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
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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[out] WORK
145: *> \verbatim
146: *> WORK is DOUBLE PRECISION array, dimension (5*N).
147: *> Workspace.
148: *> \endverbatim
149: *>
150: *> \param[out] 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: *> \ingroup doubleGBcomputational
165: *
166: * =====================================================================
167: DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB,
168: $ AFB, LDAFB, IPIV, CMODE, C,
169: $ INFO, WORK, IWORK )
170: *
171: * -- LAPACK computational routine --
172: * -- LAPACK is a software package provided by Univ. of Tennessee, --
173: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
174: *
175: * .. Scalar Arguments ..
176: CHARACTER TRANS
177: INTEGER N, LDAB, LDAFB, INFO, KL, KU, CMODE
178: * ..
179: * .. Array Arguments ..
180: INTEGER IWORK( * ), IPIV( * )
181: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
182: $ C( * )
183: * ..
184: *
185: * =====================================================================
186: *
187: * .. Local Scalars ..
188: LOGICAL NOTRANS
189: INTEGER KASE, I, J, KD, KE
190: DOUBLE PRECISION AINVNM, TMP
191: * ..
192: * .. Local Arrays ..
193: INTEGER ISAVE( 3 )
194: * ..
195: * .. External Functions ..
196: LOGICAL LSAME
197: EXTERNAL LSAME
198: * ..
199: * .. External Subroutines ..
200: EXTERNAL DLACN2, DGBTRS, XERBLA
201: * ..
202: * .. Intrinsic Functions ..
203: INTRINSIC ABS, MAX
204: * ..
205: * .. Executable Statements ..
206: *
207: DLA_GBRCOND = 0.0D+0
208: *
209: INFO = 0
210: NOTRANS = LSAME( TRANS, 'N' )
211: IF ( .NOT. NOTRANS .AND. .NOT. LSAME(TRANS, 'T')
212: $ .AND. .NOT. LSAME(TRANS, 'C') ) THEN
213: INFO = -1
214: ELSE IF( N.LT.0 ) THEN
215: INFO = -2
216: ELSE IF( KL.LT.0 .OR. KL.GT.N-1 ) THEN
217: INFO = -3
218: ELSE IF( KU.LT.0 .OR. KU.GT.N-1 ) THEN
219: INFO = -4
220: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
221: INFO = -6
222: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
223: INFO = -8
224: END IF
225: IF( INFO.NE.0 ) THEN
226: CALL XERBLA( 'DLA_GBRCOND', -INFO )
227: RETURN
228: END IF
229: IF( N.EQ.0 ) THEN
230: DLA_GBRCOND = 1.0D+0
231: RETURN
232: END IF
233: *
234: * Compute the equilibration matrix R such that
235: * inv(R)*A*C has unit 1-norm.
236: *
237: KD = KU + 1
238: KE = KL + 1
239: IF ( NOTRANS ) THEN
240: DO I = 1, N
241: TMP = 0.0D+0
242: IF ( CMODE .EQ. 1 ) THEN
243: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
244: TMP = TMP + ABS( AB( KD+I-J, J ) * C( J ) )
245: END DO
246: ELSE IF ( CMODE .EQ. 0 ) THEN
247: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
248: TMP = TMP + ABS( AB( KD+I-J, J ) )
249: END DO
250: ELSE
251: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
252: TMP = TMP + ABS( AB( KD+I-J, J ) / C( J ) )
253: END DO
254: END IF
255: WORK( 2*N+I ) = TMP
256: END DO
257: ELSE
258: DO I = 1, N
259: TMP = 0.0D+0
260: IF ( CMODE .EQ. 1 ) THEN
261: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
262: TMP = TMP + ABS( AB( KE-I+J, I ) * C( J ) )
263: END DO
264: ELSE IF ( CMODE .EQ. 0 ) THEN
265: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
266: TMP = TMP + ABS( AB( KE-I+J, I ) )
267: END DO
268: ELSE
269: DO J = MAX( I-KL, 1 ), MIN( I+KU, N )
270: TMP = TMP + ABS( AB( KE-I+J, I ) / C( J ) )
271: END DO
272: END IF
273: WORK( 2*N+I ) = TMP
274: END DO
275: END IF
276: *
277: * Estimate the norm of inv(op(A)).
278: *
279: AINVNM = 0.0D+0
280:
281: KASE = 0
282: 10 CONTINUE
283: CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
284: IF( KASE.NE.0 ) THEN
285: IF( KASE.EQ.2 ) THEN
286: *
287: * Multiply by R.
288: *
289: DO I = 1, N
290: WORK( I ) = WORK( I ) * WORK( 2*N+I )
291: END DO
292:
293: IF ( NOTRANS ) THEN
294: CALL DGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
295: $ IPIV, WORK, N, INFO )
296: ELSE
297: CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV,
298: $ WORK, N, INFO )
299: END IF
300: *
301: * Multiply by inv(C).
302: *
303: IF ( CMODE .EQ. 1 ) THEN
304: DO I = 1, N
305: WORK( I ) = WORK( I ) / C( I )
306: END DO
307: ELSE IF ( CMODE .EQ. -1 ) THEN
308: DO I = 1, N
309: WORK( I ) = WORK( I ) * C( I )
310: END DO
311: END IF
312: ELSE
313: *
314: * Multiply by inv(C**T).
315: *
316: IF ( CMODE .EQ. 1 ) THEN
317: DO I = 1, N
318: WORK( I ) = WORK( I ) / C( I )
319: END DO
320: ELSE IF ( CMODE .EQ. -1 ) THEN
321: DO I = 1, N
322: WORK( I ) = WORK( I ) * C( I )
323: END DO
324: END IF
325:
326: IF ( NOTRANS ) THEN
327: CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV,
328: $ WORK, N, INFO )
329: ELSE
330: CALL DGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB,
331: $ IPIV, WORK, N, INFO )
332: END IF
333: *
334: * Multiply by R.
335: *
336: DO I = 1, N
337: WORK( I ) = WORK( I ) * WORK( 2*N+I )
338: END DO
339: END IF
340: GO TO 10
341: END IF
342: *
343: * Compute the estimate of the reciprocal condition number.
344: *
345: IF( AINVNM .NE. 0.0D+0 )
346: $ DLA_GBRCOND = ( 1.0D+0 / AINVNM )
347: *
348: RETURN
349: *
350: * End of DLA_GBRCOND
351: *
352: END
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