1: SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
2: $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3: $ RCOND, FERR, BERR, WORK, IWORK, INFO )
4: *
5: * -- LAPACK driver routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER EQUED, FACT, TRANS
12: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
13: DOUBLE PRECISION RCOND
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * ), IWORK( * )
17: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18: $ BERR( * ), C( * ), FERR( * ), R( * ),
19: $ WORK( * ), X( LDX, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DGBSVX uses the LU factorization to compute the solution to a real
26: * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
27: * where A is a band matrix of order N with KL subdiagonals and KU
28: * superdiagonals, and X 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 by this subroutine:
37: *
38: * 1. If FACT = 'E', real scaling factors are computed to equilibrate
39: * the system:
40: * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
41: * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
42: * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
43: * Whether or not the system will be equilibrated depends on the
44: * scaling of the matrix A, but if equilibration is used, A is
45: * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
46: * or diag(C)*B (if TRANS = 'T' or 'C').
47: *
48: * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
49: * matrix A (after equilibration if FACT = 'E') as
50: * A = L * U,
51: * where L is a product of permutation and unit lower triangular
52: * matrices with KL subdiagonals, and U is upper triangular with
53: * KL+KU superdiagonals.
54: *
55: * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
56: * returns with INFO = i. Otherwise, the factored form of A is used
57: * to estimate the condition number of the matrix A. If the
58: * reciprocal of the condition number is less than machine precision,
59: * INFO = N+1 is returned as a warning, but the routine still goes on
60: * to solve for X and compute error bounds as described below.
61: *
62: * 4. The system of equations is solved for X using the factored form
63: * of A.
64: *
65: * 5. Iterative refinement is applied to improve the computed solution
66: * matrix and calculate error bounds and backward error estimates
67: * for it.
68: *
69: * 6. If equilibration was used, the matrix X is premultiplied by
70: * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
71: * that it solves the original system before equilibration.
72: *
73: * Arguments
74: * =========
75: *
76: * FACT (input) CHARACTER*1
77: * Specifies whether or not the factored form of the matrix A is
78: * supplied on entry, and if not, whether the matrix A should be
79: * equilibrated before it is factored.
80: * = 'F': On entry, AFB and IPIV contain the factored form of
81: * A. If EQUED is not 'N', the matrix A has been
82: * equilibrated with scaling factors given by R and C.
83: * AB, AFB, and IPIV are not modified.
84: * = 'N': The matrix A will be copied to AFB and factored.
85: * = 'E': The matrix A will be equilibrated if necessary, then
86: * copied to AFB and factored.
87: *
88: * TRANS (input) CHARACTER*1
89: * Specifies the form of the system of equations.
90: * = 'N': A * X = B (No transpose)
91: * = 'T': A**T * X = B (Transpose)
92: * = 'C': A**H * X = B (Transpose)
93: *
94: * N (input) INTEGER
95: * The number of linear equations, i.e., the order of the
96: * matrix A. N >= 0.
97: *
98: * KL (input) INTEGER
99: * The number of subdiagonals within the band of A. KL >= 0.
100: *
101: * KU (input) INTEGER
102: * The number of superdiagonals within the band of A. KU >= 0.
103: *
104: * NRHS (input) INTEGER
105: * The number of right hand sides, i.e., the number of columns
106: * of the matrices B and X. NRHS >= 0.
107: *
108: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
109: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
110: * The j-th column of A is stored in the j-th column of the
111: * array AB as follows:
112: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
113: *
114: * If FACT = 'F' and EQUED is not 'N', then A must have been
115: * equilibrated by the scaling factors in R and/or C. AB is not
116: * modified if FACT = 'F' or 'N', or if FACT = 'E' and
117: * EQUED = 'N' on exit.
118: *
119: * On exit, if EQUED .ne. 'N', A is scaled as follows:
120: * EQUED = 'R': A := diag(R) * A
121: * EQUED = 'C': A := A * diag(C)
122: * EQUED = 'B': A := diag(R) * A * diag(C).
123: *
124: * LDAB (input) INTEGER
125: * The leading dimension of the array AB. LDAB >= KL+KU+1.
126: *
127: * AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
128: * If FACT = 'F', then AFB is an input argument and on entry
129: * contains details of the LU factorization of the band matrix
130: * A, as computed by DGBTRF. U is stored as an upper triangular
131: * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
132: * and the multipliers used during the factorization are stored
133: * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
134: * the factored form of the equilibrated matrix A.
