1: *> \brief \b DGBEQU
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGBEQU + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbequ.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGBEQU( 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 AB( LDAB, * ), C( * ), R( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DGBEQU computes row and column scalings intended to equilibrate an
39: *> M-by-N band matrix A and reduce its condition number. R returns the
40: *> row scale factors and C the column scale factors, chosen to try to
41: *> make 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] KL
66: *> \verbatim
67: *> KL is INTEGER
68: *> The number of subdiagonals within the band of A. KL >= 0.
69: *> \endverbatim
70: *>
71: *> \param[in] KU
72: *> \verbatim
73: *> KU is INTEGER
74: *> The number of superdiagonals within the band of A. KU >= 0.
75: *> \endverbatim
76: *>
77: *> \param[in] AB
78: *> \verbatim
79: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
80: *> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
81: *> column of A is stored in the j-th column of the array AB as
82: *> follows:
83: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
84: *> \endverbatim
85: *>
86: *> \param[in] LDAB
87: *> \verbatim
88: *> LDAB is INTEGER
89: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
90: *> \endverbatim
91: *>
92: *> \param[out] R
93: *> \verbatim
94: *> R is DOUBLE PRECISION array, dimension (M)
95: *> If INFO = 0, or INFO > M, R contains the row scale factors
96: *> for A.
97: *> \endverbatim
98: *>
99: *> \param[out] C
100: *> \verbatim
101: *> C is DOUBLE PRECISION array, dimension (N)
102: *> If INFO = 0, C contains the column scale factors for A.
103: *> \endverbatim
104: *>
105: *> \param[out] ROWCND
106: *> \verbatim
107: *> ROWCND is DOUBLE PRECISION
108: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
109: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
110: *> AMAX is neither too large nor too small, it is not worth
111: *> scaling by R.
112: *> \endverbatim
113: *>
114: *> \param[out] COLCND
115: *> \verbatim
116: *> COLCND is DOUBLE PRECISION
117: *> If INFO = 0, COLCND contains the ratio of the smallest
118: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
119: *> worth scaling by C.
120: *> \endverbatim
121: *>
122: *> \param[out] AMAX
123: *> \verbatim
124: *> AMAX is DOUBLE PRECISION
125: *> Absolute value of largest matrix element. If AMAX is very
126: *> close to overflow or very close to underflow, the matrix
127: *> should be scaled.
128: *> \endverbatim
129: *>
130: *> \param[out] INFO
131: *> \verbatim
132: *> INFO is INTEGER
133: *> = 0: successful exit
134: *> < 0: if INFO = -i, the i-th argument had an illegal value
135: *> > 0: if INFO = i, and i is
136: *> <= M: the i-th row of A is exactly zero
137: *> > M: the (i-M)-th column of A is exactly zero
138: *> \endverbatim
139: *
140: * Authors:
141: * ========
142: *
143: *> \author Univ. of Tennessee
144: *> \author Univ. of California Berkeley
145: *> \author Univ. of Colorado Denver
146: *> \author NAG Ltd.
147: *
148: *> \ingroup doubleGBcomputational
149: *
150: * =====================================================================
151: SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
152: $ AMAX, INFO )
153: *
154: * -- LAPACK computational routine --
155: * -- LAPACK is a software package provided by Univ. of Tennessee, --
156: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157: *
158: * .. Scalar Arguments ..
159: INTEGER INFO, KL, KU, LDAB, M, N
160: DOUBLE PRECISION AMAX, COLCND, ROWCND
161: * ..
162: * .. Array Arguments ..
163: DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
164: * ..
165: *
166: * =====================================================================
167: *
168: * .. Parameters ..
169: DOUBLE PRECISION ONE, ZERO
170: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
171: * ..
172: * .. Local Scalars ..
173: INTEGER I, J, KD
174: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
175: * ..
176: * .. External Functions ..
177: DOUBLE PRECISION DLAMCH
178: EXTERNAL DLAMCH
179: * ..
180: * .. External Subroutines ..
181: EXTERNAL XERBLA
182: * ..
183: * .. Intrinsic Functions ..
184: INTRINSIC ABS, MAX, MIN
185: * ..
186: * .. Executable Statements ..
187: *
188: * Test the input parameters
189: *
190: INFO = 0
191: IF( M.LT.0 ) THEN
192: INFO = -1
193: ELSE IF( N.LT.0 ) THEN
194: INFO = -2
195: ELSE IF( KL.LT.0 ) THEN
196: INFO = -3
197: ELSE IF( KU.LT.0 ) THEN
198: INFO = -4
199: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
200: INFO = -6
201: END IF
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'DGBEQU', -INFO )
204: RETURN
205: END IF
206: *
207: * Quick return if possible
208: *
209: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
210: ROWCND = ONE
211: COLCND = ONE
212: AMAX = ZERO
213: RETURN
214: END IF
215: *
216: * Get machine constants.
217: *
218: SMLNUM = DLAMCH( 'S' )
219: BIGNUM = ONE / SMLNUM
220: *
221: * Compute row scale factors.
222: *
223: DO 10 I = 1, M
224: R( I ) = ZERO
225: 10 CONTINUE
226: *
227: * Find the maximum element in each row.
228: *
229: KD = KU + 1
230: DO 30 J = 1, N
231: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
232: R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
233: 20 CONTINUE
234: 30 CONTINUE
235: *
236: * Find the maximum and minimum scale factors.
237: *
238: RCMIN = BIGNUM
239: RCMAX = ZERO
240: DO 40 I = 1, M
241: RCMAX = MAX( RCMAX, R( I ) )
242: RCMIN = MIN( RCMIN, R( I ) )
243: 40 CONTINUE
244: AMAX = RCMAX
245: *
246: IF( RCMIN.EQ.ZERO ) THEN
247: *
248: * Find the first zero scale factor and return an error code.
249: *
250: DO 50 I = 1, M
251: IF( R( I ).EQ.ZERO ) THEN
252: INFO = I
253: RETURN
254: END IF
255: 50 CONTINUE
256: ELSE
257: *
258: * Invert the scale factors.
259: *
260: DO 60 I = 1, M
261: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
262: 60 CONTINUE
263: *
264: * Compute ROWCND = min(R(I)) / max(R(I))
265: *
266: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
267: END IF
268: *
269: * Compute column scale factors
270: *
271: DO 70 J = 1, N
272: C( J ) = ZERO
273: 70 CONTINUE
274: *
275: * Find the maximum element in each column,
276: * assuming the row scaling computed above.
277: *
278: KD = KU + 1
279: DO 90 J = 1, N
280: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
281: C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
282: 80 CONTINUE
283: 90 CONTINUE
284: *
285: * Find the maximum and minimum scale factors.
286: *
287: RCMIN = BIGNUM
288: RCMAX = ZERO
289: DO 100 J = 1, N
290: RCMIN = MIN( RCMIN, C( J ) )
291: RCMAX = MAX( RCMAX, C( J ) )
292: 100 CONTINUE
293: *
294: IF( RCMIN.EQ.ZERO ) THEN
295: *
296: * Find the first zero scale factor and return an error code.
297: *
298: DO 110 J = 1, N
299: IF( C( J ).EQ.ZERO ) THEN
300: INFO = M + J
301: RETURN
302: END IF
303: 110 CONTINUE
304: ELSE
305: *
306: * Invert the scale factors.
307: *
308: DO 120 J = 1, N
309: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
310: 120 CONTINUE
311: *
312: * Compute COLCND = min(C(J)) / max(C(J))
313: *
314: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
315: END IF
316: *
317: RETURN
318: *
319: * End of DGBEQU
320: *
321: END
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