1: *> \brief \b ZGBEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGBEQUB + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbequb.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
22: * AMAX, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, KL, KU, LDAB, M, N
26: * DOUBLE PRECISION AMAX, COLCND, ROWCND
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION C( * ), R( * )
30: * COMPLEX*16 AB( LDAB, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZGBEQUB computes row and column scalings intended to equilibrate an
40: *> M-by-N matrix A and reduce its condition number. R returns the row
41: *> scale factors and C the column scale factors, chosen to try to make
42: *> the largest element in each row and column of the matrix B with
43: *> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
44: *> the radix.
45: *>
46: *> R(i) and C(j) are restricted to be a power of the radix between
47: *> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
48: *> of these scaling factors is not guaranteed to reduce the condition
49: *> number of A but works well in practice.
50: *>
51: *> This routine differs from ZGEEQU by restricting the scaling factors
52: *> to a power of the radix. Baring over- and underflow, scaling by
53: *> these factors introduces no additional rounding errors. However, the
54: *> scaled entries' magnitured are no longer approximately 1 but lie
55: *> between sqrt(radix) and 1/sqrt(radix).
56: *> \endverbatim
57: *
58: * Arguments:
59: * ==========
60: *
61: *> \param[in] M
62: *> \verbatim
63: *> M is INTEGER
64: *> The number of rows of the matrix A. M >= 0.
65: *> \endverbatim
66: *>
67: *> \param[in] N
68: *> \verbatim
69: *> N is INTEGER
70: *> The number of columns of the matrix A. N >= 0.
71: *> \endverbatim
72: *>
73: *> \param[in] KL
74: *> \verbatim
75: *> KL is INTEGER
76: *> The number of subdiagonals within the band of A. KL >= 0.
77: *> \endverbatim
78: *>
79: *> \param[in] KU
80: *> \verbatim
81: *> KU is INTEGER
82: *> The number of superdiagonals within the band of A. KU >= 0.
83: *> \endverbatim
84: *>
85: *> \param[in] AB
86: *> \verbatim
87: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
88: *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
89: *> The j-th column of A is stored in the j-th column of the
90: *> array AB as follows:
91: *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
92: *> \endverbatim
93: *>
94: *> \param[in] LDAB
95: *> \verbatim
96: *> LDAB is INTEGER
97: *> The leading dimension of the array A. LDAB >= max(1,M).
98: *> \endverbatim
99: *>
100: *> \param[out] R
101: *> \verbatim
102: *> R is DOUBLE PRECISION array, dimension (M)
103: *> If INFO = 0 or INFO > M, R contains the row scale factors
104: *> for A.
105: *> \endverbatim
106: *>
107: *> \param[out] C
108: *> \verbatim
109: *> C is DOUBLE PRECISION array, dimension (N)
110: *> If INFO = 0, C contains the column scale factors for A.
111: *> \endverbatim
112: *>
113: *> \param[out] ROWCND
114: *> \verbatim
115: *> ROWCND is DOUBLE PRECISION
116: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
117: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
118: *> AMAX is neither too large nor too small, it is not worth
119: *> scaling by R.
120: *> \endverbatim
121: *>
122: *> \param[out] COLCND
123: *> \verbatim
124: *> COLCND is DOUBLE PRECISION
125: *> If INFO = 0, COLCND contains the ratio of the smallest
126: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
127: *> worth scaling by C.
128: *> \endverbatim
129: *>
130: *> \param[out] AMAX
131: *> \verbatim
132: *> AMAX is DOUBLE PRECISION
133: *> Absolute value of largest matrix element. If AMAX is very
134: *> close to overflow or very close to underflow, the matrix
135: *> should be scaled.
136: *> \endverbatim
137: *>
138: *> \param[out] INFO
139: *> \verbatim
140: *> INFO is INTEGER
141: *> = 0: successful exit
142: *> < 0: if INFO = -i, the i-th argument had an illegal value
143: *> > 0: if INFO = i, and i is
144: *> <= M: the i-th row of A is exactly zero
145: *> > M: the (i-M)-th column of A is exactly zero
146: *> \endverbatim
147: *
148: * Authors:
149: * ========
150: *
151: *> \author Univ. of Tennessee
152: *> \author Univ. of California Berkeley
153: *> \author Univ. of Colorado Denver
154: *> \author NAG Ltd.
155: *
156: *> \date November 2011
157: *
158: *> \ingroup complex16GBcomputational
159: *
160: * =====================================================================
161: SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
162: $ AMAX, INFO )
163: *
164: * -- LAPACK computational routine (version 3.4.0) --
165: * -- LAPACK is a software package provided by Univ. of Tennessee, --
166: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167: * November 2011
168: *
169: * .. Scalar Arguments ..
170: INTEGER INFO, KL, KU, LDAB, M, N
171: DOUBLE PRECISION AMAX, COLCND, ROWCND
172: * ..
