1: SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
2: $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
3: $ RCOND, FERR, BERR, WORK, RWORK, 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( * )
17: DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
18: $ RWORK( * )
19: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
20: $ WORK( * ), X( LDX, * )
21: * ..
22: *
23: * Purpose
24: * =======
25: *
26: * ZGBSVX uses the LU factorization to compute the solution to a complex
27: * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
28: * where A is a band matrix of order N with KL subdiagonals and KU
29: * superdiagonals, and X 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 by this subroutine:
38: *
39: * 1. If FACT = 'E', real scaling factors are computed to equilibrate
40: * the system:
41: * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42: * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43: * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44: * Whether or not the system will be equilibrated depends on the
45: * scaling of the matrix A, but if equilibration is used, A is
46: * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47: * or diag(C)*B (if TRANS = 'T' or 'C').
48: *
49: * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50: * matrix A (after equilibration if FACT = 'E') as
51: * A = L * U,
52: * where L is a product of permutation and unit lower triangular
53: * matrices with KL subdiagonals, and U is upper triangular with
54: * KL+KU superdiagonals.
55: *
56: * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
57: * returns with INFO = i. Otherwise, the factored form of A is used
58: * to estimate the condition number of the matrix A. If the
59: * reciprocal of the condition number is less than machine precision,
60: * INFO = N+1 is returned as a warning, but the routine still goes on
61: * to solve for X and compute error bounds as described below.
62: *
63: * 4. The system of equations is solved for X using the factored form
64: * of A.
65: *
66: * 5. Iterative refinement is applied to improve the computed solution
67: * matrix and calculate error bounds and backward error estimates
68: * for it.
69: *
70: * 6. If equilibration was used, the matrix X is premultiplied by
71: * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
72: * that it solves the original system before equilibration.
73: *
74: * Arguments
75: * =========
76: *
77: * FACT (input) CHARACTER*1
78: * Specifies whether or not the factored form of the matrix A is
79: * supplied on entry, and if not, whether the matrix A should be
80: * equilibrated before it is factored.
81: * = 'F': On entry, AFB and IPIV contain the factored form of
82: * A. If EQUED is not 'N', the matrix A has been
83: * equilibrated with scaling factors given by R and C.
84: * AB, AFB, and IPIV are not modified.
85: * = 'N': The matrix A will be copied to AFB and factored.
86: * = 'E': The matrix A will be equilibrated if necessary, then
87: * copied to AFB and factored.
88: *
89: * TRANS (input) CHARACTER*1
90: * Specifies the form of the system of equations.
91: * = 'N': A * X = B (No transpose)
92: * = 'T': A**T * X = B (Transpose)
93: * = 'C': A**H * X = B (Conjugate transpose)
94: *
95: * N (input) INTEGER
96: * The number of linear equations, i.e., the order of the
97: * matrix A. N >= 0.
98: *
99: * KL (input) INTEGER
100: * The number of subdiagonals within the band of A. KL >= 0.
101: *
102: * KU (input) INTEGER
103: * The number of superdiagonals within the band of A. KU >= 0.
104: *
105: * NRHS (input) INTEGER
106: * The number of right hand sides, i.e., the number of columns
107: * of the matrices B and X. NRHS >= 0.
108: *
109: * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
110: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
111: * The j-th column of A is stored in the j-th column of the
112: * array AB as follows:
113: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
114: *
115: * If FACT = 'F' and EQUED is not 'N', then A must have been
116: * equilibrated by the scaling factors in R and/or C. AB is not
117: * modified if FACT = 'F' or 'N', or if FACT = 'E' and
118: * EQUED = 'N' on exit.
119: *
120: * On exit, if EQUED .ne. 'N', A is scaled as follows:
121: * EQUED = 'R': A := diag(R) * A
122: * EQUED = 'C': A := A * diag(C)
123: * EQUED = 'B': A := diag(R) * A * diag(C).
124: *
125: * LDAB (input) INTEGER
126: * The leading dimension of the array AB. LDAB >= KL+KU+1.
127: *
128: * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N)
129: * If FACT = 'F', then AFB is an input argument and on entry
130: * contains details of the LU factorization of the band matrix
131: * A, as computed by ZGBTRF. U is stored as an upper triangular
132: * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
133: * and the multipliers used during the factorization are stored
134: * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
135: * the factored form of the equilibrated matrix A.
