LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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zgbsvxx.f
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1*> \brief <b> ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
22* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
23* RCOND, RPVGRW, BERR, N_ERR_BNDS,
24* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
25* WORK, RWORK, INFO )
26*
27* .. Scalar Arguments ..
28* CHARACTER EQUED, FACT, TRANS
29* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
30* \$ N_ERR_BNDS
31* DOUBLE PRECISION RCOND, RPVGRW
32* ..
33* .. Array Arguments ..
34* INTEGER IPIV( * )
35* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
36* \$ X( LDX , * ),WORK( * )
37* DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
38* \$ ERR_BNDS_NORM( NRHS, * ),
39* \$ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
40* ..
41*
42*
43*> \par Purpose:
44* =============
45*>
46*> \verbatim
47*>
48*> ZGBSVXX uses the LU factorization to compute the solution to a
49*> complex*16 system of linear equations A * X = B, where A is an
50*> N-by-N matrix and X and B are N-by-NRHS matrices.
51*>
52*> If requested, both normwise and maximum componentwise error bounds
53*> are returned. ZGBSVXX will return a solution with a tiny
54*> guaranteed error (O(eps) where eps is the working machine
55*> precision) unless the matrix is very ill-conditioned, in which
56*> case a warning is returned. Relevant condition numbers also are
57*> calculated and returned.
58*>
59*> ZGBSVXX accepts user-provided factorizations and equilibration
60*> factors; see the definitions of the FACT and EQUED options.
61*> Solving with refinement and using a factorization from a previous
62*> ZGBSVXX call will also produce a solution with either O(eps)
63*> errors or warnings, but we cannot make that claim for general
64*> user-provided factorizations and equilibration factors if they
65*> differ from what ZGBSVXX would itself produce.
66*> \endverbatim
67*
68*> \par Description:
69* =================
70*>
71*> \verbatim
72*>
73*> The following steps are performed:
74*>
75*> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
76*> the system:
77*>
78*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
79*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
80*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
81*>
82*> Whether or not the system will be equilibrated depends on the
83*> scaling of the matrix A, but if equilibration is used, A is
84*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
85*> or diag(C)*B (if TRANS = 'T' or 'C').
86*>
87*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
88*> the matrix A (after equilibration if FACT = 'E') as
89*>
90*> A = P * L * U,
91*>
92*> where P is a permutation matrix, L is a unit lower triangular
93*> matrix, and U is upper triangular.
94*>
95*> 3. If some U(i,i)=0, so that U is exactly singular, then the
96*> routine returns with INFO = i. Otherwise, the factored form of A
97*> is used to estimate the condition number of the matrix A (see
98*> argument RCOND). If the reciprocal of the condition number is less
99*> than machine precision, the routine still goes on to solve for X
100*> and compute error bounds as described below.
101*>
102*> 4. The system of equations is solved for X using the factored form
103*> of A.
104*>
105*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
106*> the routine will use iterative refinement to try to get a small
107*> error and error bounds. Refinement calculates the residual to at
108*> least twice the working precision.
109*>
110*> 6. If equilibration was used, the matrix X is premultiplied by
111*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
112*> that it solves the original system before equilibration.
113*> \endverbatim
114*
115* Arguments:
116* ==========
117*
118*> \verbatim
119*> Some optional parameters are bundled in the PARAMS array. These
120*> settings determine how refinement is performed, but often the
121*> defaults are acceptable. If the defaults are acceptable, users
122*> can pass NPARAMS = 0 which prevents the source code from accessing
123*> the PARAMS argument.
124*> \endverbatim
125*>
126*> \param[in] FACT
127*> \verbatim
128*> FACT is CHARACTER*1
129*> Specifies whether or not the factored form of the matrix A is
130*> supplied on entry, and if not, whether the matrix A should be
131*> equilibrated before it is factored.
132*> = 'F': On entry, AF and IPIV contain the factored form of A.
133*> If EQUED is not 'N', the matrix A has been
134*> equilibrated with scaling factors given by R and C.