135: *
136: * If FACT = 'N', then AFB is an output argument and on exit
137: * returns details of the LU factorization of A.
138: *
139: * If FACT = 'E', then AFB is an output argument and on exit
140: * returns details of the LU factorization of the equilibrated
141: * matrix A (see the description of AB for the form of the
142: * equilibrated matrix).
143: *
144: * LDAFB (input) INTEGER
145: * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
146: *
147: * IPIV (input or output) INTEGER array, dimension (N)
148: * If FACT = 'F', then IPIV is an input argument and on entry
149: * contains the pivot indices from the factorization A = L*U
150: * as computed by DGBTRF; row i of the matrix was interchanged
151: * with row IPIV(i).
152: *
153: * If FACT = 'N', then IPIV is an output argument and on exit
154: * contains the pivot indices from the factorization A = L*U
155: * of the original matrix A.
156: *
157: * If FACT = 'E', then IPIV is an output argument and on exit
158: * contains the pivot indices from the factorization A = L*U
159: * of the equilibrated matrix A.
160: *
161: * EQUED (input or output) CHARACTER*1
162: * Specifies the form of equilibration that was done.
163: * = 'N': No equilibration (always true if FACT = 'N').
164: * = 'R': Row equilibration, i.e., A has been premultiplied by
165: * diag(R).
166: * = 'C': Column equilibration, i.e., A has been postmultiplied
167: * by diag(C).
168: * = 'B': Both row and column equilibration, i.e., A has been
169: * replaced by diag(R) * A * diag(C).
170: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
171: * output argument.
172: *
173: * R (input or output) DOUBLE PRECISION array, dimension (N)
174: * The row scale factors for A. If EQUED = 'R' or 'B', A is
175: * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
176: * is not accessed. R is an input argument if FACT = 'F';
177: * otherwise, R is an output argument. If FACT = 'F' and
178: * EQUED = 'R' or 'B', each element of R must be positive.
179: *
180: * C (input or output) DOUBLE PRECISION array, dimension (N)
181: * The column scale factors for A. If EQUED = 'C' or 'B', A is
182: * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
183: * is not accessed. C is an input argument if FACT = 'F';
184: * otherwise, C is an output argument. If FACT = 'F' and
185: * EQUED = 'C' or 'B', each element of C must be positive.
186: *
187: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
188: * On entry, the right hand side matrix B.
189: * On exit,
190: * if EQUED = 'N', B is not modified;
191: * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
192: * diag(R)*B;
193: * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
194: * overwritten by diag(C)*B.
195: *
196: * LDB (input) INTEGER
197: * The leading dimension of the array B. LDB >= max(1,N).
198: *
199: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
200: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
201: * to the original system of equations. Note that A and B are
202: * modified on exit if EQUED .ne. 'N', and the solution to the
203: * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
204: * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
205: * and EQUED = 'R' or 'B'.
206: *
207: * LDX (input) INTEGER
208: * The leading dimension of the array X. LDX >= max(1,N).
209: *
210: * RCOND (output) DOUBLE PRECISION
211: * The estimate of the reciprocal condition number of the matrix
212: * A after equilibration (if done). If RCOND is less than the
213: * machine precision (in particular, if RCOND = 0), the matrix
214: * is singular to working precision. This condition is
215: * indicated by a return code of INFO > 0.
216: *
217: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
218: * The estimated forward error bound for each solution vector
219: * X(j) (the j-th column of the solution matrix X).
220: * If XTRUE is the true solution corresponding to X(j), FERR(j)
221: * is an estimated upper bound for the magnitude of the largest
222: * element in (X(j) - XTRUE) divided by the magnitude of the
223: * largest element in X(j). The estimate is as reliable as
224: * the estimate for RCOND, and is almost always a slight
225: * overestimate of the true error.
226: *
227: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
228: * The componentwise relative backward error of each solution
229: * vector X(j) (i.e., the smallest relative change in
230: * any element of A or B that makes X(j) an exact solution).
231: *
232: * WORK (workspace/output) DOUBLE PRECISION array, dimension (3*N)
233: * On exit, WORK(1) contains the reciprocal pivot growth
234: * factor norm(A)/norm(U). The "max absolute element" norm is
235: * used. If WORK(1) is much less than 1, then the stability
236: * of the LU factorization of the (equilibrated) matrix A
237: * could be poor. This also means that the solution X, condition
238: * estimator RCOND, and forward error bound FERR could be
239: * unreliable. If factorization fails with 0<INFO<=N, then
240: * WORK(1) contains the reciprocal pivot growth factor for the
241: * leading INFO columns of A.
