LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
cla_gbrcond_x.f
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1 *> \brief \b CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * REAL FUNCTION CLA_GBRCOND_X( TRANS, N, KL, KU, AB, LDAB, AFB,
22 * LDAFB, IPIV, X, INFO, WORK, RWORK )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER TRANS
26 * INTEGER N, KL, KU, KD, KE, LDAB, LDAFB, INFO
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * )
30 * COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
31 * \$ X( * )
32 * REAL RWORK( * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> CLA_GBRCOND_X Computes the infinity norm condition number of
42 *> op(A) * diag(X) where X is a COMPLEX vector.
43 *> \endverbatim
44 *
45 * Arguments:
46 * ==========
47 *
48 *> \param[in] TRANS
49 *> \verbatim
50 *> TRANS is CHARACTER*1
51 *> Specifies the form of the system of equations:
52 *> = 'N': A * X = B (No transpose)
53 *> = 'T': A**T * X = B (Transpose)
54 *> = 'C': A**H * X = B (Conjugate Transpose = Transpose)
55 *> \endverbatim
56 *>
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The number of linear equations, i.e., the order of the
61 *> matrix A. N >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in] KL
65 *> \verbatim
66 *> KL is INTEGER
67 *> The number of subdiagonals within the band of A. KL >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in] KU
71 *> \verbatim
72 *> KU is INTEGER
73 *> The number of superdiagonals within the band of A. KU >= 0.
74 *> \endverbatim
75 *>
76 *> \param[in] AB
77 *> \verbatim
78 *> AB is COMPLEX array, dimension (LDAB,N)
79 *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
80 *> The j-th column of A is stored in the j-th column of the
81 *> array AB as follows:
82 *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
83 *> \endverbatim
84 *>
85 *> \param[in] LDAB
86 *> \verbatim
87 *> LDAB is INTEGER
88 *> The leading dimension of the array AB. LDAB >= KL+KU+1.
89 *> \endverbatim
90 *>
91 *> \param[in] AFB
92 *> \verbatim
93 *> AFB is COMPLEX array, dimension (LDAFB,N)
94 *> Details of the LU factorization of the band matrix A, as
95 *> computed by CGBTRF. U is stored as an upper triangular
96 *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
97 *> and the multipliers used during the factorization are stored
98 *> in rows KL+KU+2 to 2*KL+KU+1.
99 *> \endverbatim
100 *>
101 *> \param[in] LDAFB
102 *> \verbatim
103 *> LDAFB is INTEGER
104 *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
105 *> \endverbatim
106 *>
107 *> \param[in] IPIV
108 *> \verbatim
109 *> IPIV is INTEGER array, dimension (N)
110 *> The pivot indices from the factorization A = P*L*U
111 *> as computed by CGBTRF; row i of the matrix was interchanged
112 *> with row IPIV(i).
113 *> \endverbatim
114 *>
115 *> \param[in] X
116 *> \verbatim
117 *> X is COMPLEX array, dimension (N)
118 *> The vector X in the formula op(A) * diag(X).
119 *> \endverbatim
120 *>
121 *> \param[out] INFO
122 *> \verbatim
123 *> INFO is INTEGER
124 *> = 0: Successful exit.
125 *> i > 0: The ith argument is invalid.
126 *> \endverbatim
127 *>
128 *> \param[in] WORK
129 *> \verbatim
130 *> WORK is COMPLEX array, dimension (2*N).
131 *> Workspace.
132 *> \endverbatim
133 *>
134 *> \param[in] RWORK
135 *> \verbatim
136 *> RWORK is REAL array, dimension (N).
137 *> Workspace.
138 *> \endverbatim
139 *
140 * Authors:
141 * ========
142 *
143 *> \author Univ. of Tennessee
144 *> \author Univ. of California Berkeley
145 *> \author Univ. of Colorado Denver
146 *> \author NAG Ltd.
147 *
148 *> \date September 2012
149 *
150 *> \ingroup complexGBcomputational
151 *
152 * =====================================================================
153  REAL FUNCTION cla_gbrcond_x( TRANS, N, KL, KU, AB, LDAB, AFB,
154  \$ ldafb, ipiv, x, info, work, rwork )
155 *
156 * -- LAPACK computational routine (version 3.4.2) --
157 * -- LAPACK is a software package provided by Univ. of Tennessee, --
158 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159 * September 2012
160 *
161 * .. Scalar Arguments ..
162  CHARACTER trans
163  INTEGER n, kl, ku, kd, ke, ldab, ldafb, info
164 * ..
165 * .. Array Arguments ..
166  INTEGER ipiv( * )
167  COMPLEX ab( ldab, * ), afb( ldafb, * ), work( * ),
168  \$ x( * )
169  REAL rwork( * )
170 * ..
171 *
172 * =====================================================================
173 *
174 * .. Local Scalars ..
