I, J )
348 20 CONTINUE
349 30 CONTINUE
350 END IF
351 *
352 IF( NOFACT .OR. EQUIL ) THEN
353 *
354 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
355 *
356 IF( UPPER ) THEN
357 DO 40 J = 1, N
358 J1 = MAX( J-KD, 1 )
359 CALL ZCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
360 $ AFB( KD+1-J+J1, J ), 1 )
361 40 CONTINUE
362 ELSE
363 DO 50 J = 1, N
364 J2 = MIN( J+KD, N )
365 CALL ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
366 50 CONTINUE
367 END IF
368 *
369 CALL ZPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
370 *
371 * Return if INFO is non-zero.
372 *
373 IF( INFO.GT.0 )THEN
374 RCOND = ZERO
375 RETURN
376 END IF
377 END IF
378 *
379 * Compute the norm of the matrix A.
380 *
381 ANORM = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK )
382 *
383 * Compute the reciprocal of the condition number of A.
384 *
385 CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK,
386 $ INFO )
387 *
388 * Compute the solution matrix X.
389 *
390 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
391 CALL ZPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
392 *
393 * Use iterative refinement to improve the computed solution and
394 * compute error bounds and backward error estimates for it.
395 *
396 CALL ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
397 $ LDX, FERR, BERR, WORK, RWORK, INFO )
398 *
399 * Transform the solution matrix X to a solution of the original
400 * system.
401 *
402 IF( RCEQU ) THEN
403 DO 70 J = 1, NRHS
404 DO 60 I = 1, N
405 X( I, J ) = S( I )*X( I, J )
406 60 CONTINUE
407 70 CONTINUE
408 DO 80 J = 1, NRHS
409 FERR( J ) = FERR( J ) / SCOND
410 80 CONTINUE
411 END IF
412 *
413 * Set INFO = N+1 if the matrix is singular to working precision.
414 *
415 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
416 $ INFO = N + 1
417 *
418 RETURN
419 *
420 * End of ZPBSVX
421 *
422 END
2 $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, UPLO
12 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
17 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
25 * compute the solution to a complex system of linear equations
26 * A * X = B,
27 * where A is an N-by-N Hermitian positive definite band matrix and X
28 * and B are N-by-NRHS matrices.
29 *
30 * Error bounds on the solution and a condition estimate are also
31 * provided.
32 *
33 * Description
34 * ===========
35 *
36 * The following steps are performed:
37 *
38 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
39 * the system:
40 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
41 * Whether or not the system will be equilibrated depends on the
42 * scaling of the matrix A, but if equilibration is used, A is
43 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
44 *
45 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
46 * factor the matrix A (after equilibration if FACT = 'E') as
47 * A = U**H * U, if UPLO = 'U', or
48 * A = L * L**H, if UPLO = 'L',
49 * where U is an upper triangular band matrix, and L is a lower
50 * triangular band matrix.
51 *
52 * 3. If the leading i-by-i principal minor is not positive definite,
53 * then the routine returns with INFO = i. Otherwise, the factored
54 * form of A is used to estimate the condition number of the matrix
55 * A. If the reciprocal of the condition number is less than machine
56 * precision, INFO = N+1 is returned as a warning, but the routine
57 * still goes on to solve for X and compute error bounds as
58 * described below.
59 *
60 * 4. The system of equations is solved for X using the factored form
61 * of A.
62 *
63 * 5. Iterative refinement is applied to improve the computed solution
64 * matrix and calculate error bounds and backward error estimates
65 * for it.
66 *
67 * 6. If equilibration was used, the matrix X is premultiplied by
68 * diag(S) so that it solves the original system before
69 * equilibration.
70 *
71 * Arguments
72 * =========
73 *
74 * FACT (input) CHARACTER*1
75 * Specifies whether or not the factored form of the matrix A is
76 * supplied on entry, and if not, whether the matrix A should be
77 * equilibrated before it is factored.
78 * = 'F': On entry, AFB contains the factored form of A.
