1: *> \brief \b ZGBEQU
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGBEQU + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGBEQU( 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: *> ZGBEQU computes row and column scalings intended to equilibrate an
40: *> M-by-N band matrix A and reduce its condition number. R returns the
41: *> row scale factors and C the column scale factors, chosen to try to
42: *> make 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 absolute value 1.
44: *>
45: *> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
46: *> number and BIGNUM = largest safe number. Use of these scaling
47: *> factors is not guaranteed to reduce the condition number of A but
48: *> works well in practice.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] M
55: *> \verbatim
56: *> M is INTEGER
57: *> The number of rows of the matrix A. M >= 0.
58: *> \endverbatim
59: *>
60: *> \param[in] N
61: *> \verbatim
62: *> N is INTEGER
63: *> The number of columns of the matrix A. N >= 0.
64: *> \endverbatim
65: *>
66: *> \param[in] KL
67: *> \verbatim
68: *> KL is INTEGER
69: *> The number of subdiagonals within the band of A. KL >= 0.
70: *> \endverbatim
71: *>
72: *> \param[in] KU
73: *> \verbatim
74: *> KU is INTEGER
75: *> The number of superdiagonals within the band of A. KU >= 0.
76: *> \endverbatim
77: *>
78: *> \param[in] AB
79: *> \verbatim
80: *> AB is COMPLEX*16 array, dimension (LDAB,N)
81: *> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
82: *> column of A is stored in the j-th column of the array AB as
83: *> follows:
84: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
85: *> \endverbatim
86: *>
87: *> \param[in] LDAB
88: *> \verbatim
89: *> LDAB is INTEGER
90: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
91: *> \endverbatim
92: *>
93: *> \param[out] R
94: *> \verbatim
95: *> R is DOUBLE PRECISION array, dimension (M)
96: *> If INFO = 0, or INFO > M, R contains the row scale factors
97: *> for A.
98: *> \endverbatim
99: *>
100: *> \param[out] C
101: *> \verbatim
102: *> C is DOUBLE PRECISION array, dimension (N)
103: *> If INFO = 0, C contains the column scale factors for A.
104: *> \endverbatim
105: *>
106: *> \param[out] ROWCND
107: *> \verbatim
108: *> ROWCND is DOUBLE PRECISION
109: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
110: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
111: *> AMAX is neither too large nor too small, it is not worth
112: *> scaling by R.
113: *> \endverbatim
114: *>
115: *> \param[out] COLCND
116: *> \verbatim
117: *> COLCND is DOUBLE PRECISION
118: *> If INFO = 0, COLCND contains the ratio of the smallest
119: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
120: *> worth scaling by C.
121: *> \endverbatim
122: *>
123: *> \param[out] AMAX
124: *> \verbatim
125: *> AMAX is DOUBLE PRECISION
126: *> Absolute value of largest matrix element. If AMAX is very
127: *> close to overflow or very close to underflow, the matrix
128: *> should be scaled.
129: *> \endverbatim
130: *>
131: *> \param[out] INFO
132: *> \verbatim
133: *> INFO is INTEGER
134: *> = 0: successful exit
135: *> < 0: if INFO = -i, the i-th argument had an illegal value
136: *> > 0: if INFO = i, and i is
137: *> <= M: the i-th row of A is exactly zero
138: *> > M: the (i-M)-th column of A is exactly zero
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: SUBROUTINE ZGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
153: $ AMAX, INFO )
154: *
155: * -- LAPACK computational routine --
156: * -- LAPACK is a software package provided by Univ. of Tennessee, --
157: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
158: *
159: * .. Scalar Arguments ..
160: INTEGER INFO, KL, KU, LDAB, M, N
161: DOUBLE PRECISION AMAX, COLCND, ROWCND
162: * ..
163: * .. Array Arguments ..
164: DOUBLE PRECISION C( * ), R( * )
165: COMPLEX*16 AB( LDAB, * )
166: * ..
167: *
168: * =====================================================================
169: *
170: * .. Parameters ..
171: DOUBLE PRECISION ONE, ZERO
172: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
173: * ..
174: * .. Local Scalars ..