173: * .. Array Arguments ..
174: DOUBLE PRECISION C( * ), R( * )
175: COMPLEX*16 AB( LDAB, * )
176: * ..
177: *
178: * =====================================================================
179: *
180: * .. Parameters ..
181: DOUBLE PRECISION ONE, ZERO
182: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
183: * ..
184: * .. Local Scalars ..
185: INTEGER I, J, KD
186: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
187: $ LOGRDX
188: COMPLEX*16 ZDUM
189: * ..
190: * .. External Functions ..
191: DOUBLE PRECISION DLAMCH
192: EXTERNAL DLAMCH
193: * ..
194: * .. External Subroutines ..
195: EXTERNAL XERBLA
196: * ..
197: * .. Intrinsic Functions ..
198: INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
199: * ..
200: * .. Statement Functions ..
201: DOUBLE PRECISION CABS1
202: * ..
203: * .. Statement Function definitions ..
204: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
205: * ..
206: * .. Executable Statements ..
207: *
208: * Test the input parameters.
209: *
210: INFO = 0
211: IF( M.LT.0 ) THEN
212: INFO = -1
213: ELSE IF( N.LT.0 ) THEN
214: INFO = -2
215: ELSE IF( KL.LT.0 ) THEN
216: INFO = -3
217: ELSE IF( KU.LT.0 ) THEN
218: INFO = -4
219: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
220: INFO = -6
221: END IF
222: IF( INFO.NE.0 ) THEN
223: CALL XERBLA( 'ZGBEQUB', -INFO )
224: RETURN
225: END IF
226: *
227: * Quick return if possible.
228: *
229: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
230: ROWCND = ONE
231: COLCND = ONE
232: AMAX = ZERO
233: RETURN
234: END IF
235: *
236: * Get machine constants. Assume SMLNUM is a power of the radix.
237: *
238: SMLNUM = DLAMCH( 'S' )
239: BIGNUM = ONE / SMLNUM
240: RADIX = DLAMCH( 'B' )
241: LOGRDX = LOG(RADIX)
242: *
243: * Compute row scale factors.
244: *
245: DO 10 I = 1, M
246: R( I ) = ZERO
247: 10 CONTINUE
248: *
249: * Find the maximum element in each row.
250: *
251: KD = KU + 1
252: DO 30 J = 1, N
253: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
254: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
255: 20 CONTINUE
256: 30 CONTINUE
257: DO I = 1, M
258: IF( R( I ).GT.ZERO ) THEN
259: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
260: END IF
261: END DO
262: *
263: * Find the maximum and minimum scale factors.
264: *
265: RCMIN = BIGNUM
266: RCMAX = ZERO
267: DO 40 I = 1, M
268: RCMAX = MAX( RCMAX, R( I ) )
269: RCMIN = MIN( RCMIN, R( I ) )
270: 40 CONTINUE
271: AMAX = RCMAX
272: *
273: IF( RCMIN.EQ.ZERO ) THEN
274: *
275: * Find the first zero scale factor and return an error code.
276: *
277: DO 50 I = 1, M
278: IF( R( I ).EQ.ZERO ) THEN
279: INFO = I
280: RETURN
281: END IF
282: 50 CONTINUE
283: ELSE
284: *
285: * Invert the scale factors.
286: *
287: DO 60 I = 1, M
288: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
289: 60 CONTINUE
290: *
291: * Compute ROWCND = min(R(I)) / max(R(I)).
292: *
293: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
294: END IF
295: *
296: * Compute column scale factors.
297: *
298: DO 70 J = 1, N
299: C( J ) = ZERO
300: 70 CONTINUE
301: *
302: * Find the maximum element in each column,
303: * assuming the row scaling computed above.
304: *
305: DO 90 J = 1, N
306: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
307: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
308: 80 CONTINUE
309: IF( C( J ).GT.ZERO ) THEN
310: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
311: END IF
312: 90 CONTINUE
313: *
314: * Find the maximum and minimum scale factors.
315: *
316: RCMIN = BIGNUM
317: RCMAX = ZERO
318: DO 100 J = 1, N
319: RCMIN = MIN( RCMIN, C( J ) )
320: RCMAX = MAX( RCMAX, C( J ) )
321: 100 CONTINUE
322: *
323: IF( RCMIN.EQ.ZERO ) THEN
324: *
325: * Find the first zero scale factor and return an error code.
326: *
327: DO 110 J = 1, N
328: IF( C( J ).EQ.ZERO ) THEN
329: INFO = M + J
330: RETURN
331: END IF
332: 110 CONTINUE
333: ELSE
334: *
335: * Invert the scale factors.
336: *
337: DO 120 J = 1, N
338: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
339: 120 CONTINUE
340: *
341: * Compute COLCND = min(C(J)) / max(C(J)).
342: *
343: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
344: END IF
345: *
346: RETURN
347: *
348: * End of ZGBEQUB
349: *
350: END
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