Annotation of rpl/lapack/lapack/dgbequb.f, revision 1.4
1.1 bertrand 1: SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
2: $ AMAX, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
6: * -- Jason Riedy of Univ. of California Berkeley. --
7: * -- November 2008 --
8: *
9: * -- LAPACK is a software package provided by Univ. of Tennessee, --
10: * -- Univ. of California Berkeley and NAG Ltd. --
11: *
12: IMPLICIT NONE
13: * ..
14: * .. Scalar Arguments ..
15: INTEGER INFO, KL, KU, LDAB, M, N
16: DOUBLE PRECISION AMAX, COLCND, ROWCND
17: * ..
18: * .. Array Arguments ..
19: DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DGBEQUB computes row and column scalings intended to equilibrate an
26: * M-by-N matrix A and reduce its condition number. R returns the row
27: * scale factors and C the column scale factors, chosen to try to make
28: * the largest element in each row and column of the matrix B with
29: * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
30: * the radix.
31: *
32: * R(i) and C(j) are restricted to be a power of the radix between
33: * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
34: * of these scaling factors is not guaranteed to reduce the condition
35: * number of A but works well in practice.
36: *
37: * This routine differs from DGEEQU by restricting the scaling factors
38: * to a power of the radix. Baring over- and underflow, scaling by
39: * these factors introduces no additional rounding errors. However, the
40: * scaled entries' magnitured are no longer approximately 1 but lie
41: * between sqrt(radix) and 1/sqrt(radix).
42: *
43: * Arguments
44: * =========
45: *
46: * M (input) INTEGER
47: * The number of rows of the matrix A. M >= 0.
48: *
49: * N (input) INTEGER
50: * The number of columns of the matrix A. N >= 0.
51: *
52: * KL (input) INTEGER
53: * The number of subdiagonals within the band of A. KL >= 0.
54: *
55: * KU (input) INTEGER
56: * The number of superdiagonals within the band of A. KU >= 0.
57: *
58: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
59: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
60: * The j-th column of A is stored in the j-th column of the
61: * array AB as follows:
62: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
63: *
64: * LDAB (input) INTEGER
65: * The leading dimension of the array A. LDAB >= max(1,M).
66: *
67: * R (output) DOUBLE PRECISION array, dimension (M)
68: * If INFO = 0 or INFO > M, R contains the row scale factors
69: * for A.
70: *
71: * C (output) DOUBLE PRECISION array, dimension (N)
72: * If INFO = 0, C contains the column scale factors for A.
73: *
74: * ROWCND (output) DOUBLE PRECISION
75: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
76: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
77: * AMAX is neither too large nor too small, it is not worth
78: * scaling by R.
79: *
80: * COLCND (output) DOUBLE PRECISION
81: * If INFO = 0, COLCND contains the ratio of the smallest
82: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
83: * worth scaling by C.
84: *
85: * AMAX (output) DOUBLE PRECISION
86: * Absolute value of largest matrix element. If AMAX is very
87: * close to overflow or very close to underflow, the matrix
88: * should be scaled.
89: *
90: * INFO (output) INTEGER
91: * = 0: successful exit
92: * < 0: if INFO = -i, the i-th argument had an illegal value
93: * > 0: if INFO = i, and i is
94: * <= M: the i-th row of A is exactly zero
95: * > M: the (i-M)-th column of A is exactly zero
96: *
97: * =====================================================================
98: *
99: * .. Parameters ..
100: DOUBLE PRECISION ONE, ZERO
101: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
102: * ..
103: * .. Local Scalars ..
104: INTEGER I, J, KD
105: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
106: * ..
107: * .. External Functions ..
108: DOUBLE PRECISION DLAMCH
109: EXTERNAL DLAMCH
110: * ..
111: * .. External Subroutines ..
112: EXTERNAL XERBLA
113: * ..
114: * .. Intrinsic Functions ..
115: INTRINSIC ABS, MAX, MIN, LOG
116: * ..
