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