136: *
137: * If FACT = 'N', then AFB is an output argument and on exit
138: * returns details of the LU factorization of A.
139: *
140: * If FACT = 'E', then AFB is an output argument and on exit
141: * returns details of the LU factorization of the equilibrated
142: * matrix A (see the description of AB for the form of the
143: * equilibrated matrix).
144: *
145: * LDAFB (input) INTEGER
146: * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
147: *
148: * IPIV (input or output) INTEGER array, dimension (N)
149: * If FACT = 'F', then IPIV is an input argument and on entry
150: * contains the pivot indices from the factorization A = L*U
151: * as computed by ZGBTRF; row i of the matrix was interchanged
152: * with row IPIV(i).
153: *
154: * If FACT = 'N', then IPIV is an output argument and on exit
155: * contains the pivot indices from the factorization A = L*U
156: * of the original matrix A.
157: *
158: * If FACT = 'E', then IPIV is an output argument and on exit
159: * contains the pivot indices from the factorization A = L*U
160: * of the equilibrated matrix A.
161: *
162: * EQUED (input or output) CHARACTER*1
163: * Specifies the form of equilibration that was done.
164: * = 'N': No equilibration (always true if FACT = 'N').
165: * = 'R': Row equilibration, i.e., A has been premultiplied by
166: * diag(R).
167: * = 'C': Column equilibration, i.e., A has been postmultiplied
168: * by diag(C).
169: * = 'B': Both row and column equilibration, i.e., A has been
170: * replaced by diag(R) * A * diag(C).
171: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
172: * output argument.
173: *
174: * R (input or output) DOUBLE PRECISION array, dimension (N)
175: * The row scale factors for A. If EQUED = 'R' or 'B', A is
176: * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
177: * is not accessed. R is an input argument if FACT = 'F';
178: * otherwise, R is an output argument. If FACT = 'F' and
179: * EQUED = 'R' or 'B', each element of R must be positive.
180: *
181: * C (input or output) DOUBLE PRECISION array, dimension (N)
182: * The column scale factors for A. If EQUED = 'C' or 'B', A is
183: * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
184: * is not accessed. C is an input argument if FACT = 'F';
185: * otherwise, C is an output argument. If FACT = 'F' and
186: * EQUED = 'C' or 'B', each element of C must be positive.
187: *
188: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
189: * On entry, the right hand side matrix B.
190: * On exit,
191: * if EQUED = 'N', B is not modified;
192: * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
193: * diag(R)*B;
194: * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
195: * overwritten by diag(C)*B.
196: *
197: * LDB (input) INTEGER
198: * The leading dimension of the array B. LDB >= max(1,N).
199: *
200: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
201: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
202: * to the original system of equations. Note that A and B are
203: * modified on exit if EQUED .ne. 'N', and the solution to the
204: * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
205: * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
206: * and EQUED = 'R' or 'B'.
207: *
208: * LDX (input) INTEGER
209: * The leading dimension of the array X. LDX >= max(1,N).
210: *
211: * RCOND (output) DOUBLE PRECISION
212: * The estimate of the reciprocal condition number of the matrix
213: * A after equilibration (if done). If RCOND is less than the
214: * machine precision (in particular, if RCOND = 0), the matrix
215: * is singular to working precision. This condition is
216: * indicated by a return code of INFO > 0.
217: *
218: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
219: * The estimated forward error bound for each solution vector
220: * X(j) (the j-th column of the solution matrix X).
221: * If XTRUE is the true solution corresponding to X(j), FERR(j)
222: * is an estimated upper bound for the magnitude of the largest
223: * element in (X(j) - XTRUE) divided by the magnitude of the
224: * largest element in X(j). The estimate is as reliable as
225: * the estimate for RCOND, and is almost always a slight
226: * overestimate of the true error.
227: *
228: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
229: * The componentwise relative backward error of each solution
230: * vector X(j) (i.e., the smallest relative change in
231: * any element of A or B that makes X(j) an exact solution).
232: *
233: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
234: *
235: * RWORK (workspace/output) DOUBLE PRECISION array, dimension (N)
236: * On exit, RWORK(1) contains the reciprocal pivot growth
237: * factor norm(A)/norm(U). The "max absolute element" norm is
238: * used. If RWORK(1) is much less than 1, then the stability
239: * of the LU factorization of the (equilibrated) matrix A
240: * could be poor. This also means that the solution X, condition
241: * estimator RCOND, and forward error bound FERR could be
242: * unreliable. If factorization fails with 0<INFO<=N, then
243: * RWORK(1) contains the reciprocal pivot growth factor for the
244: * leading INFO columns of A.