135*> A, AF, and IPIV are not modified.
136*> = 'N': The matrix A will be copied to AF and factored.
137*> = 'E': The matrix A will be equilibrated if necessary, then
138*> copied to AF and factored.
139*> \endverbatim
140*>
141*> \param[in] TRANS
142*> \verbatim
143*> TRANS is CHARACTER*1
144*> Specifies the form of the system of equations:
145*> = 'N': A * X = B (No transpose)
146*> = 'T': A**T * X = B (Transpose)
147*> = 'C': A**H * X = B (Conjugate Transpose = Transpose)
148*> \endverbatim
149*>
150*> \param[in] N
151*> \verbatim
152*> N is INTEGER
153*> The number of linear equations, i.e., the order of the
154*> matrix A. N >= 0.
155*> \endverbatim
156*>
157*> \param[in] KL
158*> \verbatim
159*> KL is INTEGER
160*> The number of subdiagonals within the band of A. KL >= 0.
161*> \endverbatim
162*>
163*> \param[in] KU
164*> \verbatim
165*> KU is INTEGER
166*> The number of superdiagonals within the band of A. KU >= 0.
167*> \endverbatim
168*>
169*> \param[in] NRHS
170*> \verbatim
171*> NRHS is INTEGER
172*> The number of right hand sides, i.e., the number of columns
173*> of the matrices B and X. NRHS >= 0.
174*> \endverbatim
175*>
176*> \param[in,out] AB
177*> \verbatim
178*> AB is COMPLEX*16 array, dimension (LDAB,N)
179*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
180*> The j-th column of A is stored in the j-th column of the
181*> array AB as follows:
182*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
183*>
184*> If FACT = 'F' and EQUED is not 'N', then AB must have been
185*> equilibrated by the scaling factors in R and/or C. AB is not
186*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
187*> EQUED = 'N' on exit.
188*>
189*> On exit, if EQUED .ne. 'N', A is scaled as follows:
190*> EQUED = 'R': A := diag(R) * A
191*> EQUED = 'C': A := A * diag(C)
192*> EQUED = 'B': A := diag(R) * A * diag(C).
193*> \endverbatim
194*>
195*> \param[in] LDAB
196*> \verbatim
197*> LDAB is INTEGER
198*> The leading dimension of the array AB. LDAB >= KL+KU+1.
199*> \endverbatim
200*>
201*> \param[in,out] AFB
202*> \verbatim
203*> AFB is COMPLEX*16 array, dimension (LDAFB,N)
204*> If FACT = 'F', then AFB is an input argument and on entry
205*> contains details of the LU factorization of the band matrix
206*> A, as computed by ZGBTRF. U is stored as an upper triangular
207*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
208*> and the multipliers used during the factorization are stored
209*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
210*> the factored form of the equilibrated matrix A.
211*>
212*> If FACT = 'N', then AF is an output argument and on exit
213*> returns the factors L and U from the factorization A = P*L*U
214*> of the original matrix A.
215*>
216*> If FACT = 'E', then AF is an output argument and on exit
217*> returns the factors L and U from the factorization A = P*L*U
218*> of the equilibrated matrix A (see the description of A for
219*> the form of the equilibrated matrix).
220*> \endverbatim
221*>
222*> \param[in] LDAFB
223*> \verbatim
224*> LDAFB is INTEGER
225*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
226*> \endverbatim
227*>
228*> \param[in,out] IPIV
229*> \verbatim
230*> IPIV is INTEGER array, dimension (N)
231*> If FACT = 'F', then IPIV is an input argument and on entry
232*> contains the pivot indices from the factorization A = P*L*U
233*> as computed by ZGETRF; row i of the matrix was interchanged
234*> with row IPIV(i).
235*>
236*> If FACT = 'N', then IPIV is an output argument and on exit
237*> contains the pivot indices from the factorization A = P*L*U
238*> of the original matrix A.
239*>
240*> If FACT = 'E', then IPIV is an output argument and on exit
241*> contains the pivot indices from the factorization A = P*L*U
242*> of the equilibrated matrix A.