79 * If EQUED = 'Y', the matrix A has been equilibrated
80 * with scaling factors given by S. AB and AFB will not
81 * be modified.
82 * = 'N': The matrix A will be copied to AFB and factored.
83 * = 'E': The matrix A will be equilibrated if necessary, then
84 * copied to AFB and factored.
85 *
86 * UPLO (input) CHARACTER*1
87 * = 'U': Upper triangle of A is stored;
88 * = 'L': Lower triangle of A is stored.
89 *
90 * N (input) INTEGER
91 * The number of linear equations, i.e., the order of the
92 * matrix A. N >= 0.
93 *
94 * KD (input) INTEGER
95 * The number of superdiagonals of the matrix A if UPLO = 'U',
96 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
97 *
98 * NRHS (input) INTEGER
99 * The number of right-hand sides, i.e., the number of columns
100 * of the matrices B and X. NRHS >= 0.
101 *
102 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
103 * On entry, the upper or lower triangle of the Hermitian band
104 * matrix A, stored in the first KD+1 rows of the array, except
105 * if FACT = 'F' and EQUED = 'Y', then A must contain the
106 * equilibrated matrix diag(S)*A*diag(S). The j-th column of A
107 * is stored in the j-th column of the array AB as follows:
108 * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
109 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
110 * See below for further details.
111 *
112 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
113 * diag(S)*A*diag(S).
114 *
115 * LDAB (input) INTEGER
116 * The leading dimension of the array A. LDAB >= KD+1.
117 *
118 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
119 * If FACT = 'F', then AFB is an input argument and on entry
120 * contains the triangular factor U or L from the Cholesky
121 * factorization A = U**H *U or A = L*L**H of the band matrix
122 * A, in the same storage format as A (see AB). If EQUED = 'Y',
123 * then AFB is the factored form of the equilibrated matrix A.
124 *
125 * If FACT = 'N', then AFB is an output argument and on exit
126 * returns the triangular factor U or L from the Cholesky
127 * factorization A = U**H *U or A = L*L**H.
128 *
129 * If FACT = 'E', then AFB is an output argument and on exit
130 * returns the triangular factor U or L from the Cholesky
131 * factorization A = U**H *U or A = L*L**H of the equilibrated
132 * matrix A (see the description of A for the form of the
133 * equilibrated matrix).
134 *
135 * LDAFB (input) INTEGER
136 * The leading dimension of the array AFB. LDAFB >= KD+1.
137 *
138 * EQUED (input or output) CHARACTER*1
139 * Specifies the form of equilibration that was done.
140 * = 'N': No equilibration (always true if FACT = 'N').
141 * = 'Y': Equilibration was done, i.e., A has been replaced by
142 * diag(S) * A * diag(S).
143 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
144 * output argument.
145 *
146 * S (input or output) DOUBLE PRECISION array, dimension (N)
147 * The scale factors for A; not accessed if EQUED = 'N'. S is
148 * an input argument if FACT = 'F'; otherwise, S is an output
149 * argument. If FACT = 'F' and EQUED = 'Y', each element of S
150 * must be positive.
151 *
152 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
153 * On entry, the N-by-NRHS right hand side matrix B.
154 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
155 * B is overwritten by diag(S) * B.
156 *
157 * LDB (input) INTEGER
158 * The leading dimension of the array B. LDB >= max(1,N).
159 *
160 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
161 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
162 * the original system of equations. Note that if EQUED = 'Y',
163 * A and B are modified on exit, and the solution to the
164 * equilibrated system is inv(diag(S))*X.
165 *
166 * LDX (input) INTEGER
167 * The leading dimension of the array X. LDX >= max(1,N).
168 *
169 * RCOND (output) DOUBLE PRECISION
170 * The estimate of the reciprocal condition number of the matrix
171 * A after equilibration (if done). If RCOND is less than the
172 * machine precision (in particular, if RCOND = 0), the matrix
173 * is singular to working precision. This condition is
174 * indicated by a return code of INFO > 0.
