LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cppequ.f
Go to the documentation of this file.
1 *> \brief \b CPPEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CPPEQU + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cppequ.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cppequ.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cppequ.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, N
26 * REAL AMAX, SCOND
27 * ..
28 * .. Array Arguments ..
29 * REAL S( * )
30 * COMPLEX AP( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CPPEQU computes row and column scalings intended to equilibrate a
40 *> Hermitian positive definite matrix A in packed storage and reduce
41 *> its condition number (with respect to the two-norm). S contains the
42 *> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
43 *> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
44 *> This choice of S puts the condition number of B within a factor N of
45 *> the 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 triangle of A is stored;
56 *> = 'L': Lower triangle 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] AP
66 *> \verbatim
67 *> AP is COMPLEX array, dimension (N*(N+1)/2)
68 *> The upper or lower triangle of the Hermitian matrix A, packed
69 *> columnwise in a linear array. The j-th column of A is stored
70 *> in the array AP as follows:
71 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
72 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
73 *> \endverbatim
74 *>
75 *> \param[out] S
76 *> \verbatim
77 *> S is REAL array, dimension (N)
78 *> If INFO = 0, S contains the scale factors for A.
79 *> \endverbatim
80 *>
81 *> \param[out] SCOND
82 *> \verbatim
83 *> SCOND is REAL
84 *> If INFO = 0, S contains the ratio of the smallest S(i) to
85 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
86 *> large nor too small, it is not worth scaling by S.
87 *> \endverbatim
88 *>
89 *> \param[out] AMAX
90 *> \verbatim
91 *> AMAX is REAL
92 *> Absolute value of largest matrix element. If AMAX is very
93 *> close to overflow or very close to underflow, the matrix
94 *> should be scaled.
95 *> \endverbatim
96 *>
97 *> \param[out] INFO
98 *> \verbatim
99 *> INFO is INTEGER
100 *> = 0: successful exit
101 *> < 0: if INFO = -i, the i-th argument had an illegal value
102 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
103 *> \endverbatim
104 *
105 * Authors:
106 * ========
107 *
108 *> \author Univ. of Tennessee
109 *> \author Univ. of California Berkeley
110 *> \author Univ. of Colorado Denver
111 *> \author NAG Ltd.
112 *
113 *> \ingroup complexOTHERcomputational
114 *
115 * =====================================================================
116  SUBROUTINE cppequ( UPLO, N, AP, S, SCOND, AMAX, INFO )
117 *
118 * -- LAPACK computational routine --
119 * -- LAPACK is a software package provided by Univ. of Tennessee, --
120 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
121 *
122 * .. Scalar Arguments ..
123  CHARACTER UPLO
124  INTEGER INFO, N
125  REAL AMAX, SCOND
126 * ..
127 * .. Array Arguments ..
128  REAL S( * )
129  COMPLEX AP( * )
130 * ..
131 *
132 * =====================================================================
133 *
134 * .. Parameters ..
135  REAL ONE, ZERO
136  parameter( one = 1.0e+0, zero = 0.0e+0 )
137 * ..
138 * .. Local Scalars ..
139  LOGICAL UPPER
140  INTEGER I, JJ
141  REAL SMIN
142 * ..
143 * .. External Functions ..
144  LOGICAL LSAME
145  EXTERNAL lsame
146 * ..
147 * .. External Subroutines ..
148  EXTERNAL xerbla
149 * ..
150 * .. Intrinsic Functions ..
151  INTRINSIC max, min, real, sqrt
152 * ..
153 * .. Executable Statements ..
154 *
155 * Test the input parameters.
156 *
157  info = 0
158  upper = lsame( uplo, 'U' )
159  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
160  info = -1
161  ELSE IF( n.LT.0 ) THEN
162  info = -2
163  END IF
164  IF( info.NE.0 ) THEN
165  CALL xerbla( 'CPPEQU', -info )
166  RETURN
167  END IF
168 *
169 * Quick return if possible
170 *
171  IF( n.EQ.0 ) THEN
172  scond = one
173  amax = zero
174  RETURN
175  END IF
176 *
177 * Initialize SMIN and AMAX.
178 *
179  s( 1 ) = real( ap( 1 ) )
180  smin = s( 1 )
181  amax = s( 1 )
182 *
183  IF( upper ) THEN
184 *
185 * UPLO = 'U': Upper triangle of A is stored.
186 * Find the minimum and maximum diagonal elements.
187 *
188  jj = 1
189  DO 10 i = 2, n
190  jj = jj + i
191  s( i ) = real( ap( jj ) )
192  smin = min( smin, s( i ) )
193  amax = max( amax, s( i ) )
194  10 CONTINUE
195 *
196  ELSE
197 *
198 * UPLO = 'L': Lower triangle of A is stored.
199 * Find the minimum and maximum diagonal elements.
200 *
201  jj = 1
202  DO 20 i = 2, n
203  jj = jj + n - i + 2
204  s( i ) = real( ap( jj ) )
205  smin = min( smin, s( i ) )
206  amax = max( amax, s( i ) )
207  20 CONTINUE
208  END IF
209 *
210  IF( smin.LE.zero ) THEN
211 *
212 * Find the first non-positive diagonal element and return.
213 *
214  DO 30 i = 1, n
215  IF( s( i ).LE.zero ) THEN
216  info = i
217  RETURN
218  END IF
219  30 CONTINUE
220  ELSE
221 *
222 * Set the scale factors to the reciprocals
223 * of the diagonal elements.
224 *
225  DO 40 i = 1, n
226  s( i ) = one / sqrt( s( i ) )
227  40 CONTINUE
228 *
229 * Compute SCOND = min(S(I)) / max(S(I))
230 *
231  scond = sqrt( smin ) / sqrt( amax )
232  END IF
233  RETURN
234 *
235 * End of CPPEQU
236 *
237  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cppequ(UPLO, N, AP, S, SCOND, AMAX, INFO)
CPPEQU
Definition: cppequ.f:117