1: SUBROUTINE ZGBEQUB( 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 C( * ), R( * )
20: COMPLEX*16 AB( LDAB, * )
21: * ..
22: *
23: * Purpose
24: * =======
25: *
26: * ZGBEQUB computes row and column scalings intended to equilibrate an
27: * M-by-N matrix A and reduce its condition number. R returns the row
28: * scale factors and C the column scale factors, chosen to try to make
29: * the largest element in each row and column of the matrix B with
30: * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
31: * the radix.
32: *
33: * R(i) and C(j) are restricted to be a power of the radix between
34: * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
35: * of these scaling factors is not guaranteed to reduce the condition
36: * number of A but works well in practice.
37: *
38: * This routine differs from ZGEEQU by restricting the scaling factors
39: * to a power of the radix. Baring over- and underflow, scaling by
40: * these factors introduces no additional rounding errors. However, the
41: * scaled entries' magnitured are no longer approximately 1 but lie
42: * between sqrt(radix) and 1/sqrt(radix).
43: *
44: * Arguments
45: * =========
46: *
47: * M (input) INTEGER
48: * The number of rows of the matrix A. M >= 0.
49: *
50: * N (input) INTEGER
51: * The number of columns of the matrix A. N >= 0.
52: *
53: * KL (input) INTEGER
54: * The number of subdiagonals within the band of A. KL >= 0.
55: *
56: * KU (input) INTEGER
57: * The number of superdiagonals within the band of A. KU >= 0.
58: *
59: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
60: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
61: * The j-th column of A is stored in the j-th column of the
62: * array AB as follows:
63: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
64: *
65: * LDAB (input) INTEGER
66: * The leading dimension of the array A. LDAB >= max(1,M).
67: *
68: * R (output) DOUBLE PRECISION array, dimension (M)
69: * If INFO = 0 or INFO > M, R contains the row scale factors
70: * for A.
71: *
72: * C (output) DOUBLE PRECISION array, dimension (N)
73: * If INFO = 0, C contains the column scale factors for A.
74: *
75: * ROWCND (output) DOUBLE PRECISION
76: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
77: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
78: * AMAX is neither too large nor too small, it is not worth
79: * scaling by R.
80: *
81: * COLCND (output) DOUBLE PRECISION
82: * If INFO = 0, COLCND contains the ratio of the smallest
83: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
84: * worth scaling by C.
85: *
86: * AMAX (output) DOUBLE PRECISION
87: * Absolute value of largest matrix element. If AMAX is very
88: * close to overflow or very close to underflow, the matrix
89: * should be scaled.
90: *
91: * INFO (output) INTEGER
92: * = 0: successful exit
93: * < 0: if INFO = -i, the i-th argument had an illegal value
94: * > 0: if INFO = i, and i is
95: * <= M: the i-th row of A is exactly zero
96: * > M: the (i-M)-th column of A is exactly zero
97: *
98: * =====================================================================
99: *
100: * .. Parameters ..
101: DOUBLE PRECISION ONE, ZERO
102: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
103: * ..
104: * .. Local Scalars ..
105: INTEGER I, J, KD
106: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
107: $ LOGRDX
108: COMPLEX*16 ZDUM
109: * ..
110: * .. External Functions ..
111: DOUBLE PRECISION DLAMCH
112: EXTERNAL DLAMCH
113: * ..
114: * .. External Subroutines ..
115: EXTERNAL XERBLA
116: * ..
117: * .. Intrinsic Functions ..
118: INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
119: * ..
120: * .. Statement Functions ..
121: DOUBLE PRECISION CABS1
122: * ..
123: * .. Statement Function definitions ..
124: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
125: * ..
126: * .. Executable Statements ..
127: *
128: * Test the input parameters.
129: *
130: INFO = 0
131: IF( M.LT.0 ) THEN
132: INFO = -1
133: ELSE IF( N.LT.0 ) THEN
134: INFO = -2
135: ELSE IF( KL.LT.0 ) THEN
136: INFO = -3
137: ELSE IF( KU.LT.0 ) THEN
138: INFO = -4
139: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
140: INFO = -6
141: END IF
142: IF( INFO.NE.0 ) THEN
143: CALL XERBLA( 'ZGBEQUB', -INFO )
144: RETURN
145: END IF
146: *
147: * Quick return if possible.
148: *
149: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
150: ROWCND = ONE
151: COLCND = ONE
152: AMAX = ZERO
153: RETURN
154: END IF
155: *
156: * Get machine constants. Assume SMLNUM is a power of the radix.
157: *
158: SMLNUM = DLAMCH( 'S' )
159: BIGNUM = ONE / SMLNUM
160: RADIX = DLAMCH( 'B' )
161: LOGRDX = LOG(RADIX)
162: *
163: * Compute row scale factors.
164: *
165: DO 10 I = 1, M
166: R( I ) = ZERO
167: 10 CONTINUE
168: *
169: * Find the maximum element in each row.
170: *
171: KD = KU + 1
172: DO 30 J = 1, N
173: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
174: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
175: 20 CONTINUE
176: 30 CONTINUE
177: DO I = 1, M
178: IF( R( I ).GT.ZERO ) THEN
179: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
180: END IF
181: END DO
182: *
183: * Find the maximum and minimum scale factors.
184: *
185: RCMIN = BIGNUM
186: RCMAX = ZERO
187: DO 40 I = 1, M
188: RCMAX = MAX( RCMAX, R( I ) )
189: RCMIN = MIN( RCMIN, R( I ) )
190: 40 CONTINUE
191: AMAX = RCMAX
192: *
193: IF( RCMIN.EQ.ZERO ) THEN
194: *
195: * Find the first zero scale factor and return an error code.
196: *
197: DO 50 I = 1, M
198: IF( R( I ).EQ.ZERO ) THEN
199: INFO = I
200: RETURN
201: END IF
202: 50 CONTINUE
203: ELSE
204: *
205: * Invert the scale factors.
206: *
207: DO 60 I = 1, M
208: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
209: 60 CONTINUE
210: *
211: * Compute ROWCND = min(R(I)) / max(R(I)).
212: *
213: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
214: END IF
215: *
216: * Compute column scale factors.
217: *
218: DO 70 J = 1, N
219: C( J ) = ZERO
220: 70 CONTINUE
221: *
222: * Find the maximum element in each column,
223: * assuming the row scaling computed above.
224: *
225: DO 90 J = 1, N
226: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
227: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
228: 80 CONTINUE
229: IF( C( J ).GT.ZERO ) THEN
230: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
231: END IF
232: 90 CONTINUE
233: *
234: * Find the maximum and minimum scale factors.
235: *
236: RCMIN = BIGNUM
237: RCMAX = ZERO
238: DO 100 J = 1, N
239: RCMIN = MIN( RCMIN, C( J ) )
240: RCMAX = MAX( RCMAX, C( J ) )
241: 100 CONTINUE
242: *
243: IF( RCMIN.EQ.ZERO ) THEN
244: *
245: * Find the first zero scale factor and return an error code.
246: *
247: DO 110 J = 1, N
248: IF( C( J ).EQ.ZERO ) THEN
249: INFO = M + J
250: RETURN
251: END IF
252: 110 CONTINUE
253: ELSE
254: *
255: * Invert the scale factors.
256: *
257: DO 120 J = 1, N
258: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
259: 120 CONTINUE
260: *
261: * Compute COLCND = min(C(J)) / max(C(J)).
262: *
263: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
264: END IF
265: *
266: RETURN
267: *
268: * End of ZGBEQUB
269: *
270: END
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