Annotation of rpl/lapack/lapack/dgeequ.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b DGEEQU
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DGEEQU + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequ.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGEEQU( 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 A( LDA, * ), C( * ), R( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DGEEQU computes row and column scalings intended to equilibrate an
! 39: *> M-by-N matrix A and reduce its condition number. R returns the row
! 40: *> scale factors and C the column scale factors, chosen to try to make
! 41: *> the largest element in each row and column of the matrix B with
! 42: *> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
! 43: *>
! 44: *> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
! 45: *> number and BIGNUM = largest safe number. Use of these scaling
! 46: *> factors is not guaranteed to reduce the condition number of A but
! 47: *> works well in practice.
! 48: *> \endverbatim
! 49: *
! 50: * Arguments:
! 51: * ==========
! 52: *
! 53: *> \param[in] M
! 54: *> \verbatim
! 55: *> M is INTEGER
! 56: *> The number of rows of the matrix A. M >= 0.
! 57: *> \endverbatim
! 58: *>
! 59: *> \param[in] N
! 60: *> \verbatim
! 61: *> N is INTEGER
! 62: *> The number of columns of the matrix A. N >= 0.
! 63: *> \endverbatim
! 64: *>
! 65: *> \param[in] A
! 66: *> \verbatim
! 67: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 68: *> The M-by-N matrix whose equilibration factors are
! 69: *> to be computed.
! 70: *> \endverbatim
! 71: *>
! 72: *> \param[in] LDA
! 73: *> \verbatim
! 74: *> LDA is INTEGER
! 75: *> The leading dimension of the array A. LDA >= max(1,M).
! 76: *> \endverbatim
! 77: *>
! 78: *> \param[out] R
! 79: *> \verbatim
! 80: *> R is DOUBLE PRECISION array, dimension (M)
! 81: *> If INFO = 0 or INFO > M, R contains the row scale factors
! 82: *> for A.
! 83: *> \endverbatim
! 84: *>
! 85: *> \param[out] C
! 86: *> \verbatim
! 87: *> C is DOUBLE PRECISION array, dimension (N)
! 88: *> If INFO = 0, C contains the column scale factors for A.
! 89: *> \endverbatim
! 90: *>
! 91: *> \param[out] ROWCND
! 92: *> \verbatim
! 93: *> ROWCND is DOUBLE PRECISION
! 94: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 95: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 96: *> AMAX is neither too large nor too small, it is not worth
! 97: *> scaling by R.
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[out] COLCND
! 101: *> \verbatim
! 102: *> COLCND is DOUBLE PRECISION
! 103: *> If INFO = 0, COLCND contains the ratio of the smallest
! 104: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 105: *> worth scaling by C.
! 106: *> \endverbatim
! 107: *>
! 108: *> \param[out] AMAX
! 109: *> \verbatim
! 110: *> AMAX is DOUBLE PRECISION
! 111: *> Absolute value of largest matrix element. If AMAX is very
! 112: *> close to overflow or very close to underflow, the matrix
! 113: *> should be scaled.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[out] INFO
! 117: *> \verbatim
! 118: *> INFO is INTEGER
! 119: *> = 0: successful exit
! 120: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 121: *> > 0: if INFO = i, and i is
! 122: *> <= M: the i-th row of A is exactly zero
! 123: *> > M: the (i-M)-th column of A is exactly zero
! 124: *> \endverbatim
! 125: *
! 126: * Authors:
! 127: * ========
! 128: *
! 129: *> \author Univ. of Tennessee
! 130: *> \author Univ. of California Berkeley
! 131: *> \author Univ. of Colorado Denver
! 132: *> \author NAG Ltd.
! 133: *
! 134: *> \date November 2011
! 135: *
! 136: *> \ingroup doubleGEcomputational
! 137: *
! 138: * =====================================================================
1.1 bertrand 139: SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
140: $ INFO )
141: *
1.8 ! bertrand 142: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 143: * -- LAPACK is a software package provided by Univ. of Tennessee, --
144: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 145: * November 2011
1.1 bertrand 146: *
147: * .. Scalar Arguments ..
148: INTEGER INFO, LDA, M, N
149: DOUBLE PRECISION AMAX, COLCND, ROWCND
150: * ..
