LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dpbequ.f
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1 *> \brief \b DPBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KD, LDAB, N
26 * DOUBLE PRECISION AMAX, SCOND
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION AB( LDAB, * ), S( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> DPBEQU computes row and column scalings intended to equilibrate a
39 *> symmetric positive definite band matrix A and reduce its condition
40 *> number (with respect to the two-norm). S contains the scale factors,
41 *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
42 *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
43 *> choice of S puts the condition number of B within a factor N of the
44 *> smallest possible condition number over all possible diagonal
45 *> scalings.
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] UPLO
52 *> \verbatim
53 *> UPLO is CHARACTER*1
54 *> = 'U': Upper triangular of A is stored;
55 *> = 'L': Lower triangular of A is stored.
56 *> \endverbatim
57 *>
58 *> \param[in] N
59 *> \verbatim
60 *> N is INTEGER
61 *> The order of the matrix A. N >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in] KD
65 *> \verbatim
66 *> KD is INTEGER
67 *> The number of superdiagonals of the matrix A if UPLO = 'U',
68 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
69 *> \endverbatim
70 *>
71 *> \param[in] AB
72 *> \verbatim
73 *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
74 *> The upper or lower triangle of the symmetric band matrix A,
75 *> stored in the first KD+1 rows of the array. The j-th column
76 *> of A is stored in the j-th column of the array AB as follows:
77 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
78 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
79 *> \endverbatim
80 *>
81 *> \param[in] LDAB
82 *> \verbatim
83 *> LDAB is INTEGER
84 *> The leading dimension of the array A. LDAB >= KD+1.
85 *> \endverbatim
86 *>
87 *> \param[out] S
88 *> \verbatim
89 *> S is DOUBLE PRECISION array, dimension (N)
90 *> If INFO = 0, S contains the scale factors for A.
91 *> \endverbatim
92 *>
93 *> \param[out] SCOND
94 *> \verbatim
95 *> SCOND is DOUBLE PRECISION
96 *> If INFO = 0, S contains the ratio of the smallest S(i) to
97 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
98 *> large nor too small, it is not worth scaling by S.
99 *> \endverbatim
100 *>
101 *> \param[out] AMAX
102 *> \verbatim
103 *> AMAX is DOUBLE PRECISION
104 *> Absolute value of largest matrix element. If AMAX is very
105 *> close to overflow or very close to underflow, the matrix
106 *> should be scaled.
107 *> \endverbatim
108 *>
109 *> \param[out] INFO
110 *> \verbatim
111 *> INFO is INTEGER
112 *> = 0: successful exit
113 *> < 0: if INFO = -i, the i-th argument had an illegal value.
114 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
115 *> \endverbatim
116 *
117 * Authors:
118 * ========
119 *
120 *> \author Univ. of Tennessee
121 *> \author Univ. of California Berkeley
122 *> \author Univ. of Colorado Denver
123 *> \author NAG Ltd.
124 *
125 *> \date November 2011
126 *
127 *> \ingroup doubleOTHERcomputational
128 *
129 * =====================================================================
130  SUBROUTINE dpbequ( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
131 *
132 * -- LAPACK computational routine (version 3.4.0) --
133 * -- LAPACK is a software package provided by Univ. of Tennessee, --
134 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
135 * November 2011
136 *
137 * .. Scalar Arguments ..
138  CHARACTER uplo
139  INTEGER info, kd, ldab, n
140  DOUBLE PRECISION amax, scond
141 * ..
142 * .. Array Arguments ..
143  DOUBLE PRECISION ab( ldab, * ), s( * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149  DOUBLE PRECISION zero, one
150  parameter( zero = 0.0d+0, one = 1.0d+0 )
151 * ..
152 * .. Local Scalars ..
153  LOGICAL upper
154  INTEGER i, j
155  DOUBLE PRECISION smin
156 * ..
157 * .. External Functions ..
158  LOGICAL lsame
159  EXTERNAL lsame
160 * ..
161 * .. External Subroutines ..
162  EXTERNAL xerbla
163 * ..
164 * .. Intrinsic Functions ..
165  INTRINSIC max, min, sqrt
166 * ..
167 * .. Executable Statements ..
168 *
169 * Test the input parameters.
170 *
171  info = 0
172  upper = lsame( uplo, 'U' )
173  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
174  info = -1
175  ELSE IF( n.LT.0 ) THEN
176  info = -2
177  ELSE IF( kd.LT.0 ) THEN
178  info = -3
179  ELSE IF( ldab.LT.kd+1 ) THEN
180  info = -5
181  END IF
182  IF( info.NE.0 ) THEN
183  CALL xerbla( 'DPBEQU', -info )
184  return
185  END IF
186 *
187 * Quick return if possible
188 *
189  IF( n.EQ.0 ) THEN
190  scond = one
191  amax = zero
192  return
193  END IF
194 *
195  IF( upper ) THEN
196  j = kd + 1
197  ELSE
198  j = 1
199  END IF
200 *
201 * Initialize SMIN and AMAX.
202 *
203  s( 1 ) = ab( j, 1 )
204  smin = s( 1 )
205  amax = s( 1 )
206 *
207 * Find the minimum and maximum diagonal elements.
208 *
209  DO 10 i = 2, n
210  s( i ) = ab( j, i )
211  smin = min( smin, s( i ) )
212  amax = max( amax, s( i ) )
213  10 continue
214 *
215  IF( smin.LE.zero ) THEN
216 *
217 * Find the first non-positive diagonal element and return.
218 *
219  DO 20 i = 1, n
220  IF( s( i ).LE.zero ) THEN
221  info = i
222  return
223  END IF
224  20 continue
225  ELSE
226 *
227 * Set the scale factors to the reciprocals
228 * of the diagonal elements.
229 *
230  DO 30 i = 1, n
231  s( i ) = one / sqrt( s( i ) )
232  30 continue
233 *
234 * Compute SCOND = min(S(I)) / max(S(I))
235 *
236  scond = sqrt( smin ) / sqrt( amax )
237  END IF
238  return
239 *
240 * End of DPBEQU
241 *
242  END