175  LOGICAL notrans
176  INTEGER kase, i, j
177  REAL ainvnm, anorm, tmp
178  COMPLEX zdum
179 * ..
180 * .. Local Arrays ..
181  INTEGER isave( 3 )
182 * ..
183 * .. External Functions ..
184  LOGICAL lsame
185  EXTERNAL lsame
186 * ..
187 * .. External Subroutines ..
188  EXTERNAL clacn2, cgbtrs, xerbla
189 * ..
190 * .. Intrinsic Functions ..
191  INTRINSIC abs, max
192 * ..
193 * .. Statement Functions ..
194  REAL cabs1
195 * ..
196 * .. Statement Function Definitions ..
197  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
198 * ..
199 * .. Executable Statements ..
200 *
201  cla_gbrcond_x = 0.0e+0
202 *
203  info = 0
204  notrans = lsame( trans, 'N' )
205  IF ( .NOT. notrans .AND. .NOT. lsame(trans, 'T') .AND. .NOT.
206  \$ lsame( trans, 'C' ) ) THEN
207  info = -1
208  ELSE IF( n.LT.0 ) THEN
209  info = -2
210  ELSE IF( kl.LT.0 .OR. kl.GT.n-1 ) THEN
211  info = -3
212  ELSE IF( ku.LT.0 .OR. ku.GT.n-1 ) THEN
213  info = -4
214  ELSE IF( ldab.LT.kl+ku+1 ) THEN
215  info = -6
216  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
217  info = -8
218  END IF
219  IF( info.NE.0 ) THEN
220  CALL xerbla( 'CLA_GBRCOND_X', -info )
221  return
222  END IF
223 *
224 * Compute norm of op(A)*op2(C).
225 *
226  kd = ku + 1
227  ke = kl + 1
228  anorm = 0.0
229  IF ( notrans ) THEN
230  DO i = 1, n
231  tmp = 0.0e+0
232  DO j = max( i-kl, 1 ), min( i+ku, n )
233  tmp = tmp + cabs1( ab( kd+i-j, j) * x( j ) )
234  END DO
235  rwork( i ) = tmp
236  anorm = max( anorm, tmp )
237  END DO
238  ELSE
239  DO i = 1, n
240  tmp = 0.0e+0
241  DO j = max( i-kl, 1 ), min( i+ku, n )
242  tmp = tmp + cabs1( ab( ke-i+j, i ) * x( j ) )
243  END DO
244  rwork( i ) = tmp
245  anorm = max( anorm, tmp )
246  END DO
247  END IF
248 *
249 * Quick return if possible.
250 *
251  IF( n.EQ.0 ) THEN
252  cla_gbrcond_x = 1.0e+0
253  return
254  ELSE IF( anorm .EQ. 0.0e+0 ) THEN
255  return
256  END IF
257 *
258 * Estimate the norm of inv(op(A)).
259 *
260  ainvnm = 0.0e+0
261 *
262  kase = 0
263  10 continue
264  CALL clacn2( n, work( n+1 ), work, ainvnm, kase, isave )
265  IF( kase.NE.0 ) THEN
266  IF( kase.EQ.2 ) THEN
267 *
268 * Multiply by R.
269 *
270  DO i = 1, n
271  work( i ) = work( i ) * rwork( i )
272  END DO
273 *
274  IF ( notrans ) THEN
275  CALL cgbtrs( 'No transpose', n, kl, ku, 1, afb, ldafb,
276  \$ ipiv, work, n, info )
277  ELSE
278  CALL cgbtrs( 'Conjugate transpose', n, kl, ku, 1, afb,
279  \$ ldafb, ipiv, work, n, info )
280  ENDIF
281 *
282 * Multiply by inv(X).
283 *
284  DO i = 1, n
285  work( i ) = work( i ) / x( i )
286  END DO
287  ELSE
288 *
289 * Multiply by inv(X**H).
290 *
291  DO i = 1, n
292  work( i ) = work( i ) / x( i )
293  END DO
294 *
295  IF ( notrans ) THEN
296  CALL cgbtrs( 'Conjugate transpose', n, kl, ku, 1, afb,
297  \$ ldafb, ipiv, work, n, info )
298  ELSE
299  CALL cgbtrs( 'No transpose', n, kl, ku, 1, afb, ldafb,
300  \$ ipiv, work, n, info )
301  END IF
302 *
303 * Multiply by R.
304 *
305  DO i = 1, n
306  work( i ) = work( i ) * rwork( i )
307  END DO
308  END IF
309  go to 10
310  END IF
311 *
312 * Compute the estimate of the reciprocal condition number.
313 *
314  IF( ainvnm .NE. 0.0e+0 )
315  \$ cla_gbrcond_x = 1.0e+0 / ainvnm
316 *
317  return
318 *
319  END