151: * .. Array Arguments ..
152: DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
153: * ..
154: *
155: * =====================================================================
156: *
157: * .. Parameters ..
158: DOUBLE PRECISION ONE, ZERO
159: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
160: * ..
161: * .. Local Scalars ..
162: INTEGER I, J
163: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
164: * ..
165: * .. External Functions ..
166: DOUBLE PRECISION DLAMCH
167: EXTERNAL DLAMCH
168: * ..
169: * .. External Subroutines ..
170: EXTERNAL XERBLA
171: * ..
172: * .. Intrinsic Functions ..
173: INTRINSIC ABS, MAX, MIN
174: * ..
175: * .. Executable Statements ..
176: *
177: * Test the input parameters.
178: *
179: INFO = 0
180: IF( M.LT.0 ) THEN
181: INFO = -1
182: ELSE IF( N.LT.0 ) THEN
183: INFO = -2
184: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
185: INFO = -4
186: END IF
187: IF( INFO.NE.0 ) THEN
188: CALL XERBLA( 'DGEEQU', -INFO )
189: RETURN
190: END IF
191: *
192: * Quick return if possible
193: *
194: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
195: ROWCND = ONE
196: COLCND = ONE
197: AMAX = ZERO
198: RETURN
199: END IF
200: *
201: * Get machine constants.
202: *
203: SMLNUM = DLAMCH( 'S' )
204: BIGNUM = ONE / SMLNUM
205: *
206: * Compute row scale factors.
207: *
208: DO 10 I = 1, M
209: R( I ) = ZERO
210: 10 CONTINUE
211: *
212: * Find the maximum element in each row.
213: *
214: DO 30 J = 1, N
215: DO 20 I = 1, M
216: R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
217: 20 CONTINUE
218: 30 CONTINUE
219: *
220: * Find the maximum and minimum scale factors.
221: *
222: RCMIN = BIGNUM
223: RCMAX = ZERO
224: DO 40 I = 1, M
225: RCMAX = MAX( RCMAX, R( I ) )
226: RCMIN = MIN( RCMIN, R( I ) )
227: 40 CONTINUE
228: AMAX = RCMAX
229: *
230: IF( RCMIN.EQ.ZERO ) THEN
231: *
232: * Find the first zero scale factor and return an error code.
233: *
234: DO 50 I = 1, M
235: IF( R( I ).EQ.ZERO ) THEN
236: INFO = I
237: RETURN
238: END IF
239: 50 CONTINUE
240: ELSE
241: *
242: * Invert the scale factors.
243: *
244: DO 60 I = 1, M
245: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
246: 60 CONTINUE
247: *
248: * Compute ROWCND = min(R(I)) / max(R(I))
249: *
250: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
251: END IF
252: *
253: * Compute column scale factors
254: *
255: DO 70 J = 1, N
256: C( J ) = ZERO
257: 70 CONTINUE
258: *
259: * Find the maximum element in each column,
260: * assuming the row scaling computed above.
261: *
262: DO 90 J = 1, N
263: DO 80 I = 1, M
264: C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
265: 80 CONTINUE
266: 90 CONTINUE
267: *
268: * Find the maximum and minimum scale factors.
269: *
270: RCMIN = BIGNUM
271: RCMAX = ZERO
272: DO 100 J = 1, N
273: RCMIN = MIN( RCMIN, C( J ) )
274: RCMAX = MAX( RCMAX, C( J ) )
275: 100 CONTINUE
276: *
277: IF( RCMIN.EQ.ZERO ) THEN
278: *
279: * Find the first zero scale factor and return an error code.
280: *
281: DO 110 J = 1, N
282: IF( C( J ).EQ.ZERO ) THEN
283: INFO = M + J
284: RETURN
285: END IF
286: 110 CONTINUE
287: ELSE
288: *
289: * Invert the scale factors.
290: *
291: DO 120 J = 1, N
292: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
293: 120 CONTINUE
294: *
295: * Compute COLCND = min(C(J)) / max(C(J))
296: *
297: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
298: END IF
299: *
300: RETURN
301: *
302: * End of DGEEQU
303: *
304: END
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