243*> \endverbatim
244*>
245*> \param[in,out] EQUED
246*> \verbatim
247*> EQUED is CHARACTER*1
248*> Specifies the form of equilibration that was done.
249*> = 'N': No equilibration (always true if FACT = 'N').
250*> = 'R': Row equilibration, i.e., A has been premultiplied by
251*> diag(R).
252*> = 'C': Column equilibration, i.e., A has been postmultiplied
253*> by diag(C).
254*> = 'B': Both row and column equilibration, i.e., A has been
255*> replaced by diag(R) * A * diag(C).
256*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
257*> output argument.
258*> \endverbatim
259*>
260*> \param[in,out] R
261*> \verbatim
262*> R is DOUBLE PRECISION array, dimension (N)
263*> The row scale factors for A. If EQUED = 'R' or 'B', A is
264*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
265*> is not accessed. R is an input argument if FACT = 'F';
266*> otherwise, R is an output argument. If FACT = 'F' and
267*> EQUED = 'R' or 'B', each element of R must be positive.
268*> If R is output, each element of R is a power of the radix.
269*> If R is input, each element of R should be a power of the radix
270*> to ensure a reliable solution and error estimates. Scaling by
271*> powers of the radix does not cause rounding errors unless the
272*> result underflows or overflows. Rounding errors during scaling
273*> lead to refining with a matrix that is not equivalent to the
274*> input matrix, producing error estimates that may not be
275*> reliable.
276*> \endverbatim
277*>
278*> \param[in,out] C
279*> \verbatim
280*> C is DOUBLE PRECISION array, dimension (N)
281*> The column scale factors for A. If EQUED = 'C' or 'B', A is
282*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
283*> is not accessed. C is an input argument if FACT = 'F';
284*> otherwise, C is an output argument. If FACT = 'F' and
285*> EQUED = 'C' or 'B', each element of C must be positive.
286*> If C is output, each element of C is a power of the radix.
287*> If C is input, each element of C should be a power of the radix
288*> to ensure a reliable solution and error estimates. Scaling by
289*> powers of the radix does not cause rounding errors unless the
290*> result underflows or overflows. Rounding errors during scaling
291*> lead to refining with a matrix that is not equivalent to the
292*> input matrix, producing error estimates that may not be
293*> reliable.
294*> \endverbatim
295*>
296*> \param[in,out] B
297*> \verbatim
298*> B is COMPLEX*16 array, dimension (LDB,NRHS)
299*> On entry, the N-by-NRHS right hand side matrix B.
300*> On exit,
301*> if EQUED = 'N', B is not modified;
302*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
303*> diag(R)*B;
304*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
305*> overwritten by diag(C)*B.
306*> \endverbatim
307*>
308*> \param[in] LDB
309*> \verbatim
310*> LDB is INTEGER
311*> The leading dimension of the array B. LDB >= max(1,N).
312*> \endverbatim
313*>
314*> \param[out] X
315*> \verbatim
316*> X is COMPLEX*16 array, dimension (LDX,NRHS)
317*> If INFO = 0, the N-by-NRHS solution matrix X to the original
318*> system of equations. Note that A and B are modified on exit
319*> if EQUED .ne. 'N', and the solution to the equilibrated system is
320*> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
321*> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
322*> \endverbatim
323*>
324*> \param[in] LDX
325*> \verbatim
326*> LDX is INTEGER
327*> The leading dimension of the array X. LDX >= max(1,N).
328*> \endverbatim
329*>
330*> \param[out] RCOND
331*> \verbatim
332*> RCOND is DOUBLE PRECISION
333*> Reciprocal scaled condition number. This is an estimate of the
334*> reciprocal Skeel condition number of the matrix A after
335*> equilibration (if done). If this is less than the machine
336*> precision (in particular, if it is zero), the matrix is singular
337*> to working precision. Note that the error may still be small even
338*> if this number is very small and the matrix appears ill-
339*> conditioned.