242: *
243: * IWORK (workspace) INTEGER array, dimension (N)
244: *
245: * INFO (output) INTEGER
246: * = 0: successful exit
247: * < 0: if INFO = -i, the i-th argument had an illegal value
248: * > 0: if INFO = i, and i is
249: * <= N: U(i,i) is exactly zero. The factorization
250: * has been completed, but the factor U is exactly
251: * singular, so the solution and error bounds
252: * could not be computed. RCOND = 0 is returned.
253: * = N+1: U is nonsingular, but RCOND is less than machine
254: * precision, meaning that the matrix is singular
255: * to working precision. Nevertheless, the
256: * solution and error bounds are computed because
257: * there are a number of situations where the
258: * computed solution can be more accurate than the
259: * value of RCOND would suggest.
260: *
261: * =====================================================================
262: *
263: * .. Parameters ..
264: DOUBLE PRECISION ZERO, ONE
265: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
266: * ..
267: * .. Local Scalars ..
268: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
269: CHARACTER NORM
270: INTEGER I, INFEQU, J, J1, J2
271: DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
272: $ ROWCND, RPVGRW, SMLNUM
273: * ..
274: * .. External Functions ..
275: LOGICAL LSAME
276: DOUBLE PRECISION DLAMCH, DLANGB, DLANTB
277: EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB
278: * ..
279: * .. External Subroutines ..
280: EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS,
281: $ DLACPY, DLAQGB, XERBLA
282: * ..
283: * .. Intrinsic Functions ..
284: INTRINSIC ABS, MAX, MIN
285: * ..
286: * .. Executable Statements ..
287: *
288: INFO = 0
289: NOFACT = LSAME( FACT, 'N' )
290: EQUIL = LSAME( FACT, 'E' )
291: NOTRAN = LSAME( TRANS, 'N' )
292: IF( NOFACT .OR. EQUIL ) THEN
293: EQUED = 'N'
294: ROWEQU = .FALSE.
295: COLEQU = .FALSE.
296: ELSE
297: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
298: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
299: SMLNUM = DLAMCH( 'Safe minimum' )
300: BIGNUM = ONE / SMLNUM
301: END IF
302: *
303: * Test the input parameters.
304: *
305: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
306: $ THEN
307: INFO = -1
308: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
309: $ LSAME( TRANS, 'C' ) ) THEN
310: INFO = -2
311: ELSE IF( N.LT.0 ) THEN
312: INFO = -3
313: ELSE IF( KL.LT.0 ) THEN
314: INFO = -4
315: ELSE IF( KU.LT.0 ) THEN
316: INFO = -5
317: ELSE IF( NRHS.LT.0 ) THEN
318: INFO = -6
319: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
320: INFO = -8
321: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
322: INFO = -10
323: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
324: $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
325: INFO = -12
326: ELSE
327: IF( ROWEQU ) THEN
328: RCMIN = BIGNUM
329: RCMAX = ZERO
330: DO 10 J = 1, N
331: RCMIN = MIN( RCMIN, R( J ) )
332: RCMAX = MAX( RCMAX, R( J ) )
333: 10 CONTINUE
334: IF( RCMIN.LE.ZERO ) THEN
335: INFO = -13
336: ELSE IF( N.GT.0 ) THEN
337: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
338: ELSE
339: ROWCND = ONE
340: END IF
341: END IF
342: IF( COLEQU .AND. INFO.EQ.0 ) THEN
343: RCMIN = BIGNUM
344: RCMAX = ZERO
345: DO 20 J = 1, N
346: RCMIN = MIN( RCMIN, C( J ) )
347: RCMAX = MAX( RCMAX, C( J ) )
348: 20 CONTINUE
349: IF( RCMIN.LE.ZERO ) THEN
350: INFO = -14
351: ELSE IF( N.GT.0 ) THEN
352: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
353: ELSE
354: COLCND = ONE
355: END IF
356: END IF
357: IF( INFO.EQ.0 ) THEN
358: IF( LDB.LT.MAX( 1, N ) ) THEN
359: INFO = -16
360: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
361: INFO = -18
362: END IF
363: END IF
364: END IF
365: *
366: IF( INFO.NE.0 ) THEN
367: CALL XERBLA( 'DGBSVX', -INFO )
368: RETURN
369: END IF
370: *
371: IF( EQUIL ) THEN
372: *
373: * Compute row and column scalings to equilibrate the matrix A.
