Annotation of rpl/lapack/lapack/zgbequ.f, revision 1.8
1.8 ! bertrand 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
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbequ.f">
! 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: *> \date November 2011
! 150: *
! 151: *> \ingroup complex16GBcomputational
! 152: *
! 153: * =====================================================================
1.1 bertrand 154: SUBROUTINE ZGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
155: $ AMAX, INFO )
156: *
1.8 ! bertrand 157: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 158: * -- LAPACK is a software package provided by Univ. of Tennessee, --
159: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 160: * November 2011
1.1 bertrand 161: *
162: * .. Scalar Arguments ..
163: INTEGER INFO, KL, KU, LDAB, M, N
164: DOUBLE PRECISION AMAX, COLCND, ROWCND
165: * ..
166: * .. Array Arguments ..
167: DOUBLE PRECISION C( * ), R( * )
168: COMPLEX*16 AB( LDAB, * )
169: * ..
170: *
171: * =====================================================================
172: *
173: * .. Parameters ..
174: DOUBLE PRECISION ONE, ZERO
175: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
176: * ..
177: * .. Local Scalars ..
178: INTEGER I, J, KD
179: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
180: COMPLEX*16 ZDUM
181: * ..
182: * .. External Functions ..
183: DOUBLE PRECISION DLAMCH
184: EXTERNAL DLAMCH
185: * ..
186: * .. External Subroutines ..
187: EXTERNAL XERBLA
188: * ..
189: * .. Intrinsic Functions ..
190: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
191: * ..
192: * .. Statement Functions ..
193: DOUBLE PRECISION CABS1
194: * ..
195: * .. Statement Function definitions ..
196: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
197: * ..
198: * .. Executable Statements ..
199: *
200: * Test the input parameters
201: *
202: INFO = 0
203: IF( M.LT.0 ) THEN
204: INFO = -1
205: ELSE IF( N.LT.0 ) THEN
206: INFO = -2
207: ELSE IF( KL.LT.0 ) THEN
208: INFO = -3
209: ELSE IF( KU.LT.0 ) THEN
210: INFO = -4
211: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
212: INFO = -6
213: END IF
214: IF( INFO.NE.0 ) THEN
215: CALL XERBLA( 'ZGBEQU', -INFO )
216: RETURN
217: END IF
218: *
219: * Quick return if possible
220: *
221: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
222: ROWCND = ONE
223: COLCND = ONE
224: AMAX = ZERO
225: RETURN
226: END IF
227: *
228: * Get machine constants.
229: *
230: SMLNUM = DLAMCH( 'S' )
231: BIGNUM = ONE / SMLNUM
232: *
233: * Compute row scale factors.
234: *
235: DO 10 I = 1, M
236: R( I ) = ZERO
237: 10 CONTINUE
238: *
239: * Find the maximum element in each row.
240: *
241: KD = KU + 1
242: DO 30 J = 1, N
243: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
244: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
245: 20 CONTINUE
246: 30 CONTINUE
247: *
248: * Find the maximum and minimum scale factors.
249: *
250: RCMIN = BIGNUM
251: RCMAX = ZERO
252: DO 40 I = 1, M
253: RCMAX = MAX( RCMAX, R( I ) )
254: RCMIN = MIN( RCMIN, R( I ) )
255: 40 CONTINUE
256: AMAX = RCMAX
257: *
258: IF( RCMIN.EQ.ZERO ) THEN
259: *
260: * Find the first zero scale factor and return an error code.
261: *
262: DO 50 I = 1, M
263: IF( R( I ).EQ.ZERO ) THEN
264: INFO = I
265: RETURN
266: END IF
267: 50 CONTINUE
268: ELSE
269: *
270: * Invert the scale factors.
271: *
272: DO 60 I = 1, M
273: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
274: 60 CONTINUE
275: *
276: * Compute ROWCND = min(R(I)) / max(R(I))
277: *
278: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
279: END IF
280: *
281: * Compute column scale factors
282: *
283: DO 70 J = 1, N
284: C( J ) = ZERO
285: 70 CONTINUE
286: *
287: * Find the maximum element in each column,
288: * assuming the row scaling computed above.
289: *
290: KD = KU + 1
291: DO 90 J = 1, N
292: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
293: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
294: 80 CONTINUE
295: 90 CONTINUE
296: *
297: * Find the maximum and minimum scale factors.
298: *
299: RCMIN = BIGNUM
300: RCMAX = ZERO
301: DO 100 J = 1, N
302: RCMIN = MIN( RCMIN, C( J ) )
303: RCMAX = MAX( RCMAX, C( J ) )
304: 100 CONTINUE
305: *
306: IF( RCMIN.EQ.ZERO ) THEN
307: *
308: * Find the first zero scale factor and return an error code.
309: *
310: DO 110 J = 1, N
311: IF( C( J ).EQ.ZERO ) THEN
312: INFO = M + J
313: RETURN
314: END IF
315: 110 CONTINUE
316: ELSE
317: *
318: * Invert the scale factors.
319: *
320: DO 120 J = 1, N
321: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
322: 120 CONTINUE
323: *
324: * Compute COLCND = min(C(J)) / max(C(J))
325: *
326: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
327: END IF
328: *
329: RETURN
330: *
331: * End of ZGBEQU
332: *
333: END
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