340*> \endverbatim
341*>
342*> \param[out] RPVGRW
343*> \verbatim
344*> RPVGRW is DOUBLE PRECISION
345*> Reciprocal pivot growth. On exit, this contains the reciprocal
346*> pivot growth factor norm(A)/norm(U). The "max absolute element"
347*> norm is used. If this is much less than 1, then the stability of
348*> the LU factorization of the (equilibrated) matrix A could be poor.
349*> This also means that the solution X, estimated condition numbers,
350*> and error bounds could be unreliable. If factorization fails with
351*> 0<INFO<=N, then this contains the reciprocal pivot growth factor
352*> for the leading INFO columns of A. In ZGESVX, this quantity is
353*> returned in RWORK(1).
354*> \endverbatim
355*>
356*> \param[out] BERR
357*> \verbatim
358*> BERR is DOUBLE PRECISION array, dimension (NRHS)
359*> Componentwise relative backward error. This is the
360*> componentwise relative backward error of each solution vector X(j)
361*> (i.e., the smallest relative change in any element of A or B that
362*> makes X(j) an exact solution).
363*> \endverbatim
364*>
365*> \param[in] N_ERR_BNDS
366*> \verbatim
367*> N_ERR_BNDS is INTEGER
368*> Number of error bounds to return for each right hand side
369*> and each type (normwise or componentwise). See ERR_BNDS_NORM and
370*> ERR_BNDS_COMP below.
371*> \endverbatim
372*>
373*> \param[out] ERR_BNDS_NORM
374*> \verbatim
375*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
376*> For each right-hand side, this array contains information about
377*> various error bounds and condition numbers corresponding to the
378*> normwise relative error, which is defined as follows:
379*>
380*> Normwise relative error in the ith solution vector:
381*> max_j (abs(XTRUE(j,i) - X(j,i)))
382*> ------------------------------
383*> max_j abs(X(j,i))
384*>
385*> The array is indexed by the type of error information as described
386*> below. There currently are up to three pieces of information
387*> returned.
388*>
389*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
390*> right-hand side.
391*>
392*> The second index in ERR_BNDS_NORM(:,err) contains the following
393*> three fields:
394*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
395*> reciprocal condition number is less than the threshold
396*> sqrt(n) * dlamch('Epsilon').
397*>
398*> err = 2 "Guaranteed" error bound: The estimated forward error,
399*> almost certainly within a factor of 10 of the true error
400*> so long as the next entry is greater than the threshold
401*> sqrt(n) * dlamch('Epsilon'). This error bound should only
402*> be trusted if the previous boolean is true.
403*>
404*> err = 3 Reciprocal condition number: Estimated normwise
405*> reciprocal condition number. Compared with the threshold
406*> sqrt(n) * dlamch('Epsilon') to determine if the error
407*> estimate is "guaranteed". These reciprocal condition
408*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
409*> appropriately scaled matrix Z.
410*> Let Z = S*A, where S scales each row by a power of the
411*> radix so all absolute row sums of Z are approximately 1.
412*>
413*> See Lapack Working Note 165 for further details and extra
414*> cautions.
415*> \endverbatim
416*>
417*> \param[out] ERR_BNDS_COMP
418*> \verbatim
419*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
420*> For each right-hand side, this array contains information about
421*> various error bounds and condition numbers corresponding to the
422*> componentwise relative error, which is defined as follows:
423*>
424*> Componentwise relative error in the ith solution vector:
425*> abs(XTRUE(j,i) - X(j,i))
426*> max_j ----------------------
427*> abs(X(j,i))
428*>
429*> The array is indexed by the right-hand side i (on which the
430*> componentwise relative error depends), and the type of error
431*> information as described below. There currently are up to three
432*> pieces of information returned for each right-hand side. If
433*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
434*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
435*> the first (:,N_ERR_BNDS) entries are returned.
436*>
437*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
438*> right-hand side.