175 *
176 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
177 * The estimated forward error bound for each solution vector
178 * X(j) (the j-th column of the solution matrix X).
179 * If XTRUE is the true solution corresponding to X(j), FERR(j)
180 * is an estimated upper bound for the magnitude of the largest
181 * element in (X(j) - XTRUE) divided by the magnitude of the
182 * largest element in X(j). The estimate is as reliable as
183 * the estimate for RCOND, and is almost always a slight
184 * overestimate of the true error.
185 *
186 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
187 * The componentwise relative backward error of each solution
188 * vector X(j) (i.e., the smallest relative change in
189 * any element of A or B that makes X(j) an exact solution).
190 *
191 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
192 *
193 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
194 *
195 * INFO (output) INTEGER
196 * = 0: successful exit
197 * < 0: if INFO = -i, the i-th argument had an illegal value
198 * > 0: if INFO = i, and i is
199 * <= N: the leading minor of order i of A is
200 * not positive definite, so the factorization
201 * could not be completed, and the solution has not
202 * been computed. RCOND = 0 is returned.
203 * = N+1: U is nonsingular, but RCOND is less than machine
204 * precision, meaning that the matrix is singular
205 * to working precision. Nevertheless, the
206 * solution and error bounds are computed because
207 * there are a number of situations where the
208 * computed solution can be more accurate than the
209 * value of RCOND would suggest.
210 *
211 * Further Details
212 * ===============
213 *
214 * The band storage scheme is illustrated by the following example, when
215 * N = 6, KD = 2, and UPLO = 'U':
216 *
217 * Two-dimensional storage of the Hermitian matrix A:
218 *
219 * a11 a12 a13
220 * a22 a23 a24
221 * a33 a34 a35
222 * a44 a45 a46
223 * a55 a56
224 * (aij=conjg(aji)) a66
225 *
226 * Band storage of the upper triangle of A:
227 *
228 * * * a13 a24 a35 a46
229 * * a12 a23 a34 a45 a56
230 * a11 a22 a33 a44 a55 a66
231 *
232 * Similarly, if UPLO = 'L' the format of A is as follows:
233 *
234 * a11 a22 a33 a44 a55 a66
235 * a21 a32 a43 a54 a65 *
236 * a31 a42 a53 a64 * *
237 *
238 * Array elements marked * are not used by the routine.
239 *
240 * =====================================================================
241 *
242 * .. Parameters ..
243 DOUBLE PRECISION ZERO, ONE
244 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
245 * ..
246 * .. Local Scalars ..
247 LOGICAL EQUIL, NOFACT, RCEQU, UPPER
248 INTEGER I, INFEQU, J, J1, J2
249 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
250 * ..
251 * .. External Functions ..
252 LOGICAL LSAME
253 DOUBLE PRECISION DLAMCH, ZLANHB
254 EXTERNAL LSAME, DLAMCH, ZLANHB
255 * ..
256 * .. External Subroutines ..
257 EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU,
258 $ ZPBRFS, ZPBTRF, ZPBTRS
259 * ..
260 * .. Intrinsic Functions ..
261 INTRINSIC MAX, MIN
262 * ..
263 * .. Executable Statements ..
264 *
265 INFO = 0
266 NOFACT = LSAME( FACT, 'N' )
267 EQUIL = LSAME( FACT, 'E' )
268 UPPER = LSAME( UPLO, 'U' )
269 IF( NOFACT .OR. EQUIL ) THEN
270 EQUED = 'N'
271 RCEQU = .FALSE.
272 ELSE
273 RCEQU = LSAME( EQUED, 'Y' )
274 SMLNUM = DLAMCH( 'Safe minimum' )
275 BIGNUM = ONE / SMLNUM
276 END IF
277 *
278 * Test the input parameters.