245: *
246: * INFO (output) INTEGER
247: * = 0: successful exit
248: * < 0: if INFO = -i, the i-th argument had an illegal value
249: * > 0: if INFO = i, and i is
250: * <= N: U(i,i) is exactly zero. The factorization
251: * has been completed, but the factor U is exactly
252: * singular, so the solution and error bounds
253: * could not be computed. RCOND = 0 is returned.
254: * = N+1: U is nonsingular, but RCOND is less than machine
255: * precision, meaning that the matrix is singular
256: * to working precision. Nevertheless, the
257: * solution and error bounds are computed because
258: * there are a number of situations where the
259: * computed solution can be more accurate than the
260: * value of RCOND would suggest.
261: *
262: * =====================================================================
263: * Moved setting of INFO = N+1 so INFO does not subsequently get
264: * overwritten. Sven, 17 Mar 05.
265: * =====================================================================
266: *
267: * .. Parameters ..
268: DOUBLE PRECISION ZERO, ONE
269: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
270: * ..
271: * .. Local Scalars ..
272: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
273: CHARACTER NORM
274: INTEGER I, INFEQU, J, J1, J2
275: DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
276: $ ROWCND, RPVGRW, SMLNUM
277: * ..
278: * .. External Functions ..
279: LOGICAL LSAME
280: DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
281: EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB
282: * ..
283: * .. External Subroutines ..
284: EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
285: $ ZGBTRS, ZLACPY, ZLAQGB
286: * ..
287: * .. Intrinsic Functions ..
288: INTRINSIC ABS, MAX, MIN
289: * ..
290: * .. Executable Statements ..
291: *
292: INFO = 0
293: NOFACT = LSAME( FACT, 'N' )
294: EQUIL = LSAME( FACT, 'E' )
295: NOTRAN = LSAME( TRANS, 'N' )
296: IF( NOFACT .OR. EQUIL ) THEN
297: EQUED = 'N'
298: ROWEQU = .FALSE.
299: COLEQU = .FALSE.
300: ELSE
301: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
302: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
303: SMLNUM = DLAMCH( 'Safe minimum' )
304: BIGNUM = ONE / SMLNUM
305: END IF
306: *
307: * Test the input parameters.
308: *
309: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
310: $ THEN
311: INFO = -1
312: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
313: $ LSAME( TRANS, 'C' ) ) THEN
314: INFO = -2
315: ELSE IF( N.LT.0 ) THEN
316: INFO = -3
317: ELSE IF( KL.LT.0 ) THEN
318: INFO = -4
319: ELSE IF( KU.LT.0 ) THEN
320: INFO = -5
321: ELSE IF( NRHS.LT.0 ) THEN
322: INFO = -6
323: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
324: INFO = -8
325: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
326: INFO = -10
327: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
328: $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
329: INFO = -12
330: ELSE
331: IF( ROWEQU ) THEN
332: RCMIN = BIGNUM
333: RCMAX = ZERO
334: DO 10 J = 1, N
335: RCMIN = MIN( RCMIN, R( J ) )
336: RCMAX = MAX( RCMAX, R( J ) )
337: 10 CONTINUE
338: IF( RCMIN.LE.ZERO ) THEN
339: INFO = -13
340: ELSE IF( N.GT.0 ) THEN
341: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
342: ELSE
343: ROWCND = ONE
344: END IF
345: END IF
346: IF( COLEQU .AND. INFO.EQ.0 ) THEN
347: RCMIN = BIGNUM
348: RCMAX = ZERO
349: DO 20 J = 1, N
350: RCMIN = MIN( RCMIN, C( J ) )
351: RCMAX = MAX( RCMAX, C( J ) )
352: 20 CONTINUE
353: IF( RCMIN.LE.ZERO ) THEN
354: INFO = -14
355: ELSE IF( N.GT.0 ) THEN
356: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
357: ELSE
358: COLCND = ONE
359: END IF
360: END IF
361: IF( INFO.EQ.0 ) THEN
362: IF( LDB.LT.MAX( 1, N ) ) THEN
363: INFO = -16
364: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
365: INFO = -18
366: END IF
367: END IF
368: END IF
369: *
370: IF( INFO.NE.0 ) THEN
371: CALL XERBLA( 'ZGBSVX', -INFO )
372: RETURN
373: END IF
374: *
375: IF( EQUIL ) THEN
376: *
377: * Compute row and column scalings to equilibrate the matrix A.
