Annotation of rpl/lapack/lapack/zgeequb.f, revision 1.12
1.5 bertrand 1: *> \brief \b ZGEEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.11 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.5 bertrand 7: *
8: *> \htmlonly
1.11 bertrand 9: *> Download ZGEEQUB + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeequb.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeequb.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeequb.f">
1.5 bertrand 15: *> [TXT]</a>
1.11 bertrand 16: *> \endhtmlonly
1.5 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
22: * INFO )
1.11 bertrand 23: *
1.5 bertrand 24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, M, N
26: * DOUBLE PRECISION AMAX, COLCND, ROWCND
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION C( * ), R( * )
30: * COMPLEX*16 A( LDA, * )
31: * ..
1.11 bertrand 32: *
1.5 bertrand 33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZGEEQUB 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
1.11 bertrand 52: *> to a power of the radix. Barring over- and underflow, scaling by
1.5 bertrand 53: *> these factors introduces no additional rounding errors. However, the
1.11 bertrand 54: *> scaled entries' magnitudes are no longer approximately 1 but lie
1.5 bertrand 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] A
74: *> \verbatim
75: *> A is COMPLEX*16 array, dimension (LDA,N)
76: *> The M-by-N matrix whose equilibration factors are
77: *> to be computed.
78: *> \endverbatim
79: *>
80: *> \param[in] LDA
81: *> \verbatim
82: *> LDA is INTEGER
83: *> The leading dimension of the array A. LDA >= max(1,M).
84: *> \endverbatim
85: *>
86: *> \param[out] R
87: *> \verbatim
88: *> R is DOUBLE PRECISION array, dimension (M)
89: *> If INFO = 0 or INFO > M, R contains the row scale factors
90: *> for A.
91: *> \endverbatim
92: *>
93: *> \param[out] C
94: *> \verbatim
95: *> C is DOUBLE PRECISION array, dimension (N)
96: *> If INFO = 0, C contains the column scale factors for A.
97: *> \endverbatim
98: *>
99: *> \param[out] ROWCND
100: *> \verbatim
101: *> ROWCND is DOUBLE PRECISION
102: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
103: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
104: *> AMAX is neither too large nor too small, it is not worth
105: *> scaling by R.
106: *> \endverbatim
107: *>
108: *> \param[out] COLCND
109: *> \verbatim
110: *> COLCND is DOUBLE PRECISION
111: *> If INFO = 0, COLCND contains the ratio of the smallest
112: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
113: *> worth scaling by C.
114: *> \endverbatim
115: *>
116: *> \param[out] AMAX
117: *> \verbatim
118: *> AMAX is DOUBLE PRECISION
119: *> Absolute value of largest matrix element. If AMAX is very
120: *> close to overflow or very close to underflow, the matrix
121: *> should be scaled.
122: *> \endverbatim
123: *>
124: *> \param[out] INFO
125: *> \verbatim
126: *> INFO is INTEGER
127: *> = 0: successful exit
128: *> < 0: if INFO = -i, the i-th argument had an illegal value
129: *> > 0: if INFO = i, and i is
130: *> <= M: the i-th row of A is exactly zero
131: *> > M: the (i-M)-th column of A is exactly zero
132: *> \endverbatim
133: *
134: * Authors:
135: * ========
136: *
1.11 bertrand 137: *> \author Univ. of Tennessee
138: *> \author Univ. of California Berkeley
139: *> \author Univ. of Colorado Denver
140: *> \author NAG Ltd.
1.5 bertrand 141: *
1.11 bertrand 142: *> \date December 2016
1.5 bertrand 143: *
144: *> \ingroup complex16GEcomputational
145: *
146: * =====================================================================
1.1 bertrand 147: SUBROUTINE ZGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
148: $ INFO )
149: *
1.11 bertrand 150: * -- LAPACK computational routine (version 3.7.0) --
1.5 bertrand 151: * -- LAPACK is a software package provided by Univ. of Tennessee, --
152: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.11 bertrand 153: * December 2016
1.1 bertrand 154: *
155: * .. Scalar Arguments ..
