LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
zppequ.f
Go to the documentation of this file.
1 *> \brief \b ZPPEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zppequ.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zppequ.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zppequ.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, N
26 * DOUBLE PRECISION AMAX, SCOND
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION S( * )
30 * COMPLEX*16 AP( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> ZPPEQU 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 *> \date November 2011
114 *
115 *> \ingroup complex16OTHERcomputational
116 *
117 * =====================================================================
118  SUBROUTINE zppequ( UPLO, N, AP, S, SCOND, AMAX, INFO )
119 *
120 * -- LAPACK computational routine (version 3.4.0) --
121 * -- LAPACK is a software package provided by Univ. of Tennessee, --
122 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
123 * November 2011
124 *
125 * .. Scalar Arguments ..
126  CHARACTER UPLO
127  INTEGER INFO, N
128  DOUBLE PRECISION AMAX, SCOND
129 * ..
130 * .. Array Arguments ..
131  DOUBLE PRECISION S( * )
132  COMPLEX*16 AP( * )
133 * ..
134 *
135 * =====================================================================
136 *
137 * .. Parameters ..
138  DOUBLE PRECISION ONE, ZERO
139  parameter ( one = 1.0d+0, zero = 0.0d+0 )
140 * ..
141 * .. Local Scalars ..
142  LOGICAL UPPER
143  INTEGER I, JJ
144  DOUBLE PRECISION SMIN
145 * ..
146 * .. External Functions ..
147  LOGICAL LSAME
148  EXTERNAL lsame
149 * ..
150 * .. External Subroutines ..
151  EXTERNAL xerbla
152 * ..
153 * .. Intrinsic Functions ..
154  INTRINSIC dble, max, min, sqrt
155 * ..
156 * .. Executable Statements ..
157 *
158 * Test the input parameters.
159 *
160  info = 0
161  upper = lsame( uplo, 'U' )
162  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
163  info = -1
164  ELSE IF( n.LT.0 ) THEN
165  info = -2
166  END IF
167  IF( info.NE.0 ) THEN
168  CALL xerbla( 'ZPPEQU', -info )
169  RETURN
170  END IF
171 *
172 * Quick return if possible
173 *
174  IF( n.EQ.0 ) THEN
175  scond = one
176  amax = zero
177  RETURN
178  END IF
179 *
180 * Initialize SMIN and AMAX.
181 *
182  s( 1 ) = dble( ap( 1 ) )
183  smin = s( 1 )
184  amax = s( 1 )
185 *
186  IF( upper ) THEN
187 *
188 * UPLO = 'U': Upper triangle of A is stored.
189 * Find the minimum and maximum diagonal elements.
190 *
191  jj = 1
192  DO 10 i = 2, n
193  jj = jj + i
194  s( i ) = dble( ap( jj ) )
195  smin = min( smin, s( i ) )
196  amax = max( amax, s( i ) )
197  10 CONTINUE
198 *
199  ELSE
200 *
201 * UPLO = 'L': Lower triangle of A is stored.
202 * Find the minimum and maximum diagonal elements.
203 *
204  jj = 1
205  DO 20 i = 2, n
206  jj = jj + n - i + 2
207  s( i ) = dble( ap( jj ) )
208  smin = min( smin, s( i ) )
209  amax = max( amax, s( i ) )
210  20 CONTINUE
211  END IF
212 *
213  IF( smin.LE.zero ) THEN
214 *
215 * Find the first non-positive diagonal element and return.
216 *
217  DO 30 i = 1, n
218  IF( s( i ).LE.zero ) THEN
219  info = i
220  RETURN
221  END IF
222  30 CONTINUE
223  ELSE
224 *
225 * Set the scale factors to the reciprocals
226 * of the diagonal elements.
227 *
228  DO 40 i = 1, n
229  s( i ) = one / sqrt( s( i ) )
230  40 CONTINUE
231 *
232 * Compute SCOND = min(S(I)) / max(S(I))
233 *
234  scond = sqrt( smin ) / sqrt( amax )
235  END IF
236  RETURN
237 *
238 * End of ZPPEQU
239 *
240  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zppequ(UPLO, N, AP, S, SCOND, AMAX, INFO)
ZPPEQU
Definition: zppequ.f:119