439*>
440*> The second index in ERR_BNDS_COMP(:,err) contains the following
441*> three fields:
442*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
443*> reciprocal condition number is less than the threshold
444*> sqrt(n) * dlamch('Epsilon').
445*>
446*> err = 2 "Guaranteed" error bound: The estimated forward error,
447*> almost certainly within a factor of 10 of the true error
448*> so long as the next entry is greater than the threshold
449*> sqrt(n) * dlamch('Epsilon'). This error bound should only
450*> be trusted if the previous boolean is true.
451*>
452*> err = 3 Reciprocal condition number: Estimated componentwise
453*> reciprocal condition number. Compared with the threshold
454*> sqrt(n) * dlamch('Epsilon') to determine if the error
455*> estimate is "guaranteed". These reciprocal condition
456*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
457*> appropriately scaled matrix Z.
458*> Let Z = S*(A*diag(x)), where x is the solution for the
459*> current right-hand side and S scales each row of
460*> A*diag(x) by a power of the radix so all absolute row
461*> sums of Z are approximately 1.
462*>
463*> See Lapack Working Note 165 for further details and extra
464*> cautions.
465*> \endverbatim
466*>
467*> \param[in] NPARAMS
468*> \verbatim
469*> NPARAMS is INTEGER
470*> Specifies the number of parameters set in PARAMS. If <= 0, the
471*> PARAMS array is never referenced and default values are used.
472*> \endverbatim
473*>
474*> \param[in,out] PARAMS
475*> \verbatim
476*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
477*> Specifies algorithm parameters. If an entry is < 0.0, then
478*> that entry will be filled with default value used for that
479*> parameter. Only positions up to NPARAMS are accessed; defaults
480*> are used for higher-numbered parameters.
481*>
482*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
483*> refinement or not.
484*> Default: 1.0D+0
485*> = 0.0: No refinement is performed, and no error bounds are
486*> computed.
487*> = 1.0: Use the extra-precise refinement algorithm.
488*> (other values are reserved for future use)
489*>
490*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
491*> computations allowed for refinement.
492*> Default: 10
493*> Aggressive: Set to 100 to permit convergence using approximate
494*> factorizations or factorizations other than LU. If
495*> the factorization uses a technique other than
496*> Gaussian elimination, the guarantees in
497*> err_bnds_norm and err_bnds_comp may no longer be
498*> trustworthy.
499*>
500*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
501*> will attempt to find a solution with small componentwise
502*> relative error in the double-precision algorithm. Positive
503*> is true, 0.0 is false.
504*> Default: 1.0 (attempt componentwise convergence)
505*> \endverbatim
506*>
507*> \param[out] WORK
508*> \verbatim
509*> WORK is COMPLEX*16 array, dimension (2*N)
510*> \endverbatim
511*>
512*> \param[out] RWORK
513*> \verbatim
514*> RWORK is DOUBLE PRECISION array, dimension (2*N)
515*> \endverbatim
516*>
517*> \param[out] INFO
518*> \verbatim
519*> INFO is INTEGER
520*> = 0: Successful exit. The solution to every right-hand side is
521*> guaranteed.
522*> < 0: If INFO = -i, the i-th argument had an illegal value
523*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
524*> has been completed, but the factor U is exactly singular, so
525*> the solution and error bounds could not be computed. RCOND = 0
526*> is returned.
527*> = N+J: The solution corresponding to the Jth right-hand side is
528*> not guaranteed. The solutions corresponding to other right-
529*> hand sides K with K > J may not be guaranteed as well, but
530*> only the first such right-hand side is reported. If a small
531*> componentwise error is not requested (PARAMS(3) = 0.0) then
532*> the Jth right-hand side is the first with a normwise error
533*> bound that is not guaranteed (the smallest J such
534*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
535*> the Jth right-hand side is the first with either a normwise or
536*> componentwise error bound that is not guaranteed (the smallest
537*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
538*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
539*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
540*> about all of the right-hand sides check ERR_BNDS_NORM or
541*> ERR_BNDS_COMP.
