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1: *> \brief \b DLA_GBRCOND estimates the Skeel condition number for a general banded matrix.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLA_GBRCOND + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_gbrcond.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_gbrcond.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gbrcond.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB,
22: * AFB, LDAFB, IPIV, CMODE, C,
23: * INFO, WORK, IWORK )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER TRANS
27: * INTEGER N, LDAB, LDAFB, INFO, KL, KU, CMODE
28: * ..
29: * .. Array Arguments ..
30: * INTEGER IWORK( * ), IPIV( * )
31: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
32: * $ C( * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DLA_GBRCOND Estimates the Skeel condition number of op(A) * op2(C)
42: *> where op2 is determined by CMODE as follows
43: *> CMODE = 1 op2(C) = C
44: *> CMODE = 0 op2(C) = I
45: *> CMODE = -1 op2(C) = inv(C)
46: *> The Skeel condition number cond(A) = norminf( |inv(A)||A| )
47: *> is computed by computing scaling factors R such that
48: *> diag(R)*A*op2(C) is row equilibrated and computing the standard
49: *> infinity-norm condition number.
50: *> \endverbatim
51: *
52: * Arguments:
53: * ==========
54: *
55: *> \param[in] TRANS
56: *> \verbatim
57: *> TRANS is CHARACTER*1
58: *> Specifies the form of the system of equations:
59: *> = 'N': A * X = B (No transpose)
60: *> = 'T': A**T * X = B (Transpose)
61: *> = 'C': A**H * X = B (Conjugate Transpose = Transpose)
62: *> \endverbatim
63: *>
64: *> \param[in] N
65: *> \verbatim
66: *> N is INTEGER
67: *> The number of linear equations, i.e., the order of the
68: *> matrix A. N >= 0.
69: *> \endverbatim
70: *>
71: *> \param[in] KL
72: *> \verbatim
73: *> KL is INTEGER
74: *> The number of subdiagonals within the band of A. KL >= 0.
75: *> \endverbatim
76: *>
77: *> \param[in] KU
78: *> \verbatim
79: *> KU is INTEGER
80: *> The number of superdiagonals within the band of A. KU >= 0.
81: *> \endverbatim
82: *>
83: *> \param[in] AB
84: *> \verbatim
85: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
86: *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
87: *> The j-th column of A is stored in the j-th column of the
88: *> array AB as follows:
89: *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
90: *> \endverbatim
91: *>
92: *> \param[in] LDAB
93: *> \verbatim
94: *> LDAB is INTEGER
95: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
96: *> \endverbatim
97: *>
98: *> \param[in] AFB
99: *> \verbatim
100: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
101: *> Details of the LU factorization of the band matrix A, as
102: *> computed by DGBTRF. U is stored as an upper triangular
103: *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
104: *> and the multipliers used during the factorization are stored
105: *> in rows KL+KU+2 to 2*KL+KU+1.
106: *> \endverbatim
107: *>
108: *> \param[in] LDAFB
109: *> \verbatim
110: *> LDAFB is INTEGER
111: *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
112: *> \endverbatim
113: *>
114: *> \param[in] IPIV
115: *> \verbatim
116: *> IPIV is INTEGER array, dimension (N)
117: *> The pivot indices from the factorization A = P*L*U
118: *> as computed by DGBTRF; row i of the matrix was interchanged
119: *> with row IPIV(i).
120: *> \endverbatim
121: *>
122: *> \param[in] CMODE
123: *> \verbatim
124: *> CMODE is INTEGER
125: *> Determines op2(C) in the formula op(A) * op2(C) as follows:
126: *> CMODE = 1 op2(C) = C
127: *> CMODE = 0 op2(C) = I
128: *> CMODE = -1 op2(C) = inv(C)
129: *> \endverbatim
130: *>
131: *> \param[in] C
132: *> \verbatim
133: *> C is DOUBLE PRECISION array, dimension (N)
134: *> The vector C in the formula op(A) * op2(C).
135: *> \endverbatim
136: *>
137: *> \param[out] INFO
138: *> \verbatim
139: *> INFO is INTEGER
140: *> = 0: Successful exit.
141: *> i > 0: The ith argument is invalid.
142: *> \endverbatim
143: *>
144: *> \param[in] WORK
145: *> \verbatim
146: *> WORK is DOUBLE PRECISION array, dimension (5*N).
147: *> Workspace.
148: *> \endverbatim
149: *>
150: *> \param[in] IWORK
151: *> \verbatim
152: *> IWORK is INTEGER array, dimension (N).
153: *> Workspace.
154: *> \endverbatim
155: *
156: * Authors:
157: * ========
158: *
159: *> \author Univ. of Tennessee
160: *> \author Univ. of California Berkeley
161: *> \author Univ. of Colorado Denver
162: *> \author NAG Ltd.
163: *
164: *> \date September 2012
165: *
166: *> \ingroup doubleGBcomputational
167: *
168: * =====================================================================
169: DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB,
170: $ AFB, LDAFB, IPIV, CMODE, C,
171: $ INFO, WORK, IWORK )
172: *
173: * -- LAPACK computational routine (version 3.4.2) --
174: * -- LAPACK is a software package provided by Univ. of Tennessee, --
175: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
176: * September 2012
177: *
178: * .. Scalar Arguments ..
179: CHARACTER TRANS
180: INTEGER N, LDAB, LDAFB, INFO, KL, KU, CMODE
181: * ..
182: * .. Array Arguments ..