374: *
375: CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
376: $ AMAX, INFEQU )
377: IF( INFEQU.EQ.0 ) THEN
378: *
379: * Equilibrate the matrix.
380: *
381: CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
382: $ AMAX, EQUED )
383: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
384: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
385: END IF
386: END IF
387: *
388: * Scale the right hand side.
389: *
390: IF( NOTRAN ) THEN
391: IF( ROWEQU ) THEN
392: DO 40 J = 1, NRHS
393: DO 30 I = 1, N
394: B( I, J ) = R( I )*B( I, J )
395: 30 CONTINUE
396: 40 CONTINUE
397: END IF
398: ELSE IF( COLEQU ) THEN
399: DO 60 J = 1, NRHS
400: DO 50 I = 1, N
401: B( I, J ) = C( I )*B( I, J )
402: 50 CONTINUE
403: 60 CONTINUE
404: END IF
405: *
406: IF( NOFACT .OR. EQUIL ) THEN
407: *
408: * Compute the LU factorization of the band matrix A.
409: *
410: DO 70 J = 1, N
411: J1 = MAX( J-KU, 1 )
412: J2 = MIN( J+KL, N )
413: CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
414: $ AFB( KL+KU+1-J+J1, J ), 1 )
415: 70 CONTINUE
416: *
417: CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
418: *
419: * Return if INFO is non-zero.
420: *
421: IF( INFO.GT.0 ) THEN
422: *
423: * Compute the reciprocal pivot growth factor of the
424: * leading rank-deficient INFO columns of A.
425: *
426: ANORM = ZERO
427: DO 90 J = 1, INFO
428: DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
429: ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
430: 80 CONTINUE
431: 90 CONTINUE
432: RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
433: $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
434: $ WORK )
435: IF( RPVGRW.EQ.ZERO ) THEN
436: RPVGRW = ONE
437: ELSE
438: RPVGRW = ANORM / RPVGRW
439: END IF
440: WORK( 1 ) = RPVGRW
441: RCOND = ZERO
442: RETURN
443: END IF
444: END IF
445: *
446: * Compute the norm of the matrix A and the
447: * reciprocal pivot growth factor RPVGRW.
448: *
449: IF( NOTRAN ) THEN
450: NORM = '1'
451: ELSE
452: NORM = 'I'
453: END IF
454: ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
455: RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
456: IF( RPVGRW.EQ.ZERO ) THEN
457: RPVGRW = ONE
458: ELSE
459: RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
460: END IF
461: *
462: * Compute the reciprocal of the condition number of A.
463: *
464: CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
465: $ WORK, IWORK, INFO )
466: *
467: * Compute the solution matrix X.
468: *
469: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
470: CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
471: $ INFO )
472: *
473: * Use iterative refinement to improve the computed solution and
474: * compute error bounds and backward error estimates for it.
475: *
476: CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
477: $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
478: *
479: * Transform the solution matrix X to a solution of the original
480: * system.
481: *
482: IF( NOTRAN ) THEN
483: IF( COLEQU ) THEN
484: DO 110 J = 1, NRHS
485: DO 100 I = 1, N
486: X( I, J ) = C( I )*X( I, J )
487: 100 CONTINUE
488: 110 CONTINUE
489: DO 120 J = 1, NRHS
490: FERR( J ) = FERR( J ) / COLCND
491: 120 CONTINUE
492: END IF
493: ELSE IF( ROWEQU ) THEN
494: DO 140 J = 1, NRHS
495: DO 130 I = 1, N
496: X( I, J ) = R( I )*X( I, J )
497: 130 CONTINUE
498: 140 CONTINUE
499: DO 150 J = 1, NRHS
500: FERR( J ) = FERR( J ) / ROWCND
501: 150 CONTINUE
502: END IF
503: *
504: * Set INFO = N+1 if the matrix is singular to working precision.
505: *
506: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
507: $ INFO = N + 1
508: *
509: WORK( 1 ) = RPVGRW
510: RETURN
511: *
512: * End of DGBSVX
513: *
514: END
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