LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
cpbequ.f
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1 *> \brief \b CPBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KD, LDAB, N
26 * REAL AMAX, SCOND
27 * ..
28 * .. Array Arguments ..
29 * REAL S( * )
30 * COMPLEX AB( LDAB, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CPBEQU computes row and column scalings intended to equilibrate a
40 *> Hermitian positive definite band matrix A and reduce its condition
41 *> number (with respect to the two-norm). S contains the scale factors,
42 *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
43 *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
44 *> choice of S puts the condition number of B within a factor N of the
45 *> smallest possible condition number over all possible diagonal
46 *> scalings.
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[in] UPLO
53 *> \verbatim
54 *> UPLO is CHARACTER*1
55 *> = 'U': Upper triangular of A is stored;
56 *> = 'L': Lower triangular of A is stored.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The order of the matrix A. N >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] KD
66 *> \verbatim
67 *> KD is INTEGER
68 *> The number of superdiagonals of the matrix A if UPLO = 'U',
69 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
70 *> \endverbatim
71 *>
72 *> \param[in] AB
73 *> \verbatim
74 *> AB is COMPLEX array, dimension (LDAB,N)
75 *> The upper or lower triangle of the Hermitian band matrix A,
76 *> stored in the first KD+1 rows of the array. The j-th column
77 *> of A is stored in the j-th column of the array AB as follows:
78 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
79 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
80 *> \endverbatim
81 *>
82 *> \param[in] LDAB
83 *> \verbatim
84 *> LDAB is INTEGER
85 *> The leading dimension of the array A. LDAB >= KD+1.
86 *> \endverbatim
87 *>
88 *> \param[out] S
89 *> \verbatim
90 *> S is REAL array, dimension (N)
91 *> If INFO = 0, S contains the scale factors for A.
92 *> \endverbatim
93 *>
94 *> \param[out] SCOND
95 *> \verbatim
96 *> SCOND is REAL
97 *> If INFO = 0, S contains the ratio of the smallest S(i) to
98 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
99 *> large nor too small, it is not worth scaling by S.
100 *> \endverbatim
101 *>
102 *> \param[out] AMAX
103 *> \verbatim
104 *> AMAX is REAL
105 *> Absolute value of largest matrix element. If AMAX is very
106 *> close to overflow or very close to underflow, the matrix
107 *> should be scaled.
108 *> \endverbatim
109 *>
110 *> \param[out] INFO
111 *> \verbatim
112 *> INFO is INTEGER
113 *> = 0: successful exit
114 *> < 0: if INFO = -i, the i-th argument had an illegal value.
115 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
116 *> \endverbatim
117 *
118 * Authors:
119 * ========
120 *
121 *> \author Univ. of Tennessee
122 *> \author Univ. of California Berkeley
123 *> \author Univ. of Colorado Denver
124 *> \author NAG Ltd.
125 *
126 *> \date November 2011
127 *
128 *> \ingroup complexOTHERcomputational
129 *
130 * =====================================================================
131  SUBROUTINE cpbequ( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
132 *
133 * -- LAPACK computational routine (version 3.4.0) --
134 * -- LAPACK is a software package provided by Univ. of Tennessee, --
135 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136 * November 2011
137 *
138 * .. Scalar Arguments ..
139  CHARACTER UPLO
140  INTEGER INFO, KD, LDAB, N
141  REAL AMAX, SCOND
142 * ..
143 * .. Array Arguments ..
144  REAL S( * )
145  COMPLEX AB( ldab, * )
146 * ..
147 *
148 * =====================================================================
149 *
150 * .. Parameters ..
151  REAL ZERO, ONE
152  parameter ( zero = 0.0e+0, one = 1.0e+0 )
153 * ..
154 * .. Local Scalars ..
155  LOGICAL UPPER
156  INTEGER I, J
157  REAL SMIN
158 * ..
159 * .. External Functions ..
160  LOGICAL LSAME
161  EXTERNAL lsame
162 * ..
163 * .. External Subroutines ..
164  EXTERNAL xerbla
165 * ..
166 * .. Intrinsic Functions ..
167  INTRINSIC max, min, REAL, SQRT
168 * ..
169 * .. Executable Statements ..
170 *
171 * Test the input parameters.
172 *
173  info = 0
174  upper = lsame( uplo, 'U' )
175  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
176  info = -1
177  ELSE IF( n.LT.0 ) THEN
178  info = -2
179  ELSE IF( kd.LT.0 ) THEN
180  info = -3
181  ELSE IF( ldab.LT.kd+1 ) THEN
182  info = -5
183  END IF
184  IF( info.NE.0 ) THEN
185  CALL xerbla( 'CPBEQU', -info )
186  RETURN
187  END IF
188 *
189 * Quick return if possible
190 *
191  IF( n.EQ.0 ) THEN
192  scond = one
193  amax = zero
194  RETURN
195  END IF
196 *
197  IF( upper ) THEN
198  j = kd + 1
199  ELSE
200  j = 1
201  END IF
202 *
203 * Initialize SMIN and AMAX.
204 *
205  s( 1 ) = REAL( AB( J, 1 ) )
206  smin = s( 1 )
207  amax = s( 1 )
208 *
209 * Find the minimum and maximum diagonal elements.
210 *
211  DO 10 i = 2, n
212  s( i ) = REAL( AB( J, I ) )
213  smin = min( smin, s( i ) )
214  amax = max( amax, s( i ) )
215  10 CONTINUE
216 *
217  IF( smin.LE.zero ) THEN
218 *
219 * Find the first non-positive diagonal element and return.
220 *
221  DO 20 i = 1, n
222  IF( s( i ).LE.zero ) THEN
223  info = i
224  RETURN
225  END IF
226  20 CONTINUE
227  ELSE
228 *
229 * Set the scale factors to the reciprocals
230 * of the diagonal elements.
231 *
232  DO 30 i = 1, n
233  s( i ) = one / sqrt( s( i ) )
234  30 CONTINUE
235 *
236 * Compute SCOND = min(S(I)) / max(S(I))
237 *
238  scond = sqrt( smin ) / sqrt( amax )
239  END IF
240  RETURN
241 *
242 * End of CPBEQU
243 *
244  END
subroutine cpbequ(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
CPBEQU
Definition: cpbequ.f:132
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62