Annotation of rpl/lapack/lapack/zgeequb.f, revision 1.5
1.5 ! bertrand 1: *> \brief \b ZGEEQUB
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 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">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
! 22: * INFO )
! 23: *
! 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: * ..
! 32: *
! 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
! 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] 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: *
! 137: *> \author Univ. of Tennessee
! 138: *> \author Univ. of California Berkeley
! 139: *> \author Univ. of Colorado Denver
! 140: *> \author NAG Ltd.
! 141: *
! 142: *> \date November 2011
! 143: *
! 144: *> \ingroup complex16GEcomputational
! 145: *
! 146: * =====================================================================
1.1 bertrand 147: SUBROUTINE ZGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
148: $ INFO )
149: *
1.5 ! bertrand 150: * -- LAPACK computational routine (version 3.4.0) --
! 151: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 152: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 153: * November 2011
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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