183: INTEGER IWORK( * ), IPIV( * )
184: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
185: $ C( * )
186: * ..
187: *
188: * =====================================================================
189: *
190: * .. Local Scalars ..
191: LOGICAL NOTRANS
192: INTEGER KASE, I, J, KD, KE
193: DOUBLE PRECISION AINVNM, TMP
194: * ..
195: * .. Local Arrays ..
196: INTEGER ISAVE( 3 )
197: * ..
198: * .. External Functions ..
199: LOGICAL LSAME
200: EXTERNAL LSAME
201: * ..
202: * .. External Subroutines ..
203: EXTERNAL DLACN2, DGBTRS, XERBLA
204: * ..
205: * .. Intrinsic Functions ..
206: INTRINSIC ABS, MAX
207: * ..
208: * .. Executable Statements ..
209: *
210: DLA_GBRCOND = 0.0D+0
211: *
212: INFO = 0
213: NOTRANS = LSAME( TRANS, 'N' )
214: IF ( .NOT. NOTRANS .AND. .NOT. LSAME(TRANS, 'T')
215: $ .AND. .NOT. LSAME(TRANS, 'C') ) THEN
216: INFO = -1
217: ELSE IF( N.LT.0 ) THEN
218: INFO = -2
219: ELSE IF( KL.LT.0 .OR. KL.GT.N-1 ) THEN
220: INFO = -3
221: ELSE IF( KU.LT.0 .OR. KU.GT.N-1 ) THEN
222: INFO = -4
223: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
224: INFO = -6
225: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
226: INFO = -8
227: END IF
228: IF( INFO.NE.0 ) THEN
229: CALL XERBLA( 'DLA_GBRCOND', -INFO )
230: RETURN
231: END IF
232: IF( N.EQ.0 ) THEN
233: DLA_GBRCOND = 1.0D+0
234: RETURN
235: END IF
236: *
237: * Compute the equilibration matrix R such that
238: * inv(R)*A*C has unit 1-norm.
239: *
240: KD = KU + 1
241: KE = KL + 1
242: IF ( NOTRANS ) THEN
243: DO I = 1, N
244: TMP = 0.0D+0
245: IF ( CMODE .EQ. 1 ) THEN
246: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
247: TMP = TMP + ABS( AB( KD+I-J, J ) * C( J ) )
248: END DO
249: ELSE IF ( CMODE .EQ. 0 ) THEN
250: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
251: TMP = TMP + ABS( AB( KD+I-J, J ) )
252: END DO
253: ELSE
254: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
255: TMP = TMP + ABS( AB( KD+I-J, J ) / C( J ) )
256: END DO
257: END IF
258: WORK( 2*N+I ) = TMP
259: END DO
260: ELSE
261: DO I = 1, N
262: TMP = 0.0D+0
263: IF ( CMODE .EQ. 1 ) THEN
264: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
265: TMP = TMP + ABS( AB( KE-I+J, I ) * C( J ) )
266: END DO
267: ELSE IF ( CMODE .EQ. 0 ) THEN
268: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
269: TMP = TMP + ABS( AB( KE-I+J, I ) )
270: END DO
271: ELSE
272: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
273: TMP = TMP + ABS( AB( KE-I+J, I ) / C( J ) )
274: END DO
275: END IF
276: WORK( 2*N+I ) = TMP
277: END DO
278: END IF
279: *
280: * Estimate the norm of inv(op(A)).
281: *
282: AINVNM = 0.0D+0
283:
284: KASE = 0
285: 10 CONTINUE
286: CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
287: IF( KASE.NE.0 ) THEN
288: IF( KASE.EQ.2 ) THEN
289: *
290: * Multiply by R.
291: *
292: DO I = 1, N
293: WORK( I ) = WORK( I ) * WORK( 2*N+I )
294: END DO
295:
296: IF ( NOTRANS ) THEN
297: CALL DGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
298: $ IPIV, WORK, N, INFO )
299: ELSE
300: CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV,
301: $ WORK, N, INFO )
302: END IF
303: *
304: * Multiply by inv(C).
305: *
306: IF ( CMODE .EQ. 1 ) THEN
307: DO I = 1, N
308: WORK( I ) = WORK( I ) / C( I )
309: END DO
310: ELSE IF ( CMODE .EQ. -1 ) THEN
311: DO I = 1, N
312: WORK( I ) = WORK( I ) * C( I )
313: END DO
314: END IF
315: ELSE
316: *
317: * Multiply by inv(C**T).
318: *
319: IF ( CMODE .EQ. 1 ) THEN
320: DO I = 1, N
321: WORK( I ) = WORK( I ) / C( I )
322: END DO
323: ELSE IF ( CMODE .EQ. -1 ) THEN
324: DO I = 1, N
325: WORK( I ) = WORK( I ) * C( I )
326: END DO
327: END IF
328:
329: IF ( NOTRANS ) THEN
330: CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV,
331: $ WORK, N, INFO )
332: ELSE
333: CALL DGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
334: $ IPIV, WORK, N, INFO )
335: END IF
336: *
337: * Multiply by R.
338: *
339: DO I = 1, N
340: WORK( I ) = WORK( I ) * WORK( 2*N+I )
341: END DO
342: END IF
343: GO TO 10
344: END IF
345: *
346: * Compute the estimate of the reciprocal condition number.
347: *
348: IF( AINVNM .NE. 0.0D+0 )
349: $ DLA_GBRCOND = ( 1.0D+0 / AINVNM )
350: *
351: RETURN
352: *
353: END
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