542*> \endverbatim
543*
544* Authors:
545* ========
546*
547*> \author Univ. of Tennessee
548*> \author Univ. of California Berkeley
549*> \author Univ. of Colorado Denver
550*> \author NAG Ltd.
551*
552*> \ingroup complex16GBsolve
553*
554* =====================================================================
555 SUBROUTINE zgbsvxx( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
556 \$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
557 \$ RCOND, RPVGRW, BERR, N_ERR_BNDS,
558 \$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
559 \$ WORK, RWORK, INFO )
560*
561* -- LAPACK driver routine --
562* -- LAPACK is a software package provided by Univ. of Tennessee, --
563* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
564*
565* .. Scalar Arguments ..
566 CHARACTER EQUED, FACT, TRANS
567 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
568 \$ N_ERR_BNDS
569 DOUBLE PRECISION RCOND, RPVGRW
570* ..
571* .. Array Arguments ..
572 INTEGER IPIV( * )
573 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
574 \$ X( LDX , * ),WORK( * )
575 DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
576 \$ err_bnds_norm( nrhs, * ),
577 \$ err_bnds_comp( nrhs, * ), rwork( * )
578* ..
579*
580* ==================================================================
581*
582* .. Parameters ..
583 DOUBLE PRECISION ZERO, ONE
584 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
585 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
586 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
587 INTEGER CMP_ERR_I, PIV_GROWTH_I
588 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
589 \$ berr_i = 3 )
590 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
591 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
592 \$ piv_growth_i = 9 )
593* ..
594* .. Local Scalars ..
595 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
596 INTEGER INFEQU, I, J, KL, KU
597 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
598 \$ ROWCND, SMLNUM
599* ..
600* .. External Functions ..
601 EXTERNAL lsame, dlamch, zla_gbrpvgrw
602 LOGICAL LSAME
603 DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW
604* ..
605* .. External Subroutines ..
606 EXTERNAL zgbequb, zgbtrf, zgbtrs, zlacpy, zlaqgb,
608* ..
609* .. Intrinsic Functions ..
610 INTRINSIC max, min
611* ..
612* .. Executable Statements ..
613*
614 info = 0
615 nofact = lsame( fact, 'N' )
616 equil = lsame( fact, 'E' )
617 notran = lsame( trans, 'N' )
618 smlnum = dlamch( 'Safe minimum' )
619 bignum = one / smlnum
620 IF( nofact .OR. equil ) THEN
621 equed = 'N'
622 rowequ = .false.
623 colequ = .false.
624 ELSE
625 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
626 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
627 END IF
628*
629* Default is failure. If an input parameter is wrong or
630* factorization fails, make everything look horrible. Only the
631* pivot growth is set here, the rest is initialized in ZGBRFSX.
632*
633 rpvgrw = zero
634*
635* Test the input parameters. PARAMS is not tested until ZGERFSX.
636*
637 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
638 \$ lsame( fact, 'F' ) ) THEN
639 info = -1
640 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
641 \$ lsame( trans, 'C' ) ) THEN
642 info = -2
643 ELSE IF( n.LT.0 ) THEN
644 info = -3
645 ELSE IF( kl.LT.0 ) THEN
646 info = -4
647 ELSE IF( ku.LT.0 ) THEN
648 info = -5
649 ELSE IF( nrhs.LT.0 ) THEN
650 info = -6
651 ELSE IF( ldab.LT.kl+ku+1 ) THEN
652 info = -8
653 ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
654 info = -10
655 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
656 \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
657 info = -12
658 ELSE
659 IF( rowequ ) THEN
660 rcmin = bignum
661 rcmax = zero
662 DO 10 j = 1, n
663 rcmin = min( rcmin, r( j ) )
664 rcmax = max( rcmax, r( j ) )
665 10 CONTINUE
666 IF( rcmin.LE.zero ) THEN
667 info = -13
668 ELSE IF( n.GT.0 ) THEN
669 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
670 ELSE
671 rowcnd = one
672 END IF
673 END IF
674 IF( colequ .AND. info.EQ.0 ) THEN
675 rcmin = bignum
676 rcmax = zero
677 DO 20 j = 1, n
678 rcmin = min( rcmin, c( j ) )
679 rcmax = max( rcmax, c( j ) )
680 20 CONTINUE
681 IF( rcmin.LE.zero ) THEN
682 info = -14
683 ELSE IF( n.GT.0 ) THEN
684 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
685 ELSE
686 colcnd = one
687 END IF
688 END IF
689 IF( info.EQ.0 ) THEN
690 IF( ldb.LT.max( 1, n ) ) THEN
691 info = -15
692 ELSE IF( ldx.LT.max( 1, n ) ) THEN
693 info = -16
694 END IF
695 END IF
696 END IF
697*
698 IF( info.NE.0 ) THEN
699 CALL xerbla( 'ZGBSVXX', -info )
700 RETURN
701 END IF
702*
703 IF( equil ) THEN
704*
705* Compute row and column scalings to equilibrate the matrix A.
