1: *> \brief \b ZLA_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
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLA_GBRCOND_X + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_gbrcond_x.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_gbrcond_x.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_gbrcond_x.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * DOUBLE PRECISION FUNCTION ZLA_GBRCOND_X( TRANS, N, KL, KU, AB,
22: * LDAB, AFB, LDAFB, IPIV,
23: * X, INFO, WORK, RWORK )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER TRANS
27: * INTEGER N, KL, KU, KD, KE, LDAB, LDAFB, INFO
28: * ..
29: * .. Array Arguments ..
30: * INTEGER IPIV( * )
31: * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
32: * $ X( * )
33: * DOUBLE PRECISION RWORK( * )
34: *
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZLA_GBRCOND_X Computes the infinity norm condition number of
43: *> op(A) * diag(X) where X is a COMPLEX*16 vector.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] TRANS
50: *> \verbatim
51: *> TRANS is CHARACTER*1
52: *> Specifies the form of the system of equations:
53: *> = 'N': A * X = B (No transpose)
54: *> = 'T': A**T * X = B (Transpose)
55: *> = 'C': A**H * X = B (Conjugate Transpose = Transpose)
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The number of linear equations, i.e., the order of the
62: *> matrix A. N >= 0.
63: *> \endverbatim
64: *>
65: *> \param[in] KL
66: *> \verbatim
67: *> KL is INTEGER
68: *> The number of subdiagonals within the band of A. KL >= 0.
69: *> \endverbatim
70: *>
71: *> \param[in] KU
72: *> \verbatim
73: *> KU is INTEGER
74: *> The number of superdiagonals within the band of A. KU >= 0.
75: *> \endverbatim
76: *>
77: *> \param[in] AB
78: *> \verbatim
79: *> AB is COMPLEX*16 array, dimension (LDAB,N)
80: *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
81: *> The j-th column of A is stored in the j-th column of the
82: *> array AB as follows:
83: *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
84: *> \endverbatim
85: *>
86: *> \param[in] LDAB
87: *> \verbatim
88: *> LDAB is INTEGER
89: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
90: *> \endverbatim
91: *>
92: *> \param[in] AFB
93: *> \verbatim
94: *> AFB is COMPLEX*16 array, dimension (LDAFB,N)
95: *> Details of the LU factorization of the band matrix A, as
96: *> computed by ZGBTRF. U is stored as an upper triangular
97: *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
98: *> and the multipliers used during the factorization are stored
99: *> in rows KL+KU+2 to 2*KL+KU+1.
100: *> \endverbatim
101: *>
102: *> \param[in] LDAFB
103: *> \verbatim
104: *> LDAFB is INTEGER
105: *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
106: *> \endverbatim
107: *>
108: *> \param[in] IPIV
109: *> \verbatim
110: *> IPIV is INTEGER array, dimension (N)
111: *> The pivot indices from the factorization A = P*L*U
112: *> as computed by ZGBTRF; row i of the matrix was interchanged
113: *> with row IPIV(i).
114: *> \endverbatim
115: *>
116: *> \param[in] X
117: *> \verbatim
118: *> X is COMPLEX*16 array, dimension (N)
119: *> The vector X in the formula op(A) * diag(X).
120: *> \endverbatim
121: *>
122: *> \param[out] INFO
123: *> \verbatim
124: *> INFO is INTEGER
125: *> = 0: Successful exit.
126: *> i > 0: The ith argument is invalid.
127: *> \endverbatim
128: *>
129: *> \param[out] WORK
130: *> \verbatim
131: *> WORK is COMPLEX*16 array, dimension (2*N).
132: *> Workspace.
133: *> \endverbatim
134: *>
135: *> \param[out] RWORK
136: *> \verbatim
137: *> RWORK is DOUBLE PRECISION array, dimension (N).
138: *> Workspace.
139: *> \endverbatim
140: *
141: * Authors:
142: * ========
143: *
144: *> \author Univ. of Tennessee
145: *> \author Univ. of California Berkeley
146: *> \author Univ. of Colorado Denver
147: *> \author NAG Ltd.
148: *
149: *> \ingroup complex16GBcomputational
150: *
151: * =====================================================================
152: DOUBLE PRECISION FUNCTION ZLA_GBRCOND_X( TRANS, N, KL, KU, AB,
153: $ LDAB, AFB, LDAFB, IPIV,
154: $ X, INFO, WORK, RWORK )
155: *
156: * -- LAPACK computational routine --
157: * -- LAPACK is a software package provided by Univ. of Tennessee, --
158: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159: *
160: * .. Scalar Arguments ..
161: CHARACTER TRANS
162: INTEGER N, KL, KU, KD, KE, LDAB, LDAFB, INFO
163: * ..
164: * .. Array Arguments ..
165: INTEGER IPIV( * )
166: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
167: $ X( * )
168: DOUBLE PRECISION RWORK( * )
169: *
170: *
171: * =====================================================================
172: *
173: * .. Local Scalars ..