175: INTEGER I, J, KD
176: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
177: COMPLEX*16 ZDUM
178: * ..
179: * .. External Functions ..
180: DOUBLE PRECISION DLAMCH
181: EXTERNAL DLAMCH
182: * ..
183: * .. External Subroutines ..
184: EXTERNAL XERBLA
185: * ..
186: * .. Intrinsic Functions ..
187: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
188: * ..
189: * .. Statement Functions ..
190: DOUBLE PRECISION CABS1
191: * ..
192: * .. Statement Function definitions ..
193: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
194: * ..
195: * .. Executable Statements ..
196: *
197: * Test the input parameters
198: *
199: INFO = 0
200: IF( M.LT.0 ) THEN
201: INFO = -1
202: ELSE IF( N.LT.0 ) THEN
203: INFO = -2
204: ELSE IF( KL.LT.0 ) THEN
205: INFO = -3
206: ELSE IF( KU.LT.0 ) THEN
207: INFO = -4
208: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
209: INFO = -6
210: END IF
211: IF( INFO.NE.0 ) THEN
212: CALL XERBLA( 'ZGBEQU', -INFO )
213: RETURN
214: END IF
215: *
216: * Quick return if possible
217: *
218: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
219: ROWCND = ONE
220: COLCND = ONE
221: AMAX = ZERO
222: RETURN
223: END IF
224: *
225: * Get machine constants.
226: *
227: SMLNUM = DLAMCH( 'S' )
228: BIGNUM = ONE / SMLNUM
229: *
230: * Compute row scale factors.
231: *
232: DO 10 I = 1, M
233: R( I ) = ZERO
234: 10 CONTINUE
235: *
236: * Find the maximum element in each row.
237: *
238: KD = KU + 1
239: DO 30 J = 1, N
240: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
241: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
242: 20 CONTINUE
243: 30 CONTINUE
244: *
245: * Find the maximum and minimum scale factors.
246: *
247: RCMIN = BIGNUM
248: RCMAX = ZERO
249: DO 40 I = 1, M
250: RCMAX = MAX( RCMAX, R( I ) )
251: RCMIN = MIN( RCMIN, R( I ) )
252: 40 CONTINUE
253: AMAX = RCMAX
254: *
255: IF( RCMIN.EQ.ZERO ) THEN
256: *
257: * Find the first zero scale factor and return an error code.
258: *
259: DO 50 I = 1, M
260: IF( R( I ).EQ.ZERO ) THEN
261: INFO = I
262: RETURN
263: END IF
264: 50 CONTINUE
265: ELSE
266: *
267: * Invert the scale factors.
268: *
269: DO 60 I = 1, M
270: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
271: 60 CONTINUE
272: *
273: * Compute ROWCND = min(R(I)) / max(R(I))
274: *
275: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
276: END IF
277: *
278: * Compute column scale factors
279: *
280: DO 70 J = 1, N
281: C( J ) = ZERO
282: 70 CONTINUE
283: *
284: * Find the maximum element in each column,
285: * assuming the row scaling computed above.
286: *
287: KD = KU + 1
288: DO 90 J = 1, N
289: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
290: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
291: 80 CONTINUE
292: 90 CONTINUE
293: *
294: * Find the maximum and minimum scale factors.
295: *
296: RCMIN = BIGNUM
297: RCMAX = ZERO
298: DO 100 J = 1, N
299: RCMIN = MIN( RCMIN, C( J ) )
300: RCMAX = MAX( RCMAX, C( J ) )
301: 100 CONTINUE
302: *
303: IF( RCMIN.EQ.ZERO ) THEN
304: *
305: * Find the first zero scale factor and return an error code.
306: *
307: DO 110 J = 1, N
308: IF( C( J ).EQ.ZERO ) THEN
309: INFO = M + J
310: RETURN
311: END IF
312: 110 CONTINUE
313: ELSE
314: *
315: * Invert the scale factors.
316: *
317: DO 120 J = 1, N
318: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
319: 120 CONTINUE
320: *
321: * Compute COLCND = min(C(J)) / max(C(J))
322: *
323: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
324: END IF
325: *
326: RETURN
327: *
328: * End of ZGBEQU
329: *
330: END
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