706*
707 CALL zgbequb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
708 \$ amax, infequ )
709 IF( infequ.EQ.0 ) THEN
710*
711* Equilibrate the matrix.
712*
713 CALL zlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
714 \$ amax, equed )
715 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
716 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
717 END IF
718*
719* If the scaling factors are not applied, set them to 1.0.
720*
721 IF ( .NOT.rowequ ) THEN
722 DO j = 1, n
723 r( j ) = 1.0d+0
724 END DO
725 END IF
726 IF ( .NOT.colequ ) THEN
727 DO j = 1, n
728 c( j ) = 1.0d+0
729 END DO
730 END IF
731 END IF
732*
733* Scale the right-hand side.
734*
735 IF( notran ) THEN
736 IF( rowequ ) CALL zlascl2( n, nrhs, r, b, ldb )
737 ELSE
738 IF( colequ ) CALL zlascl2( n, nrhs, c, b, ldb )
739 END IF
740*
741 IF( nofact .OR. equil ) THEN
742*
743* Compute the LU factorization of A.
744*
745 DO 40, j = 1, n
746 DO 30, i = kl+1, 2*kl+ku+1
747 afb( i, j ) = ab( i-kl, j )
748 30 CONTINUE
749 40 CONTINUE
750 CALL zgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
751*
752* Return if INFO is non-zero.
753*
754 IF( info.GT.0 ) THEN
755*
756* Pivot in column INFO is exactly 0
757* Compute the reciprocal pivot growth factor of the
758* leading rank-deficient INFO columns of A.
759*
760 rpvgrw = zla_gbrpvgrw( n, kl, ku, info, ab, ldab, afb,
761 \$ ldafb )
762 RETURN
763 END IF
764 END IF
765*
766* Compute the reciprocal pivot growth factor RPVGRW.
767*
768 rpvgrw = zla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
769*
770* Compute the solution matrix X.
771*
772 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
773 CALL zgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
774 \$ info )
775*
776* Use iterative refinement to improve the computed solution and
777* compute error bounds and backward error estimates for it.
778*
779 CALL zgbrfsx( trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
780 \$ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
781 \$ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
782 \$ work, rwork, info )
783
784*
785* Scale solutions.
786*
787 IF ( colequ .AND. notran ) THEN
788 CALL zlascl2( n, nrhs, c, x, ldx )
789 ELSE IF ( rowequ .AND. .NOT.notran ) THEN
790 CALL zlascl2( n, nrhs, r, x, ldx )
791 END IF
792*
793 RETURN
794*
795* End of ZGBSVXX
796*
797 END
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: zlaqgb.f:160
subroutine zgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQUB
Definition: zgbequb.f:161
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:144
double precision function zla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
Definition: zla_gbrpvgrw.f:117
subroutine zgbrfsx(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZGBRFSX
Definition: zgbrfsx.f:440
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:138
subroutine zgbsvxx(FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices
Definition: zgbsvxx.f:560
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a matrix.
Definition: zlascl2.f:91