378: *
379: CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
380: $ AMAX, INFEQU )
381: IF( INFEQU.EQ.0 ) THEN
382: *
383: * Equilibrate the matrix.
384: *
385: CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
386: $ AMAX, EQUED )
387: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
388: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
389: END IF
390: END IF
391: *
392: * Scale the right hand side.
393: *
394: IF( NOTRAN ) THEN
395: IF( ROWEQU ) THEN
396: DO 40 J = 1, NRHS
397: DO 30 I = 1, N
398: B( I, J ) = R( I )*B( I, J )
399: 30 CONTINUE
400: 40 CONTINUE
401: END IF
402: ELSE IF( COLEQU ) THEN
403: DO 60 J = 1, NRHS
404: DO 50 I = 1, N
405: B( I, J ) = C( I )*B( I, J )
406: 50 CONTINUE
407: 60 CONTINUE
408: END IF
409: *
410: IF( NOFACT .OR. EQUIL ) THEN
411: *
412: * Compute the LU factorization of the band matrix A.
413: *
414: DO 70 J = 1, N
415: J1 = MAX( J-KU, 1 )
416: J2 = MIN( J+KL, N )
417: CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
418: $ AFB( KL+KU+1-J+J1, J ), 1 )
419: 70 CONTINUE
420: *
421: CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
422: *
423: * Return if INFO is non-zero.
424: *
425: IF( INFO.GT.0 ) THEN
426: *
427: * Compute the reciprocal pivot growth factor of the
428: * leading rank-deficient INFO columns of A.
429: *
430: ANORM = ZERO
431: DO 90 J = 1, INFO
432: DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
433: ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
434: 80 CONTINUE
435: 90 CONTINUE
436: RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
437: $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
438: $ RWORK )
439: IF( RPVGRW.EQ.ZERO ) THEN
440: RPVGRW = ONE
441: ELSE
442: RPVGRW = ANORM / RPVGRW
443: END IF
444: RWORK( 1 ) = RPVGRW
445: RCOND = ZERO
446: RETURN
447: END IF
448: END IF
449: *
450: * Compute the norm of the matrix A and the
451: * reciprocal pivot growth factor RPVGRW.
452: *
453: IF( NOTRAN ) THEN
454: NORM = '1'
455: ELSE
456: NORM = 'I'
457: END IF
458: ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
459: RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
460: IF( RPVGRW.EQ.ZERO ) THEN
461: RPVGRW = ONE
462: ELSE
463: RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
464: END IF
465: *
466: * Compute the reciprocal of the condition number of A.
467: *
468: CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
469: $ WORK, RWORK, INFO )
470: *
471: * Compute the solution matrix X.
472: *
473: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
474: CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
475: $ INFO )
476: *
477: * Use iterative refinement to improve the computed solution and
478: * compute error bounds and backward error estimates for it.
479: *
480: CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
481: $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
482: *
483: * Transform the solution matrix X to a solution of the original
484: * system.
485: *
486: IF( NOTRAN ) THEN
487: IF( COLEQU ) THEN
488: DO 110 J = 1, NRHS
489: DO 100 I = 1, N
490: X( I, J ) = C( I )*X( I, J )
491: 100 CONTINUE
492: 110 CONTINUE
493: DO 120 J = 1, NRHS
494: FERR( J ) = FERR( J ) / COLCND
495: 120 CONTINUE
496: END IF
497: ELSE IF( ROWEQU ) THEN
498: DO 140 J = 1, NRHS
499: DO 130 I = 1, N
500: X( I, J ) = R( I )*X( I, J )
501: 130 CONTINUE
502: 140 CONTINUE
503: DO 150 J = 1, NRHS
504: FERR( J ) = FERR( J ) / ROWCND
505: 150 CONTINUE
506: END IF
507: *
508: * Set INFO = N+1 if the matrix is singular to working precision.
509: *
510: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
511: $ INFO = N + 1
512: *
513: RWORK( 1 ) = RPVGRW
514: RETURN
515: *
516: * End of ZGBSVX
517: *
518: END
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