156: INTEGER INFO, LDA, M, N
157: DOUBLE PRECISION AMAX, COLCND, ROWCND
158: * ..
159: * .. Array Arguments ..
160: DOUBLE PRECISION C( * ), R( * )
161: COMPLEX*16 A( LDA, * )
162: * ..
163: *
164: * =====================================================================
165: *
166: * .. Parameters ..
167: DOUBLE PRECISION ONE, ZERO
168: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
169: * ..
170: * .. Local Scalars ..
171: INTEGER I, J
172: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
173: COMPLEX*16 ZDUM
174: * ..
175: * .. External Functions ..
176: DOUBLE PRECISION DLAMCH
177: EXTERNAL DLAMCH
178: * ..
179: * .. External Subroutines ..
180: EXTERNAL XERBLA
181: * ..
182: * .. Intrinsic Functions ..
1.5 bertrand 183: INTRINSIC ABS, MAX, MIN, LOG, DBLE, DIMAG
1.1 bertrand 184: * ..
185: * .. Statement Functions ..
186: DOUBLE PRECISION CABS1
187: * ..
188: * .. Statement Function definitions ..
189: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
190: * ..
191: * .. Executable Statements ..
192: *
193: * Test the input parameters.
194: *
195: INFO = 0
196: IF( M.LT.0 ) THEN
197: INFO = -1
198: ELSE IF( N.LT.0 ) THEN
199: INFO = -2
200: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
201: INFO = -4
202: END IF
203: IF( INFO.NE.0 ) THEN
204: CALL XERBLA( 'ZGEEQUB', -INFO )
205: RETURN
206: END IF
207: *
208: * Quick return if possible.
209: *
210: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
211: ROWCND = ONE
212: COLCND = ONE
213: AMAX = ZERO
214: RETURN
215: END IF
216: *
217: * Get machine constants. Assume SMLNUM is a power of the radix.
218: *
219: SMLNUM = DLAMCH( 'S' )
220: BIGNUM = ONE / SMLNUM
221: RADIX = DLAMCH( 'B' )
222: LOGRDX = LOG( RADIX )
223: *
224: * Compute row scale factors.
225: *
226: DO 10 I = 1, M
227: R( I ) = ZERO
228: 10 CONTINUE
229: *
230: * Find the maximum element in each row.
231: *
232: DO 30 J = 1, N
233: DO 20 I = 1, M
234: R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
235: 20 CONTINUE
236: 30 CONTINUE
237: DO I = 1, M
238: IF( R( I ).GT.ZERO ) THEN
239: R( I ) = RADIX**INT( LOG(R( I ) ) / LOGRDX )
240: END IF
241: END DO
242: *
243: * Find the maximum and minimum scale factors.
244: *
245: RCMIN = BIGNUM
246: RCMAX = ZERO
247: DO 40 I = 1, M
248: RCMAX = MAX( RCMAX, R( I ) )
249: RCMIN = MIN( RCMIN, R( I ) )
250: 40 CONTINUE
251: AMAX = RCMAX
252: *
253: IF( RCMIN.EQ.ZERO ) THEN
254: *
255: * Find the first zero scale factor and return an error code.
256: *
257: DO 50 I = 1, M
258: IF( R( I ).EQ.ZERO ) THEN
259: INFO = I
260: RETURN
261: END IF
262: 50 CONTINUE
263: ELSE
264: *
265: * Invert the scale factors.
266: *
267: DO 60 I = 1, M
268: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
269: 60 CONTINUE
270: *
271: * Compute ROWCND = min(R(I)) / max(R(I)).
272: *
273: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
274: END IF
275: *
276: * Compute column scale factors.
277: *
278: DO 70 J = 1, N
279: C( J ) = ZERO
280: 70 CONTINUE
281: *
282: * Find the maximum element in each column,
283: * assuming the row scaling computed above.
284: *
285: DO 90 J = 1, N
286: DO 80 I = 1, M
287: C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
288: 80 CONTINUE
289: IF( C( J ).GT.ZERO ) THEN
290: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
291: END IF
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 ZGEEQUB
329: *
330: END
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