Annotation of rpl/lapack/lapack/zgeequ.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b ZGEEQU
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGEEQU + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeequ.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGEEQU( 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: *> ZGEEQU 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 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] A
! 67: *> \verbatim
! 68: *> A is COMPLEX*16 array, dimension (LDA,N)
! 69: *> The M-by-N matrix whose equilibration factors are
! 70: *> to be computed.
! 71: *> \endverbatim
! 72: *>
! 73: *> \param[in] LDA
! 74: *> \verbatim
! 75: *> LDA is INTEGER
! 76: *> The leading dimension of the array A. LDA >= max(1,M).
! 77: *> \endverbatim
! 78: *>
! 79: *> \param[out] R
! 80: *> \verbatim
! 81: *> R is DOUBLE PRECISION array, dimension (M)
! 82: *> If INFO = 0 or INFO > M, R contains the row scale factors
! 83: *> for A.
! 84: *> \endverbatim
! 85: *>
! 86: *> \param[out] C
! 87: *> \verbatim
! 88: *> C is DOUBLE PRECISION array, dimension (N)
! 89: *> If INFO = 0, C contains the column scale factors for A.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[out] ROWCND
! 93: *> \verbatim
! 94: *> ROWCND is DOUBLE PRECISION
! 95: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 96: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 97: *> AMAX is neither too large nor too small, it is not worth
! 98: *> scaling by R.
! 99: *> \endverbatim
! 100: *>
! 101: *> \param[out] COLCND
! 102: *> \verbatim
! 103: *> COLCND is DOUBLE PRECISION
! 104: *> If INFO = 0, COLCND contains the ratio of the smallest
! 105: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 106: *> worth scaling by C.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] AMAX
! 110: *> \verbatim
! 111: *> AMAX is DOUBLE PRECISION
! 112: *> Absolute value of largest matrix element. If AMAX is very
! 113: *> close to overflow or very close to underflow, the matrix
! 114: *> should be scaled.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[out] INFO
! 118: *> \verbatim
! 119: *> INFO is INTEGER
! 120: *> = 0: successful exit
! 121: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 122: *> > 0: if INFO = i, and i is
! 123: *> <= M: the i-th row of A is exactly zero
! 124: *> > M: the (i-M)-th column of A is exactly zero
! 125: *> \endverbatim
! 126: *
! 127: * Authors:
! 128: * ========
! 129: *
! 130: *> \author Univ. of Tennessee
! 131: *> \author Univ. of California Berkeley
! 132: *> \author Univ. of Colorado Denver
! 133: *> \author NAG Ltd.
! 134: *
! 135: *> \date November 2011
! 136: *
! 137: *> \ingroup complex16GEcomputational
! 138: *
! 139: * =====================================================================
1.1 bertrand 140: SUBROUTINE ZGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
141: $ INFO )
142: *
1.8 ! bertrand 143: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 144: * -- LAPACK is a software package provided by Univ. of Tennessee, --
145: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 146: * November 2011
1.1 bertrand 147: *
148: * .. Scalar Arguments ..
149: INTEGER INFO, LDA, M, N
150: DOUBLE PRECISION AMAX, COLCND, ROWCND
151: * ..
152: * .. Array Arguments ..
153: DOUBLE PRECISION C( * ), R( * )
154: COMPLEX*16 A( LDA, * )
155: * ..
156: *
157: * =====================================================================
158: *
159: * .. Parameters ..
160: DOUBLE PRECISION ONE, ZERO
161: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
162: * ..
163: * .. Local Scalars ..
164: INTEGER I, J
165: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
166: COMPLEX*16 ZDUM
167: * ..
168: * .. External Functions ..
169: DOUBLE PRECISION DLAMCH
170: EXTERNAL DLAMCH
171: * ..
172: * .. External Subroutines ..
173: EXTERNAL XERBLA
174: * ..
175: * .. Intrinsic Functions ..
176: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
177: * ..
178: * .. Statement Functions ..
179: DOUBLE PRECISION CABS1
180: * ..
181: * .. Statement Function definitions ..
182: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
183: * ..
184: * .. Executable Statements ..
185: *
186: * Test the input parameters.
187: *
188: INFO = 0
189: IF( M.LT.0 ) THEN
190: INFO = -1
191: ELSE IF( N.LT.0 ) THEN
192: INFO = -2
193: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
194: INFO = -4
195: END IF
196: IF( INFO.NE.0 ) THEN
197: CALL XERBLA( 'ZGEEQU', -INFO )
198: RETURN
199: END IF
200: *
201: * Quick return if possible
202: *
203: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
204: ROWCND = ONE
205: COLCND = ONE
206: AMAX = ZERO
207: RETURN
208: END IF
209: *
210: * Get machine constants.
211: *
212: SMLNUM = DLAMCH( 'S' )
213: BIGNUM = ONE / SMLNUM
214: *
215: * Compute row scale factors.
216: *
217: DO 10 I = 1, M
218: R( I ) = ZERO
219: 10 CONTINUE
220: *
221: * Find the maximum element in each row.
222: *
223: DO 30 J = 1, N
224: DO 20 I = 1, M
225: R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
226: 20 CONTINUE
227: 30 CONTINUE
228: *
229: * Find the maximum and minimum scale factors.
230: *
231: RCMIN = BIGNUM
232: RCMAX = ZERO
233: DO 40 I = 1, M
234: RCMAX = MAX( RCMAX, R( I ) )
235: RCMIN = MIN( RCMIN, R( I ) )
236: 40 CONTINUE
237: AMAX = RCMAX
238: *
239: IF( RCMIN.EQ.ZERO ) THEN
240: *
241: * Find the first zero scale factor and return an error code.
242: *
243: DO 50 I = 1, M
244: IF( R( I ).EQ.ZERO ) THEN
245: INFO = I
246: RETURN
247: END IF
248: 50 CONTINUE
249: ELSE
250: *
251: * Invert the scale factors.
252: *
253: DO 60 I = 1, M
254: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
255: 60 CONTINUE
256: *
257: * Compute ROWCND = min(R(I)) / max(R(I))
258: *
259: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
260: END IF
261: *
262: * Compute column scale factors
263: *
264: DO 70 J = 1, N
265: C( J ) = ZERO
266: 70 CONTINUE
267: *
268: * Find the maximum element in each column,
269: * assuming the row scaling computed above.
270: *
271: DO 90 J = 1, N
272: DO 80 I = 1, M
273: C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
274: 80 CONTINUE
275: 90 CONTINUE
276: *
277: * Find the maximum and minimum scale factors.
278: *
279: RCMIN = BIGNUM
280: RCMAX = ZERO
281: DO 100 J = 1, N
282: RCMIN = MIN( RCMIN, C( J ) )
283: RCMAX = MAX( RCMAX, C( J ) )
284: 100 CONTINUE
285: *
286: IF( RCMIN.EQ.ZERO ) THEN
287: *
288: * Find the first zero scale factor and return an error code.
289: *
290: DO 110 J = 1, N
291: IF( C( J ).EQ.ZERO ) THEN
292: INFO = M + J
293: RETURN
294: END IF
295: 110 CONTINUE
296: ELSE
297: *
298: * Invert the scale factors.
299: *
300: DO 120 J = 1, N
301: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
302: 120 CONTINUE
303: *
304: * Compute COLCND = min(C(J)) / max(C(J))
305: *
306: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
307: END IF
308: *
309: RETURN
310: *
311: * End of ZGEEQU
312: *
313: END
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