279 *
280 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
281 $ THEN
282 INFO = -1
283 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
284 INFO = -2
285 ELSE IF( N.LT.0 ) THEN
286 INFO = -3
287 ELSE IF( KD.LT.0 ) THEN
288 INFO = -4
289 ELSE IF( NRHS.LT.0 ) THEN
290 INFO = -5
291 ELSE IF( LDAB.LT.KD+1 ) THEN
292 INFO = -7
293 ELSE IF( LDAFB.LT.KD+1 ) THEN
294 INFO = -9
295 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
296 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
297 INFO = -10
298 ELSE
299 IF( RCEQU ) THEN
300 SMIN = BIGNUM
301 SMAX = ZERO
302 DO 10 J = 1, N
303 SMIN = MIN( SMIN, S( J ) )
304 SMAX = MAX( SMAX, S( J ) )
305 10 CONTINUE
306 IF( SMIN.LE.ZERO ) THEN
307 INFO = -11
308 ELSE IF( N.GT.0 ) THEN
309 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
310 ELSE
311 SCOND = ONE
312 END IF
313 END IF
314 IF( INFO.EQ.0 ) THEN
315 IF( LDB.LT.MAX( 1, N ) ) THEN
316 INFO = -13
317 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
318 INFO = -15
319 END IF
320 END IF
321 END IF
322 *
323 IF( INFO.NE.0 ) THEN
324 CALL XERBLA( 'ZPBSVX', -INFO )
325 RETURN
326 END IF
327 *
328 IF( EQUIL ) THEN
329 *
330 * Compute row and column scalings to equilibrate the matrix A.
331 *
332 CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
333 IF( INFEQU.EQ.0 ) THEN
334 *
335 * Equilibrate the matrix.
336 *
337 CALL ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
338 RCEQU = LSAME( EQUED, 'Y' )
339 END IF
340 END IF
341 *
342 * Scale the right-hand side.
343 *
344 IF( RCEQU ) THEN
345 DO 30 J = 1, NRHS
346 DO 20 I = 1, N
347 B( I, J ) = S( I )*B( I, J )
348 20 CONTINUE
349 30 CONTINUE
350 END IF
351 *
352 IF( NOFACT .OR. EQUIL ) THEN
353 *
354 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
355 *
356 IF( UPPER ) THEN
357 DO 40 J = 1, N
358 J1 = MAX( J-KD, 1 )
359 CALL ZCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
360 $ AFB( KD+1-J+J1, J ), 1 )
361 40 CONTINUE
362 ELSE
363 DO 50 J = 1, N
364 J2 = MIN( J+KD, N )
365 CALL ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
366 50 CONTINUE
367 END IF
368 *
369 CALL ZPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
370 *
371 * Return if INFO is non-zero.
372 *
373 IF( INFO.GT.0 )THEN
374 RCOND = ZERO
375 RETURN
376 END IF
377 END IF
378 *
379 * Compute the norm of the matrix A.
380 *
381 ANORM = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK )
382 *
383 * Compute the reciprocal of the condition number of A.
384 *
385 CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK,
386 $ INFO )
387 *
388 * Compute the solution matrix X.
389 *
390 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
391 CALL ZPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
392 *
393 * Use iterative refinement to improve the computed solution and
394 * compute error bounds and backward error estimates for it.
395 *
396 CALL ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
397 $ LDX, FERR, BERR, WORK, RWORK, INFO )
398 *
399 * Transform the solution matrix X to a solution of the original
400 * system.
401 *
402 IF( RCEQU ) THEN
403 DO 70 J = 1, NRHS
404 DO 60 I = 1, N
405 X( I, J ) = S( I )*X( I, J )
406 60 CONTINUE
407 70 CONTINUE
408 DO 80 J = 1, NRHS
409 FERR( J ) = FERR( J ) / SCOND
410 80 CONTINUE
411 END IF
412 *
413 * Set INFO = N+1 if the matrix is singular to working precision.
414 *
415 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
416 $ INFO = N + 1
417 *
418 RETURN
419 *
420 * End of ZPBSVX
421 *
422 END