174: LOGICAL NOTRANS
175: INTEGER KASE, I, J
176: DOUBLE PRECISION AINVNM, ANORM, TMP
177: COMPLEX*16 ZDUM
178: * ..
179: * .. Local Arrays ..
180: INTEGER ISAVE( 3 )
181: * ..
182: * .. External Functions ..
183: LOGICAL LSAME
184: EXTERNAL LSAME
185: * ..
186: * .. External Subroutines ..
187: EXTERNAL ZLACN2, ZGBTRS, XERBLA
188: * ..
189: * .. Intrinsic Functions ..
190: INTRINSIC ABS, MAX
191: * ..
192: * .. Statement Functions ..
193: DOUBLE PRECISION CABS1
194: * ..
195: * .. Statement Function Definitions ..
196: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
197: * ..
198: * .. Executable Statements ..
199: *
200: ZLA_GBRCOND_X = 0.0D+0
201: *
202: INFO = 0
203: NOTRANS = LSAME( TRANS, 'N' )
204: IF ( .NOT. NOTRANS .AND. .NOT. LSAME(TRANS, 'T') .AND. .NOT.
205: $ LSAME( TRANS, 'C' ) ) THEN
206: INFO = -1
207: ELSE IF( N.LT.0 ) THEN
208: INFO = -2
209: ELSE IF( KL.LT.0 .OR. KL.GT.N-1 ) THEN
210: INFO = -3
211: ELSE IF( KU.LT.0 .OR. KU.GT.N-1 ) THEN
212: INFO = -4
213: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
214: INFO = -6
215: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
216: INFO = -8
217: END IF
218: IF( INFO.NE.0 ) THEN
219: CALL XERBLA( 'ZLA_GBRCOND_X', -INFO )
220: RETURN
221: END IF
222: *
223: * Compute norm of op(A)*op2(C).
224: *
225: KD = KU + 1
226: KE = KL + 1
227: ANORM = 0.0D+0
228: IF ( NOTRANS ) THEN
229: DO I = 1, N
230: TMP = 0.0D+0
231: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
232: TMP = TMP + CABS1( AB( KD+I-J, J) * X( J ) )
233: END DO
234: RWORK( I ) = TMP
235: ANORM = MAX( ANORM, TMP )
236: END DO
237: ELSE
238: DO I = 1, N
239: TMP = 0.0D+0
240: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
241: TMP = TMP + CABS1( AB( KE-I+J, I ) * X( J ) )
242: END DO
243: RWORK( I ) = TMP
244: ANORM = MAX( ANORM, TMP )
245: END DO
246: END IF
247: *
248: * Quick return if possible.
249: *
250: IF( N.EQ.0 ) THEN
251: ZLA_GBRCOND_X = 1.0D+0
252: RETURN
253: ELSE IF( ANORM .EQ. 0.0D+0 ) THEN
254: RETURN
255: END IF
256: *
257: * Estimate the norm of inv(op(A)).
258: *
259: AINVNM = 0.0D+0
260: *
261: KASE = 0
262: 10 CONTINUE
263: CALL ZLACN2( N, WORK( N+1 ), WORK, AINVNM, KASE, ISAVE )
264: IF( KASE.NE.0 ) THEN
265: IF( KASE.EQ.2 ) THEN
266: *
267: * Multiply by R.
268: *
269: DO I = 1, N
270: WORK( I ) = WORK( I ) * RWORK( I )
271: END DO
272: *
273: IF ( NOTRANS ) THEN
274: CALL ZGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
275: $ IPIV, WORK, N, INFO )
276: ELSE
277: CALL ZGBTRS( 'Conjugate transpose', N, KL, KU, 1, AFB,
278: $ LDAFB, IPIV, WORK, N, INFO )
279: ENDIF
280: *
281: * Multiply by inv(X).
282: *
283: DO I = 1, N
284: WORK( I ) = WORK( I ) / X( I )
285: END DO
286: ELSE
287: *
288: * Multiply by inv(X**H).
289: *
290: DO I = 1, N
291: WORK( I ) = WORK( I ) / X( I )
292: END DO
293: *
294: IF ( NOTRANS ) THEN
295: CALL ZGBTRS( 'Conjugate transpose', N, KL, KU, 1, AFB,
296: $ LDAFB, IPIV, WORK, N, INFO )
297: ELSE
298: CALL ZGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
299: $ IPIV, WORK, N, INFO )
300: END IF
301: *
302: * Multiply by R.
303: *
304: DO I = 1, N
305: WORK( I ) = WORK( I ) * RWORK( I )
306: END DO
307: END IF
308: GO TO 10
309: END IF
310: *
311: * Compute the estimate of the reciprocal condition number.
312: *
313: IF( AINVNM .NE. 0.0D+0 )
314: $ ZLA_GBRCOND_X = 1.0D+0 / AINVNM
315: *
316: RETURN
317: *
318: * End of ZLA_GBRCOND_X
319: *
320: END
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