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. Barring over- and underflow, scaling by
53: *> these factors introduces no additional rounding errors. However, the
54: *> scaled entries' magnitudes 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 COMPLEX*16 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: *> \ingroup complex16GBcomputational
157: *
158: * =====================================================================
159: SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
160: $ AMAX, INFO )
161: *
162: * -- LAPACK computational routine --
163: * -- LAPACK is a software package provided by Univ. of Tennessee, --
164: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165: *
166: * .. Scalar Arguments ..
167: INTEGER INFO, KL, KU, LDAB, M, N
168: DOUBLE PRECISION AMAX, COLCND, ROWCND
169: * ..
170: * .. Array Arguments ..
171: DOUBLE PRECISION C( * ), R( * )
172: COMPLEX*16 AB( LDAB, * )
173: * ..
174: *
175: * =====================================================================
176: *
177: * .. Parameters ..
178: DOUBLE PRECISION ONE, ZERO
179: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
180: * ..
181: * .. Local Scalars ..
182: INTEGER I, J, KD
183: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
184: $ LOGRDX
185: COMPLEX*16 ZDUM
186: * ..
187: * .. External Functions ..
188: DOUBLE PRECISION DLAMCH
189: EXTERNAL DLAMCH
190: * ..
191: * .. External Subroutines ..
192: EXTERNAL XERBLA
193: * ..
194: * .. Intrinsic Functions ..
195: INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
196: * ..
197: * .. Statement Functions ..
198: DOUBLE PRECISION CABS1
199: * ..
200: * .. Statement Function definitions ..
201: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
202: * ..
203: * .. Executable Statements ..
204: *
205: * Test the input parameters.
206: *
207: INFO = 0
208: IF( M.LT.0 ) THEN
209: INFO = -1
210: ELSE IF( N.LT.0 ) THEN
211: INFO = -2
212: ELSE IF( KL.LT.0 ) THEN
213: INFO = -3
214: ELSE IF( KU.LT.0 ) THEN
215: INFO = -4
216: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
217: INFO = -6
218: END IF
219: IF( INFO.NE.0 ) THEN
220: CALL XERBLA( 'ZGBEQUB', -INFO )
221: RETURN
222: END IF
223: *
224: * Quick return if possible.
225: *
226: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
227: ROWCND = ONE
228: COLCND = ONE
229: AMAX = ZERO
230: RETURN
231: END IF
232: *
233: * Get machine constants. Assume SMLNUM is a power of the radix.
234: *
235: SMLNUM = DLAMCH( 'S' )
236: BIGNUM = ONE / SMLNUM
237: RADIX = DLAMCH( 'B' )
238: LOGRDX = LOG(RADIX)
239: *
240: * Compute row scale factors.
241: *
242: DO 10 I = 1, M
243: R( I ) = ZERO
244: 10 CONTINUE
245: *
246: * Find the maximum element in each row.
247: *
248: KD = KU + 1
249: DO 30 J = 1, N
250: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
251: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
252: 20 CONTINUE
253: 30 CONTINUE
254: DO I = 1, M
255: IF( R( I ).GT.ZERO ) THEN
256: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
257: END IF
258: END DO
259: *
260: * Find the maximum and minimum scale factors.
261: *
262: RCMIN = BIGNUM
263: RCMAX = ZERO
264: DO 40 I = 1, M
265: RCMAX = MAX( RCMAX, R( I ) )
266: RCMIN = MIN( RCMIN, R( I ) )
267: 40 CONTINUE
268: AMAX = RCMAX
269: *
270: IF( RCMIN.EQ.ZERO ) THEN
271: *
272: * Find the first zero scale factor and return an error code.
273: *
274: DO 50 I = 1, M
275: IF( R( I ).EQ.ZERO ) THEN
276: INFO = I
277: RETURN
278: END IF
279: 50 CONTINUE
280: ELSE
281: *
282: * Invert the scale factors.
283: *
284: DO 60 I = 1, M
285: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
286: 60 CONTINUE
287: *
288: * Compute ROWCND = min(R(I)) / max(R(I)).
289: *
290: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
291: END IF
292: *
293: * Compute column scale factors.
294: *
295: DO 70 J = 1, N
296: C( J ) = ZERO
297: 70 CONTINUE
298: *
299: * Find the maximum element in each column,
300: * assuming the row scaling computed above.
301: *
302: DO 90 J = 1, N
303: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
304: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
305: 80 CONTINUE
306: IF( C( J ).GT.ZERO ) THEN
307: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
308: END IF
309: 90 CONTINUE
310: *
311: * Find the maximum and minimum scale factors.
312: *
313: RCMIN = BIGNUM
314: RCMAX = ZERO
315: DO 100 J = 1, N
316: RCMIN = MIN( RCMIN, C( J ) )
317: RCMAX = MAX( RCMAX, C( J ) )
318: 100 CONTINUE
319: *
320: IF( RCMIN.EQ.ZERO ) THEN
321: *
322: * Find the first zero scale factor and return an error code.
323: *
324: DO 110 J = 1, N
325: IF( C( J ).EQ.ZERO ) THEN
326: INFO = M + J
327: RETURN
328: END IF
329: 110 CONTINUE
330: ELSE
331: *
332: * Invert the scale factors.
333: *
334: DO 120 J = 1, N
335: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
336: 120 CONTINUE
337: *
338: * Compute COLCND = min(C(J)) / max(C(J)).
339: *
340: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
341: END IF
342: *
343: RETURN
344: *
345: * End of ZGBEQUB
346: *
347: END
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