117: * .. Executable Statements ..
118: *
119: * Test the input parameters.
120: *
121: INFO = 0
122: IF( M.LT.0 ) THEN
123: INFO = -1
124: ELSE IF( N.LT.0 ) THEN
125: INFO = -2
126: ELSE IF( KL.LT.0 ) THEN
127: INFO = -3
128: ELSE IF( KU.LT.0 ) THEN
129: INFO = -4
130: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
131: INFO = -6
132: END IF
133: IF( INFO.NE.0 ) THEN
134: CALL XERBLA( 'DGBEQUB', -INFO )
135: RETURN
136: END IF
137: *
138: * Quick return if possible.
139: *
140: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
141: ROWCND = ONE
142: COLCND = ONE
143: AMAX = ZERO
144: RETURN
145: END IF
146: *
147: * Get machine constants. Assume SMLNUM is a power of the radix.
148: *
149: SMLNUM = DLAMCH( 'S' )
150: BIGNUM = ONE / SMLNUM
151: RADIX = DLAMCH( 'B' )
152: LOGRDX = LOG(RADIX)
153: *
154: * Compute row scale factors.
155: *
156: DO 10 I = 1, M
157: R( I ) = ZERO
158: 10 CONTINUE
159: *
160: * Find the maximum element in each row.
161: *
162: KD = KU + 1
163: DO 30 J = 1, N
164: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
165: R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
166: 20 CONTINUE
167: 30 CONTINUE
168: DO I = 1, M
169: IF( R( I ).GT.ZERO ) THEN
170: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
171: END IF
172: END DO
173: *
174: * Find the maximum and minimum scale factors.
175: *
176: RCMIN = BIGNUM
177: RCMAX = ZERO
178: DO 40 I = 1, M
179: RCMAX = MAX( RCMAX, R( I ) )
180: RCMIN = MIN( RCMIN, R( I ) )
181: 40 CONTINUE
182: AMAX = RCMAX
183: *
184: IF( RCMIN.EQ.ZERO ) THEN
185: *
186: * Find the first zero scale factor and return an error code.
187: *
188: DO 50 I = 1, M
189: IF( R( I ).EQ.ZERO ) THEN
190: INFO = I
191: RETURN
192: END IF
193: 50 CONTINUE
194: ELSE
195: *
196: * Invert the scale factors.
197: *
198: DO 60 I = 1, M
199: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
200: 60 CONTINUE
201: *
202: * Compute ROWCND = min(R(I)) / max(R(I)).
203: *
204: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
205: END IF
206: *
207: * Compute column scale factors.
208: *
209: DO 70 J = 1, N
210: C( J ) = ZERO
211: 70 CONTINUE
212: *
213: * Find the maximum element in each column,
214: * assuming the row scaling computed above.
215: *
216: DO 90 J = 1, N
217: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
218: C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
219: 80 CONTINUE
220: IF( C( J ).GT.ZERO ) THEN
221: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
222: END IF
223: 90 CONTINUE
224: *
225: * Find the maximum and minimum scale factors.
226: *
227: RCMIN = BIGNUM
228: RCMAX = ZERO
229: DO 100 J = 1, N
230: RCMIN = MIN( RCMIN, C( J ) )
231: RCMAX = MAX( RCMAX, C( J ) )
232: 100 CONTINUE
233: *
234: IF( RCMIN.EQ.ZERO ) THEN
235: *
236: * Find the first zero scale factor and return an error code.
237: *
238: DO 110 J = 1, N
239: IF( C( J ).EQ.ZERO ) THEN
240: INFO = M + J
241: RETURN
242: END IF
243: 110 CONTINUE
244: ELSE
245: *
246: * Invert the scale factors.
247: *
248: DO 120 J = 1, N
249: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
250: 120 CONTINUE
251: *
252: * Compute COLCND = min(C(J)) / max(C(J)).
253: *
254: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
255: END IF
256: *
257: RETURN
258: *
259: * End of DGBEQUB
260: *
261: END
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